Patent Publication Number: US-9893922-B2

Title: System and method for implementing orthogonal time frequency space communications using OFDM

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of U.S. application Ser. No. 15/194,494, entitled ORTHOGONAL TIME FREQUENCY SPACE COMMUNICATION SYSTEM COMPATIBLE WITH OFDM, filed Jun. 27, 2016, which claims the benefit of priority under 35 U.S.C. § 119(e) of U.S. Provisional Application No. 62/185,617, entitled ORTHOGONAL TIME FREQUENCY SPACE MODULATION, filed Jun. 27, 2015, of U.S. Provisional Application No. 62/185,619, entitled ORTHOGONAL TIME FREQUENCY SPACE MODULATION, filed Jun. 27, 2015, of U.S. Provisional Application No. 62/185,643, entitled ORTHOGONAL TIME FREQUENCY SPACE MODULATION, filed Jun. 28, 2015, and of U.S. Provisional Application No. 62/265,352, entitled OTFS COMMUNICATION SYSTEM USING MULTICARRIER MODULATORS AND DEMODULATORS, filed Dec. 9, 2015, the contents of each of which are hereby incorporated by reference in their entirety for all purposes. The present application is a continuation-in-part of U.S. application Ser. No. 15/188,946, entitled SYMPLECTIC ORTHOGONAL TIME FREQUENCY SPACE MODULATION SYSTEM, filed Jun. 21, 2016, which is a continuation-in-part of U.S. application Ser. No. 15/152,464, entitled ORTHOGONAL TIME FREQUENCY SPACE MODULATION SYSTEM, filed May 11, 2016, which claims the benefit of priority under 35 U.S.C. § 119(e) of U.S. Provisional Application No. 62/159,853, entitled ORTHOGONAL TIME FREQUENCY SPACE OTFS MODULATION, filed May 11, 2015, of U.S. Provisional Application No. 62/160,257, entitled SYSTEMS AND METHODS FOR SYMPLECTIC ORTHOGONAL TIME FREQUENCY SHIFTING MODULATION AND TRANSMISSION OF DATA, filed May 12, 2015, of U.S. Provisional Application No. 62/173,801, entitled SYSTEMS AND METHODS FOR SYMPLECTIC ORTHOGONAL TIME FREQUENCY SHIFTING MODULATION AND TRANSMISSION OF DATA, filed Jun. 10, 2015, of U.S. Provisional Application No. 62/182,372, entitled OTFS A NEW MODULATION FOR 5G, filed Jun. 19, 2015, and of U.S. Provisional Application No. 62/215,663, entitled ORTHOGONAL TIME FREQUENCY SPACE COMMUNICATION SYSTEM AND METHOD, filed Sep. 8, 2015, the contents of each of which are hereby incorporated by reference in their entirety for all purposes. The present application is also a continuation-in-part of U.S. application Ser. No. 14/709,377, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed May 11, 2015, which is a continuation of U.S. patent application Ser. No. 13/927,086, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed on Jun. 25, 2013, which claims the benefit of priority under 35 U.S.C. § 119(e) of U.S. Provisional Application Ser. No. 61/664,020, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Jun. 25, 2012, of U.S. Provisional Application Ser. No. 61/801,398, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. Provisional Application Ser. No. 61/801,366, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. Provisional Application Ser. No. 61/801,435, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. Provisional Application Ser. No. 61/801,495, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar. 15, 2013, of U.S. Provisional Application Ser. No. 61/801,994, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar. 15, 2013, and of U.S. Provisional Application Ser. No. 61/801,968, entitled MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM, filed Mar. 15, 2013, the contents of each of which are hereby incorporated by reference in their entirety for all purposes. 
    
    
     FIELD 
     This disclosure generally relates to communications protocols and methods, and more particularly relates to methods for modulation and related processing of signals used for wireless and other forms of communication. 
     BACKGROUND 
     Fourth generation (4G) wireless networks have served the public well, providing ubiquitous access to the Internet and enabling the explosion of mobile apps, smartphones and sophisticated data intensive applications like mobile video. This continues the evolution of cellular technologies, where each new generation brings substantial benefits to the public, enabling significant gains in productivity, convenience, and quality of life. 
     Looking ahead to the demands that the ever increasing and diverse data usage is placing on existing networks, it is becoming clear to the industry that current 4G networks will not be able to support the foreseen needs in data usage. This is in part because data traffic volume has been, and continues to, increase at an exponential rate. Moreover, new applications such as, for example, immersive reality and remote robotic operation, coupled with the ongoing expansion of mobile video, are expected to overwhelm the carrying capacity of current network systems. One of the goals of 5G system design is to be able to economically scale the capabilities of networks in dense urban settings (e.g., to 750 Gbps per sq. Km), which is not possible using technology which has been commercially deployed. 
     In addition to being able to handle larger volumes of data, next generation systems will need to improve the quality of data delivery in order to support desired future applications. The public is increasingly coming to expect that wireless networks provide a near “wireline” experience to the untethered user. This may translate to, for example, a requirement of 50+ Mbps throughout coverage areas (i.e., even at cell edges), which will require advanced interference mitigation technologies to be realized. 
     Another aspect of the quality of user experience is mobility. The throughput of current wireless networks tends to be dramatically reduced in tandem with increased mobile speeds due to Doppler effects. Future 5G systems aim to not only increase supported speeds up to 500 Km/h for high speed trains and aviation, but to also support a host of new automotive applications for vehicle-to-vehicle and vehicle-to-infrastructure communications. 
     While the support of increased and higher quality data traffic is necessary for wireless networks to continue supporting user needs, carriers are also exploring new applications that will enable new revenues and innovative use cases. These include the automotive and smart infrastructure applications discussed above. Other desired applications include the deployment of public safety ultra-reliable networks, the use of cellular networks to support the sunset of the PSTN, and the like. Moreover, it is anticipated the 5G networks will usher in the deployment of large numbers of Internet connected devices, also known as the Internet of Things (IoT). However, existing networks are not designed to support a very large number of connected devices with very low traffic per device. 
     SUMMARY 
     In one aspect the disclosure is directed to a method of transmitting data over a communication channel. The method includes receiving a plurality of information symbols. The method further includes encoding an N×M array containing the plurality of information symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of information symbols with respect to both time and frequency. The two-dimensional array of modulation symbols is then transmitted using M mutually orthogonal waveforms included within M frequency sub-bands. 
     The encoding may further include transforming the N×M array into an array of filtered OFDM symbols using at least one Fourier transform and a filtering process and transforming the array of filtered OFDM symbols into an array of OTFS symbols using at least one two-dimensional Fourier transform. 
     The encoding may also be performed in accordance with the following relationship: 
               X   ⁡     [     n   ,   m     ]       =       1   MN     ⁢       W   tr     ⁡     [     n   ,   m     ]       ⁢       ∑     k   =   0       N   -   1       ⁢       ∑     l   =   0       M   -   1       ⁢       x   ⁡     [     l   ,   k     ]       ⁢       b     k   ,   l       ⁡     [     n   ,   m     ]                               b     k   ,   l       ⁡     [     n   ,   m     ]       =     e     j   ⁢           ⁢   2   ⁢           ⁢     π   ⁡     (         m   ⁢           ⁢   l     M     -     nk   N       )                 
where x[l,k], k=0, . . . , N−1, l=0, . . . , M−1 represents the the N×M array containing the plurality of information symbols, X[n,m], n=0, . . . , N−1, m=0, . . . , M−1 represents the two-dimensional array of modulation symbols, W tr [n,m] is a windowing function, and b k,l [n,m] represent a set of basis functions.
 
     The disclosure also pertains to an automated method of wireless communication over an impaired data channel. The method includes receiving a plurality of data symbols. The method further includes encoding an N×M two-dimensional array containing the plurality of data symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of data symbols using a set of cyclically time-shifted and frequency-shifted basis functions. The two-dimensional array of modulation symbols is transmitted using M mutually orthogonal wireless waveforms included within M frequency sub-bands. 
     The encoding may further include transforming the N×M array into an array of filtered OFDM symbols using at least one Fourier transform and a filtering process and transforming the array of filtered OFDM symbols into an array of OTFS symbols using at least one two-dimensional Fourier transform. 
     The encoding may further include encoding the at least one N×M two-dimensional array of data symbols onto at least one symplectic-like analysis compatible manifold distributed over a column time axis of length T and row frequency axis of length F, thereby producing at least one Information manifold. In addition, the encoding may include transforming the at least one Information manifold in accordance with a two-dimensional symplectic-like Fourier transform, thereby producing at least one two-dimensional Fourier transformed Information manifold. 
     In another aspect the disclosure relates to a method of generating a waveform for transmission over a communication channel. The method includes receiving a plurality of information symbols and creating a plurality of modulation symbols by using each of the plurality information symbols to modulate one of a plurality of two-dimensional basis functions on a time-frequency plane Each of the plurality of two-dimensional basis functions is uniquely associated with one of the plurality of information symbols. The method further includes generating a transmit waveform comprised of a plurality of pulse waveforms, each of the plurality of pulse waveforms corresponding to a combination of one of the plurality of modulation symbols and one of a plurality of time-translated and frequency-modulated versions of a fundamental transmit pulse. 
     In one implementation each of the plurality of time-translated and frequency-modulated versions of the fundamental transmit pulse is associated with a time translation by one of N multiples of T and a frequency modulation by one of M multiples of Δƒ, where the transmit waveform is of a total duration of NT seconds and a total bandwidth of MΔƒ Hz. The fundamental transmit pulse may have the property of being orthogonal to translations by a multiple of time T and modulation by a multiple of Δƒ. 
     In another aspect the disclosure relates to a communication device including a wireless transmitter, a processor and a memory including program code executable by the processor. The program code includes code for causing the processor to receive a plurality of information symbols. The program code further includes code for causing the processor to create a plurality of modulation symbols by using each of the plurality information symbols to modulate one of a plurality of two-dimensional basis functions on a time-frequency plane. Each of the plurality of two-dimensional basis functions is uniquely associated with one of the plurality of information symbols. The program code further causes the processor to generate a transmit waveform comprised of a plurality of pulse waveforms, each of the plurality of pulse waveforms corresponding to a combination of one of the plurality of modulation symbols and one of a plurality of time-translated and frequency-modulated versions of a fundamental transmit pulse. The program code further includes code for causing the processor to provide the transmit waveform to the wireless transmitter. 
     The disclosure is also directed to a method for receiving, at a communication receiver, one or more modulated waveforms. The method further includes performing matched filtering of samples of the one or more modulated waveforms with respect to a receive pulse to produce estimated time-frequency modulation symbols. Each of the estimated time-frequency modulation symbols corresponds to modulation of one of a plurality of orthogonal two-dimensional basis functions by one of a plurality of information symbols. The method further includes projecting the estimates of the time-frequency modulation symbols on the plurality of orthogonal two-dimensional basis functions in order to obtain estimates of the plurality of information symbols. 
     In one implementation the method may further include performing windowing and periodization operations with respect to the estimated time-frequency modulation symbols. In addition, the projecting operation may include performing a symplectic Fourier transform operation with respect to a periodic sequence comprised of the estimates of the time-frequency modulation symbols. 
     In yet another aspect the disclosure pertains to a communication device including a wireless receiver configured to receive one or more modulated waveforms, a processor and a memory including program code executable by the processor. The program code includes code for causing the processor to receive, from the wireless receiver, samples of one or more modulated waveforms. The code further includes code for causing the processor to matched filter samples of the one or more modulated waveforms with respect to a receive pulse to produce estimated time-frequency modulation symbols. Each of the estimated time-frequency modulation symbols corresponds to modulation of one of a plurality of orthogonal two-dimensional basis functions by one of a plurality of information symbols. The program code further includes code for causing the processor to project the estimated time-frequency modulation symbols on the plurality of orthogonal two-dimensional basis functions in order to obtain estimates of the plurality of information symbols. 
     In one implementation the program code may further include code for performing windowing and periodization operations with respect to the estimated time-frequency modulation symbols. In addition, the code may include code for causing the processor to perform a symplectic Fourier transform operation with respect to a periodic sequence comprised of the estimated time-frequency modulation symbols. 
     The disclosure is further directed to a method of providing a modulated signal useable in a signal transmission system. The method includes performing a two dimensional time-frequency transformation of a data frame including a plurality of information symbols into a plane of time-frequency modulation symbols. The method further includes generating the modulated signal by performing a Heisenberg transform using the plane of time-frequency modulation symbols. In one implementation performing the two dimensional time-frequency transformation entails performing an inverse time-frequency symplectic transformation and a windowing operation. 
     In another aspect the disclosure pertains to a signal modulation method in which a data frame including a plurality of information symbols is transformed into a plane of time-frequency modulation symbols using a two dimensional time-frequency transformation. The method includes generating the modulated signal based upon the plane of time-frequency modulation symbols using an OFDM modulator. In one implementation performing the two dimensional time-frequency transformation entails performing an inverse time-frequency symplectic transformation and a windowing operation. 
     The disclosure also pertains to a method of providing a modulated signal useable in a signal transmission system. The method includes performing a two dimensional time-frequency transformation of a data frame including a plurality of information symbols into a plane of time-frequency modulation symbols. The method further includes generating the modulated signal based upon the plane of time-frequency modulation symbols using a multicarrier filter bank (MCFB) modulator. In one implementation performing the two dimensional time-frequency transformation includes performing an inverse time-frequency symplectic transformation and a windowing operation. 
     In another aspect, the disclosure describes a transmitter apparatus including an OTFS pre-processing unit. The OTFS pre-processing unit is configured to perform a two dimensional time-frequency transformation of a data frame including a plurality of information symbols into a plane of time-frequency modulation symbols. The transmitter apparatus includes an OFDM or multicarrier filter bank modulator configured to generate the modulated signal based upon the plane of time-frequency modulation symbols. In one implementation the two dimensional transformation comprises an inverse time-frequency symplectic transformation and a windowing operation. 
     The disclosure further pertains to a method of receiving a modulated signal. The method includes receiving a plurality of signal components of the modulated signal and generating, by performing an OFDM demodulation operation using the plurality of signal components, a plane of estimated time-frequency modulation symbols. The method further includes providing an estimated data frame by performing an inverse of a two dimensional time-frequency transformation with respect to the plane of estimated time-frequency modulation symbols. In one implementation performing the inverse of the two dimensional time-frequency transformation includes performing a windowing operation and a symplectic Fourier transform. 
     In a further aspect the disclosure is directed to a method of receiving a modulated signal. The method includes receiving a plurality of signal components of the modulated signal and generating, by performing one of an OFDM and MCFB demodulation operation using the plurality of signal components, a plane of estimated time-frequency modulation symbols. The method further includes providing an estimated data frame by performing an inverse of a two dimensional time-frequency transformation with respect to the plane of estimated time-frequency modulation symbols. In one implementation performing the inverse of the two dimensional time-frequency transformation includes performing a windowing operation and a symplectic Fourier transform. 
     In an additional aspect the disclosure relates to a receiver apparatus including a receiver front end configured to receive a plurality of signal components of a modulated signal. The receiver includes one of an OFDM and MCFB demodulator configured to generate a plane of estimated time-frequency modulation symbols based upon the plurality of signal components. An OTFS post-processing unit is operative to provide an estimated data frame by performing an inverse of a two dimensional time-frequency transformation with respect to the plane of estimated time-frequency modulation symbols. In one implementation the inverse of the two dimensional time-frequency transformation corresponds to a windowing operation and a symplectic Fourier transform. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a better understanding of the nature and objects of various embodiments of the invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings, wherein: 
         FIG. 1A  illustrates an example of a wireless communication system that may exhibit time/frequency selective fading. 
         FIG. 1B  provides a high-level representation of a conventional transceiver which could be utilized in the wireless communication system of  FIG. 1A . 
         FIG. 2A  shows the time-varying impulse response for an accelerating reflector in a channel represented by a one-dimensional channel model in a (τ,t) coordinate system. 
         FIG. 2B  shows the same channel represented using a time invariant impulse response in a delay-Doppler (τ,ν) coordinate system. 
         FIG. 3  is a block diagram of components of an exemplary OTFS communication system. 
         FIG. 4  represents a conceptual implementation of a Heisenberg transform in an OTFS transmitter and a Wigner transform in an OTFS receiver. 
         FIG. 5A  illustratively represents an exemplary embodiment of OTFS modulation, including the transformation of the time-frequency plane to the Doppler-delay plane. 
         5 B indicates relationships between sampling rate, delay resolution and time resolution in an OTFS communication system. 
         FIG. 5C  illustrates signaling between various domains in an OTFS communication system. 
         FIG. 5D  depicts exemplary notations used in denoting signals present at various stages of processing in an OTFS transmitter and receiver. 
         FIG. 6  shows a discrete impulse in the OTFS domain which is used for purposes of channel estimation. 
         FIGS. 7A and 7B  illustrate two different basis functions belonging to different users, each of which spans the entire time-frequency frame. 
         FIGS. 7C and 7D  provide an illustration of two-dimensional spreading with respect to both time and frequency in an OTFS communication system. 
         FIG. 7E  illustrates members of a potential set of such mutually orthogonal two-dimensional basis functions. 
         FIGS. 8 and 9  illustrate multiplexing multiple users in the time-frequency domain by allocating different resource blocks or subframes to different users in an interleaved manner. 
         FIG. 10  illustrates components of an exemplary OTFS transceiver. 
         FIG. 11  illustrates a comparison of bit error rates (BER) predicted by a simulation of a TDMA system and an OTFS system. 
         FIG. 12  is a flowchart representative of the operations performed by an exemplary OTFS transceiver. 
         FIG. 13  illustrates functioning of an OTFS modulator as an orthogonal map disposed to transform a two-dimensional time-frequency matrix into a transmitted waveform. 
         FIG. 14  illustrates operation of an OTFS demodulator in transforming a received waveform into a two-dimensional time-frequency matrix in accordance with an orthogonal map. 
         FIG. 15  illustratively represents a pulse train included within a pulse waveform produced by an OTFS modulator. 
         FIG. 16  depicts a two-dimensional decision feedback equalizer configured to perform a least means square (LMS) equalization procedure. 
         FIGS. 17A-17D  depict an OTFS transmitter and receiver and the operation of each with respect to associated time-frequency grids. 
         FIGS. 18A and 18B  illustratively represent OTFS communication over a communication channel characterized by a two-dimensional delay-Doppler impulse response. 
         FIG. 19A  illustrates transmission of a two-dimensional Fourier transformed information manifold represented by an N×M structure over M frequency bands during N time periods of duration Tμ. 
         FIG. 19B  illustrates another perspective on image domain and transform domain dual grids that may be used for the symplectic OTFS methods described herein. 
         FIG. 20  shows an example of M filtered OTFS frequency bands being simultaneously transmitted according to various smaller time slices Tμ. 
         FIG. 21  provides an additional example of OTFS waveforms being transmitted according to various smaller time slices Tμ. 
         FIG. 22  provides a block diagrammatic representation of an exemplary process of OTFS transmission and reception. 
         FIG. 23  illustrates represents an exemplary structure of finite OTFS modulation map. 
         FIGS. 24A and 24B  respectively illustrate a standard communication lattice and the reciprocal of the standard communication lattice. 
         FIG. 25  illustratively represents a standard communication torus. 
         FIG. 26  illustratively represents a standard communication finite torus. 
         FIG. 27  illustrates an exemplary structure of an OTFS modulation map. 
         FIG. 28  illustrates a frequency domain interpretation of an OTFS modulation block. 
         FIGS. 29A and 29B  illustrate one manner in which symplectic OTFS methods can operate in a transmitter and receiver system. 
         FIG. 29C  illustrates characteristics of OTFS pre-processing enabling compatibility with OFDM modulation systems. 
         FIG. 29D  illustrates further details of an OTFS pre-processing operation compatible with OFDM modulation systems. 
         FIG. 30  shows an alternative method of transmitting and receiving data over a channel. 
         FIG. 31  shows the impact of channel caused Doppler and time delays on the image domain and transform domain dual grids. 
         FIG. 32  shows one example of interleaving 
         FIG. 33  shows another example of interleaving, in which same size frames are interleaved on a frequency staggered basis. 
         FIG. 34  shows another example of interleaving, in which variable size frames are interleaved on a time basis. 
         FIG. 35  depicts an OTFS pre-processing step within a transmitter module and an OTFS post-processing step within a receiver module. 
         FIG. 36  provides a block diagram of an OTFS transmitter according to an embodiment. 
         FIG. 37  depicts an OTFS receiver configured to demodulate OTFS-modulated data received over a wireless link. 
         FIG. 38  shows an example of how an active OTFS relay system may operate between an OTFS transmitter and receiver. 
     
    
    
     DETAILED DESCRIPTION 
     As is discussed below, embodiments of orthogonal time frequency space (OTFS) modulation involve transmitting each information symbol by modulating a two-dimensional (2D) basis function on the time-frequency plane. In exemplary embodiments the modulation basis function set is specifically derived to best represent the dynamics of the time varying multipath channel. In this way OTFS transforms the time-varying multipath channel into a time invariant delay-Doppler two dimensional convolution channel. This effectively eliminates the difficulties in tracking time-varying fading in, for example, communications involving high speed vehicles. 
     OTFS increases the coherence time of the channel by orders of magnitude. It simplifies signaling over the channel using well studied AWGN codes over the average channel SNR. More importantly, it enables linear scaling of throughput with the number of antennas in moving vehicle applications due to the inherently accurate and efficient estimation of channel state information (CSI). In addition, since the delay-Doppler channel representation is very compact, OTFS enables massive MIMO and beamforming with CSI at the transmitter for four, eight, and more antennas in moving vehicle applications. The CSI information needed in OTFS is a fraction of what is needed to track a time varying channel. 
     As will be appreciated from the discussion below, one characteristic of OTFS is that a single QAM symbol may be spread over multiple time and/or frequency points. This is a key technique to increase processing gain and in building penetration capabilities for IoT deployment and PSTN replacement applications. Spreading in the OTFS domain allows spreading over wider bandwidth and time durations while maintaining a stationary channel that does not need to be tracked over time. 
     These benefits of OTFS will become apparent once the basic concepts behind OTFS are understood. There is a rich mathematical foundation of OTFS that leads to several variations; for example it can be combined with OFDM or with multicarrier filter banks. Prior to proceeding to a detailed discussion of OTFS, various drawbacks of communication systems predicated on one-dimensional channel models are first described. 
       FIG. 1A  illustrates an example of a wireless communication system  100  that may exhibit time/frequency selective fading. The system  100  includes a transmitter  110  (e.g., a cell phone tower) and a receiver  120  (e.g., a cell phone). The scenario illustrated in  FIG. 1  includes multiple pathways (multi-path) that the signal transmitted from the transmitter  100  travels through before arriving at the receiver  100 . A first pathway  130  reflects through a tree  132 , second pathway  140  reflects off of a building  142  and a third pathway  150  reflects off of a second building  152 . A fourth pathway  160  reflects off of a moving car  162 . Because each of the pathways  130 ,  140 ,  150  and  160  travels a different distance, and is attenuated or faded at a different level and at a different frequency, when conventionally configured the receiver  120  may drop a call or at least suffer low throughput due to destructive interference of the multi-path signals. 
     Turning now to  FIG. 1B , a high-level representation is provided of a conventional transceiver  200  which could be utilized in the wireless communication system  100  of  FIG. 1A . The transceiver  200  could, for example, operate in accordance with established protocols for time-division multiple access (TDMA), code-division multiple access (CDMA) or orthogonal frequency-division multiple access (OFDM) systems. In conventional wireless communication systems such as TDMA, CDMA, and OFDM) systems, the multipath communication channel  210  between a transmitter  204  and a receiver  208  is represented by a one-dimensional model. In these systems channel distortion is characterized using a one-dimensional representation of the impulse response of the communication channel. The transceiver  200  may include a one-dimensional equalizer  220  configured to at least partially remove this estimated channel distortion from the one-dimensional output data stream  230  produced by the receiver  208 . 
     Unfortunately, use of a one-dimensional channel model presents a number of fundamental problems. First, the one-dimensional channel models employed in existing communication systems are non-stationary; that is, the symbol-distorting influence of the communication channel changes from symbol to symbol. In addition, when a channel is modeled in only one dimension it is likely and possible that certain received symbols will be significantly lower in energy than others due to “channel fading”. Finally, one-dimensional channel state information (CSI) appears random and much of it is estimated by interpolating between channel measurements taken at specific points, thus rendering the information inherently inaccurate. These problems are only exacerbated in multi-antenna (MIMO) communication systems. As is discussed below, embodiments of the OTFS method described herein can be used to substantially overcome the fundamental problems arising from use of a one-dimensional channel model. 
     The multipath fading channel is commonly modeled one-dimensionally in the baseband as a convolution channel with a time varying impulse response
 
 r ( t )=∫ (τ, t ) s ( t −τ) dτ   (1)
 
where s(t) and r(t) represent the complex baseband channel input and output respectively and where   (τ,t) is the complex baseband time varying channel response.
 
     This representation, while general, does not give us insight into the behavior and variations of the time varying impulse response. A more useful and insightful model, which is also commonly used for Doppler multipath doubly fading channels is
 
 r ( t )=∫∫ h (τ,ν) e   j2πν(t−τ)   s ( t −τ) dνdτ   (2)
 
     In this representation, the received signal is a superposition of reflected copies of the transmitted signal, where each copy is delayed by the path delay τ, frequency shifted by the Doppler shift ν and weighted by the time-independent delay-Doppler impulse response h(τ,ν) for that τ and ν. In addition to the intuitive nature of this representation, Eq. (2) maintains the generality of Eq. (1). In other words it can represent complex Doppler trajectories, like accelerating vehicles, reflectors etc. This can be seen if we express the time varying impulse response as a Fourier expansion with respect to the time variable t
 
 (τ, t )=∫ h (τ,ν) e   j2πνt   dt   (3)
 
     Substituting (3) in (1) we obtain Eq. (2) after some manipulation. As an example,  FIG. 2A  shows the time-varying impulse response for an accelerating reflector in the (τ,t) coordinate system, while  FIG. 2B  shows the same channel represented as a time invariant impulse response in the (τ,ν) coordinate system. 
     An important feature revealed by these two figures is how compact the (τ,ν) representation is compared to the (τ,t) representation. This has important implications for channel estimation, equalization and tracking as will be discussed later. 
     Notice that while h(τ,ν) is, in fact, time-independent, the operation on s(t) is still time varying, as can be seen by the effect of the explicit complex exponential function of time in Eq. (2). In implementation the disclosed modulation scheme contemplates an appropriate choice of orthogonal basis functions that render the effects of this channel to become truly time-independent in the domain defined by those basis functions. The proposed scheme has the following high level outline. 
     First, let us consider a set of orthonormal basis functions φ τ,ν (t) indexed by τ,ν which are orthogonal to translation and modulation, i.e.,
 
φ τ,ν ( t−τ   0 )=φ τ+τ     0     ,ν ( t )
 
 e   j2πν     0     t φ τ,ν ( t )=φ τ,ν−ν     0   ( t )  (4)
 
and let us consider the transmitted signal as a superposition of these basis functions
 
 s ( t )=∫∫ x (τ,ν)φ τ,ν ( t ) dτdν   (5)
 
where the weights x(τ,ν) represent the information bearing signal to be transmitted. After the transmitted signal of (5) goes through the time varying channel of Eq. (2) we obtain a superposition of delayed and modulated versions of the basis functions, which due to (4) results in
 
                           r   ⁡     (   t   )       =       ⁢     ∫     ∫       h   ⁡     (     τ   ,   v     )       ⁢     e     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     v   ⁡     (     t   -   τ     )           ⁢     s   ⁡     (     t   -   τ     )       ⁢   dvd   ⁢           ⁢   τ                     =       ⁢     ∫     ∫         ϕ     τ   ,   v       ⁡     (   t   )       ⁢     {       h   ⁡     (     τ   ,   v     )       *     x   ⁡     (     τ   ,   v     )         }     ⁢   d   ⁢           ⁢   τ   ⁢           ⁢   dv                       (   6   )               
where * denotes two dimensional convolution. Eq. (6) can be thought of as a generalization of the convolution relationship for linear time invariant systems, using one dimensional exponentials as basis functions. Notice that the term in brackets can be recovered at the receiver by matched filtering against each basis function φ τ,ν (t). In this way a two dimensional channel relationship is established in the (τ,ν) domain
 
 y (τ,ν)= h (τ,ν)* x (τ,ν)  (7)
 
where y(τ,ν) is the receiver two dimensional matched filter output. Notice also, that in this domain the channel is described by a time invariant two-dimensional convolution.
 
     A final different interpretation of the wireless channel will also be useful in what follows. Let us consider s(t) and r(t) as elements of the Hilbert space of square integrable functions  . Then Eq. (2) can be interpreted as a linear operator on   acting on the input s(t), parametrized by the impulse response h(τ,ν), and producing the output r(t): 
     
       
         
           
             
               
                 
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                             h 
                           
                           ⁡ 
                           
                             ( 
                             · 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         r 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     ∈ 
                     
                       ℋ 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Notice that although the operator is linear, it is not time-invariant. If there is no Doppler, i.e., if h(ν,τ)=h(0,τ)δ(ν), then Eq. (2) reduces to a time invariant convolution. Also notice that while for time invariant systems the impulse response is parameterized by one dimension, in the time varying case we have a two dimensional impulse response. While in the time invariant case the convolution operator produces a superposition of delays of the input s(t), (hence the parameterization is along the one dimensional delay axis) in the time varying case we have a superposition of delay-and-modulate operations as seen in Eq. (2) (hence the parameterization is along the two dimensional delay and Doppler axes). This is a major difference which makes the time varying representation non-commutative (in contrast to the convolution operation which is commutative), and complicates the treatment of time varying systems. 
     One important point of Eq. (8) is that the operator Π h (•) can be compactly parametrized by a two dimensional function h(τ,ν), providing an efficient, time-independent description of the channel. Typical channel delay spreads and Doppler spreads are a very small fraction of the symbol duration and subcarrier spacing of multicarrier systems. 
     The representation of time varying systems defined by equations (2) and (8) may be characterized as a Heisenberg representation. In this regard it may be shown that every linear operator (eq. (8)) can be parameterized by some impulse response as in equation (2). 
     OTFS Modulation Over the Doppler Multipath Channel 
     The time variation of the channel introduces significant difficulties in wireless communications related to channel acquisition, tracking, equalization and transmission of channel state information (CSI) to the transmit side for beamforming and MIMO processing. We herein develop a modulation domain based on a set of orthonormal basis functions over which we can transmit the information symbols, and over which the information symbols experience a static, time invariant, two dimensional channel for the duration of the packet or burst transmission. In that modulation domain, the channel coherence time is increased by orders of magnitude and the issues associated with channel fading in the time or frequency domain in SISO or MIMO systems are significantly reduced. 
       FIG. 3  is a block diagram of components of an exemplary OTFS communication system  300 . As shown, the system  300  includes a transmitter  310  and a receiver  330 . The transmitting device  310  and the receiving device  330  include first and second OTFS transceivers  315 - 1  and  315 - 2 , respectively. The OTFS transceivers  315 - 1  and  315 - 2  communicate, either unidirectionally or bidirectionally, via communication channel  320  in the manner described herein. Although in the exemplary embodiments described herein the system  300  may comprise a wireless communication system, in other embodiments the communication channel may comprise a wired communication channel such as, for example, a communication channel within a fiber optic or coaxial cable. As was described above, the communication channel  320  may include multiple pathways and be characterized by time/frequency selective fading. 
     The components of the OTFS transceiver may be implemented in hardware, software, or a combination thereof. For a hardware implementation, the processing units may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, other electronic units designed to perform the functions described above, and/or a combination thereof. 
     Referring now to  FIG. 3B , there is provided a pictorial view of the two transformations that constitute an exemplary form of OTFS modulation. It shows at a high level the signal processing steps that are required at a transmitter, such as the transmitter  310 , and a receiver, such as the receiver  330 . It also includes the parameters that define each step, which will become apparent as we further expose each step. Further,  FIG. 3C  shows a block diagram of the different processing stages at the transmitter and receiver and establishes the notation that will be used for the various signals. 
     We initially describe the transform which relates the waveform domain to the time-frequency domain. 
     The Heisenberg Transform 
     Our purpose in this section is to construct an appropriate transmit waveform which carries information provided by symbols on a grid in the time-frequency plane. Our intent in developing this modulation scheme is to transform the channel operation to an equivalent operation on the time-frequency domain with two important properties: (i) the channel is orthogonalized on the time-frequency grid; and (ii) the channel time variation is simplified on the time-frequency grid and can be addressed with an additional transform. Fortunately, these goals can be accomplished with a scheme that is very close to well-known multicarrier modulation techniques, as explained next. We will start with a general framework for multicarrier modulation and then give examples of OFDM and multicarrier filter bank implementations. 
     Let us consider the following components of a time frequency modulation:
         A lattice or grid on the time frequency plane, that is a sampling of the time axis with sampling period T and the frequency axis with sampling period Δƒ.
 
Λ={( nT,m Δƒ), n,mε     }   (9)
   A packet burst with total duration NT secs and total bandwidth MΔƒ Hz   A set of modulation symbols X[n,m], n=0, . . . , N−1, m=0, . . . , M−1 we wish to transmit over this burst   A transmit pulse g tr (t) with the property of being orthogonal to translations by T and modulations by Δƒ (generally required if the receiver uses the same pulse as the transmitter)       

     
       
         
           
             
               
                 
                   
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                         ⁡ 
                         
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                         ⁡ 
                         
                           ( 
                           m 
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                       ⁢ 
                       
                         δ 
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Given the above components, the time-frequency modulator is a Heisenberg operator on the lattice Λ, that is, it maps the two dimensional symbols X[n,m] to a transmitted waveform, via a superposition of delay-and-modulate operations on the pulse waveform g tr (t) 
     
       
         
           
             
               
                 
                   
                     s 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         
                           
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                           / 
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                           / 
                           2 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
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                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           X 
                           ⁡ 
                           
                             [ 
                             
                               n 
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           
                             g 
                             tr 
                           
                           ⁡ 
                           
                             ( 
                             
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                             ⁢ 
                             
                                 
                             
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                             ⁢ 
                             
                                 
                             
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                             ⁢ 
                             
                                 
                             
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                             ⁢ 
                             
                                 
                             
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                             ⁢ 
                             
                                 
                             
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                               ⁡ 
                               
                                 ( 
                                 
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                                   - 
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                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     More formally 
                   x   =           Π   X     ⁡     (     g   tr     )       ⁢     :     ⁢       g   tr     ⁡     (   t   )         ∈     ℋ   ⁢     →       Π   X     ⁡     (   ·   )         ⁢     y   ⁡     (   t   )         ∈   ℋ             (   12   )               
where we denote by Π X (•) the “discrete” Heisenberg operator, parameterized by discrete values X[n,m].
 
     Notice the similarity of (12) with the channel equation (8). This is not by coincidence, but rather because we apply a modulation effect that mimics the channel effect, so that the end effect of the cascade of modulation and channel is more tractable at the receiver. It is not uncommon practice; for example, linear modulation (aimed at time invariant channels) is in its simplest form a convolution of the transmit pulse g(t) with a delta train of QAM information symbols sampled at the Baud rate T. 
     
       
         
           
             
               
                 
                   
                     s 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
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                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         X 
                         ⁡ 
                         
                           [ 
                           n 
                           ] 
                         
                       
                       ⁢ 
                       
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                         ⁡ 
                         
                           ( 
                           
                             t 
                             - 
                             
                               n 
                               ⁢ 
                               T 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In the present case, aimed at the time varying channel, we convolve-and-modulate the transmit pulse (c.f. the channel Eq. (2)) with a two dimensional delta train which samples the time frequency domain at a certain Baud rate and subcarrier spacing. 
     The sampling rate in the time-frequency domain is related to the bandwidth and time duration of the pulse g tr (t); namely, its time-frequency localization. In order for the orthogonality condition of (10) to hold for a frequency spacing Δƒ, the time spacing must be T≧1/Δƒ. The critical sampling case of T=1/Δƒ is generally not practical and refers to limiting cases, for example to OFDM systems with cyclic prefix length equal to zero or to filter banks with g tr (t) equal to the ideal Nyquist pulse. 
     Some examples illustrate these principles: 
     EXAMPLE 1 
     OFDM Modulation 
     Let us consider an OFDM system with M subcarriers, symbol length T OFDM , cyclic prefix length T CP  and subcarrier spacing 1/T OFDM . If we substitute in Equation (11) symbol duration T=T OFDM +T CP , number of symbols N=1, subcarrier spacing Δƒ=1/T OFDM  and g tr (t) a square window that limits the duration of the subcarriers to the symbol length T 
                       g   tr     ⁡     (   t   )       =     {             1   /       T   -     T   CP           ,             -     T   CP       &lt;   t   &lt;     T   -     T   CP                   0   ,         else                   (   14   )               
then we obtain the OFDM formula
 
     
       
         
           
             
               
                 
                   
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                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         
                           
                             - 
                             M 
                           
                           / 
                           2 
                         
                       
                       
                         
                           M 
                           / 
                           2 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         X 
                         ⁡ 
                         
                           [ 
                           
                             n 
                             , 
                             m 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
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                           tr 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       
                         e 
                         
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                           ⁢ 
                           
                               
                           
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                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ft 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Technically, the pulse of Eq. (14) is not orthonormal but is orthogonal to the receive filter (where the CP samples are discarded). 
     EXAMPLE 2 
     Single Carrier Modulation 
     Equation (11) reduces to single carrier modulation if we substitute M=1 subcarrier, T equal to the Baud period and g tr (t) equal to a square root raised cosine Nyquist pulse. 
     EXAMPLE 3 
     Multicarrier Filter Banks (MCFB) 
     Equation (11) describes a MCFB if g tr (t) is a square root raised cosine Nyquist pulse with excess bandwidth α, T is equal to the Baud period and Δƒ=(1+α)/T. 
     Expressing the modulation operation as a Heisenberg transform as in Eq. (12) may be counterintuitive. That is, modulation is usually perceived as a transformation of the modulation symbols X[m,n] to a transmit waveform s(t). The Heisenberg transform instead, uses X[m,n] as weights/parameters of an operator that produces s(t) when applied to the prototype transmit filter response g tr (t) c.f. Eq. (12). While counterintuitive, this formulation is useful in pursuing an abstraction of the modulation-channel-demodulation cascade effects in a two dimensional domain where the channel can be described as time invariant. 
     Attention is turned next to the processing on the receiver side needed to go back from the waveform domain to the time-frequency domain. Since the received signal has undergone the cascade of two Heisenberg transforms (one by the modulation effect and one by the channel effect), it is natural to inquire what the end-to-end effect of this cascade is. The answer to this question is given by the following result: 
     Proposition 1: Let two Heisenberg transforms as defined by Eqs. (8), (2) be parametrized by impulse responses h 1 (τ,ν), h 2  (τ,ν) and be applied in cascade to a waveform g(t) ε  . Then
 
Π h     2   (Π h     1   ( g ( t )))=Π h ( g ( t ))  (16)
 
where h(τ,ν)=h 2 (τ,ν)⊙h 1 (τ,ν) is the “twisted” convolution of h 1 (τ,ν), h 2  (τ,ν) defined by the following convolve-and-modulate operation
 
 h (τ,ν)=∫∫ h   2 (τ′,ν′) h   1 (τ−τ′,ν−ν′) e   j2πν′(τ−τ′)t   dτ′dν′   (17)
 
     Applying the above result to the cascade of the modulation and channel Heisenberg transforms of (12) and (8), one can show that the received signal is given by the Heisenberg transform
 
 r ( t )=Π ƒ ( g   tr ( t ))+ν( t )=∫∫ƒ(τ,ν) e   j2πν(t−τ)   g   tr ( t −τ) dνd τ+ν( t )  (18)
 
where ν(t) is additive noise and ƒ(τ,ν), the impulse response of the combined transform, is given by the twisted convolution of X[n,m] and h(τ,ν)
 
     
       
         
           
             
               
                 
                   
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                         , 
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                         ⁡ 
                         
                           ( 
                           
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                             v 
                           
                           ) 
                         
                       
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                         ⁡ 
                         
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                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     This result can be considered an extension of the single carrier modulation case, where the received signal through a time invariant channel is given by the convolution of the QAM symbols with a composite pulse, that pulse being the convolution of the transmitter pulse and the channel impulse response. 
     With this result established we are ready to examine exemplary receiver processing steps. 
     Receiver Processing and the Wigner Transform 
     Typical communication system design generally requires that the receiver perform a matched filtering operation, taking the inner product of the received waveform with the transmitter pulse, appropriately delayed or otherwise distorted by the channel. In the present case, we have used a collection of delayed and modulated transmit pulses, and a matched filtering operation is typically performed with respect to each one of them. 
       FIG. 4  provides a conceptual view of this processing. On the transmitter, we modulate a set of M subcarriers for each symbol we transmit, while on the receiver we perform matched filtering on each of those subcarrier pulses. We define a receiver pulse g r (t) and take the inner product with a collection of delayed and modulated versions of it. The receiver pulse g r (t) is in many cases identical to the transmitter pulse, but we keep the separate notation to cover some cases where it is not (most notably in OFDM where the CP samples have to be discarded). 
     While this approach will yield the sufficient statistics for data detection in the case of an ideal channel, a concern can be raised here for the case of non-ideal channel effects. In this case, the sufficient statistics for symbol detection are obtained by matched filtering with the channel-distorted, information-carrying pulses (assuming that the additive noise is white and Gaussian). In many well designed multicarrier systems however (e.g., OFDM and MCFB), the channel distorted version of each subcarrier signal is only a scalar version of the transmitted signal, allowing for a matched filter design that is independent of the channel and uses the original transmitted subcarrier pulse. We will make these statements more precise shortly and examine the required conditions for this to be true. 
     In actual embodiments of an OTFS receiver, this matched filtering may be implemented in the digital domain using an FFT or a polyphase transform for OFDM and MCFB respectively. However, for purposes of the present discussion, we will consider a generalization of this matched filtering by taking the inner product &lt;g r (t−τ)e j2πν(t−τ) ,r(t)&gt; of the received waveform with the delayed and modulated versions of the receiver pulse for arbitrary time and frequency offset (τ,ν). While likely not necessarily a practical implementation, it allows us to view the operations of  FIG. 4  as a two dimensional sampling of this more general inner product. 
     Let us define the inner product
 
 A   g     r     ,r (τ,ν)=&lt; g   r ( t −τ) e   j2πν(t−τ)   ,r ( t )&gt;=∫ g   r   * ( t −τ) e   −j2πν(t−τ)   r ( t ) dt   (20)
 
     The function A g     r     ,r (τ,ν) is known as the cross-ambiguity function and yields the matched filter output if sampled at τ=nT, ν=mΔƒ (on the lattice Λ), i.e.,
 
 Y[n,m]=A   g     r     ,r (τ,ν)| τ=nT,ν=mΔƒ   (21)
 
     The ambiguity function is related to the inverse of the Heisenberg transform, namely the Wigner transform.  FIG. 4  provides an intuitive feel for that, as the receiver appears to invert the operations of the transmitter. More formally, if we take the cross-ambiguity or the transmit and receive pulses A g     r     ,g     tr   (τ,ν), and use it as the impulse response of the Heisenberg operator, then we obtain the orthogonal cross-projection operator
 
Π A     gr,gtr   ( y ( t ))=g tr ( t )&lt;g r ( t ), y ( t )&gt;
 
     In words, the coefficients that come out of the matched filter, if used in a Heisenberg representation, will provide the best approximation to the original y(t) in the sense of minimum square error. 
     One key question to be addressed is the relationship is between the matched filter output Y[n,m] (or more generally Y(τ,ν)) and the transmitter input X[n,m]. We have already established in (18) that the input to the matched filter r(t) can be expressed as a Heisenberg representation with impulse response ƒ(τ,ν) (plus noise). The output of the matched filter then has two contributions
 
 Y (τ,ν)= A   g     r     ,r (τ,ν)= A   g     r     ,[Π     ƒ     (g     tr     )+ν] (τ,ν)= A   g     r     ,Π     ƒ     (g     tr     ) (τ,ν)+ A   g     r     ,ν (τ,ν)  (22)
 
     The last term is the contribution of noise, which we will denote V(τ,ν)=A g     r     ,ν (τ,ν). The first term on the right hand side is the matched filter output to the (noiseless) input comprising of a superposition of delayed and modulated versions of the transmit pulse. We next establish that this term can be expressed as the twisted convolution of the two dimensional impulse response ƒ(τ,ν) with the cross-ambiguity function (or two dimensional cross correlation) of the transmit and receive pulses. 
     The following theorem summarizes the key result. 
     Theorem 1: (Fundamental time-frequency domain channel equation). If the received signal can be expressed as
 
Π ƒ ( g   tr ( t ))=∫∫ƒ(τ,ν) e   j2πν(t−τ)   g   tr ( t −τ) dνdτ   (23)
 
     Then the cross-ambiguity of that signal with the receive pulse g tr (t) can be expressed as
 
 A   g     r     ,Π     ƒ     (g     tr     ) (τ,ν)=ƒ(τ,ν)⊙ A   g     r     ,g     tr   (τ,ν)  (24)
 
     Recall from (19) that ƒ(τ,ν)=h(τ,ν)⊙X[n,m], that is, the composite impulse response is itself a twisted convolution of the channel response and the modulation symbols. 
     Substituting ƒ(τ,ν) from (19) into (22) we obtain the end-to-end channel description in the time frequency domain 
                           Y   ⁡     (     τ   ,   v     )       =       ⁢         A       g   ⁢           ⁢   r     ,       Π   r     ⁡     (     g   tr     )           ⁡     (     τ   ,   v     )       +     V   ⁡     (     τ   ,   v     )                     =       ⁢         h   ⁡     (     τ   ,   v     )       ⊙     X   ⁡     [     n   ,   m     ]       ⊙       A       g   r     ,     g   tr         ⁡     (     τ   ,   v     )         +     V   ⁡     (     τ   ,   v     )                       (   25   )               
where V(τ,ν) is the additive noise term. Eq. (25) provides an abstraction of the time varying channel on the time-frequency plane. It states that the matched filter output at any time and frequency point (τ,ν) is given by the delay-Doppler impulse response of the channel twist-convolved with the impulse response of the modulation operator twist-convolved with the cross-ambiguity (or two dimensional cross correlation) function of the transmit and receive pulses.
 
     Evaluating Eq. (25) on the lattice Λ we obtain the matched filter output modulation symbol estimates
 
 {circumflex over (X)} [ m,n ] =Y [ n,m ] =Y (τ,ν)| τ=nT,ν=mΔƒ   (26)
 
     In order to get more intuition on Equations (25), (26). let us first consider the case of an ideal channel, i.e., h(τ,ν)=δ(τ)δ(ν). In this case by direct substitution we get the convolution relationship 
     
       
         
           
             
               
                 
                   
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                                   ⁢ 
                                   f 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     + 
                     
                       V 
                       ⁡ 
                       
                         [ 
                         
                           m 
                           , 
                           n 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     In order to simplify Eq. (27) we will use the orthogonality properties of the ambiguity function. Since we use a different transmit and receive pulses we will modify the orthogonality condition on the design of the transmit pulse we stated in (10) to a bi-orthogonality condition 
     
       
         
           
             
               
                 
                   
                     〈 
                     
                       
                         
                           g 
                           tr 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       , 
                       
                         
                           
                             g 
                             r 
                           
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 n 
                                 ⁢ 
                                 T 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           e 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   nT 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     〉 
                   
                   = 
                   
                     
                       ∫ 
                       
                         
                           
                             g 
                             tr 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             g 
                             r 
                           
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 n 
                                 ⁢ 
                                 T 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           e 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   nT 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         dt 
                       
                     
                     = 
                     
                       
                         δ 
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     Under this condition, only one term survives in (27) and we obtain
 
 Y [ n,m ] =X [ n,m ] +V [ n,m ]  (29)
 
where V [n,m] is the additive white noise. Eq. (29) shows that the matched filter output does recover the transmitted symbols (plus noise) under ideal channel conditions. Of more interest of course is the case of non-ideal time varying channel effects. We next show that even in this case, the channel orthogonalization is maintained (no intersymbol or intercarrier interference), while the channel complex gain distortion has a closed form expression.
 
     The following theorem summarizes the result as a generalization of (29). 
     Theorem 2: (End-to-end time-frequency domain channel equation): 
     If h(τ,ν) has finite support bounded by (τ max ,ν max ) and if A g     r     ,g     tr   (τ,ν)=0 for τε(nT−τ max ,nT+τ max ), νε(mΔƒ−ν max ,mΔƒ+ν max ), that is, the ambiguity function bi-orthogonality property of (28) is true in a neighborhood of each grid point (mΔƒ,nT) of the lattice Λ at least as large as the support of the channel response h(τ,ν), then the following equation holds
 
 Y [ n,m ] =H [ n,m ] X [ n,m ]
 
 H [ n,m ] =∫∫h (τ,ν) e   j2πνnT   e   −j2π(ν+mΔƒ)τ   dνdτ   (30)
 
     If the ambiguity function is only approximately bi-orthogonal in the neighborhood of Λ (by continuity), then (30) is only approximately true. Eq. (30) is a fundamental equation that describes the channel behavior in the time-frequency domain. It is the basis for understanding the nature of the channel and its variations along the time and frequency dimensions. 
     Some observations are now in order on Eq. (30).As mentioned before, there is no interference across X[n,m] in either time n or frequency m.
         The end-to-end channel distortion in the modulation domain is a (complex) scalar that needs to be equalized.   If there is no Doppler, i.e. h(τ,ν)=h(τ,0)δ(ν), then Eq. (30) becomes       

                           Y   ⁡     [     n   ,   m     ]       =       ⁢       X   ⁡     [     n   ,   m     ]       ⁢     ∫       h   ⁡     (     τ   ,   0     )       ⁢     e       -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   m   ⁢           ⁢   Δ   ⁢           ⁢   fτ       ⁢   d   ⁢           ⁢   τ                     =       ⁢       X   ⁡     [     n   ,   m     ]       ⁢     H   ⁡     (     0   ,     m   ⁢           ⁢   Δ   ⁢           ⁢   f       )                       (   31   )               
which is the well-known multicarrier result, that each subcarrier symbol is multiplied by the frequency response of the time invariant channel evaluated at the frequency of that subcarrier.
         If there is no multipath, i.e. h(τ,ν)=h(0,ν)δ(τ), then Eq. (30) becomes
 
 Y [ n,m ] =X [ n,m ] ∫h (ν,0) e   j2πνnT   dτ   (32)
       

     Notice that the fading each subcarrier experiences as a function of time nT has a complicated expression as a weighted superposition of exponentials. This is a major complication in the design of wireless systems with mobility like LTE; it necessitates the transmission of pilots and the continuous tracking of the channel, which becomes more difficult the higher the vehicle speed or Doppler bandwidth is. 
     Some examples of this general framework are provided below. 
     EXAMPLE 3 
     OFDM Modulation 
     In this case the fundamental transmit pulse is given by (14) and the fundamental receive pulse is 
                       g   r     ⁡     (   t   )       =     {         0           -     T   CP       &lt;   t   &lt;   0               1       T   -     T   CP                 0   &lt;   t   &lt;     T   -     T   CP                 0       else                   (   33   )               
i.e., the receiver zeroes out the CP samples and applies a square window to the symbols comprising the OFDM symbol. It is worth noting that in this case, the bi-orthogonality property holds exactly along the time dimension.
 
     EXAMPLE 4 
     MCFB Modulation 
     In the case of multicarrier filter banks g tr (t)=g r (t)=g(t). There are several designs for the fundamental pulse g(t). A square root raised cosine pulse provides good localization along the frequency dimension at the expense of less localization along the time dimension. If T is much larger than the support of the channel in the time dimension, then each subchannel sees a flat channel and the bi-orthogonality property holds approximately. 
     In summary, one of the two transforms defining OTFS has now been described. Specifically, an explanation has been provided of how the transmitter and receiver apply appropriate operators on the fundamental transmit and receive pulses and orthogonalize the channel according to Eq. (30). Examples have also been provided to illustrate how the choice of the fundamental pulse affects the time and frequency localization of the transmitted modulation symbols and the quality of the channel orthogonalization that is achieved. However, Eq. (30) shows that the channel in this domain, while free of intersymbol interference, suffers from fading across both the time and the frequency dimensions via a complicated superposition of linear phase factors. 
     In what follows we start from Eq. (30) and describe the second transform that defines OTFS; we will show how that transform defines an information domain where the channel does not fade in either dimension. 
     The 2D OTFS Transform 
     Notice that the time-frequency response H[n,m] in (30) is related to the channel delay-Doppler response h(τ,ν) by an expression that resembles a Fourier transform. However, there are two important differences: (i) the transform is two dimensional (along delay and Doppler) and (ii) the exponentials defining the transforms for the two dimensions have opposing signs. Despite these difficulties, Eq. (30) points in the direction of using complex exponentials as basis functions on which to modulate the information symbols; and only transmit on the time-frequency domain the superposition of those modulated complex exponential bases. As is discussed below, this approach exploits Fourier transform properties and effectively translates a multiplicative channel in one Fourier domain to a convolution channel in the other Fourier domain. 
     Given the difficulties of Eq. (30) mentioned above, we need to develop a suitable version of Fourier transform and associated sampling theory results. Let us start with the following definitions: 
     Definition 1: Symplectic Discrete Fourier Transform: Given a square summable two dimensional sequence X[m,n]ε (Λ) we define 
     
       
         
           
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       
                         τ 
                         , 
                         v 
                       
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           m 
                           , 
                           n 
                         
                       
                       ⁢ 
                       
                         
                           X 
                           ⁡ 
                           
                             [ 
                             
                               n 
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 vnT 
                                 - 
                                 
                                   τ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   f 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       = 
                       △ 
                     
                     ⁢ 
                     
                       S 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       D 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       F 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             X 
                             ⁡ 
                             
                               [ 
                               
                                 n 
                                 , 
                                 m 
                               
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Notice that the above 2D Fourier transform (known as the Symplectic Discrete Fourier Transform) differs from the more well known Cartesian Fourier transform in that the exponential functions across each of the two dimensions have opposing signs. This is necessary in this case, as it matches the behavior of the channel equation. 
     Further notice that the resulting x(τ,ν) is periodic with periods (1/Δƒ,1/T). This transform defines a new two dimensional plane, which we will call the delay-Doppler plane, and which can represent a max delay of 1/Δƒ and a max Doppler of 1/T. A one dimensional periodic function is also called a function on a circle, while a 2D periodic function is called a function on a torus (or donut). In this case x(τ,ν) is defined on a torus Z with circumferences (dimensions) (1/Δƒ,1/T). 
     The periodicity of x(τ,ν) (or sampling rate of the time-frequency plane) also defines a lattice on the delay-Doppler plane, which we will call the reciprocal lattice 
     
       
         
           
             
               
                 
                   
                     Λ 
                     ⊥ 
                   
                   = 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             m 
                             ⁢ 
                             
                               1 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 f 
                               
                             
                           
                           , 
                           
                             n 
                             ⁢ 
                             
                               1 
                               T 
                             
                           
                         
                         ) 
                       
                       , 
                       n 
                       , 
                       
                         m 
                         ∈ 
                         ℤ 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     The points on the reciprocal lattice have the property of making the exponent in (34), an integer multiple of 2π. 
     The inverse transform is given by: 
     
       
         
           
             
               
                 
                   
                     X 
                     ⁡ 
                     
                       [ 
                       
                         n 
                         , 
                         m 
                       
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         1 
                         c 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           
                             1 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               f 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                               1 
                               T 
                             
                           
                           ⁢ 
                           
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 
                                   τ 
                                   , 
                                   v 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       vnT 
                                       - 
                                       
                                         τ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                             ⁢ 
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             v 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             τ 
                           
                         
                       
                     
                     ⁢ 
                     
                       = 
                       △ 
                     
                     ⁢ 
                     
                       S 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       D 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       F 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           T 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             x 
                             ⁡ 
                             
                               ( 
                               
                                 τ 
                                 , 
                                 v 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     where c=TΔƒ. 
     We next define a sampled version of x(τ,ν). In particular, we wish to take M samples on the delay dimension (spaced at 1/MΔƒ) and N samples on the Doppler dimension (spaced at 1/NT). More formally, a denser version of the reciprocal lattice is defined so that Λ ⊥   ⊂ Λ 0   ⊥ . 
     
       
         
           
             
               
                 
                   
                     Λ 
                     0 
                     ⊥ 
                   
                   = 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             m 
                             ⁢ 
                             
                               1 
                               
                                 M 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 f 
                               
                             
                           
                           , 
                           
                             n 
                             ⁢ 
                             
                               1 
                               NT 
                             
                           
                         
                         ) 
                       
                       , 
                       n 
                       , 
                       
                         m 
                         ∈ 
                         ℤ 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     We define discrete periodic functions on this dense lattice with period (1/Δƒ,1/T), or equivalently we define functions on a discrete torus with these dimensions 
     
       
         
           
             
               
                 
                   
                     Z 
                     0 
                     ⊥ 
                   
                   = 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             m 
                             ⁢ 
                             
                               1 
                               
                                 M 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 f 
                               
                             
                           
                           , 
                           
                             n 
                             ⁢ 
                             
                               1 
                               NT 
                             
                           
                         
                         ) 
                       
                       , 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         = 
                         0 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         M 
                         - 
                         1 
                       
                       , 
                       
                           
                       
                       ⁢ 
                       
                         n 
                         = 
                         0 
                       
                       , 
                       
                         
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           N 
                         
                         - 
                         1 
                       
                       , 
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     These functions are related via Fourier transform relationships to discrete periodic functions on the lattice Λ, or equivalently, functions on the discrete torus
 
 Z   0 ={( nT,m Δƒ), m= 0, . . . , M− 1, n= 0, . . .  N− 1,}  (39)
 
     We wish to develop an expression for sampling Eq. (34) on the lattice of (38). First, we start with the following definition. 
     Definition 2: Symplectic Finite Fourier Transform: If X p  [k,l] is periodic with period (N,M), then we define 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             x 
                             p 
                           
                           ⁡ 
                           
                             [ 
                             
                               m 
                               , 
                               n 
                             
                             ] 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 
                                   - 
                                   
                                     M 
                                     2 
                                   
                                 
                               
                               
                                 
                                   M 
                                   2 
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   X 
                                   p 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     
                                       j 
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         nk 
                                         N 
                                       
                                       - 
                                       
                                         ml 
                                         M 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           Δ 
                         
                         ⁢ 
                           
                         ⁢ 
                         
                           SFFT 
                           ⁡ 
                           
                             ( 
                             
                               X 
                               ⁡ 
                               
                                 [ 
                                 
                                   k 
                                   , 
                                   l 
                                 
                                 ] 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     Notice that x p  [m,n] is also periodic with period [M, N] or, equivalently, it is defined on the discrete torus Z 0   ⊥ . Formally, the SFFT(X[n,m]) is a linear transformation from  (Z 0 )→ (Z 0   ⊥ ). 
     Let us now consider generating x p  [m,n] as a sampled version of (34), i.e., 
     
       
         
           
             
               
                 x 
                 p 
               
               ⁡ 
               
                 [ 
                 
                   m 
                   , 
                   n 
                 
                 ] 
               
             
             = 
             
               
                 x 
                 ⁡ 
                 
                   [ 
                   
                     m 
                     , 
                     n 
                   
                   ] 
                 
               
               = 
               
                 
                   x 
                   ⁡ 
                   
                     ( 
                     
                       τ 
                       , 
                       v 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   ❘ 
                   
                     
                       τ 
                       = 
                       
                         m 
                         
                           M 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                       
                     
                     , 
                     
                       v 
                       = 
                       
                         n 
                         NT 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
     Then we can show that (40) still holds where x p [m,n] is a periodization of X[n,m] with period (N,M) 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       p 
                     
                     ⁡ 
                     
                       [ 
                       
                         n 
                         , 
                         m 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         l 
                         , 
                         
                           k 
                           = 
                           
                             - 
                             ∞ 
                           
                         
                       
                       ∞ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       X 
                       ⁡ 
                       
                         [ 
                         
                           
                             n 
                             - 
                             kN 
                           
                           , 
                           
                             m 
                             - 
                             lM 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     This is similar to the result that sampling in one Fourier domain creates aliasing in the other domain. 
     The inverse discrete (symplectic) Fourier transform is given by 
                             X   p     ⁡     [     n   ,   m     ]       =       ⁢       1   MN     ⁢       ∑     l   ,   k       ⁢           ⁢       x   ⁡     [     l   ,   k     ]       ⁢     e       j   ⁢   2     ⁢           ⁢   π   ⁢           ⁢     (       nk   N     -     ml   M       )                             =   Δ     ⁢       ⁢       SFFT     -   1       ⁡     (     x   ⁡     [     l   ,   k     ]       )                     (   42   )               
where l=0, . . . , M−1, k=0, . . . , N−1. If the support of X[n,m] is time-frequency limited to Z 0  (no aliasing in (41)), then X p [n,m]=X[n,m] for n,mεZ 0 , and the inverse transform (42) recovers the original signal.
 
     The SDFT is termed “discrete” because it represents a signal using a discrete set of exponentials, while the SFFT is called “finite” because it represents a signal using a finite set of exponentials. 
     In the present context, an important property of the symplectic Fourier transform is that it transforms a multiplicative channel effect in one domain to a circular convolution effect in the transformed domain. This is summarized in the following proposition: 
     Proposition 2: Let X 1 [n,m]ε (Z 0 ), X 2 [n,m]ε (Z 0 ) be periodic 2D sequences. Then
 
SFFT( X   1 [ n,m ] *X   2 [ n,m ])=SFFT( X   1 [ n,m ])·SFFT( X   2[   n,m ])  (43)
 
where * denotes two dimensional circular convolution. With this framework established we are ready to define the OTFS modulation.
 
     Discrete OTFS modulation: Consider a set of NM QAM information symbols arranged on a 2D grid x[l,k], k=0, . . . , N 1, l=0, . . . , M−1 we wish to transmit. We will consider x[l,k] to be two dimensional periodic with period [N,M]. Further, assume a multicarrier modulation system defined by
         A lattice on the time frequency plane, that is a sampling of the time axis with sampling period T and the frequency axis with sampling period Δƒ (c.f. Eq. (9)).   A packet burst with total duration NT secs and total bandwidth MΔƒ Hz.   Transmit and receive pulses g tr (t), g tr (t)εL 2 ( ) satisfying the bi-orthogonality property of (28)   A transmit windowing square summable function W tr [n,m]ε (Λ) multiplying the modulation symbols in the time-frequency domain   A set of modulation symbols X[n,m], n=0, . . . , N−1, m=0, . . . , M−1 related to the information symbols x[k,l] by a set of basis functions b k,l [n,m]       

     
       
         
           
             
               
                 
                   
                     
                       X 
                       ⁡ 
                       
                         [ 
                         
                           n 
                           , 
                           m 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         1 
                         MN 
                       
                       ⁢ 
                       
                         
                           W 
                           tr 
                         
                         ⁡ 
                         
                           [ 
                           
                             n 
                             , 
                             m 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               x 
                               ⁡ 
                               
                                 [ 
                                 
                                   l 
                                   , 
                                   k 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               
                                 b 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   n 
                                   , 
                                   m 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         b 
                         
                           k 
                           , 
                           l 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           n 
                           , 
                           m 
                         
                         ] 
                       
                     
                     = 
                     
                       e 
                       
                         
                           j 
                           ⁢ 
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               ml 
                               M 
                             
                             - 
                             
                               nk 
                               N 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     where the basis functions b k,l [n,m] are related to the inverse symplectic Fourier transform (c.f., Eq. (42)) 
     Given the above components, we define the discrete OTFS modulation via the following two steps
 
 X [ n,m ] =W   tr[   n,m ]SFFT −1 ( x[k,l ])
 
 s ( t )=Π X ( g   tr ( t ))  (45)
 
     The first equation in (45) describes the OTFS transform, which combines an inverse symplectic transform with a windowing operation. The second equation describes the transmission of the modulation symbols X[n,m] via a Heisenberg transform of g tr (t) parameterized by X[n,m]. More explicit formulas for the modulation steps are given by Equations (42) and (11). 
     While the expression of the OTFS modulation via the symplectic Fourier transform reveals important properties, it is easier to understand the modulation via Eq. (44), that is, transmitting each information symbol x[k,l] by modulating a 2D basis function b k,l [n,m] on the time-frequency plane. 
     Discrete OTFS demodulation: Let us assume that the transmitted signal s(t) undergoes channel distortion according to (8), (2) yielding r(t) at the receiver. Further, let the receiver employ a receive windowing square summable function W r [n,m]. Then, the demodulation operation consists of the following steps:
         (i) Matched filtering with the receive pulse, or more formally, evaluating the ambiguity function on Λ (Wigner transform) to obtain estimates of the time-frequency modulation symbols
 
 Y [ n,m ] =A   g     r     ,y (τ,ν)| τ=nT,ν=mΔƒ   (46)
   (ii) windowing and periodization of Y[n,m]       

     
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         w 
                       
                       ⁡ 
                       
                         [ 
                         
                           n 
                           , 
                           m 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           W 
                           r 
                         
                         ⁡ 
                         
                           [ 
                           
                             n 
                             , 
                             m 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           
                             n 
                             , 
                             m 
                           
                           ] 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Y 
                         p 
                       
                       ⁡ 
                       
                         [ 
                         
                           n 
                           , 
                           m 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           k 
                           , 
                           
                             l 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                         
                         ∞ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Y 
                           w 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               n 
                               - 
                               kN 
                             
                             , 
                             
                               m 
                               - 
                               lM 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
         
         
           
             (iii) and applying the symplectic Fourier transform on the periodic sequence Y p [n,m]
 
 {circumflex over (x)} [ l,k ] =y [ l,k ]=SFFT( Y   p [ n,m ])  (48)
 
           
         
       
    
     The first step of the demodulation operation can be interpreted as a matched filtering operation on the time-frequency domain as we discussed earlier. The second step is there to ensure that the input to the SFFT is a periodic sequence. If the trivial window is used, this step can be skipped. The third step can also be interpreted as a projection of the time-frequency modulation symbols on the orthogonal basis functions 
     
       
         
           
             
               
                 
                   
                     
                       
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                       ⁡ 
                       
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                           l 
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                           0 
                         
                         
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                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
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                             = 
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                         ⁢ 
                         
                           
                             
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                               ^ 
                             
                             ⁡ 
                             
                               ( 
                               
                                 n 
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                             ⁡ 
                             
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                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         b 
                         
                           k 
                           , 
                           l 
                         
                         * 
                       
                       ⁡ 
                       
                         ( 
                         
                           n 
                           , 
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                         ) 
                       
                     
                     = 
                     
                       e 
                       
                         
                           - 
                           
                             j 
                             ⁢ 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               lm 
                               L 
                             
                             - 
                             
                               kn 
                               K 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
     The discrete OTFS modulation defined above points to efficient implementation via discrete-and-periodic FFT type processing. However, it potentially does not provide insight into the time and bandwidth resolution of these operations in the context of two dimensional Fourier sampling theory. We next introduce continuous OTFS modulation and relate the more practical discrete OTFS as a sampled version of the continuous modulation. 
     Continuous OTFS Modulation: Consider a two dimensional periodic function x(τ,ν) with period [1/Δƒ,1/T] that we wish to transmit. The choice of the period may seem arbitrary at this point, but the rationale for its choice will become apparent after the discussion below. Further, assume a multicarrier modulation system defined by
         A lattice on the time frequency plane, that is a sampling of the time axis with sampling period T and the frequency axis with sampling period Δƒ (c.f. Eq. (9)).   Transmit and receive pulses g tr (t), g tr (t)εL 2 ( ) satisfying the bi-orthogonality property of (28)   A transmit windowing function W tr [n,m]ε (Λ) multiplying the modulation symbols in the time-frequency domain       

     Given the above components, we define the continuous OTFS modulation via the following two steps
 
 X [ n,m ] =W   tr [ n,m ]SDFT −1 ( x (τ,ν))
 
 s ( t )=Π X ( g   tr ( t ))  (50)
 
     The first equation describes the inverse discrete time-frequency symplectic Fourier transform [c.f. Eq. (36)] and the windowing function, while the second equation describes the transmission of the modulation symbols via a Heisenberg transform [c.f. Eq. (11)]. 
     Continuous OTFS demodulation: Let us assume that the transmitted signal s(t) undergoes channel distortion according to (8), (2) yielding r(t) at the receiver. Further, let the receiver employ a receive windowing function W r [n,m]ε (Λ). Then, the demodulation operation consists of two steps:
         (i) Evaluating the ambiguity function on Λ (Wigner transform) to obtain estimates of the time-frequency modulation symbols
 
 Y [ n,m ] =A   g     r     ,y (τ,ν)| τ=nT,ν=mΔƒ   (51)
   (ii) Windowing and applying the symplectic Fourier transform on the modulation symbols
 
{circumflex over ( x )}(τ,ν)=SDFT( W   r [ n,m ] Y [ n,m ])  (52)
       

     Notice that in (51), (52) there is no periodization of Y[n,m], since the SDFT is defined on aperiodic square summable sequences. The periodization step needed in discrete OTFS can be understood as follows. Suppose we wish to recover the transmitted information symbols by performing a continuous OTFS demodulation and then sampling on the delay-Doppler grid 
     
       
         
           
             
               
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                 ( 
                 
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             = 
             
               
                 
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                 ⁡ 
                 
                   ( 
                   
                     τ 
                     , 
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                   ) 
                 
               
               ⁢ 
               
                 ❘ 
                 
                   
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                     = 
                     
                       m 
                       
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                         ⁢ 
                         
                             
                         
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                         f 
                       
                     
                   
                   , 
                   
                     v 
                     = 
                     
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     Since performing a continuous symplectic Fourier transform is generally not practical we consider whether the same result can be obtained using SFFT. The answer is that SFFT processing will produce exactly the samples we are looking for if the input sequence is first periodized (aliased). See also equation (40) and (41). 
     We have now described each of the steps of an exemplary form of OTFS modulation. We have also discussed how the Wigner transform at the receiver inverts the Heisenberg transform at the transmitter [c.f. Eqs. (27), (29)], and similarly for the forward and inverse symplectic Fourier transforms. 
       FIG. 5A  illustratively represents operations involved in OTFS-based communication, including the transformation of the time-frequency plane to the Doppler-delay plane. In addition,  FIGS. 5A and 5B  indicates relationships between sampling rate, delay resolution and time resolution. Referring to  FIG. 5A , in a first operation a Heisenberg transform translates a time-varying convolution channel in the waveform domain to an orthogonal but still time varying channel in the time frequency domain. For a total bandwidth BW and M subcarriers the frequency resolution is Δƒ=BW/M. For a total frame duration T ƒ  and N symbols the time resolution is T=T ƒ /N. 
     In a second operation a SFFT transform translates the time-varying channel in the time-frequency domain to a time invariant one in the delay-Doppler domain. The Doppler resolution is 1/T ƒ  and the delay resolution is 1/BW, with the Doppler and delay resolutions being mutually decoupled. The choice of window can provide a tradeoff between main lobe width (resolution) and side lobe suppression, as in classical spectral analysis. 
     Referring to  FIG. 5C , an illustration is provided of signaling between various domains in an OTFS communication system. Specifically,  FIG. 5C  illustrates signaling over: the (i) actual physical channel with a signaling waveform, (ii) the time-frequency domain, and (iii) the delay-Doppler domain. 
       FIG. 5D  illustrates exemplary notations used in denoting signals present at various stages of processing (e.g., encoding, modulation, decoding, demodulation) in an OTFS transmitter and receiver. 
     Channel Equation in the OTFS Domain 
     A mathematical characterization of the end-to-end signal relationship in an OTFS system when a non-ideal channel is between the transmitter and receiver will now be provided. Specifically, this section demonstrates how the time varying channel in (2), (8), is transformed to a time invariant convolution channel in the delay Doppler domain. 
     Proposition 3: Consider a set of NM QAM information symbols arranged in a 2D periodic sequence x[l,k] with period [M, N]. The sequence x[k, l] undergoes the following transformations:
         It is modulated using the discrete OTFS modulation of Eq. (45).   It is distorted by the delay-Doppler channel of Eqs. (2), (8).   It is demodulated by the discrete OTFS demodulation of Eqs. (46), (48).       

     The estimated sequence {circumflex over (x)}[l,k] obtained after demodulation is given by the two dimensional periodic convolution 
     
       
         
           
             
               
                 
                   
                     
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                       MN 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
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                       ⁢ 
                       
                         
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                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
     of the input QAM sequence x[m,n] and a sampled version of the windowed impulse response h w (•), 
     
       
         
           
             
               
                 
                   
                     
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                         ) 
                       
                     
                     ⁢ 
                     
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                             ′ 
                           
                           = 
                           
                             
                               l 
                               - 
                               m 
                             
                             
                               M 
                               ⁢ 
                               
                                   
                               
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                               ⁢ 
                               
                                   
                               
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                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     where h w (τ′,ν′) denotes the circular convolution of the channel response with a windowing function
 
 h   w (τ′,ν′)=∫∫ e   −j2πντ   h (τ,ν) w (τ′−τ,ν′−ν) dτdν   (55)
 
     To be precise, the window w(τ,ν) is circularly convolved with a slightly modified version of the channel impulse response e −j2πντ h(τ,ν) (by a complex exponential) as can be seen in the equation. The windowing function w(τ,ν) is the symplectic Fourier transform of the time-frequency window W [n,m] 
     
       
         
           
             
               
                 
                   
                     w 
                     ⁡ 
                     
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                         τ 
                         , 
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                       ) 
                     
                   
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                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
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                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
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                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
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                           ⁡ 
                           
                             [ 
                             
                               n 
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                             ] 
                           
                         
                         ⁢ 
                         
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                               - 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
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                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 vnT 
                                 - 
                                 
                                   τ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
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                                   ⁢ 
                                   
                                       
                                   
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                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     and where W[n,m] is the product of the transmit and receive window.
 
 W [ n,m ] =W   tr [ n,m ] W   r [ n,m ]  (57)
 
     In many cases, the windows in the transmitter and receiver are matched, i.e., W tr [n,m]=W 0 [n,m] and W r [n,m]=W 0   * [n,m], hence W [n,m]=|W 0 [n,m]| 2 . 
     The window effect is to produce a blurred version of the original channel with a resolution that depends on the span of the frequency and time samples available. If one considers the rectangular (or trivial) window, i.e., W[n,m]=1, n=0, . . . , N−1, m=−M/2, . . . , M/2−1 and zero else, then its SDFT w(τ,ν) in (56) is the two dimensional Dirichlet kernel with bandwidth inversely proportional to N and M. 
     There are several other uses of the window function. The system can be designed with a window function aimed at randomizing the phases of the transmitted symbols. This randomization may be more important for pilot symbols than data carrying symbols. For example, if neighboring cells use different window functions, the problem of pilot contamination is avoided. 
     Channel Estimation in the OTFS Domain 
     There is a variety of different ways a channel estimation scheme could be designed for an OTFS system, and a variety of different implementation options and details 
     A straightforward way to perform channel estimation entails transmitting a sounding OTFS frame containing a discrete delta function in the OTFS domain or, equivalently, a set of unmodulated carriers in the time frequency domain. From a practical standpoint, the carriers may be modulated with known, say BPSK, symbols which are removed at the receiver, as is common in many OFDM systems. 
       FIG. 6  shows a discrete impulse in the OTFS domain which may be used for purposes of channel estimation, e.g., as a form of pilot signal. In the example of  FIG. 6  the remainder of M×N delay-Doppler plane includes information symbols are arranged in a grid. In a typical implementation M×N could be, for example, 1024×256 or 512×16. 
     However, this approach may be wasteful as the extent of the channel response is only a fraction of the full extent of the OTFS frame (1/T,1/Δƒ). For example, in LTE systems 1/T≈15 KHz while the maximum Doppler shift ƒ d,max  is typically one to two orders of magnitude smaller. Similarly 1/Δƒ≈67 usec, while maximum delay spread τ max  is again one to two orders of magnitude less. We therefore can have a much smaller region of the OTFS frame devoted to channel estimation while the rest of the frame carries useful data. More specifically, for a channel with support (±ƒ d,max ,±τ max ) we need an OTFS subframe of length (2ƒ d,max /T, 2τ max /Δƒ). 
     In the case of multiuser transmission, each UE can have its own channel estimation subframe positioned in different parts of the OTFS frame. However, this channel estimation subframe may be relatively limited in size. For example, if τ max  is 5% of the extent of the delay dimension and ƒ d,max  is 5% of the Doppler dimesion, the channel estimation subframe need only be 5%×5%=0.25% of the OTFS frame. 
     Importantly, although the channel estimation symbols are limited to a small part of the OTFS frame, they actually sound the whole time-frequency domain via the corresponding two-dimensional time-frequency basis functions associated with these symbols. 
     A different approach to channel estimation is to devote pilot symbols on a subgrid in the time-frequency domain. The key question in this approach is the determination of the density of pilots that is sufficient for channel estimation without introducing aliasing. Assume that the pilots occupy the subgrid (n 0 T, m 0 Δƒ) for some integers n 0 , m 0 . Recall that for this grid the SDFT will be periodic with period (1/n 0 T,1/m 0 Δƒ). Then, applying the aliasing results discussed earlier to this grid, we obtain an alias-free Nyquist channel support region of (±ƒ d,max ,±τ max )=(±1/2n 0 T,±1/2m 0 Δƒ). The density of the pilots can then be determined from this relation given the maximum support of the channel. The pilot subgrid should extend to the whole time-frequency frame, so that the resolution of the channel is not compromised. 
     OTFS Access: Multiplexing More than One User 
     There are a variety of ways to multiplex several uplink or downlink transmissions in one OTFS frame. Here we will briefly review the following multiplexing methods:
         Multiplexing in the OTFS delay-Doppler domain   Multiplexing in the time-frequency domain   Multiplexing in the code spreading domain   Multiplexing in the spatial domain       

     1. Multiplexing in the delay-Doppler domain: This is potentially the most natural multiplexing scheme for downlink transmissions. Different sets of OTFS basis functions, or sets of information symbols or resource blocks, are given to different users. Given the orthogonality of the basis functions, the users can be separated at the UE receiver. The UE need only demodulate the portion of the OTFS frame that is assigned to it. 
     Attention is directed to  FIGS. 7A and 7B , which illustrates a pair of different exemplary basis functions capable of being used in OTFS communications. In contrast to conventional communication systems, in an OTFS system even a small subframe or resource block in the OTFS domain will be transmitted over the whole time-frequency frame via two-dimensional basis functions and will experience the average channel response.  FIGS. 7A and 7B  illustrate this point by showing two different basis functions belonging to different users. Because of this, there is no compromise on channel resolution for each user, regardless of the resource block or subframe size. 
       FIGS. 7C and 7D  collectively illustrate two-dimensional spreading with respect to both time and frequency in an OTFS communication system. As shown, one or more QAM symbols  710  may be placed within the two-dimensional information domain. Each QAM symbol within the information domain multiples a two-dimensional (i.e., time and frequency domain) basis function  720 . In a typical OTFS communication system the transmitted signal is comprised of a set of QAM symbols spread across both time and frequency by a corresponding set of two-dimensional basis functions which are mutually orthogonal with respect to both time and frequency. 
       FIG. 7E  illustrates members of one potential set of such mutually orthogonal two-dimensional basis functions. 
     In the uplink direction, transmissions from different users experience different channel responses. Hence, the different subframes in the OTFS domain will experience a different convolution channel. This can potentially introduce inter-user interference at the edges where two user subframes are adjacent, and would require guard gaps to eliminate it. In order to avoid this overhead, a different multiplexing scheme can be used in the uplink as explained next. 
     2. Multiplexing in the time frequency domain: In this approach, resource blocks or subframes are allocated to different users in the time-frequency domain.  FIG. 8  illustrates this for a three user case. As shown in  FIG. 8 , a first user (U1) occupies the whole frame length but only half the available subcarriers. A second user (U2) and a third user (U3) occupy the other half subcarriers, and divide the total length of the frame between them. 
     Notice that in this case, each user employs a slightly different version of the OTFS modulation described. One difference is that each user i performs an SFFT on a subframe (N i ,M i ), N i ≦N, M i ≦M. This reduces the resolution of the channel, or in other words reduces the extent of the time-frequency plane in which each user will experience its channel variation. On the other side, this also gives the scheduler the opportunity to schedule users in parts of the time-frequency plane where their channel is best. 
     If it is desired to extract the maximum diversity of the channel and allocate users across the whole time-frequency frame, users can be multiplexed via interleaving. In this case, one user occupies a subsampled grid of the time-frequency frame, while another user occupies another subsampled grid adjacent to it.  FIG. 9  shows the same three users as in  FIG. 8 , but interleaved on the subcarrier dimension. Of course, interleaving is possible in the time dimension as well, and/or in both dimensions. The degree of interleaving, or subsampling the grid per user is only limited by the spread of the channel that must be accommodated. 
     3. Multiplexing in the time frequency spreading code domain: Assume that it is desired to design a random access PHY and MAC layer where users can access the network without having to undergo elaborate RACH and other synchronization procedures. There is a perceived need for such a system to support Internet of Things (IoT) deployments. OTFS can support such a system by assigning each user a different two-dimensional window function that is designed as a randomizer. In this embodiment the windows of different users are designed to be nearly orthogonal to each other and nearly orthogonal to time and frequency shifts. Each user then only transmits on one or a few basis functions and uses the window as a means to randomize interference and provide processing gain. This can result in a much simplified system that may be attractive for low cost, short burst type of IoT applications. 
     4. Multiplexing in the spatial domain: Finally, like other OFDM multicarrier systems, a multi-antenna OTFS system can support multiple users transmitting on the same basis functions across the whole time-frequency frame. The users are separated by appropriate transmitter and receiver beamforming operations. 
     Exemplary Implementations of OTFS Communication Systems 
     As discussed above, embodiments of Orthogonal Time Frequency Space (OTFS) modulation are comprised of a cascade of two transformations. The first transformation maps the two dimensional plane where the information symbols reside (and which may be termed the delay-Doppler plane) to the time frequency plane. The second transformation transforms the time frequency domain to the waveform time domain where the transmitted signal is actually constructed. This transform can be thought of as a generalization of multicarrier modulation schemes. 
       FIG. 10  illustrates components of an exemplary OTFS transceiver  1000 . The OTFS transceiver  1000  can be used as one or both of the exemplary OTFS transceivers  315  illustrated in the communication system  300  of  FIG. 3 . The OTFS transceiver  1000  includes a transmitter module  1005  that includes a pre-equalizer  1010 , an OTFS encoder  1020  and an OTFS modulator  1030 . The OTFS transceiver  1000  also includes a receiver module  1055  that includes a post-equalizer  1080 , an OTFS decoder  1070  and an OTFS demodulator  1060 . The components of the OTFS transceiver may be implemented in hardware, software, or a combination thereof. For a hardware implementation, the processing units may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, other electronic units designed to perform the functions described above, and/or a combination thereof. The disclosed OTFS methods will be described in view of the various components of the transceiver  1000 . 
     Referring again to  FIG. 3 , in one aspect a method of OTFS communication involves transmitting at least one frame of data ([D]) from the transmitting device  310  to the receiving device  330  through the communication channel  320 , such frame of data comprising a matrix of up to N 2  data elements, N being greater than 1. The method comprises convolving, within the OTFS transceiver  315 - 1 , the data elements of the data frame so that the value of each data element, when transmitted, is spread over a plurality of wireless waveforms, each waveform having a characteristic frequency, and each waveform carrying the convolved results from a plurality of said data elements from the data frame [D]. Further, during the transmission process, cyclically shifting the frequency of this plurality of wireless waveforms over a plurality of times so that the value of each data element is transmitted as a plurality of cyclically frequency shifted waveforms sent over a plurality of times. At the receiving device  330 , the OTFS transceiver  315 - 2  receives and deconvolves these wireless waveforms thereby reconstructing a replica of said at least one frame of data [D]. In the exemplary embodiment the convolution process is such that an arbitrary data element of an arbitrary frame of data ([D]) cannot be guaranteed to be reconstructed with full accuracy until substantially all of these wireless waveforms have been transmitted and received. 
       FIG. 11  illustrates a comparison of bit error rates (BER) predicted by a simulation of a TDMA system and an OTFS system. Both systems utilize a 16 QAM constellation. The simulation modeled a Doppler spread of 100 Hz and a delay spread of 3 microsec. As can be seen from the graphs, the OTFS system offers much lower BER than the TDMA system for the same signal-to-noise ratio (SNR). 
     Attention is now directed to  FIG. 12 , which is a flowchart representative of the operations performed by an OTFS transceiver  1200  which may be implemented as, for example, the OTFS transceiver  1000  ( FIG. 10 ). The OTFS transceiver  1200  includes a transmitter including a modulator  1210  and a receiver including a demodulator  1220  and two-dimensional equalizer  1230 . In operation, a transmitter of the OTFS transceiver  1200  receives a two-dimensional symbol stream in the form of an N×N matrix of symbols, which may hereinafter be referred to as a TF matrix:
 
 xεC   N×N  
 
     As is illustrated in  FIG. 13 , in one embodiment the modulator  1210  functions as an orthogonal map disposed to transform the two-dimensional TF matrix to the following transmitted waveform:
 
φ t   =M ( x )=Σ x ( i,j )φ i,j  φ i,j ⊥φ k,l  
 
     Referring to  FIG. 14 , the demodulator  1220  transforms the received waveform to a two-dimensional TF matrix in accordance with an orthogonal map in order to generate an output stream:
 
φ r     y=D (φ r )
 
     In one embodiment the OTFS transceiver  1200  may be characterized by a number of variable parameters including, for example, delay resolution (i.e., digital time “tick” or clock increment), Doppler resolution, processing gain factor (block size) and orthonormal basis function. Each of these variable parameters may be represented as follows. 
     Delay resolution (digital time tick): 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               T 
             
             ∈ 
             
               
                 R 
                 
                   &gt; 
                   0 
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     T 
                   
                   = 
                   
                     1 
                     Bw 
                   
                 
                 ) 
               
             
           
         
       
     
     Doppler resolution: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               F 
             
             ∈ 
             
               
                 R 
                 
                   &gt; 
                   0 
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     F 
                   
                   = 
                   
                     1 
                     Trans 
                   
                 
                 ) 
               
             
           
         
       
     
     Processing gain factor (block size):
 
 N&gt; 0
 
     Orthonormal basis of C N×1  (spectral shapes):
 
 U={u   1   ,u   2   , . . . ,u   N }
 
     As is illustrated by  FIG. 12 , during operation the modulator  1210  takes a TF matrix xεC N×N  and transforms it into a pulse waveform. In one embodiment the pulse waveform comprises a pulse train defined in terms of the Heisenberg representation and the spectral shapes: 
     
       
         
           
             
               ϕ 
               t 
             
             = 
             
               
                 M 
                 ⁡ 
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   
                     
                       Π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         x 
                         ) 
                       
                       ⁢ 
                       
                         u 
                         1 
                       
                     
                     
                       ︸ 
                       
                         b 
                         1 
                       
                     
                   
                   , 
                   
                     
                       Π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         x 
                         ) 
                       
                       ⁢ 
                       
                         u 
                         2 
                       
                     
                     
                       ︸ 
                       
                         b 
                         2 
                       
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     
                       Π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         x 
                         ) 
                       
                       ⁢ 
                       
                         u 
                         N 
                       
                     
                     
                       ︸ 
                       
                         b 
                         N 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     where b 1 , b 2  . . . b N  are illustrated in  FIG. 15  and where, in accordance with the Heisenberg relation:
 
Π( h*x )=Π( h )·Π( x ) in particular:
 
Π(δ (τ,n)*   x )= L   t ·Π( x )
 
Π(δ (0,w)   *x )= M   w ·Π( x )
 
     The Heisenberg representation provides that: 
                   Π   ⁢     :     ⁢           ⁢     C     N   ×   N         ⁢     →   ∝     ⁢       C     N   ×   N       ⁢           ⁢   given   ⁢           ⁢   by   ⁢     :     ⁢           ⁢     Π   ⁡     (   x   )           =       ∏     τ   ,     w   =   0         N   -   1       ⁢           ⁢       x   ⁡     (     τ   ,   w     )       ⁢     M   w     ⁢     L   τ           ,     x   ∈     C     N   ×   N               
where L t  and M w  are respectively representative of cyclic time and frequency shifts and may be represented as:
 
     
       
         
           
             
               
                 
                   L 
                   τ 
                 
                 ∈ 
                 
                   
                     C 
                     
                       N 
                       × 
                       N 
                     
                   
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       L 
                       τ 
                     
                     ⁡ 
                     
                       ( 
                       φ 
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     t 
                     ) 
                   
                 
               
               = 
               
                 φ 
                 ⁡ 
                 
                   ( 
                   
                     t 
                     + 
                     τ 
                   
                   ) 
                 
               
             
             , 
             
               τ 
               = 
               
                 0 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 … 
               
             
             ⁢ 
             
                 
             
             , 
             
               
                 
                   N 
                   - 
                   
                     1 
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       M 
                       w 
                     
                   
                 
                 ∈ 
                 
                   
                     C 
                     
                       N 
                       × 
                       N 
                     
                   
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       M 
                       w 
                     
                     ⁡ 
                     
                       ( 
                       φ 
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     t 
                     ) 
                   
                 
               
               = 
               
                 
                   e 
                   
                     
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         π 
                       
                       N 
                     
                     ⁢ 
                     wt 
                   
                 
                 ⁢ 
                 
                   φ 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
               
             
             , 
             
               w 
               = 
               0 
             
             , 
             … 
             ⁢ 
             
                 
             
             , 
             
               N 
               - 
               1 
             
           
         
       
     
     The demodulator  1220  takes a received waveform and transforms it into a TF matrix yεC N×N  defined in terms of the Wigner transform and the spectral shapes: 
     
       
         
           
             
               ϕ 
               r 
             
             = 
             
               ( 
               
                 
                   b 
                   1 
                 
                 , 
                 
                   b 
                   2 
                 
                 , 
                 … 
                 ⁢ 
                 
                     
                 
                 , 
                 
                   b 
                   N 
                 
               
               ) 
             
           
         
       
       
         
           
             
               y 
               ⁡ 
               
                 ( 
                 
                   τ 
                   , 
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                 ) 
               
             
             = 
             
               
                 
                   D 
                   ⁡ 
                   
                     ( 
                     
                       ϕ 
                       r 
                     
                     ) 
                   
                 
                 ⁢ 
                 
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                     τ 
                     , 
                     w 
                   
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               = 
               
                 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         〈 
                         
                           
                             
                               M 
                               w 
                             
                             ⁢ 
                             
                               L 
                               τ 
                             
                             ⁢ 
                             
                               u 
                               n 
                             
                           
                           , 
                           
                             b 
                             n 
                           
                         
                         〉 
                       
                     
                   
                   ︷ 
                 
                 
                   Wigner 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   transform 
                 
               
             
           
         
       
     
     Main property of M and D (Stone von Neumann theorem):
 
 D ( h   0   M ( x ))= h*x  where:
 
 h (τ, w )≈ a (τΔ T,wΔF )
 
     As illustrated in  FIG. 16 , the equalizer  1230  may be implemented as a two-dimensional decision feedback equalizer configured to perform a least means square (LMS) equalization procedure such that:
 
 y     {circumflex over (x)} 
 
The equalizer  1230  leverages the feature of OTFS that, in the time-frequency plane, all bits experience the same distortion during propagation through the channel. By “de-blurring” the bits using the same distortion experienced by the bits in the channel, the signal transmitted into the channel is revealed.
 
Transmitter Grid and Receiver Bin Structure
 
     Attention is now directed to  FIGS. 17A-17D , which depict an OTFS transmitter  102  and receiver  104  to which reference will be made in describing the transmission and reception of OTFS waveforms. More specifically,  FIGS. 17B-17D  illustrate the transmission and reception of OTFS waveforms relative to a time-frequency transmitter grid or series of bins and a corresponding time-frequency receiver grid or series of bins. As will be discussed below, the receiver  104  will generally operate with respect to a time-frequency receive grid of a finer mesh than that of the time-frequency transmit grid associated with the transmitter  102 . 
     Turning now to  FIG. 17A , the transmitter  102  and receiver  104  are separated by an impaired wireless data channel  100  including one or more reflectors  106 . As shown, the reflectors  106  may reflect or otherwise impair waveforms ( 112 ,  114   a ,  114   b ) as they travel through the data channel  100 . These reflectors may be inherently represented by the two-dimensional (2D) channel state of the channel  100  (see, e.g., the finite channel h eqv,f  of  FIG. 18 ). 
       FIGS. 18A and 18B  illustratively represent OTFS communication over a communication channel characterized by a two-dimensional delay-Doppler impulse response. The set of bar diagrams of  FIGS. 18A and 18B  may represent a two-dimensional impulse response realizing a finite modulation equivalent channel, a transmitted information vector x comprised of OTFS QAM symbols and a received information vector y comprised of received OTFS symbols. One advantage of embodiments of OTFS communication systems suggested by  FIGS. 18 a    and  18 B is that the two-dimensional channel model is stationary and all symbols experience the same distortion. As a consequence of being stationary for the necessary time duration, the OTFS channel model is non-fading and every symbol may be spread across the full duration and bandwidth and experiences all diversity branches of the channel. The deterministic nature of the two-dimensional OTFS channel reflects the geometry of the channel reflectors (distance and velocity). 
     In one embodiment the transmitter  102  includes a transmitter processor  102   p  to package input data into at least one N×M array of data symbols. An encoding process is then used to transmit this array of data symbols in accordance with the OTFS modulation techniques described herein. The transmitted OTFS waveforms are received by a receiver  104 , which includes a receiver processor  104   p . In one embodiment the receiver processor  104   p  utilizes information pertaining to the 2D state of the channel  100  to enable these OTFS waveforms to be decoded and recover the transmitted data symbols. Specifically, the receiver processor  104   p  may use an inverse of the OTFS encoding process to decode and extract this plurality of data symbols. Alternatively the correction of signals for data channel impairments can be done after the receiver has decoded and extracted the plurality of data symbols. 
     In some embodiments OTFS data transmission may be implemented by transforming the input N×M array of data symbols into at least one block or array of filtered OFDM symbols. This can be done, for example, using one dimensional Fourier transforms and a filtering process or algorithm. This block or array of filtered OFDM symbols may then be transformed into at least one block or array of OTFS symbols using various types of two dimensional Fourier transforms. These results will typically be stored in transmitter memory  102   m . The stored results can then be communicated over wireless frequency sub-bands by various methods. For example, in one embodiment a transmitter  102   c  that employs a series of M narrow-band filter banks may be utilized. In this implementation the transmitter  102   c  produces a series of M mutually orthogonal waveforms transmitted over at least N time intervals. 
     In one embodiment gaps or “guard bands” in both time and frequency may be imposed to minimize the possibility of inadvertent cross talk between the various narrow-band filters and time intervals prior to transmission. Depending on the characteristics of the data channel, any such gaps or guard bands can be increased or decreased or set to zero as situations warrant. 
     Alternatively, the OTFS encoding process may encode the N×M array of data symbols onto a manifold compatible with symplectic analysis. The symbols may be distributed over a column time axis of length T and row frequency axis of length F, thereby producing at least one information manifold for storage in transmitter memory  102   m.    
     The information manifold effectively holds information corresponding to the input data symbols in a form enabling them to be subsequently transformed in accordance with the desired OTFS transformation operation such as, for example, a symplectic 2D Fourier transform, a discrete symplectic 2D Fourier transform, a finite symplectic Fourier transform, and the like. In certain embodiments the data symbols may also be spread prior to being held within an information manifold. 
     The OTFS processor  102   p  may then transform the information manifold according to a 2D symplectic Fourier transform. This transformation may be effected using any of the previously discussed symplectic 2D Fourier transforms, discrete symplectic 2D Fourier transforms, and finite symplectic Fourier transforms. This operation produces at least one 2D Fourier transformed information manifold, which may be stored in transmitter memory  102   m.    
     The OTFS transmitter  102   c  will typically transmit this at least one 2D Fourier transformed information manifold as a series of “M” simultaneous narrow band waveforms, each series over consecutive time intervals, until the entire 2D Fourier transformed information manifold has been transmitted. For example, the transmitter processor  102   p  can operate, often on a one column at a time basis, over all frequencies and times of this 2D Fourier transformed information manifold. The transmitter processor  102   p  can select a given column at location n (where n can vary from 1 to N) and transmit a column with a width according to a time slice of duration proportional to Tμ, where μ=1/N. Those frequencies in the column slice of this 2D Fourier transformed information manifold (e.g. frequencies corresponding to this transmitting time slice) may then be passed through a bank of at least M different, nonoverlapping, narrow-band frequency filters. This produces M mutually orthogonal waveforms. The processor  102   p  can then cause these resulting filtered waveforms to be transmitted, over different transmitted time intervals (e.g. one column at a time), as a plurality of at least M mutually orthogonal waveforms until an entire 2D Fourier transformed information manifold has been transmitted. 
     In one embodiment gaps or “guard bands” in both time and frequency may be imposed to minimize the possibility of inadvertent cross talk between the various narrow-band filters and time intervals prior to transmission. Depending on the characteristics of the data channel, any such gaps or guard bands can be increased or decreased or set to zero as situations warrant. 
     Each OTFS receiver  104  may then receive a channel-convoluted version of the 2D Fourier transformed information manifold transmitted by the transmitter  102 . Due to distortions introduced by the channel  100 , the M narrow band waveforms originally transmitted at M original frequencies may now comprise more than M narrow band waveforms at a different range of frequencies. Moreover, due to transmitted OTFS waveforms impinging various reflectors  106 , the originally transmitted signals and reflections thereof may be received at different times. As a consequence, each receiver  104  will generally supersample or oversample the various received waveforms on a time-frequency grid having a finer mesh than that associated with the transmitter  102 . This oversampling process is represented by  FIGS. 17B-17D , which depict a receiver time-frequency grid having smaller time and frequency increments than the transmitter OTFS grid. 
     Each OTFS receiver  104  operates to receive the transmitted 2D Fourier transformed information manifold over time slices having durations that are generally less than or equal to the transmission time intervals employed by the transmitter  102 . In one embodiment the receiver  104  analyzes the received waveforms using a receiving bank of at least M different, non-overlapping, narrow-band frequency filters. The receiver will then generally store the resoling approximation (channel convoluted version) of the originally transmitted 2D Fourier transformed information manifold in receiver memory  104   m.    
     Once the waveforms transmitted by the transmitter  102  have been received, the receiver  104  then corrects for the convolution effect of the channel  100  in order to facilitate recovery of an estimate of the originally transmitted data symbols. The receiver  104  may effect these corrections in a number of ways. 
     For example, the receiver  104  may use an inverse of the 2D symplectic Fourier transform used by the transmitter  102  to transform the received waveforms into an initial approximation of the information manifold originally transmitted. Alternatively, the receiver  104  may first use information pertaining to the 2D channel state to correct the channel-convoluted approximation of the transmitted 2D Fourier transformed information manifold (stored in receiver memory). Following this correction the receiver  104  may then use the inverse of the 2D symplectic Fourier transform employed at the transmitter  102  to generate a received information manifold and subsequently extract estimated data symbols. 
     Although the OTFS methods described herein inherently spread any given data symbol over the entire time-frequency plane associated with a transmitter, in some embodiments it may be useful to implement an additional spreading operation to insure that the transmitted data symbols are uniformly distributed. This spreading operation may be carried out by the transmitter processor  102   p  either prior to or after encoding the input N×M 2D array of data symbols onto the symplectic analysis compatible manifold. A number of spreading functions such as, for example, a 2D chirp operation, may be used for this purpose. In the event such a spreading operation is implemented at the transmitter  102 , the receiver  104  will utilize an inverse of this spreading operation in order to decode and extract the data symbols from the various received information manifolds. 
       FIG. 19  illustrates transmission of a 2D Fourier transformed information manifold represented by an N×M structure over M frequency bands during N time periods of duration Tμ. In this example, each of the M frequency bands is represented by a given row and each different time period is represented by a given column. In the embodiment of  FIG. 19  it is assumed that the OTFS transmitter is configured to transmit OTFS signals during without guard intervals over the allocated bandwidth, which encompasses the M frequency bands. The bandwidth (ω 0 ) of each of the M frequency bands is, is 1/Tμ. Accordingly, if it is desired to transmit all N columns of information over a minimum time interval of N*Tμ, then M must have a bandwidth no larger than 1/Tμ and the bandwidth used by all M filtered OTFS frequency bands cannot exceed M/T, where T is the total amount of time used to transmit all N columns of the 2D Fourier transformed information manifold. 
     At the receiver  104 , the various 2D Fourier transformed information manifolds may be received using banks of different, nonoverlapping, narrow-band frequency filters that are generally similar to those used by the transmitter  102 . Again, the receiver time slices and receiving banks of filters will generally operate with finer granularity; that is, the receiver will typically operate over smaller frequency bandwidths, and shorter time slices, but over a typically broader total range of frequencies and times. Thus the receiver bin structure will preferably oversample the corresponding transmitting time slices and transmitting banks of different, non-overlapping, narrow-band frequency filters previously used by the transmitter. 
     As may be appreciated with reference to  FIG. 19 , the OTFS transmitter will typically transmit the resulting filtered waveforms (in this example over all rows and successive columns), until the entire 2D Fourier transformed information manifold has been transmitted. However the transmitter can either transmit the successive columns (time slices) continuously and contiguously that is without any time gaps in-between, as more of a series of continuous longer duration waveforms, or alternatively the transmitter can put some time spacing between the various successive columns, thus creating a more obvious series of waveform bursts. 
     Stated differently, the transmitter can transmit the resulting filtered waveforms as either: 1) a plurality of at least M simultaneously transmitted mutually orthogonal waveforms over either different consecutive transmitted time intervals; or 2) a plurality OTFS data or OTFS pilot bursts comprising at least M simultaneously transmitted mutually orthogonal waveform bursts over different transmitted intervals separated by at least one spacer time interval. 
       FIG. 20  shows an example of the M filtered OTFS frequency bands being simultaneously transmitted according to various smaller time slices Tμ. The repeating curved shapes show the center frequency for each filtered band according to g(t·e jkω     0   ). One of the transmitted bins of frequency bandwidth, which is of size 1/T and time duration T*μ, is shown in more detail. Again, as previously discussed, in a preferred embodiment the OTFS receiver will use oversampling, and thus use finer granularity bins that nonetheless may extend over a broader range of times and frequencies so as to catch signals with high degrees of delay or Doppler frequency shift. 
     Stated differently, in some embodiments, the non-overlapping, narrow-band frequency filters used at the transmitter may be configured to pass frequencies from the various 2D Fourier transformed Information manifolds that are proportional to a filter function g(t·e jkω     0   ), where j is the square root of −1, t corresponds to a given time slice of duration Tμ chosen from a 2D Fourier transformed information manifold, and k corresponds to a given row position in a given 2D Fourier transformed information manifold, where k varies between 1 and M. In this example, the bandwidth, ω 0 , in frequency units Hz, can be proportional to 1/T, and T=M/(allowed wireless bandwidth). 
     As may be appreciated from  FIGS. 19 and 20 , the various 2D Fourier transformed information manifolds can have overall dimensions NTμ according to a time axis and M/T according to a frequency axis, and each “cell” or “bin” in the various 2D Fourier transformed information manifold may have overall dimensions proportional to Tμ according to a time axis and 1/T according to a frequency axis. 
       FIG. 21  provides another example of OTFS waveforms being transmitted according to various smaller time slices Tμ. In the illustration of  FIG. 21  the amplitude or extent of modulation of the various waveforms as a function of time is also shown. 
     In some embodiments it may be useful to modulate the transmitted wireless OTFS waveforms using an underlying modulation signal that allows the receiver to distinguish where, on the original 2D time and frequency grid, a given received signal originated. This may, for example, assist an OTFS receiver in distinguishing the various types of received signals, and in distinguishing direct signals from various time delayed and/or frequency shifted reflected signals. In these embodiments grid, bin, or lattice locations of the originally transmitted OTFS waveforms may be distinguished by determining time and frequency related parameters of the received waveforms. For example, in the presently discussed “symplectic” implementations, where each “row” of the 2D Fourier transformed information manifold is passed through a narrow band filter that operates according to parameters such as g(t·e jkω     0   ), the “kω 0 ” term may enable the receiver to distinguish any given incoming OTFS waveform by its originating “column” location “t”. In this case the receiver should also be able to determine the bin (grid, lattice) location of the various received waveforms by determining both the t (time related) and k (frequency related) values of the various received waveforms. These values may then be used during subsequent deconvolution of the received signals. 
     If further distinguishability of the bin (grid lattice) originating time and frequency origins of the received OTFS signals is desired, then an additional time and/or frequency varying modulation scheme may also be imposed on the OTFS signals, prior to transmission, to allow the OTFS receiver to further distinguish the bin (grid, lattice) origin of the various received signals. 
     In alternative embodiments either the information manifold or the 2D Fourier transformed information manifolds may be sampled and modulated using Dirac comb methods. The Dirac combs utilized by these methods may be, for example, a periodic tempered distribution constructed from Dirac delta functions. 
     Attention is now directed to  FIG. 22 , which provides a block diagrammatic representation of an exemplary process  2200  of OTFS transmission and reception in accordance with the present disclosure. The process  2200  begins with the packaging of data for transmission and its optional precoding to correct for known channel impairments (stage  2210 ). This material is then processed by a 2D Fourier Transform (such as a symplectic Fourier transform, discrete symplectic Fourier transform, or finite symplectic Fourier transform) (stage  2220 ). Following this processing the results are then passed through a filter bank (FB) and transmitted over a series of time intervals Tμ (stage  2230 ). The transmitted wireless OTFS waveforms then pass through the communications or data channel (C), where they are subject to various distortions and signal impairments (stage  2240 ). At the receiver, the received waveforms are received according to a filter bank at various time intervals (stage  2250 ). The receiver filter bank (FB*) may be an oversampled filter bank (FB*) operating according to oversampled time durations that may be a fraction of the original time intervals Tμ. This oversampling enables the received signals to be better analyzed for channel caused time delays and frequency shifts at a high degree of resolution. At a stage  2260  the received material is analyzed by an inverse 2D Fourier Transform (2D-FT s ) (which again may be an inverse symplectic Fourier transform, inverse discrete symplectic Fourier transform, or inverse finite symplectic Fourier transform). The results may then be further corrected for channel distortions using, for example, 2D channel state information (stage  2270 ). In other embodiments stage  2270  may precede stage  2260 . 
     Further Mathematical Characterization of OTFS Modulation and Derivation of the Two-Dimensional (2D) Channel Model 
     In what follows we further develop the OTFS communication paradigm focusing on the central role played by the Heisenberg representation and the two dimensional symplectic Fourier transform. A principal technical result of this development is a rigorous derivation of the OTFS two-dimensional channel model. 
     0. Introduction 
     Orrthogonal time frequency space is a novel modulation scheme capable of being implemented by communication transceivers that converts the dynamic one dimensional wireless medium into a static two dimensional local ISI channel by putting the time and frequency dimensions on an equal footing. Among the primary benefits of an OTFS transceiver relative to a conventional transceiver are the following: 
     1. Fading. Elimination of fading both time and frequency selective. 
     2. Diversity. Extraction of all diversity branches in the channel. 
     3. Stationarity. All symbols experience the same distortion. 
     4. CSI. Perfect and efficient channel state information (CSI). 
     In a sense, the OTFS transceiver establishes a virtual wire through a communication medium, thus allowing the application of conventional wired DSP technologies in the wireless domain. Embodiments of the OTFS transceiver are based on principles from representation theory, generalizing constructions from classical Fourier theory. On the operational level, OTFS may be roughly characterized as an application of the two dimensional Fourier transform to a block of filtered OFDM symbols. OTFS is a true two dimensional time-frequency modulation, and may incorporate both two dimensional time-frequency filtering and two dimensional equalization techniques. In what follows we provide a formal mathematical development of the OTFS transceiver, focusing on a rigorous derivation of the two dimensional channel model. 
     OTFS and Lattices 
     We first choose an undersampled time-frequency lattice, that is, a two dimensional lattice of density smaller or equal than 1. The undersampling condition is essential for perfect reconstruction, however, it seems to limit the delay-Doppler resolution of the channel acquisition. In contrast, radar theory amounts to choosing an oversampled time frequency lattice of density greater or equal than 1 where the oversampling condition is essential for maximizing the delay-Doppler resolution of target measurement. As it turns out, the symplectic (two dimensional) Fourier transform intertwines between communication and radar lattices. The OTFS communication paradigm is to multiplex information symbols on an oversampled high resolution radar lattice and use the symplectic Fourier transform together with two dimensional filtering to convert back to communication coordinates. This allows OTFS to reap the benefits of both worlds—high resolution delay-Doppler channel state measurement without sacrificing spectral efficiency. In particular, the OTFS channel model may be thought of as a high resolution delay-Doppler radar image of the wireless medium. 
     The Wireless Channel 
     In order to understand OTFS, it is beneficial to understand the wireless channel as a mathematical object. Let H=L 2  (R) denote the vector space of “physical” waveforms defined on the time domain. The physics of the wireless medium is governed by the phenomena of multipath reflection, that is, the transmitted signal is propagating through the atmosphere and reflected from various objects in the surrounding. Some of the objects, possibly including the transmitter and the receiver, are moving at a non-zero velocity. Consequently, (under some mild “narrow band” assumption) the received signal is a superposition of time delays and Doppler shifts of the transmitted signal where the delay in time is caused by the excess distance transversed by the reflected waveform and the Doppler shift is caused by the relative velocity between the reflector and the transmitting and/or receiving antennas. Mathematically, this amounts to the fact that the wireless channel can be expressed as a linear transformation C:H→H realized as a weighted superposition of multitude of time delays and Doppler shifts, namely: 
                         C   ⁡     (   φ   )       ⁢     (   t   )       =         ∫   ∫       τ   ,   v       ⁢     h   ⁡     (     τ   ,   v     )       ⁢     e     2   ⁢   π   ⁢           ⁢     iv   ⁡     (     t   -   τ     )           ⁢     φ   ⁡     (     t   -   τ     )       ⁢   d   ⁢           ⁢   τ   ⁢           ⁢   dv       ,           (   0.1   )               
for every transmit waveform φεH. From Equation (0.1) one can see that the channel C is determined by the function h that depends on two variables τ and ν, referred to as delay and Doppler. The pair (τ,ν) can be viewed as a point in the plane V=R 2 , referred to as the delay Doppler plane. Consequently, h is a kind of a two dimensional (delay Doppler) impulse response characterizing the wireless channel. However, one should keep in mind that this terminology is misleading since the action of h given by (0.1) is not a convolution action.
 
     Fading 
     One basic physical phenomena characteristic to the wireless channel is fading. The phenomena of fading corresponds to local attenuation in the energy profile of the received signal as measured over a specific dimension. It is customary to consider two kind of fadings: time selective fading and frequency selective fading. The first is caused by destructive superposition of Doppler shifts and the second is caused by destructive superposition of time delays. Since the wireless channel consists of combination of both time delays and Doppler shifts it exhibits both types of fading. Mitigating the fading phenomena is a significant motivation behind the development of the OTFS transceiver. 
     The Heisenberg Representation 
     One key observation is that the delay Doppler channel representation given in Equation (0.1) is the application of a fundamental mathematical transform, called the Heisenberg representation, transforming between functions on the delay Doppler plane V and linear operators on the signal space H. To see this, let us denote by L τ  and M ν  are the operations of time delay by τ and Doppler shift by ν respectively, that is:
 
 L   τ (φ)( t )=φ( t −τ),
 
 M   ν (φ)( t )= e   2πiνt φ( t ),
 
     for every φεH. Using this terminology, we can rewrite channel equation (0.1) in the following form: 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         φ 
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ∫ 
                           ∫ 
                         
                         
                           τ 
                           , 
                           v 
                         
                       
                       ⁢ 
                       
                         h 
                         ⁡ 
                         
                           ( 
                           
                             τ 
                             , 
                             v 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         L 
                         τ 
                       
                       ⁢ 
                       
                         
                           M 
                           τ 
                         
                         ⁡ 
                         
                           ( 
                           φ 
                           ) 
                         
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       τ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       dv 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               ∫ 
                               ∫ 
                             
                             
                               τ 
                               , 
                               v 
                             
                           
                           ⁢ 
                           
                             h 
                             ⁡ 
                             
                               ( 
                               
                                 τ 
                                 , 
                                 v 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             L 
                             τ 
                           
                           ⁢ 
                           
                             M 
                             v 
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           dv 
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ( 
                           φ 
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.2 
                   ) 
                 
               
             
           
         
       
     
     Let us define the Heisenberg representation to be the transform taking a function a:V→C to the linear operator Π(a):H→H, given by: 
     
       
         
           
             
               
                 
                   
                     Π 
                     ⁡ 
                     
                       ( 
                       a 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         ∫ 
                       
                       
                         τ 
                         , 
                         v 
                       
                     
                     ⁢ 
                     
                       a 
                       ⁡ 
                       
                         ( 
                         
                           τ 
                           , 
                           v 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       L 
                       τ 
                     
                     ⁢ 
                     
                       M 
                       v 
                     
                     ⁢ 
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     τ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       dv 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   0.3 
                   ) 
                 
               
             
           
         
       
     
     We refer to the function a as the delay Doppler impulse response of the operator Π(a). Taking this perspective, we see that the wireless channel is an application of the Heisenberg representation to a specific function h on the delay Doppler plane. This higher level of abstraction establishes the map Π as the fundamental object underlying wireless communication. In fact, the correspondence a Π( a ) generalizes the classical correspondence between a stationary linear system and a one dimensional impulse response to the case of arbitrary time varying systems (also known as linear operators). In this regard, the main property of the Heisenberg representation is that it translates between composition of linear operators and an operation of twisted convolution between the corresponding impulse responses. In more details, if:
 
 A =Π( a ),
 
 B =Π( b ),
 
     then we have:
 
 A∘B =Π( a*   t   b ),  (0.4)
 
     where * t  is a non commutative twist of two dimensional convolution. Equation (0.4) is key to the derivation of the two dimensional channel model—the characteristic property of the OTFS transceiver. 
     The OTFS Transceiver and the 2D Channel Model 
     The OTFS transceiver provides a mathematical transformation having the effect of converting the fading wireless channel into a stationary two dimensional convolution channel. We refer to this property as the two dimensional channel model. 
     Formally, the OTFS transceiver may be characterized as a pair of linear transformations (M,D) where M is termed a modulation map and D is termed a demodulation map and is the inverse of M. According to the OTFS paradigm the information bits are encoded as a complex valued function on V which periodic with respect to a lattice Λ ⊥ ⊂V called the reciprocal communication lattice. Note that the term “reciprocal” is used to suggest a type of duality relation between Λ ⊥  and a more conventional lattice Λ, called the primal communication lattice. If we denote by C(V) Λ     ⊥    the vector space of Λ ⊥ -periodic functions on V then the OTFS modulation is a linear transformation:
 
 M:C ( V ) Λ     ⊥     →H.   (0.5)
 
     Geometrically, one can think of the information as a function on a two dimensional periodic domain (a donut) obtained by folding V with respect to the lattice Λ ⊥ . Respectively, the demodulation map is a linear transformation acting in the opposite direction, namely:
 
 D:H→C ( V ) Λ     ⊥   .  (0.6)
 
     The precise mathematical meaning of the two dimensional channel model is that given an information function xεC(V) Λ     ⊥   , we have:
 
 D∘C∘M ( x )= c*x,   (0.7)
 
     where * stands for periodic convolution on the torus and the function c is a periodization with respect to the reciprocal lattice Λ ⊥  of the delay Doppler impulse response h of the wireless channel, that is:
 
 c =per Λ     ⊥   ( h )  (0.8)
 
     Equations (0.7) and (0.8) encodes the precise manner of interaction between the OTFS transceiver and the wireless channel. 
     The remainder of this explanation of OTFS method and the OTFS transceiver is organized as follows: 
     Section 1 discusses several basic mathematical structures associated with the delay Doppler plane V. We begin by introducing the symplectic form on V which is an antisymmetric variant of the more familiar Euclidean form used in classical signal processing. We than discuss lattices which are two dimensional discrete subdomains of V. We focus our attention to the construction of the reciprocal lattice. The reciprocal lattice plays a pivotal role in the definition of the OTFS transceiver. We than proceed to discuss the dual object of a lattice, called a torus, which is a two dimensional periodic domain obtained by folding the plain with respect to a lattice. 
     Section 2 discusses the symplectic Fourier transform, which is a variant of the two dimensional Fourier transform defined in terms of the symplectic form on V. We discuss three variants of the symplectic Fourier transform: the continuos, the discrete and the finite. We explain the relationships between these variants. 
     Section 3 discusses the Heisenberg representation and its inverse—the Wigner transform. In a nutshell, the Heisenberg representation is the structure that encodes the precise algebraic relations between the operations of time delay and Doppler shifts. We relate the Wigner transform to the more familiar notions of the ambiguity function and the cross ambiguity function. We conclude with a formulation of the fundamental channel equation. 
     Section 4 discusses the continuos variant of the OTFS transceiver. We begin by specifying the parameters defining the OTFS transceiver. Then we proceed to define the modulation and demodulation maps. We conclude the section with a derivation of the two dimensional channel model from first principles. 
     Section 5 discusses the finite variant of the OTFS transceiver. In a nutshell, the finite variant is obtained from the continuos variant by sampling the reciprocal torus along finite uniformly distributed subtorus. We define the finite OTFS modulation and demodulation maps. We then formulate the finite version of the two dimensional channel model, explaining the finite two dimensional impulse response is the restriction of the continuos one to the finite subtorus. We conclude this section with an explicit interpretation of the modulation formula in terms of classical DSP operations. 
     1. The Delay-Doppler Plane 
     1.1 The symplectic plane 
     The delay Doppler plane is a two dimensional vector space over the real numbers. Concretely, we take V=R 2  where the first coordinate is delay, denoted by r and the second coordinate is Doppler, denoted by ν. The delay Doppler plane is equipped with an intrinsic geometric structure encoded by a symplectic form (also called symplectic inner product or symplectic pairing). The symplectic form is a pairing ω:V×V→R defined by the determinant formula: 
     
       
         
           
             
               
                 
                   
                     
                       ω 
                       ⁡ 
                       
                         ( 
                         
                           
                             v 
                             ′ 
                           
                           , 
                           v 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           det 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   τ 
                                 
                                 
                                   
                                     τ 
                                     ′ 
                                   
                                 
                               
                               
                                 
                                   v 
                                 
                                 
                                   
                                     v 
                                     ′ 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         
                           v 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             τ 
                             ′ 
                           
                         
                         - 
                         
                           τ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             v 
                             ′ 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1.1 
                   ) 
                 
               
             
           
         
       
     
     where ν=(τ,ν) and ν′=(τ′,ν′). Note that the symplectic form, in contrast to its Euclidean counterpart, is anti-symmetric, namely ω(ν,ν′)=ω(ν,ν′) for every ν,ν′εV. Consequently, the symplectic product of a vector with itself is always equal zero, that is ω(ν,ν)=0, for every νεV. As it turns out, the fabric of time and frequency is governed by a symplectic structure. 
     1.1.1 Functions on the plane 
     We denote by C(V) the vector space of complex valued functions on V. We denote by * the operation of linear convolution of functions on V. Given a pair of functions ƒ,gεC(V), their convolution is defined by: 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       * 
                       
                         g 
                         ⁡ 
                         
                           ( 
                           v 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           
                             
                               
                                 v 
                                 1 
                               
                               + 
                               
                                 v 
                                 2 
                               
                             
                             = 
                             v 
                           
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           ∫ 
                           
                             
                               v 
                               ′ 
                             
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 ′ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 - 
                                 
                                   v 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             dv 
                             ′ 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1.2 
                   ) 
                 
               
             
           
         
       
     
     for every νεV. 
     1.2 Lattices 
     A lattice Λ⊂V is a commutative subgroup isomorphic to Z 2  defined as follows: 
     
       
         
           
             
               Λ 
               = 
               
                 
                   
                     Zv 
                     1 
                   
                   ⊕ 
                   
                     Zv 
                     2 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         av 
                         1 
                       
                       + 
                       
                         
                           bv 
                           2 
                         
                         ⁢ 
                         
                           : 
                         
                         ⁢ 
                         a 
                       
                     
                     , 
                     
                       b 
                       ∈ 
                       Z 
                     
                   
                   } 
                 
               
             
             , 
           
         
       
     
     where ν 1 ,ν 2 εV are linear independent vectors. In words, Λ consists of all integral linear combinations of the vectors ν 1  and ν 2 . See  FIG. 23 . The vectors ν 1  and ν 2  are called generators of the lattice. The volume of Λ is, by definition, the volume of a fundamental domain. One can show that:
 
vol(Λ)=|ω(ν 1 ,ν 2 )|.  (1.3)
 
     when vol(Λ)≧1 the lattice is called undersampled and when vol(Λ)≦1 the lattice is called oversampled. Finally, in case vol(Λ)=1 the lattice is called critically sampled. 
     Example 1.1 (Standard communication lattice). Fix parameters T≧0 and μ≧1. Let: 
     
       
         
           
             
               
                 
                   
                     Λ 
                     
                       T 
                       , 
                       μ 
                     
                   
                   = 
                   
                     
                       
                         ZT 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         μ 
                       
                       ⊕ 
                       
                         Z 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           / 
                           T 
                         
                       
                     
                     = 
                     
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     K 
                                     · 
                                     T 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   μ 
                                 
                                 + 
                                 
                                   L 
                                   · 
                                   
                                     1 
                                     / 
                                     T 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               : 
                             
                             ⁢ 
                             K 
                           
                           , 
                           
                             L 
                             ∈ 
                             Z 
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   1.4 
                   ) 
                 
               
             
           
         
       
     
     We have that vol(Λ T,μ )=μ. We refer to Λ T,μ  as the standard communication lattice. 
     1.2.1 Reciprocal lattice. Given a lattice Λ⊂V, its orthogonal complement lattice is defined by:
 
Λ ⊥   ={νεV :ω(ν,λ)ε Z  forevery λεΛ}.  (1.5)
 
     In words, Λ ⊥  consists of all vectors in V such that their symplectic pairing with every vector in Λ is integral. One can show that Λ ⊥  is indeed a lattice. We refer to Λ ⊥  as the reciprocal lattice of Λ. One can show that:
 
vol(Λ ⊥ )=1/vol(Λ),  (1.6)
 
which implies that Λ is undersampled if and only if Λ ⊥  is oversampled. This means that reciprocity interchanges between coarse (undersampled) lattices and fine (oversampled) lattices. Another attribute concerns how lattice inclusion behaves under reciprocity. Given a pair consisting of a lattice Λ⊂V and a sublattice Λ 0 ⊂Λ, one can show that the the inclusion between the reciprocals is reversed, that is:
 
Λ ⊥ ⊂Λ 0   ⊥ .  (1.7)
 
     Example 1.2 Consider the standard communication lattice Λ T,μ . Its reciprocal is given by:
 
(Λ T,μ ) ⊥   =ZT⊕Z 1/ Tμ.   (1.8)
 
     See  FIGS. 24A and 24B , which respectively illustrate a standard communication lattice and the reciprocal of the standard communication lattice. Indeed, we have that: 
     
       
         
           
             
               ω 
               ⁡ 
               
                 ( 
                 
                   
                     [ 
                     
                       
                         
                           KT 
                         
                       
                       
                         
                           
                             
                               L 
                               / 
                               T 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           
                             
                               K 
                               ′ 
                             
                             ⁢ 
                             T 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                         
                       
                       
                         
                           
                             
                               L 
                               ′ 
                             
                             / 
                             T 
                           
                         
                       
                     
                     ] 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   LK 
                   ′ 
                 
                 - 
                 
                   KL 
                   ′ 
                 
               
               ∈ 
               
                 Z 
                 . 
               
             
           
         
       
     
     Note that vol(Λ T,μ ) ⊥ =1/μ which means that as the primal lattice becomes sparser, the reciprocal lattice becomes denser. 
     1.2.2 Functions on a lattice. We denote by C(Λ) the vector space of complex valued functions on the lattice. We denote by R Λ :C(V)→C(Λ) the canonical restriction map, given by:
 
 R   Λ (ƒ)(λ)=ƒ(λ),
 
for every ƒεC(V) and λεΛ. We denote by * the convolution operations between functions on Λ. Given ƒ,gεC(Λ), their convolutions is defined by:
 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       * 
                       
                         g 
                         ⁡ 
                         
                           ( 
                           λ 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             
                               
                                 λ 
                                 1 
                               
                               + 
                               
                                 λ 
                                 2 
                               
                             
                             = 
                             λ 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 λ 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 λ 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             
                               λ 
                               ′ 
                             
                             ∈ 
                             Λ 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 λ 
                                 ′ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 λ 
                                 - 
                                 
                                   λ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1.9 
                   ) 
                 
               
             
           
         
       
     
     for every λεΛ. 
     1.3 Tori A torus Z is a two dimensional periodic domain that constitutes the geometric dual of a lattice Λ. Formally, Z is the continuos group obtained as the quotient of the vector space V by the lattice Λ, namely:
 
 Z=V/Λ.   (1.10)
 
     In particular, a point zεZ is by definition a Λ-coset in V, namely:
 
 z=ν+Λ,   (1.11)
 
for some νεV. An alternative, albeit less canonical, way to construct Z is to glue opposite faces of a fundamental domain of Λ. Geometrically, Z has the shape of a “donut” obtained by folding the plane V with respect to the lattice Λ. We refer to Z as the torus associated with A or sometimes also as the dual of Λ. Note that a torus is the two dimensional counterpart of a circle, where the second is obtained by folding the line R with respect to a one dimensional lattice ZT⊂R.
 
     Example 1.3 (standard communication torus). As shown in  FIG. 25 , the torus associated with the standard communication lattice Λ T,μ  is given by: 
                                 Z     T   ,   μ       =       V   /     Λ     T   ,   μ         =         R   /   ZT     ⁢           ⁢   μ     ⊕       R   /   Z     ⁢           ⁢     1   /   T             ;     [     0   ,     T   ⁢           ⁢   μ       )     ×   0     ,     1   /   T       )     .           (   1.12   )               
Geometrically, Z T,μ  is the Cartesian product of two circles; one of diameter Tμ and the other of diameter 1/T. We refer to Z T,μ  as the standard communication torus.
 
     1.3.1 Functions on tori. We denote by C(Z) the vector space of complex valued functions on a torus Z=V/Λ. A function on Z is naturally equivalent to a function ƒ:V→C periodic with respect to translations by elements of the lattice Λ, that is:
 
ƒ(ν+λ)=ƒ(ν),  (1.13)
 
for every νεV and λεΛ. Hence, the vector space of functions on Z coincides with the subspace of Λ periodic functions on V, that is, C(Z)=C(V) Λ . Consequently, we have a natural periodization map R Λ :C(V)→C(Z), given by:
 
                           R   Λ     ⁡     (   f   )       ⁢     (   v   )       =       ∑     λ   ∈   Λ       ⁢           ⁢     f   ⁡     (     v   +   λ     )           ,           (   1.14   )               
for every ƒεC(V) and νεV. We denote by * the operation of cyclic convolution of functions on Z. Given a pair of functions ƒ,gεC(Z), their convolution is defined by:
 
                       f   *     g   ⁡     (   v   )         =         ∫         v   1     +     v   2       =   v       ⁢       f   ⁡     (     v   1     )       ⁢     g   ⁡     (     v   2     )           =       ∫       v   ′     ∈   Z       ⁢       f   ⁡     (     v   ′     )       ⁢     g   ⁡     (     v   -     v   ′       )       ⁢     dv   ′             ,           (   1.15   )               
for every νεV. Note that integration over the torus Z amounts to integration over a fundamental domain of the lattice Λ.
 
     1.4 Finite Tori A finite torus Z 0  is a domain associated with a pair consisting of a lattice Λ⊂V and a sublattice Λ 0 ⊂Λ. Formally, Z 0  is the finite group defined by the quotient of the lattice A by the sublattice Λ 0 , that is:
 
 Z   0 =Λ/Λ 0 .  (1.16)
 
     In particular, a point zεZ 0  is a Λ 0 -coset in Λ, namely:
 
 z=λ+Λ   0 ,  (1.17)
 
for some λεΛ. Geometrically, Z 0  is a finite uniform sampling of the continuos torus Z=V/Λ 0  as we have a natural inclusion:
 
Λ/Λ 0   °V/Λ   0 .  (1.18)
 
     Example 1.4 (The standard communication finite torus). Consider the standard communication lattice Λ T,μ . Fix positive integers n,mεN ≧1 . Let (Λ T,μ ) n,m  be the sublattice defined by:
 
(Λ T,μ ) n,m   =ZnTμ⊕Zm/T   (1.19)
 
     The finite torus associated with (Λ T,μ ) n,m ⊂Λ T,μ  is given by (see  FIG. 26 ): 
                           Z     T   ,   μ       m   ,   n       =       Λ     T   ,   μ       ⁢     /     ⁢       (     Λ     T   ,   μ       )       n   ,   m                       =     ZT   ⁢           ⁢   μ   ⁢     /     ⁢   ZnT   ⁢           ⁢   μ   ×   Z   ⁢           ⁢   1   ⁢     /     ⁢   T   ⁢     /     ⁢   Zm   ⁢     /     ⁢   T       ;     Z   ⁢     /     ⁢   nZ   ×   Z   ⁢     /     ⁢     mZ   .                     (   1.20   )               
Concluding, the finite torus Z T,μ   m,n  is isomorphic to the Cartesian product of two cyclic groups; one of order n and the other of order m. We refer to Z T,μ   m,n  as the standard communication finite torus.
 
     1.4.1 Functions on finite tori. We denote by C(Z 0 ) the vector space of complex valued functions on a finite torus Z 0 =Λ/Λ 0 . A function on Z 0  is naturally equivalent to a function ƒ:Λ→C that is periodic with respect to translations by the sublattice Λ 0 , that is:
 
ƒ(λ+λ 0 )=ƒ(λ),  (1.21)
 
for every λεΛ and λ 0 εΛ 0 . Hence, the vector space C(Z 0 ) coincides with the subspace of Λ 0  periodic functions on Λ, that is, C(Z 0 )=C(Λ) Λ     0   . Consequently, we have a natural periodization map R Λ     0   :C(Λ)→C(Z 0 ) given by:
 
                           R     Λ   0       ⁡     (   f   )       ⁢     (   λ   )       =       ∑       λ   0     ∈   Λ0       ⁢     f   ⁡     (     λ   +     λ   0       )           ,           (   1.22   )               
for every ƒεC(Λ) and λεΛ. We denote by * the operation of finite cyclic convolution of functions on Z 0 . Given a pair of functions ƒ,gεC(Z 0 ), their convolution is defined by:
 
                           f   ⋆     g   ⁡     (   λ   )         =       ∑         λ   1     +     λ   2       =   λ       ⁢       f   ⁡     (     λ   1     )       ⁢     g   ⁡     (     λ   2     )                       =       ∑       λ   ′     ∈   Z       ⁢       f   ⁡     (     λ   ′     )       ⁢     g   ⁡     (     λ   -     λ   ′       )                         (   1.23   )               
for every νεV. Note that summation over the finite torus Z 0  amounts to summation over a fundamental domain of the sublattice Λ 0  in the superlattice Λ.
 
     1.4.2 Reciprocity between finite tori. Given a finite torus Z 0 =Λ/Λ 0 , we denote by Z ⊥  the finite torus associated with the reciprocal pair Λ ⊥ ⊂Λ 0   ⊥ , that is:
 
 Z   0   ⊥ =Λ 0   ⊥ /Λ ⊥   (1.24)
 
     We refer to Z 0   ⊥  as the reciprocal finite torus. Although different as sets, one can show that, in fact, Z 0  and Z 0   ⊥  are isomorphic as finite groups. 
     Example 1.5 Consider the pair consisting of the standard communication lattice Λ T,μ  and the sublattice (Λ T,μ ) m,n ⊂Λ T,μ . As shown above, the finite torus associated with (Λ T,μ ) n,m ⊂Λ T,μ  is isomorphic to:
 
 Z   0   ;Z/Zn×Z/Zm.  
 
     The reciprocal lattices are given by:
 
(Λ T,μ ) ⊥   =ZT⊕Z 1/ Tμ,  
 
(Λ T,μ ) m,n   ⊥   =ZT/m⊕Z 1/ nTμ.  
 
     Consequently, the reciprocal finite torus is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Z 
                           0 
                           ⊥ 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 Λ 
                                 
                                   T 
                                   , 
                                   μ 
                                 
                               
                               ) 
                             
                             
                               m 
                               , 
                               n 
                             
                             ⊥ 
                           
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 Λ 
                                 
                                   T 
                                   , 
                                   μ 
                                 
                               
                               ) 
                             
                             ⊥ 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           
                             Z 
                             ⁢ 
                             
                               ( 
                               
                                 T 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 m 
                               
                               ) 
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             ZT 
                             × 
                             
                               Z 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   nT 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   μ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             
                               Z 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   μ 
                                 
                                 ) 
                               
                             
                           
                         
                         ; 
                         
                           Z 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           mZ 
                           × 
                           Z 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           
                             nZ 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     We see that Z 0  and Z 0   i  are isomorphic as finite groups as both groups are isomorphic to the Cartesian product (albeit in different order) of two cyclic groups, one of order n and the other of order m. 
     2 The Symplectic Fourier Transform 
     In this section we introduce a variant of the two dimensional Fourier transform, associated with the symplectic form, called the symplectic Fourier transform. Let ψ:R→C x  denote the standard complex exponential function:
 
ψ( z )= e   2πiz ,  (2.1)
 
     for every zεR. 
     2.1 Properties of the Symplectic Fourier Transform 
     The symplectic Fourier transform is a variant of the two dimensional Fourier transform that is associated with the symplectic form ω. Formally, the symplectic Fourier transform is the linear transformation SF:C(V)→C(V) defined by the rule: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             SF 
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         = 
                         
                           
                             ∫ 
                             
                               v 
                               ∈ 
                               V 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   
                                     ω 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         u 
                                         , 
                                         v 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 v 
                                 ) 
                               
                             
                             ⁢ 
                             dv 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           
                             
                               ∫ 
                               
                                 τ 
                                 , 
                                 
                                   v 
                                   ∈ 
                                   R 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 ψ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     tv 
                                     - 
                                     
                                       f 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 f 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     τ 
                                     , 
                                     v 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               τ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               v 
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2.2 
                   ) 
                 
               
             
           
         
       
     
     for every ƒεC(V) and u=(t,f). We refer to the coordinates (t,f) of the transformed domain as time and frequency, respectively. 
     In general, the inverse transform of (2.2) is given by the formula: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               SF 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         = 
                         
                           
                             ∫ 
                             
                               u 
                               ∈ 
                               V 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 
                                   + 
                                   
                                     ω 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         u 
                                         , 
                                         v 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                             ⁢ 
                             du 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ∫ 
                             
                               t 
                               , 
                               
                                 f 
                                 ∈ 
                                 R 
                               
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     τ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     f 
                                   
                                   - 
                                   vt 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   f 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             dtdf 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2.3 
                   ) 
                 
               
             
           
         
       
     
     However, since ω is anti-symmetric, we have that SF −1 =SF. Namely, the symplectic Fourier transform is equal to its inverse. 
     2.1.1 Interchanging Property. The symplectic Fourier transform interchanges between function multiplication and function convolution as formulated in the following proposition. 
     Proposition 2.1 (Interchanging property). The following conditions hold:
 
SF(ƒ· g )=SF(ƒ)*SF( g ),
 
SF(ƒ* g )=SF(ƒ)·SF( g ),  (2.4)
 
     for every ƒ,gεC(V). 
     In fact, the interchanging property follows from a more fundamental property that concerns the operations of two dimensional translation and symplectic modulation.
         Translation: given a vector ν 0 εV, define translation by ν 0  to be the linear transformation L ν     0   :C(V)→C(V), given by:
 
 L   ν     0   (ƒ)(ν)=ƒ(ν−ν 0 ),  (2.5)
       

     for every ƒεC(V).
         Modulation: given a vector ν 0 εV, define symplectic modulation by ν 0  to be the linear transformation M ν     0   :C(V)→C(V), given by:
 
 M   ν     0   (ƒ)(ν)=ψ(ω(ν 0 ,ν))ƒ(ν),  (2.6)
       

     for every ƒεC(V). 
     Perhaps the most fundamental property of the symplectic Fourier transform is that it interchanges between translation and symplectic modulation. This property is formulated in the following proposition. 
     Proposition 2.2 (Interchanging translation with symplectic modulation). The following conditions hold:
 
SF∘ L   ν     0     =M   ν     0   ∘SF
 
SF∘ M   ν     0     =L   ν     0   ∘SF,
 
     for every ν 0 εV. 
     2.2 The discrete symplectic Fourier transform 
     The discrete symplectic Fourier transform relates between functions of two discrete variables and functions of two continuos periodic variables. The formal definition assumes a choice of a lattice Λ⊂V. Let Λ ⊥ ⊂V be the reciprocal lattice and let Z ⊥  denote the torus associated with Λ ⊥ , that is:
 
 Z   ⊥   =V/Λ   ⊥ .
 
     We refer to Z ⊥  as the reciprocal torus. The discrete symplectic Fourier transform is the linear transformation SF Λ :C(Λ)→C(Z ⊥ ) given by: 
                           SF   Λ     ⁡     (   f   )       ⁢     (   u   )       =     c   ·       ∑     λ   ∈   Λ       ⁢       ψ   ⁡     (     -     ω   ⁡     (     u   ,   λ     )         )       ⁢     f   ⁡     (   λ   )               ,           (   2.7   )               
for every ƒεC(A) and uεV where c is a normalization coefficient taken to be c=vol(Λ). Note, that fixing the value of λεΛ, the function ψ(−ω(u,λ))ƒ(λ) is periodic with respect to the reciprocal lattice hence is a function on the reciprocal torus. The inverse transform SF Λ   −1 :C(Z ⊥ )→C(Λ) is given by:
 
                           SF   Λ     -   1       ⁡     (   f   )       ⁢     (   λ   )       =       ∫     u   ∈     Z   ⊥         ⁢       ψ   ⁡     (     -     ω   ⁡     (     λ   ,   u     )         )       ⁢     f   ⁡     (   u   )       ⁢   du         ,           (   2.8   )               
for every ƒεC(Λ). Note that taking the integral over the torus Z ⊥  is equivalent to integrating over a fundamental domain of the lattice Λ ⊥ .
 
     2.2.1 Discrete interchanging property. The discrete symplectic Fourier transform interchanges between function multiplication and function convolution as formulated in the following proposition. 
     Proposition 2.3 (Discrete interchanging property). The following conditions hold: 
     
       
         
           
             
               
                 
                   
                     
                       
                         SF 
                         Λ 
                       
                       ⁡ 
                       
                         ( 
                         
                           f 
                           · 
                           g 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           SF 
                           Λ 
                         
                         ⁡ 
                         
                           ( 
                           f 
                           ) 
                         
                       
                       ⋆ 
                       
                         
                           SF 
                           Λ 
                         
                         ⁡ 
                         
                           ( 
                           g 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.9 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         1 
                         
                           c 
                         
                       
                       ⁢ 
                       
                         
                           SF 
                           Λ 
                         
                         ⁡ 
                         
                           ( 
                           
                             f 
                             ⋆ 
                             g 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           c 
                         
                       
                       ⁢ 
                       
                         
                           
                             SF 
                             Λ 
                           
                           ⁡ 
                           
                             ( 
                             f 
                             ) 
                           
                         
                         · 
                         
                           1 
                           
                             c 
                           
                         
                       
                       ⁢ 
                       
                         
                           SF 
                           Λ 
                         
                         ⁡ 
                         
                           ( 
                           g 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.10 
                   ) 
                 
               
             
           
         
       
     
     for every ƒ,gεC(Λ) where * stands for periodic convolution. 
     2.2.2 Compatibility with the continuous transform. The continuos and discrete symplectic Fourier transforms are compatible. The compatibility relation is formulated in the following Theorem. 
     Theorem 2.4 (discrete-continuos compatibility relation). We have:
 
SF Λ   ∘R   Λ   =R   Λ     ⊥   ∘SF,  (2.11)
 
SF Λ   −1   ∘R   Λ     ⊥     =R   Λ ∘SF −1 .  (2.12)
 
     Stated differently, Equation (2.11) provides that taking the continuos Fourier transform of a function ƒ and than periodizing with respect to translations by the reciprocal lattice Λ ⊥  is the same as first restricting ƒ to the lattice Λ and then taking the discrete Fourier transform. 
     2.3 The finite symplectic Fourier Transform 
     The finite symplectic Fourier transform relates functions of two finite periodic variables. The formal definition assumes a pair consisting of a lattice Λ⊂V and a sublattice Λ 0 ⊂Λ. We denote by Z 0  the finite torus associated with this pair, that is:
 
 Z   0 =Λ/Λ 0 .
 
     Let Λ ⊥  and Λ 0   ⊥  be the corresponding reciprocal lattices. We denote by Z ⊥  the finite torus associated with the reciprocal pair, that is:
 
 Z   0   ⊥ =Λ 0   ⊥ /Λ ⊥ .
 
     The finite symplectic Fourier transform is the linear transformation SF Z     0   :C(Z 0 )→C(Z 0   ⊥ ) defined by the rule:
 
SF Z     0   (ƒ)(μ)= c·   λεZ     0   ψ(−ω(μ,λ))ƒ(λ)  (2.13)
 
for every ƒεC(Z 0 ) and μεΛ 0   ⊥  where c is a normalization coefficient taken to be c=vol(Λ). The inverse transform SF Z     0     −1 :C(Z 0   ⊥ )→C(Z 0 ) is given by:
 
                           SF     Z   0       -   1       ⁡     (   f   )       ⁢     (   λ   )       =         1     c   0     μ   ∈     Z   0   ⊥           ·     ψ   ⁡     (     -     ω   ⁡     (     λ   ,   μ     )         )         ⁢     f   ⁡     (   μ   )           ,           (   2.14   )               
for every ƒεC(Z 0   ⊥ ) and λεΛ where c 0  is a normalization coefficient taken to be c 0 =vol(Λ 0 ).
 
     2.3.1 Finite interchanging property. The finite symplectic Fourier transform interchanges between function multiplication and function cyclic convolution as formulated in the following proposition. 
     Proposition 2.5 (Discrete interchanging property). The following conditions hold: 
     
       
         
           
             
               
                 
                   
                     
                       
                         c 
                         
                           c 
                           0 
                         
                       
                       ⁢ 
                       
                         
                           SF 
                           
                             Z 
                             0 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             f 
                             · 
                             g 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           c 
                           
                             c 
                             0 
                           
                         
                         ⁢ 
                         
                           
                             SF 
                             
                               Z 
                               0 
                             
                           
                           ⁡ 
                           
                             ( 
                             f 
                             ) 
                           
                         
                       
                       ⋆ 
                       
                         
                           c 
                           
                             c 
                             0 
                           
                         
                         ⁢ 
                         
                           
                             SF 
                             
                               Z 
                               0 
                             
                           
                           ⁡ 
                           
                             ( 
                             g 
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         1 
                         c 
                       
                       ⁢ 
                       
                         
                           SF 
                           
                             Z 
                             0 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             f 
                             ⋆ 
                             g 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         1 
                         c 
                       
                       ⁢ 
                       
                         
                           
                             SF 
                             
                               Z 
                               0 
                             
                           
                           ⁡ 
                           
                             ( 
                             f 
                             ) 
                           
                         
                         · 
                         
                           1 
                           c 
                         
                       
                       ⁢ 
                       
                         
                           SF 
                           
                             Z 
                             0 
                           
                         
                         ⁡ 
                         
                           ( 
                           g 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.16 
                   ) 
                 
               
             
           
         
       
     
     for every ƒ,gεC(Z 0 ) where * stands for finite cyclic convolution. 
     Note that the normalization coefficient c/c 0  in equation (2.15) is equal the number of points in the finite torus Z 0 . 
     2.3.2 Compatibility with the discrete transform. The discrete and finite symplectic Fourier transforms are compatible. The compatibility relation is formulated in the following Theorem. 
     Theorem 2.6. 
     We have: 
     
       
         
           
             
               
                 
                   
                     
                       
                         SF 
                         
                           Z 
                           0 
                         
                       
                       ∘ 
                       
                         R 
                         
                           Λ 
                           0 
                         
                       
                     
                     = 
                     
                       
                         R 
                         
                           Λ 
                           0 
                           ⊥ 
                         
                       
                       ∘ 
                       
                         SF 
                         Λ 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.17 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       SF 
                       
                         Z 
                         0 
                       
                       
                         - 
                         1 
                       
                     
                     ∘ 
                     
                       R 
                       
                         Λ 
                         0 
                         ⊥ 
                       
                     
                   
                   = 
                   
                     
                       R 
                       
                         Λ 
                         0 
                       
                     
                     ∘ 
                     
                       SF 
                       Λ 
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2.18 
                   ) 
                 
               
             
           
         
       
     
     In plain language, Equation (2.17) states that taking the discrete symplectic Fourier transform of a function ƒ on a lattice Λ and than restricting to the reciprocal lattice Λ 0   ⊥  is the same as first periodizing ƒ with respect to translations by the sublattice Λ 0  and than taking the finite Fourier transform. 
     Example 2.7 Consider the standard communication lattice Λ T,μ  and the sublattice (Λ T,μ ) n,m . We have the following isomorphisms:
 
 Z   0   ;Z/nZ×Z/mZ,  
 
 Z   0   ⊥   ;Z/mZ×Z/nZ.  
 
     In terms of these realizations the finite symplectic Fourier transform and its inverse take the following concrete forms: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           SF 
                           
                             Z 
                             0 
                           
                         
                         ⁡ 
                         
                           ( 
                           f 
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           k 
                           , 
                           l 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         μ 
                         
                           K 
                           = 
                           
                             0 
                             
                               L 
                               = 
                               0 
                             
                           
                         
                         
                           n 
                           - 
                           
                             1 
                             
                               m 
                               - 
                               1 
                             
                           
                         
                       
                       ⁢ 
                       
                         ψ 
                         ⁡ 
                         
                           ( 
                           
                             kL 
                             - 
                             lK 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             K 
                             , 
                             L 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           SF 
                           
                             Z 
                             0 
                           
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           f 
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           K 
                           , 
                           L 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             m 
                             - 
                             
                               1 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           mn 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             μ 
                             
                               k 
                               = 
                               
                                 0 
                                 
                                   l 
                                   = 
                                   0 
                                 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         ψ 
                         ⁡ 
                         
                           ( 
                           
                             Kl 
                             - 
                             Lk 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             k 
                             , 
                             l 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2.20 
                   ) 
                 
               
             
           
         
       
     
     where in the first equation kε[0,m−1], lε[0,n−1] and in the second equation Kε[0,n−1], Lε[0,m−1]. Note the minus sign in the Fourier exponent due to the symplectic pairing. 
     3 Heisenberg Theory 
     Let H denote the Hilbert space of square integrable complex functions on the real line R. We denote the parameter of the line by t and refer to it as time. The inner product on H is given by the standard formula:
 
 ƒ, g     =   xεR   ƒ( x )   g ( x ) dx,   (3.1)
 
     We refer to H as the signal space and to functions in the signal space as waveforms. Heisenberg theory concerns the mathematical structures underlying the intricate interaction between the time and frequency dimensions. In a nutshell, the theory study the algebraic relations between two basic operations on functions: time delay and Doppler shift. 
     3.1 Time delay and Doppler Shift 
     The operations of time delay and Doppler shift establish two one parametric families of Unitary transformations on H. 
     3.1.1 Time delay. Given a real parameter τεR the operation of time delay by τ is a linear transformation L τ :H→H given by
 
 L   τ (ƒ)( t )=ƒ( t −τ),  (3.2)
 
for every ƒεH and tεR. One can show that L τ  is a Unitary transformation, namely it preserves the inner product:
 
   L   τ   ƒ,L   τ   g     =     ƒ,g     ,  
 
     for every ƒ,gεH. More over, the family of transformation {L τ :τεR} satisfies:
 
 L   τ     1     +τ     2     =L   τ     1     ∘L   τ     2   ,
 
for every τ 1 ,τ 2 εR. In particular, the operations of time delay commute with one another, that is, L τ     1   ∘L τ     2   =L τ     2   ∘L τ     1   .
 
     3.1.2 Doppler shift. Given a real parameter νεR the operation of Doppler shift by ν is a linear transformation M ν : H given by
 
 M   ν (ƒ)( t )=ψ(ν t )ƒ( t ),  (3.3)
 
for every ƒεH and tεR. Recall that ψ stands for the standard complex exponential function ψ(z)=e 2πiz . One can show that M ν  is a Unitary transformation, namely it preserves the inner product:
 
   M   ν   ƒ,M   ν   g     =     ƒ,g     ,  
 
for every ƒ,gεH. More over, the family of transformation {M ν :νεR} satisfies:
 
 M   ν     1     +ν     2     =M   ν     1     ∘M   ν     2   ,
 
     for every ν 1 ,ν 2 εR. In particular, the operations of time delay commute with one another, that is, M ν     1   ∘M ν     2   =M ν     2   ∘M ν     1   . 
     3.2 The Heisenberg representation 
     The Heisenberg representation is a mathematical structure unifying the two operations of time delay and Doppler shift. The main difficulty is that these operations do not commute with one another, instead they satisfy the following condition:
 
 L   τ   M   ν =ψ(−τν) M   ν   L   τ .  (3.4)
 
     The starting point is to consider the unified delay-Doppler linear transformation:
 
π(τ,ν)= L   τ   M   ν ,  (3.5)
 
for every pair of real parameters τ,νεR. In this representation, the ordered pair (τ,ν) is considered as a point in the delay Doppler plane V. One can show that π(τ,ν) is a Unitary transformation as composition of such. The two dimensional family of transformations {π(ν):νεV} defines a linear transformation Π:C(V)→Hom(H,H), given by:
 
Π(ƒ)=∫ νεV ƒ(ν)π(ν) dν,   (3.6)
 
for every ƒεC(V), where the range of H is the vector space of linear transformations from H to itself which we denote by Hom(H,H). In words, the map Π takes a function on the delay Doppler plane and send it to the linear transformation given by weighted superposition of delay-Doppler transformations where the weights are specified by the values of the function. The map Π is called the Heisenberg representation. A fundamental fact which we will not prove is that the map Π is essentially an isomorphism of vector spaces. Hence it admits an inverse Π −1 :Hom(H,H)→C(V) called the Wigner transform. The Wigner transform is given by:
 
Π −1 ( A )(ν)= Tr (π(ν) H   A ).  (3.7)
 
for every AεHom(H,H) and νεV. The Heisenberg representation and the Wigner transform should be thought of as a “change of coordinates” converting between functions on the delay Doppler plane and linear transformations on the signal space (which may be represented using matrices). To summarize, a linear transformation AεHom(H,H) admits a unique expansion as a superposition of delay-Doppler transformations. The coefficients in this expansion are given by the function a=Π −1 (A). The function a is refereed to as the delay-Doppler impulse response of the transformation A. The Heisenberg formalism generalizes the classical framework of time invarinat linear systems to time varying linear systems. Note that in the former, a time invariant linear transformation admits a unique expansion as a superposition of time delays and the coefficients in the expansion constitute the classical impulse response.
 
     3.2.1 Ambiguity function. The formula of the Wigner transform of a general linear transformation, Equation (3.7), is quite abstract. Fortunately, for specific type of linear transformations the Wigner transform takes a more explicit form. Say we are given a waveform gεH, of unit norm ∥g∥=1. Let P g  denote the orthogonal projection on the one dimensional subspace spanned by g, given by:
 
 P   g (φ)= g     g,φ     ,   (3.8)
 
     for every ωεH. 
     Proposition. The Wigner transform of P g  admits the following formula:
 
Π −1 ( P   g )(ν)= π(ν) g,g     ,   (3.9)
 
     for every νεV. 
     Denote A g =Π −1 (P g ) and refer to this function as the ambiguity function of g. We have:
 
Π( A   g )= P   g .  (3.10)
 
     The above equation means that A g  is the coefficients in the delay-Doppler expansion of the operator P g —this is the Heisenberg interpretation of the ambiguity function. 
     3.2.2 Cross ambiguity function. The cross ambiguity function is a generalization of the ambiguity function to the case of two waveforms g 1 ,g 2 εH where g 1  is assumed to be of unit norm. Let P g     1     ,g     2    denote the following rank one linear transformation on H:
 
 P   g     1     ,g     2   (φ)= g   2     g   1 ,φ ,  (3.11)
 
     for every φεH. 
     Proposition. The Wigner transform of P g     1     ,g     2    admits the following formula:
 
Π −1 ( P   g     1     ,g     2   )(ν)= π(ν) g   1   ,g   2   ,  (3.12)
 
     for every νεV. 
     Denote A g     1     ,g     2   =Π −1 (P g     1     ,g     2   ) and refer to this function as the cross ambiguity function of g 1  and g 2 . We have:
 
Π( A   g     1     ,g     2   )= P   g     1     ,g     2   .  (3.13)
 
Hence, according to the Heisenberg interpretation, the cross-ambiguity function is the coefficients in the delay-Doppler expansion of the operator P g     1     ,g     2   .
 
     3.3 Heisenberg Interchanging Property 
     The main property of the Heisenberg representation is that it interchanges between the operation of composition of linear transformations on H and a twisted version of the convolution operation of functions on V. In order to define the operation of twisted convolution we consider the form β:V×V→V, given by:
 
β(ν,ν′)=ντ′,  (3.14)
 
where ν=(τ,ν) and ν′=(τ′,ν′). The form β satisfies the “polarization” condition:
 
β(ν,ν′)−β(ν′,ν)=ω(ν,ν′),  (3.15)
 
for every ν,ν′εV. Given a pair of functions ƒ,gεC(V) their twisted convolution is defined by the following rule:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           f 
                           ⁢ 
                           
                             ⋆ 
                             t 
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             ∫ 
                             
                               
                                 
                                   v 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 + 
                                 
                                   v 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               = 
                               v 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         1 
                                       
                                       , 
                                       
                                         v 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 
                                   v 
                                   1 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               g 
                               ⁡ 
                               
                                 ( 
                                 
                                   v 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ∫ 
                             
                               
                                 v 
                                 ′ 
                               
                               ∈ 
                               V 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         ′ 
                                       
                                       , 
                                       
                                         v 
                                         - 
                                         
                                           v 
                                           ′ 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 v 
                                 ) 
                               
                             
                             ⁢ 
                             
                               g 
                               ⁡ 
                               
                                 ( 
                                 
                                   v 
                                   - 
                                   
                                     v 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               dv 
                               ′ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3.16 
                   ) 
                 
               
             
           
         
       
     
     One can see that the twisted convolution operation differs from the usual convolution operation, Equation (1.2), by the multiplicative factor ψ(β(ν 1 ,ν 2 )). As a consequence of this factor, twisted convolution is a non-commutative operation in contrast with conventional convolution. This non-commutativity is intrinsic to the fabric of time and frequency. The Heisenberg interchanging property is formulated in the following Theorem. 
     Theorem 3.1 (Heisenberg Interchanging Property). We have:
 
Π(ƒ* t   g )=Π(ƒ)∘Π( g ),  (3.17)
 
     for every ƒ,gεC(V). 
     The following example is key to understanding the motivation behind the constructions presented in this section. In a nutshell, it explains why the twist in Formula (3.16) accounts for the phase in the commutation relation between the time delay and Doppler shift operations, see Equation (3.4). 
     Example 3.2 We verify Equation (3.17) in a concrete case. Let ν=(τ,ν) and ν′=(τ′,ν′). Consider the delta functions δ ν  and δ ν     ′   . On the one hand, we have:
 
Π(δ ν )= L   τ   M   ν ,
 
Π(δ ν     ′   )= L   τ     ′     M   ν     ′   ,
 
     and consequently: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Π 
                             ⁡ 
                             
                               ( 
                               
                                 δ 
                                 v 
                               
                               ) 
                             
                           
                           ∘ 
                           
                             Π 
                             ⁡ 
                             
                               ( 
                               
                                 δ 
                                 
                                   v 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             L 
                             τ 
                           
                           ⁢ 
                           
                             M 
                             v 
                           
                           ⁢ 
                           
                             L 
                             
                               τ 
                               ′ 
                             
                           
                           ⁢ 
                           
                             M 
                             
                               v 
                               ′ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   τ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             L 
                             τ 
                           
                           ⁢ 
                           
                             L 
                             
                               τ 
                               ′ 
                             
                           
                           ⁢ 
                           
                             M 
                             v 
                           
                           ⁢ 
                           
                             M 
                             
                               v 
                               ′ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   τ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             L 
                             
                               τ 
                               + 
                               
                                 τ 
                                 ′ 
                               
                             
                           
                           ⁢ 
                           
                             M 
                             
                               v 
                               + 
                               
                                 v 
                                 ′ 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   τ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             π 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 + 
                                 
                                   v 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           Π 
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   ψ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       v 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   δ 
                                   
                                     v 
                                     + 
                                     
                                       v 
                                       ′ 
                                     
                                   
                                 
                               
                               ) 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3.18 
                   ) 
                 
               
             
           
         
       
     
     On the other hand: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             δ 
                             v 
                           
                           ⁢ 
                           
                             ⋆ 
                             t 
                           
                           ⁢ 
                           
                             δ 
                             
                               v 
                               ′ 
                             
                           
                         
                         = 
                         
                           
                             
                               ψ 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       v 
                                       , 
                                       
                                         v 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               δ 
                               v 
                             
                           
                           ⋆ 
                           
                             δ 
                             
                               v 
                               ′ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   τ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               δ 
                               
                                 v 
                                 + 
                                 
                                   v 
                                   ′ 
                                 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3.19 
                   ) 
                 
               
             
           
         
       
     
     Consequently:
 
Π(δ ν * t δ ν     ′   )=ψ(ντ′)π(ν+ν′).  (3.20)
 
     Hence we verified that: Π(δ ν * t δ ν     ′   )=Π(δ ν )∘Π(δ ν     ′   ). 
     3.4 Fundamental channel equation 
     We conclude this section with formulating a fundamental equation relating the following structures: 
     1. Cross ambiguity function. 
     2. Ambiguity function 
     3. Channel transformation. 
     4. Twisted convolution. 
     This fundamental equation is pivotal to the two dimensional channel model that will be discussed in the next section. Let gεH be a waveform of unit norm. Let hεC(V). We denote by H the channel transformation:
 
 H =Π( h ).  (3.21)
 
     Theorem 3.3 (Fundamental channel equation). The following equation holds:
 
 A   g,H(g)   =h*   t   A   g .  (3.22)
 
     In words, the fundamental equation, (3.22), asserts that the cross ambiguity function of g with H(g) is the twisted convolution of h with the ambiguity function of g. 
     4. The Continuous OTFS Transceiver 
     In this section we describe a continuos variant of the OTFS transceiver. 
     4.1 Set-Up 
     The definition of the continuos OTFS transceiver assumes the following data: 
     1. Communication lattice. An undersampled lattice:
 
Λ⊂ V,  
 
where vol(Λ)=μ, for some μ≧1.
 
     2. Generator waveform. A waveform of unit norm:
 
 gεH,  
 
     satisfying the orthogonality condition A g  (λ)=0 for every non-zero element λεΛ x . 
     3. 2D filter. A window function:
 
 WεC (Λ).
 
     We note that, typically, the support of the 2D filter along the delay and Doppler dimensions is bounded by the latency and bandwidth restrictions of the communication packet respectively. 
     Example 4.1 A typical example of a communication lattice is the standard communication lattice:
 
 A   T,μ   =ZμT⊕Z 1/ T.  
 
     A typical example of a 2D filter is: 
     
       
         
           
             
               W 
               ⁡ 
               
                 [ 
                 
                   KTμ 
                   , 
                   
                     L 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     T 
                   
                 
                 ] 
               
             
             = 
             
               { 
               
                 
                   
                     1 
                   
                   
                     
                       
                         K 
                         ∈ 
                         
                           [ 
                           
                             0 
                             , 
                             
                               n 
                               - 
                               1 
                             
                           
                           ] 
                         
                       
                       , 
                       
                         L 
                         ∈ 
                         
                           [ 
                           
                             0 
                             , 
                             
                               m 
                               - 
                               1 
                             
                           
                           ] 
                         
                       
                       , 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     otherwise 
                   
                 
               
             
           
         
       
     
     where m,nεN ≧1  and TεR. The parameter T is called the symbol time. The real numbers nμT and m/T are the latency and bandwidth of the communication packet respectively. Note that a more sophisticated design of a spectral window will involve some level of tapering around the boundaries in the expense of spectral efficiency. Finally, in case μ=1 (critical sampling) a simple example of an orthogonal waveform is:
 
 g= 1 [0,T] .
 
     4.1.1 Generalized set-up. The set-up can be slightly generalized by assuming, instead of a single orthogonal waveform g, a pair consisting of transmit waveform g t εH and receive waveform g r εH satisfying the following cross orthogonality condition:
 
 A   g     r     ,g     t   (Λ)=0,  (4.1)
 
for every λεΛ x . The trade-off in using a pair where g r ≠g t  is gaining more freedom in the design of the shape of each waveform in expense of lower effective SNR at the receiver. For the sake of simplicity, in what follows we will consider only the case when g r =g t  with the understanding that all results can be easily extended to the more general case.
 
     4.2 Continuous OTFS Modulation Map. 
     Let Z ⊥  denote the torus associated with the lattice Λ ⊥  reciprocal to the communication lattice. The continuos OTFS modulation map is the linear transformation M:C(Z ⊥ )→H, given by:
 
 M ( x )=Π( W ·SF Λ   −1 ( x )) g,   (4.2)
 
for every xεC(Z ⊥ ). Roughly, the continuos OTFS modulation is the composition of the Heisenberg representation with the (inverse) discrete symplectic Fourier transform. In this regard it combines the two intrinsic structures of the delay Doppler plane. Formula (4.2) can be written more explicitly as:
 
                       M   ⁡     (   x   )       =       ∑     λ   ∈   Λ       ⁢       W   ⁡     (   λ   )       ⁢     X   ⁡     (   λ   )       ⁢     π   ⁡     (   λ   )       ⁢   g         ,           (   4.3   )               
where X=SF Λ   −1 (x).
 
       FIG. 27  illustrates an exemplary structure of the OTFS modulation map. Note that  FIG. 27  includes an additional spreading transformation given by convolution with a specifically designed function αεC(Z ⊥ ). The effect of this convolution is to spread the energy of each information symbol uniformly along the torus Z ⊥  achieving a balanced power profile of the transmit waveform depending only on the total energy of the information vector x. 
     4.3 continuous OTFS Demodulation Map 
     The continuos OTFS demodulation map is the linear transformation D:H→C(Z ⊥ ), given by:
 
 D (φ)=SF Λ (   W ·R   Λ ( A   g,φ )),  (4.4)
 
for every φεH. Roughly, the continuos OTFS demodulation map is the composition of the discrete symplectic Fourier transform with the Wigner transform. Formula (4.4) can be written more explicitly as:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         D 
                         ⁡ 
                         
                           ( 
                           φ 
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     = 
                     
                       c 
                       · 
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             Λ 
                           
                         
                         ⁢ 
                         
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               
                                 - 
                                 
                                   ω 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       u 
                                       , 
                                       λ 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               W 
                               _ 
                             
                             ⁡ 
                             
                               ( 
                               λ 
                               ) 
                             
                           
                           ⁢ 
                           
                             〈 
                             
                               
                                 
                                   π 
                                   ⁡ 
                                   
                                     ( 
                                     λ 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 g 
                               
                               , 
                               φ 
                             
                             〉 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   4.5 
                   ) 
                 
               
             
           
         
       
     
     for every φεH and uεZ ⊥ . 
     4.4 Two dimensional channel model 
     Before describing the technical details of the two-dimensional channel model for the OTFS transceiver, we will provide an overview in simplified terms. Consider first that in the standard one-dimensional physical coordinates of time (or frequency), the wireless channel is a combination of multipath moving reflectors that induce a distortion on the transmitted signal. This distortion arises due to superposition of time delay and Doppler shifts. Such a general distortion appears in standard physical coordinates as a fading non-stationary inter-symbol interference pattern. In contrast, when converted to the coordinates of the OTFS modulation torus, the distortion becomes a static two dimensional local ISI distortion. This is a novel and characteristic attribute of the OTFS transceiver. In what follows we provide a rigorous derivation of this characteristic. To this end, we begin by considering the simplest multipath channel H:H→H that is already a combination of time delay and Doppler shift. In our terminology, this channel is given by:
 
 H =Π(δ ν     0   )= L   τ     0     M   ν     0   ,  (4.6)
 
for some ν 0 =(τ 0 ,ν 0 )εV. We assume, in addition, that the vector ν 0  satisfy ∥ν 0 ∥=diam(Λ) where the diameter of the lattice is by definition the radius of its Voronoi region. Stated differently, we assume that the vector is small compared with the dimensions of the lattice. Note, that this assumption holds for most relevant scenarios in wireless applications. We proceed to derive the structure of the modulation equivalent channel. Let q:V→C be the quadratic exponential function given by:
 
 q (ν)=ψ(−β(ν,ν)),  (4.7)
 
     for every νεV. 
     Proposition 4.2 The modulation equivalent channel y=D∘H∘M(x) is a periodic convolution y=h eqv *x, where the impulse response h eqv εC(Z ⊥ ) is given by:
 
 h   eqv   =R   Λ     ⊥   ( q (ν 0 )δ ν     0   )*SF Λ   |W|   2   (4.8)
 
That is, Equation (4.8) states that the modulation equivalent channel is a periodic convolution with a periodic blurred version of q(ν 0 )δ ν     0    where the blurring pulse is given by the symplectic Fourier transform of the discrete pulse |W| 2 . This blurring results in a resolution loss which is due to the spectral truncation imposed by the filter W. As a result, the resolution improves as the window size increases (what amounts to longer latency and wider bandwidth). Granting the validity of Equation (4.8), it is straightforward to deduce the modulation equivalent of a general wireless channel:
 
 H =Π( h ),  (4.9)
 
for any function hεC(V) where we assume that the support of h is much smaller than the diameter of the lattice Λ. The general two dimensional channel model is formulated in the following theorem.
 
     Theorem (Two dimensional channel model). The modulation equivalent channel y=D∘H∘M(x) is a periodic convolution y=h eqv *x with the impulse response h eqv εC(Z ⊥ ), given by:
 
 h   eqv   =R   Λ     ⊥   ( q·h )*SF Λ   |W|   2 .  (4.10)
 
Stated differently, the modulation equivalent channel is a periodic convolution with a periodic blurred version of q·h where the blurring pulse is given by the discrete symplectic Fourier transform of the discrete pulse |W| 2 .
 
     4.4.1 Derivation of the two dimensional channel model. We proceed to derive Equation (4.8). Let xεC(Z ⊥ ). Let φ t εH denote the transmitted signal. We have: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           φ 
                           t 
                         
                         = 
                           
                         ⁢ 
                         
                           M 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             Π 
                             ⁢ 
                             
                               ( 
                               
                                 W 
                                 · 
                                 X 
                               
                               ) 
                             
                             ⁢ 
                             g 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4.11 
                   ) 
                 
               
             
           
         
       
     
     where X=SF Λ   −1 (x). Let φ r εH denote the received signal. We have: 
                           φ   r     =       ⁢     H   ⁡     (     φ   t     )                   =       ⁢       Π   ⁡     (     δ     v   0       )       ⁢     •Π   ⁡     (     W   ·   X     )       ⁢   g                   =       ⁢     Π   ⁢     (       δ     v   0       ⁢     ⋆   t     ⁢     (     W   ·   X     )       )     ⁢   g       ,                 (   4.12   )               
where the third equality follows from the Heisenberg property of the map Π (Theorem 3.1). The demodulated vector y=D(φ r ) is given by:
 
 D (φ r )=SF Λ (   W ·R   Λ ( A   g,φ     r   )).  (4.13)
 
     We now evaluate the right hand side of (4.13) term by term. Applying the fundamental channel equation (Theorem 3.3) we get:
 
 A   g,φ     r   =δ ν     0   * t ( W·X )* t   A   g .  (4.14)
 
     Considering the restriction R Λ (A g,φ     r   ) we have the following proposition. 
     Proposition. We have
 
 R   Λ ( A   g,φ     r   ),φ(ν 0 ) R   Λ (ψ ν     0   )·( WX ),  (4.15)
 
     where ψ ν     0   (ν)=ψ(ω(ν 0 ,ν)) for every νεV. 
     Combining Equations (4.13) and (4.15) we get:
 
 D (φ r ); q (ν 0 )SF Λ (└ R   Λ (ψ ν     0   )| W|   2   ┘·X )
 
=└ R   Λ     ⊥   ( q (ν 0 )δ ν     0   )*SF Λ (| W|   2 )┘* x.   (4.16)
 
     This concludes the derivation of the two dimensional channel model. 
     4.5 Explicit interpretation 
     We conclude this section by interpreting the continuos OTFS modulation map in terms of classical DSP operations. We use in the calculations the standard communication lattice Λ=Λ T,μ  from Example 1.1. Recall the definition of the continuous modulation map: 
                       M   ⁡     (   x   )       =       ∑     λ   ∈   Λ       ⁢       W   ⁡     (   λ   )       ⁢     X   ⁡     (   λ   )       ⁢     π   ⁡     (   λ   )       ⁢   g         ,           (   4.17   )               
for every xεC(Z ⊥ ), where X=SF Λ   −1 (x). Formula (4.17) can be written more explicitly as:
 
     
       
         
           
             
               
                 
                   
                     M 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           K 
                           , 
                           L 
                         
                       
                       ⁢ 
                       
                         
                           W 
                           ⁡ 
                           
                             [ 
                             
                               
                                 K 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μT 
                               
                               , 
                               
                                 L 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 T 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           X 
                           ⁡ 
                           
                             [ 
                             
                               
                                 K 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μT 
                               
                               , 
                               
                                 L 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 T 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           L 
                           
                             KT 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             M 
                             
                               L 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               T 
                             
                           
                           ⁡ 
                           
                             ( 
                             g 
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             K 
                           
                           ⁢ 
                           
                             
                               L 
                               
                                 KT 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μ 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 L 
                               
                               ⁢ 
                               
                                 
                                   
                                     W 
                                     K 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       L 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       T 
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     X 
                                     K 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       L 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       T 
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     M 
                                     
                                       L 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       T 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     g 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         K 
                       
                       ⁢ 
                       
                         
                           
                             L 
                             
                               KT 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               μ 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               K 
                             
                             ) 
                           
                         
                         . 
                         
                           
 
                         
                         ⁢ 
                         where 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4.18 
                   ) 
                 
               
             
             
               
                 
                   
                     ϕ 
                     K 
                   
                   = 
                   
                     
                       ∑ 
                       L 
                     
                     ⁢ 
                     
                       
                         
                           W 
                           K 
                         
                         ⁡ 
                         
                           [ 
                           
                             L 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             T 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           X 
                           K 
                         
                         ⁡ 
                         
                           [ 
                           
                             L 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             T 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           
                             M 
                             
                               L 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               T 
                             
                           
                           ⁡ 
                           
                             ( 
                             g 
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4.19 
                   ) 
                 
               
             
           
         
       
     
     The waveform φ K  is called the K th modulation block. 
     4.5.1 Frequency domain interpretation. Let G denote the Fourier transform of g. Equation (4.19) can be interpreted as feeding the weighted sequence W K X K  into a uniform filterbank with each subcarrier shaped by the filter G. See  FIG. 28 . 
     4.5.2 Time domain interpretation. Let w K  and x K  denote the inverse discrete Fourier transform of the discrete waveforms W K  and X K  respectively. Both waveforms are periodic with period T. We have:
 
φ K ∝( w   K   *x   K )· g,  
 
where * stands for periodic convolution. The waveform x K  can be expressed in terms of the information vector x as follows:
 
                   x   K     ⁡     (   t   )       ∝       ∫   v     ⁢       ψ   ⁡     (     KT   ⁢           ⁢   μ   ⁢           ⁢   v     )       ⁢     x   ⁡     (     t   ,   v     )       ⁢   dv         ,         
In words, x K (t) is proportional to the K th component of the inverse discrete Fourier transform of x along the Doppler dimension.
 
5 The Finite OTFS Transceiver
 
     In this section we describe a finite variant of the OTFS transceiver. This variant is obtained, via uniform sampling, from the continuous variant described previously. 
     5.1 Set-up 
     The definition of the finite OTFS transceiver assumes the following: 
     1. Communication lattice. An undersampled lattice:
 
Λ⊂ V,  
 
     where vol(Δ)=μ, for some μ≧1. 
     2. Communication sublattice. A sublattice:
 
Λ 0 ⊂Λ
 
     3. Generator waveform. A waveform of unit norm:
 
 gεH,  
 
     satisfying the orthogonality condition A g (λ)=0 for every λεΛ x . 
     4. 2D filter. A window function:
 
 WεC (Λ).
 
     Note that the support of the 2D filter is typically compatible with the configuration of the sublattice, as demonstrated in the following example. 
     Example 5.1 The standard nested pair of communication lattice and sublattice is:
 
Λ=Λ T,μ   =ZμT⊕Z 1/ T,  
 
Λ 0 =(Λ T,μ ) n m   =ZnμT⊕Zm/T,  
 
     where m,nεN ≧1  and TεR is a parameter called the symbol time. The real numbers nμT and m/T are the latency and bandwidth of the communication packet respectively. A typical compatible 2D filter is: 
               W   ⁡     [       KT   ⁢           ⁢   μ     ,     L   ⁢     /     ⁢   T       ]       =     {           1           K   ∈     [     0   ,     n   -   1       ]       ,     L   ∈     [     0   ,     m   -   1       ]                 0       otherwise         ,             
More sophisticated designs of a spectral window may involve, for example, some level of tapering around the boundaries at the expense of spectral efficiency. Finally, in case μ=1, a simple example of orthogonal waveform is:
 
 g=   1   ┌0,T┐ .
 
     5.2 Finite OTFS modulation map 
     Let Λ ⊥ ⊂Λ 0   ⊥  be the reciprocal nested pair. Let Z 0   ⊥ ⊂Z ⊥  be the finite reciprocal torus. The finite OTFS modulation map is the linear transformation M f :C(Z 0   ⊥ )→H, defined by:
 
 M   f =Π( W ·SF Z     0     −1 ( x )) g,   (5.1)
 
     for every information vector xεC(Z 0   ⊥ ). Formula (5.1) can be written more explicitly as: 
     
       
         
           
             
               
                 
                   M 
                   f 
                 
                 ⁡ 
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     λ 
                     ∈ 
                     Λ 
                   
                 
                 ⁢ 
                 
                   
                     W 
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   ⁢ 
                   
                     X 
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   ⁢ 
                   
                     π 
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   ⁢ 
                   g 
                 
               
             
             , 
           
         
       
     
     where X=SF Z     0     −1 (x). 
     5.3 Finite OTFS demodulation map 
     The finite OTFS demodulation map is the linear transformation D f :H→C(Z 0   ⊥ ), given by:
 
 D   f (φ)=SF Z     0   ( R   Λ     0   (   W ·R   Λ   A   g,φ )),  (5.2)
 
     for every φεH. Formula (5.2) can be written more explicitly as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         D 
                         f 
                       
                       ⁡ 
                       
                         ( 
                         φ 
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       μ 
                       ) 
                     
                   
                   = 
                   
                     c 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           λ 
                           ∈ 
                           
                             Z 
                             0 
                           
                         
                       
                       ⁢ 
                       
                         
                           ψ 
                           ⁡ 
                           
                             ( 
                             
                               - 
                               ω 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             μ 
                             , 
                             λ 
                           
                           ) 
                         
                       
                     
                   
                 
                 ) 
               
               ⁢ 
               
                 
                   W 
                   _ 
                 
                 ⁡ 
                 
                   ( 
                   λ 
                   ) 
                 
               
               ⁢ 
               
                 〈 
                 
                   
                     
                       π 
                       ⁡ 
                       
                         ( 
                         λ 
                         ) 
                       
                     
                     ⁢ 
                     g 
                   
                   , 
                   φ 
                 
                 〉 
               
             
             , 
           
         
       
     
     for every φεH and λεΛ 0   ⊥ . Recall that the normalization coefficient c=vol(Λ). 
     5.4 The finite two dimensional channel model 
     Let H=Π(h) be the channel transformation where hεC(V) is assumed to have small support compared with the dimensions of the communication lattice. Recall the quadratic exponential:
 
 q (ν)=ψ(−β(ν,ν)).
 
     Theorem 5.2 (Finite 2D channel model). The finite modulation equivalent channel y=D f ∘H∘M f (x) is a cyclic convolution y=h eqv,f *x with the impulse response h eqv,f εC(Z 0   ⊥ ), given by: 
     
       
         
           
             
               
                 
                   
                     h 
                     
                       eqv 
                       , 
                       f 
                     
                   
                   = 
                   
                     
                       R 
                       
                         Λ 
                         0 
                         ⊥ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             R 
                             
                               Λ 
                               ⊥ 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               · 
                               h 
                             
                             ) 
                           
                         
                         * 
                         
                           SF 
                           Λ 
                         
                         ⁢ 
                         
                           
                              
                             W 
                              
                           
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5.3 
                   ) 
                 
               
             
           
         
       
     
       FIG. 18  demonstrates the statement of this theorem. The bar diagram  1810  represents the transmitted information vector x. The bar diagram  1820  represents the received information vector y. The bar diagram  1830  represents the 2D impulse response realizing the finite modulation equivalent channel. The received vector is related to the transmit vector by 2D cyclic convolution with the 2D impulse response. Finally, we see from Equation (5.3) that the finite impulse response h eqv,f  is the sampling of the continuos impulse response h eqv  on the finite subtorus Z 0   ⊥ ⊂Z ⊥ . 
     5.5 Explicit interpretation 
     We conclude this section by interpreting the finite OTFS modulation map in terms of classical DSP operations. We use in the calculations the nested pair Λ 0 ⊂Λ from example 5.1. Recall the definition of the finite modulation map: 
     
       
         
           
             
               
                 
                   
                     
                       
                         M 
                         f 
                       
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           λ 
                           ∈ 
                           Λ 
                         
                       
                       ⁢ 
                       
                         
                           W 
                           ⁡ 
                           
                             ( 
                             λ 
                             ) 
                           
                         
                         ⁢ 
                         
                           X 
                           ⁡ 
                           
                             ( 
                             λ 
                             ) 
                           
                         
                         ⁢ 
                         
                           π 
                           ⁡ 
                           
                             ( 
                             λ 
                             ) 
                           
                         
                         ⁢ 
                         g 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   5.4 
                   ) 
                 
               
             
           
         
       
     
     for every xεC(Z 0   ⊥ ), where X=SF Z     0     −1 (x). Formula (5.4) can be written more explicitly as: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       f 
                     
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           K 
                           , 
                           L 
                         
                       
                       ⁢ 
                       
                         
                           W 
                           ⁡ 
                           
                             [ 
                             
                               
                                 K 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 T 
                               
                               , 
                               
                                 L 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 T 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           X 
                           ⁡ 
                           
                             [ 
                             
                               
                                 K 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 T 
                               
                               , 
                               
                                 L 
                                 ⁢ 
                                 
                                   / 
                                 
                                 ⁢ 
                                 T 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           L 
                           
                             KT 
                             , 
                             μ 
                           
                         
                         ⁢ 
                         
                           
                             M 
                             
                               L 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               T 
                             
                           
                           ⁡ 
                           
                             ( 
                             g 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             K 
                           
                           ⁢ 
                           
                             
                               L 
                               
                                 KT 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μ 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 L 
                               
                               ⁢ 
                               
                                 
                                   
                                     W 
                                     K 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       L 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       T 
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     X 
                                     K 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       L 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       T 
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     M 
                                     
                                       L 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       T 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     g 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         K 
                       
                       ⁢ 
                       
                         
                           
                             L 
                             
                               KT 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               μ 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               K 
                             
                             ) 
                           
                         
                         . 
                         
                           
 
                         
                         ⁢ 
                         
                           where: 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5.5 
                   ) 
                 
               
             
             
               
                 
                   
                     ϕ 
                     K 
                   
                   = 
                   
                     
                       ∑ 
                       L 
                     
                     ⁢ 
                     
                       
                         
                           W 
                           K 
                         
                         ⁡ 
                         
                           [ 
                           
                             L 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             T 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           X 
                           K 
                         
                         ⁡ 
                         
                           [ 
                           
                             L 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             T 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           M 
                           
                             L 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             T 
                           
                         
                         ⁡ 
                         
                           ( 
                           g 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5.6 
                   ) 
                 
               
             
           
         
       
     
     The waveform φ K  is called the K th modulation block. 
     5.5.1 Frequency domain interpretation. Let G denote the Fourier transform of g. Equation (5.6) can be interpreted as feeding the sequence W K X K  into a uniform filterbank with each subcarrier shaped by the filter G. 
     5.5.2 Time domain interpretation. Let w K  and x K  denote the inverse discrete Fourier transform of the discrete waveforms W K  and X K  respectively. Both waveforms are periodic with period T. We have: 
                 ϕ   K     ∝       (       w   K     ⋆       ∑   k     ⁢         x   K     ⁡     [     kT   ⁢     /     ⁢   m     ]       ⁢     δ     kT   ⁢     /     ⁢   m             )     ·   g       ,         
where * stands for periodic convolution. The waveform x K  can be expressed in terms of the information vector x as:
 
                   x   K     ⁡     (     kT   ⁢     /     ⁢   m     )       ∝       ∑     l   =   0       n   -   1       ⁢       ψ   ⁡     (   Kl   )       ⁢     x   ⁡     [       kT   m     ,     1     nT   ⁢           ⁢   μ         ]             ,         
In words, x K  is proportional to the inverse finite Fourier transform of x along the Doppler dimension.
 
     Symplectic OTFS, OFDM Compatibility, and Other Features 
       FIGS. 29A and 29B  illustrate one manner in which symplectic OTFS methods can operate in a transmitter and receiver system  2900 . Here the data on the information plane (which may be optionally subjected to predistortion  2904 ) may be then two-dimensionally transformed using an inverse 2D Fourier Transform  2910  (and also usually a 2D spreading function  2920 ) before passing through a filter bank  2930  (which may be an OFDM compatible filter bank). The various waveforms pass through the channel (C)  2940 , where they are received by a filter bank  2950  (which may be an OFDM compatible filter bank), subjected to an inverse spreading function  2960 , inverse 2D Fourier Transform  2970  (inverse of the previous IFFT  2910 ), and then equalized  2980  as needed. 
     Attention is now directed to  FIG. 29C , which illustrates characteristics of OTFS pre-processing enabling compatibility with OFDM modulation systems. As has been discussed herein and as is illustrated in  FIG. 29C , OTFS QAM symbols may be defined over a grid in the delay-Doppler domain. During an OTFS pre-processing step, these OTFS QAM symbols are transformed and spread onto a grid in the time-frequency domain, which is the same domain in which OFDM QAM symbols are defined. 
       FIG. 29D  illustrates further details of an OTFS pre-processing operation compatible with OFDM modulation systems. As shown in  FIG. 29D , an OTFS QAM symbol may be represented as a multiplication of two linear phases. In this regard frequency in the time domain corresponds to the Doppler parameter of the OTFS QAM symbol. Similarly, frequency in the frequency domain corresponds to the delay parameter of the symbol. 
       FIG. 30  shows an alternative method of transmitting and receiving data over a channel. 
     Use with alternative forms of communication 
     Note that although wireless examples have been used throughout this disclosure, these examples are not intended to be limiting. In alternative embodiments, other medium, such as electrical transmission or RF transmission over wires or cable, optical transmission over optical fibers, and other long distance communication methodology, including acoustic transmission of signals over air or water or solid material, is also contemplated. 
     Effects of channel interference 
     According to the symplectic OTFS schemes discussed herein, in the symplectic coordinate systems, channel interference such as Doppler effects will distort or transform the symplectic plane along the frequency axis as a function according to the frequency shift due to Doppler effects, while channel interference such as time delays will distort or transform the symplectic plane along the time axis as a function according to the speed of light time delays. The net effect is that on the symplectic OTFS coordinate system, channel interference time delays shows up as phase gradients in one axis, while Doppler shifts show up as amplitude modulation effects along the other axis. 
     Because symplectic OTFS methods transmit in the transformed domain, channel convolutions are much easier to deal with, because channel convolutions show up as multiplication functions, which are easier to handle. One approach is simply to sample the signals finely enough in time and frequency so as to be able to detect these channel distortion caused phase gradients and amplitude modulation effects. Once these can be detected, they can be corrected for and eliminated. 
     This helps solve a long felt problem in the area. Prior to the teachings of this disclosure there was a lack of awareness in the field as to how to correct for channel distortions such as Doppler shifts and time delays using conventional OFDM methods. The belief was that because OFDM methods relied on sending information across a plurality of relatively narrow bandwidth bands, it was infeasible to correct for such channel distortions. However, with appropriate sampling intervals to detect channel distortion caused phase gradients and amplitude modulation in the OFDM signals, such corrections are in fact possible. 
       FIG. 31  shows the impact of channel caused Doppler and time delays on the image domain and transform domain dual grids. 
     Interleaving, and compatibility with legacy OFDM methods 
     It is possible to interleave different information planes using symplectic methods. One very useful aspect of the symplectic version of OTFS is that in some embodiments the symplectic OTFS filter banks can be set up to, for example, be compatible with previous OFDM standards such as the popular cellular 4G/LTE standards. At the same time, previous OFDM standards, such as 4G/LTE, also have medium access control (MAC) protocols that allow for control over timing and interleaving. 
     Here one example of interleaving is for example, only sending a certain column time width of the entire symplectic field over a range of frequency bands during a first time interval, sending something else, and then sending another column time width of the entire symplectic field over a range of frequency bands over a later time interval. Other forms of interleaving, such as interleaving on a frequency basis, are also possible. 
       FIG. 32  shows one example of interleaving. 
       FIG. 33  shows another example of interleaving, in which same size frames are interleaved on a frequency staggered basis. 
       FIG. 34  shows another example of interleaving, in which variable size frames are interleaved on a time basis. 
     Backward compatibility with OFDM methods 
     In certain embodiments, symplectic OFDM methods can both co-exist with legacy OFDM methods on the same frequencies and times, and indeed may even be used to improve the efficiency of legacy OFDM methods. 
     In such embodiments, symplectic OTFS methods may be viewed as feeding signals to an OFDM modulator or as otherwise pre-encoding signals which are subsequently provided to an OFDM modulator. With interleaving, this same OFTM modulator may be driven using legacy OFDM symbols during some time intervals, and OTFS signals during other time intervals. In this regard, symplectic OFTS methods may be viewed as being (on the transmitting side) an improved front-end for OFDM modulators. For example, an OFTS transceiver or transmitter may be characterized and implemented as a signal pre-processing module inserted in front of an OFDM modulator within signal transmission system. Within a signal receiving system, an OTFS receiver may positioned after the OFDM receiver in order to effect signal post-processing. 
     This approach advantageously enables compatibility with popular legacy OFDM methods such as 4G/LTE to be preserved while simultaneously facilitating the use of OTFS techniques to correct for channel distortion. This makes for an easy transition from, for example a legacy 4G/LTE system to a new higher capability “5G” system based on the OTFS methods described herein. 
     OTFS is a novel modulation technique with numerous benefits and a strong mathematical foundation. From an implementation standpoint, its added benefit is the compatibility with OFDM and the need for only incremental change in the transmitter and receiver architecture. 
     More specifically, recall that an embodiment of OTFS consists of two steps. The Heisenberg transform (which takes the time-frequency domain to the waveform domain) is already implemented in today&#39;s systems in the form of OFDM/OFDMA. In the formulation used herein, this corresponds to a prototype filter g(t) which is a square pulse. Other filtered OFDM and filter bank variations have been proposed for 5G, which can also be accommodated in this general framework with different choices of g (t). 
     The second step in this embodiment of OTFS is based upon a two dimensional Fourier transform (SFFT). As is illustrated in  FIG. 35 , this may be characterized as a pre-processing step within a transmitter module and a post-processing step within a receiver module. 
     Referring to  FIG. 35  and as discussed above, the OTFS pre-processing carried out in the transmitter module may involve performing an inverse symplectic Fourier transform and a windowing operation. Similarly, the OTFS post-processing step implemented at the receiver module may include performing a symplectic Fourier transform and another windowing operation. 
     Turning now to  FIG. 36 , there is provided a block diagram of an OTFS transmitter  3600  according to an embodiment. The transmitter  3600  includes a digital processor  3604 , which may be a microprocessor, digital signal processor, or other similar device. The digital processor accepts as input a data frame  3608  that is processed by an OTFS pre-processing block  3616  in the manner discussed above in order to yield a matrix of time-frequency modulation symbols. These time-frequency modulation symbols are then provided to an OFDM or MCFB modulator  3620  and the resulting waveform is filtered by a transmitter filter  3624 . The filtered results are then accepted by a digital to analog converter (DAC)  3630 . The baseband output of the DAC  3630  is upconverted to a radio band within an RF unit  3640  in order to produce an analog radio waveform. This waveform then travels to an OTFS receiver where it is received and demodulated as will be described below with reference to  FIG. 37 . 
     Attention is now directed to  FIG. 37 , which depicts an OTFS receiver  3700  configured to demodulate OTFS-modulated data received over a wireless link. Received signals (not shown) corresponding to channel-impaired versions of radio signals transmitted by the OTFS transmitter  3600  may be received by, for example, an antenna of the OTFS receiver  3700 . The received signals will generally not comprise exact copies of the transmitted signals because of the signal artifacts, impairments, or distortions engendered by the communication channel. The received signals are amplified and downconverted from a radio band to baseband by an RF unit  3704 . The baseband signals are then digitized within an analog to digital converter (ADC)  3708  and filtered within a receiver filter  3712 . The receiver includes a digital processor  3720 , which may be a microprocessor, digital signal processor, or other similar device. The digital processor  3720  includes an OFDM or MCFB modulator  3728  which accepts the digitized waveform from the receiver filter  3712  and produces estimated time-frequency modulation symbols. The digital processor  3720  further includes an OTFS post-processing block  3736  operative to generate an estimated data frame  3750  in the manner discussed above. 
     OTFS Relay 
       FIG. 38  illustrates an example of one manner in which an active OTFS relay system  3800  may operate between an OTFS transmitter and receiver. As shown, an active OTFS relay  3810  comprised of an OTFS transceiver including and OTFS relay transmitter  3820  and OTFS relay receiver  3822  is in wireless communication between and OTFS transmitter  3830  and an OTFS receiver  3840 . During operation, OTFS-modulated signals sent by OTFS transmitter  3830  are received and demodulated by the OTFS relay receiver  3822 . The resultant recovered data is then used by the OTFS relay transmitter  3820  to generate OTFS-modulated relay signals received and demodulated by the OTFS receiver  3840 . 
     Selected Benefits of OTFS 
     As discussed above, OTFS modulation results in wireless signals experiencing stationary, deterministic and non-fading channel interaction. Specifically, all symbols experience the same channel and the two-dimensional channel impulse response reveals a deterministic channel geometry. The permits coherence and assembly of multipath channel energy so as to enable full exploitation of all diversity branches. Importantly, the deterministic nature of the channel is substantially invariant to, and highly tolerant of, mobility of transmitters and receivers within the channel. 
     In contrast to many conventional modulation techniques, OTFS requires only loose time and frequency synchronization. This is in part because time and/or frequency misalignment is captured from the acquired channel state information and may be compensated for using equalization. 
     Of particular importance with respect to next-generation 5G communication systems, OTFS systems may be efficient scaled for use with high-order MIMO techniques. This is possible in part because of the timely, precise, and compact nature of the OTFS channel state information and the low overhead required for its acquisition. OTFS is also highly suitable to high frequency (e.g., millimeter wave) spectrum of the type contemplated for use in 5G systems. For example, OTFS is insensitive to the higher relative Doppler spread and frequency offset associated with these higher frequencies. 
     OTFS also offers interleaved variable latency with adjustable frame sizes and cooperative multipoint arrangements enabled through the acquisition of accurate channel state information. Moreover, interference mitigation may be distributed rather than centralized. 
     While various embodiments have been described above, it should be understood that they have been presented by way of example only, and not limitation. They are not intended to be exhaustive or to limit the claims to the precise forms disclosed. Indeed, many modifications and variations are possible in view of the above teachings. The embodiments were chosen and described in order to best explain the principles of the described systems and methods and their practical applications, they thereby enable others skilled in the art to best utilize the described systems and methods and various embodiments with various modifications as are suited to the particular use contemplated. 
     Where methods described above indicate certain events occurring in certain order, the ordering of certain events may be modified. Additionally, certain of the events may be performed concurrently in a parallel process when possible, as well as performed sequentially as described above. Although various modules in the different devices are shown to be located in the processors of the device, they can also be located/stored in the memory of the device (e.g., software modules) and can be accessed and executed by the processors. Accordingly, the specification is intended to embrace all such modifications and variations of the disclosed embodiments that fall within the spirit and scope of the appended claims. 
     The foregoing description, for purposes of explanation, used specific nomenclature to provide a thorough understanding of the claimed systems and methods. However, it will be apparent to one skilled in the art that specific details are not required in order to practice the systems and methods described herein. Thus, the foregoing descriptions of specific embodiments of the described systems and methods are presented for purposes of illustration and description. They are not intended to be exhaustive or to limit the claims to the precise forms disclosed; obviously, many modifications and variations are possible in view of the above teachings. The embodiments were chosen and described in order to best explain the principles of the described systems and methods and their practical applications, they thereby enable others skilled in the art to best utilize the described systems and methods and various embodiments with various modifications as are suited to the particular use contemplated. It is intended that the following claims and their equivalents define the scope of the systems and methods described herein. 
     The various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. 
     Examples of computer code include, but are not limited to, micro-code or micro-instructions, machine instructions, such as produced by a compiler, code used to produce a web service, and files containing higher-level instructions that are executed by a computer using an interpreter. For example, embodiments may be implemented using imperative programming languages (e.g., C, Fortran, etc.), functional programming languages (Haskell, Erlang, etc.), logical programming languages (e.g., Prolog), object-oriented programming languages (e.g., Java, C++, etc.) or other suitable programming languages and/or development tools. Additional examples of computer code include, but are not limited to, control signals, encrypted code, and compressed code. 
     In this respect, various inventive concepts may be embodied as a computer readable storage medium (or multiple computer readable storage media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other non-transitory medium or tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments of the invention discussed above. The computer readable medium or media can be transportable, such that the program or programs stored thereon can be loaded into one or more different computers or other processors to implement various aspects of the present invention as discussed above. 
     The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as discussed above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present invention need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present invention. 
     Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments. 
     Also, data structures may be stored in computer-readable media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements. 
     Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. 
     All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms. 
     The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” 
     The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc. 
     As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law. 
     As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc. 
     In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.