Patent Publication Number: US-11050604-B2

Title: Systems, methods and apparatuses for modulation-agnostic unitary braid division multiplexing signal transformation

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a Continuation-in-Part of U.S. patent application Ser. No. 16/459,245, filed Jul. 1, 2019 and titled “Systems, Methods and Apparatus for Secure and Efficient Wireless Communication of Signals Using a Generalized Approach Within Unitary Braid Division Multiplexing,” the disclosure of which is herein incorporated by reference in its entirety for all purposes. 
     This application is related to U.S. Non-Provisional patent application Ser. No. 16/416,144, filed on May 17, 2019 and titled “COMMUNICATION SYSTEM AND METHODS USING MULTIPLE-IN-MULTIPLE-OUT (MIMO) ANTENNAS WITHIN UNITARY BRAID DIVISIONAL MULTIPLEXING (UBDM),” the disclosure of which is herein incorporated by reference in its entirety for all purposes. 
    
    
     STATEMENT REGARDING FEDERAL GOVERNMENT INTEREST 
     This United States Government holds a nonexclusive, irrevocable, royalty-free license in the invention with power to grant licenses for all United States Government purposes. 
    
    
     TECHNICAL FIELD 
     This description relates to systems and methods for transmitting wireless signals for electronic communications and, in particular, to systems and methods for securely transmitting signals using wireless communications. 
     BACKGROUND 
     In multiple access communications, multiple user devices transmit signals over a given communications channel to a receiver. These signals are superimposed, forming a combined signal that propagates over that channel. The receiver then performs a separation operation on the combined signal to recover one or more individual signals from the combined signal. For example, each user device may be a cell phone belonging to a different user and the receiver may be a cell tower. By separating signals transmitted by different user devices, the different user devices may share the same communications channel without interference. 
     A transmitter may transmit different symbols by varying a state of a carrier or subcarrier, such as by varying an amplitude, phase and/or frequency of the carrier. Each symbol may represent one or more bits. These symbols can each be mapped to a discrete value (complex number) in the complex plane, thus producing Quadrature Amplitude Modulation, or by assigning each symbol to a discrete frequency, producing Frequency Shift Keying. The symbols are then sampled at the Nyquist rate, which is at least twice the symbol transmission rate. The resulting signal is converted to analog through a digital to analog converter, and then translated up to the carrier frequency for transmission. When different user devices send symbols at the same time over the communications channel, the sine waves represented by those symbols are superimposed to form a combined signal that is received at the receiver. 
     Some known approaches to wireless signal communication include orthogonal frequency-division multiplexing (OFDM), which is a method of encoding digital data on multiple carrier frequencies. OFDM methods have been adapted to permit signal communications that cope with severe conditions of communication channels such as attenuation, interference, and frequency-selective fading. Such an approach, however, does not address a desire for a physical layer of security of signal transmission. Furthermore, the OFDM signal includes relatively smaller amplitudes over very large dynamic ranges typically resulting in the use of radio frequency (RF) amplifiers with high peak to average power ratio. 
     Thus, a need exists for improved systems, apparatuses and methods for a secure, power efficient approach to wireless communication of signals. 
     SUMMARY 
     In some embodiments, a method includes selecting a block size, via a processor of a communications system, and identifying a set of constellation points of a constellation diagram, based on a received set of bits and the constellation diagram. Identifying the set of constellation points can include mapping the received plurality of bits to the constellation diagram (e.g., using a gray code). The constellation diagram is associated with a modulation scheme. A set of symbol blocks, based on the set of constellation points, is generated. Each symbol block from the set of symbol blocks has a size equal to the block size and includes a subset of constellation points from the set of constellation points. A unitary braid division multiplexing (UBDM) transformation is applied to each symbol block from the set of symbol blocks to produce a set of complex numbers. The set of complex numbers is then sent via the processor. 
     The communications system can be configured to perform at least one of Quadrature Amplitude Modulation (QAM), Amplitude Phase Shift Keying (APSK) modulation, or Orthogonal Frequency-Division Multiplexing. In addition, the communications system can include one or more of: a wireless communication system, a wired communication system, or a fiber optic communication system. 
     In some embodiments, a method includes receiving, via a processor of a communications system, a plurality of input bits. The communications system can include one or more of: a wireless communication system, a wired communication system, or a fiber optic communication system. The method also includes converting the plurality of input bits into a plurality of complex numbers. Converting the plurality of input bits into a plurality of complex numbers includes performing bit-to-symbol mapping (e.g., using a gray code) based on the plurality of input bits and a constellation diagram, and applying a unitary braid division multiplexing (UBDM) transformation (e.g., including a plurality of nonlinear layers and a plurality of linear layers). The plurality of complex numbers is sent, via the processor and using a predetermined modulation technique, for subsequent processing (e.g., pulse shaping and/or the application of at least one filter). The predetermined modulation technique can include one or more of Quadrature Amplitude Modulation (QAM), Amplitude Phase Shift Keying (APSK) modulation, or Orthogonal Frequency-Division Multiplexing. 
     In some embodiments, a method for modulation-agnostic UBDM signal transformation includes receiving a plurality of input bits, and mapping each input bit from the plurality of input bits to a constellation using a bit-to-symbol map to identify a plurality of symbols. The constellation can be a constellation of a constellation diagram for a signal to be transmitted wirelessly, or through wired or fiber optic communication. The constellation diagram can be associated with a specific modulation scheme. Subsets of symbols from the plurality of symbols are grouped into a plurality of blocks, each block from the plurality of blocks having a size N. A UBDM transformation is applied to each block from the plurality of blocks to produce a plurality of complex numbers, and the resulting complex numbers are sent, for example, to a downstream portion of the communication system for optional subsequent processing, e.g., including pulse shaping and/or filter application. After the optical downstream processing, a signal representing the complex numbers can be transmitted, (e.g., using the modulation scheme associated with the constellation diagram). The foregoing method can result in improved security and efficiency in the generation and/or transmission of the signal over a communication channel (which may be wired, wireless and/or optical fiber). In some such implementations, the method does not include the application of an inverse Fourier transform prior to sending the transmitted signal. Alternatively or in addition, the method does not include the generation of spreading codes. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic illustration of a secure and efficient generalized Unitary Braid Divisional Multiplexing (gUBDM) system, according to an embodiment. 
         FIG. 2  is a flowchart illustrating a method of communication including a layered approach to build unitary matrices, according to an embodiment. 
         FIG. 3  is a schematic representation of a signal transmitter within a gUBDM system, according to an embodiment. 
         FIG. 4  is a schematic representation of a signal receiver within a gUBDM system, according to an embodiment 
         FIG. 5A  is a schematic representation of a processing of a signal at a signal transmitter of an OFDM system. 
         FIG. 5B  is a schematic representation of a processing of a signal at a signal transmitter of a gUBDM system, according to an embodiment. 
         FIG. 5C  is a schematic representation of a processing of a signal at a signal transmitter of a gUBDM system, according to an embodiment 
         FIG. 6  is a flowchart describing a method of processing and transmitting a signal using a gUBDM system, according to an embodiment. 
         FIG. 7  is a flowchart describing a method of processing and transmitting a signal using a gUBDM system, according to an embodiment 
         FIG. 8  is a flowchart describing a method of receiving and recovering a signal using a gUBDM system, according to an embodiment. 
         FIG. 9  is an example constellation diagram. 
         FIG. 10  is a flow diagram illustrating a first method for modulation-agnostic UBDM signal transformation, according to an embodiment. 
         FIG. 11  is a flow diagram illustrating a second method for modulation-agnostic UBDM signal transformation, according to an embodiment. 
         FIG. 12  is a flow diagram illustrating a third method for modulation-agnostic UBDM signal transformation, according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     In some embodiments, given a set of constellation points of a constellation diagram for a signal, a method performed by a system of the present disclosure includes applying a unitary braid division multiplexing (UBDM) transformation to each constellation point from the set of constellation points, in a manner that is not dependent on the modulation scheme or type of transmission that will be used for transmission of associated signals. In other words, the UBDM transformation is modulation-agnostic. The method can be used in wireless context as well as in wired or fiber optic contexts. The method can be compatible with any of a wide variety of modulation schemes, including straight digital modulation (e.g., Quadrature Amplitude Modulation (QAM), Amplitude Phase Shift Keying (APSK) modulation, or Orthogonal Frequency-Division Multiplexing). As used herein, a “constellation diagram” refers to a representation of a signal modulated by a digital modulation scheme. The constellation diagram displays the signal as a two-dimensional xy-plane scatter diagram in the complex plane at symbol sampling instants. The angle of a point, measured counterclockwise from the horizontal axis, represents the phase shift of the carrier wave from a reference phase. The distance of a point from the origin represents a measure of the amplitude or power of the signal. 
     In an example implementation, a digital point-to-point (PTP) microwave backhaul may be configured to send a direct 128-QAM constellation at a predetermined baud rate. The system is configured to receive a plurality of input bits, modulate each input bit from the plurality of input bits into the complex baseband values in the 128-QAM constellation using a bit-to-symbol mapping (e.g., a gray code) to produce a complex value, and send those complex values to another portion of the system for subsequent processing (e.g., pulse shaping, application of filters, etc.). To apply UBDM in this manner, a block size N on which to apply the UBDM transformation may first be selected. In some instances, such as OFDM, the block size N may be determined based on a number of data subcarriers, however the systems and methods set forth herein are not constrained by the modulation or transmission type, and as such, any desired block size N may be selected. The UBDM transform can be “inserted” into a given communications system in a similar manner, regardless of the block size N. 
     In some embodiments, a method for modulation-agnostic UBDM signal transformation includes receiving a plurality of input bits. Each input bit from the plurality of input bits is mapped to a constellation using a bit-to-symbol map to identify a plurality of symbols. The constellation can be a constellation of a constellation diagram for a signal to be transmitted wirelessly, or through wired or fiber optic communication. Subsets of symbols from the plurality of symbols are then grouped into a plurality of blocks (“symbol blocks”), each block from the plurality of blocks having a size N. A UBDM transformation (e.g., including a series of nonlinear layers and a series of linear layers, as described herein) is applied to each block from the plurality of blocks to produce a plurality of complex numbers, and the resulting complex numbers are sent to a downstream portion of the communication system for optional additional processing (e.g., pulse shaping, filter application, etc.). In some such embodiments, the method for modulation-agnostic UBDM signal transformation does not include the application of an inverse Fourier transform prior to transmitting the transmitted signal. Alternatively or in addition, the method for modulation-agnostic UBDM signal transformation does not include the generation of spreading codes. In some embodiments, the modulation-agnostic UBDM transformation is a generalized Unitary Braid Divisional Multiplexing system (gUBDM) transformation, as discussed below. 
     In some embodiments set forth herein, a generalized Unitary Braid Divisional Multiplexing system (gUBDM) includes a modified Orthogonal Frequency Divisional Multiplexing (OFDM) system. The modified OFDM system can include some components common to an unmodified OFDM system, but also includes a generalized version of an OFDM component (e.g., a subset of the functionality of the OFDM). The gUBDM system can be designed to implement (e.g., in hardware and/or software executed by or stored in hardware) a modified OFDM step during operation, to execute a paired operation including performing an inverse Fast Fourier Transform (iFFT) (or a Fast Fourier Transform FFT) of signals at a signal transmitter to generate transformed signals that are transmitted, and then performing a Fast Fourier Transform (FFT) (or an inverse Fourier Transform iFFT) on the transformed signals at a receiver to recover the signals. The modification includes generalizing the iFFT/FFT performed by the transmitter to an arbitrary transformation (represented by an arbitrary matrix, for example an arbitrary unitary matrix). 
     Embodiments of a gUBDM system, as described in further detail herein, and including embodiments with the above modification of an OFDM system, can impart exceptional security and efficiency in transmission of signal over wireless communication channels. Other benefits of embodiments of the gUBDM as described herein include an ability to use non-linear transformations, as well as a generalized implementation involving equiangular tight frame (ETF) transformations or nearly equiangular tight frame (NETF) transformations as an example. Standard OFDM doesn&#39;t allow for a generalization to ETF/NETF “overloading”. 
     Generalizing to an arbitrary unitary as implemented in a gUBDM system as described herein can also have the effect of spreading the energy of each symbol or vector in a signal to be transmitted out across the different subcarriers. Spreading the energy of each symbol or vector in a signal to be transmitted can reduce the Peak-to-Average-Power-Ratio (PAPR) of the signal, and provide a degree of spreading (and, therefore, interference rejection) that is comparable to systems such as Direct Sequence Spread Spectrum (DSSS) systems. Spreading the energy of each symbol or vector in a signal to be transmitted can also provide an extra degree of freedom in multiplexing. In other words, in addition to standard frequency division multiplexing and time division multiplexing, a gUBDM system introduces code division multiplexing, which adds a powerful degree of freedom for multiplexing in a signal transmission system. 
       FIG. 1  is a schematic illustration of a secure and efficient, generalized Unitary Braid Divisional Multiplexing system  100 , also referred to herein as a “gUBDM system” or “a system,” according to an embodiment. The gUBDM  100  is configured to send and/or receive wireless electronic communications in a secure and efficient manner. The gUBDM system  100  includes signal transmitters  101  and  102 , signal receivers  103  and  104 , and a communication network  106 , as illustrated in  FIG. 1 . The gUBDM system  100  is configured to process and transmit a signal from the signal transmitters  101  and  102  via one or more communication channels defined via the communication network to the signal receivers  103  and  104 . Given a signal to be transmitted from a signal transmitter  101  and/or  102  and to a signal receiver  103  and/or  104 , the gUBDM system  100  is configured such that the signal transmitter  101  and/or  102  can process the signal by applying an arbitrary transformation to generate a transformed signal that is transmitted to the signal receivers  103  and/or  104 . The arbitrary transformation can be applied using one or more of hardware, software, a field-programmable gate array (FPGA), etc. The signal transmitters  101  and/or  102  also send to the signal receivers  103  and/or  104  (e.g., before transmitting the signal) an indication of the arbitrary transformation that was applied. The signal receivers  103  and/or  104  are configured to receive the transformed signal and the indication of the arbitrary transformation applied by the signal transmitter(s) and apply an inverse of the arbitrary transformation to recover the signal from the transformed signal. While the system  100  is illustrated to include two signal transmitters  101  and  102 , and two signal receivers  103  and  104 , a similar gUBDM system can include any number of signal transmitters and/or signal receivers. 
     In some embodiments, the communication network  106  (also referred to as “the network”) can be any suitable communications network that includes one or more communication channels configured for wirelessly transferring data, operating over public and/or private networks. Although not shown, in some implementations, the signal transmitters  101 , 102  and signal receivers  103 , 104  (or portions thereof) can be configured to operate within, for example, a data center (e.g., a cloud computing environment), a computer system, one or more server/host devices, and/or so forth. In some implementations, the signal transmitters  101 , 102  and signal receivers  103 , 104  can function within various types of network environments that can include one or more devices and/or one or more server devices. For example, the network  106  can be or can include a private network, a Virtual Private Network (VPN), a Multiprotocol Label Switching (MPLS) circuit, the Internet, an intranet, a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), a worldwide interoperability for microwave access network (WiMAX®), a Bluetooth® network, a virtual network, and/or any combination thereof. In some instances, the communication network  106  can be a wireless network such as, for example, a Wi-Fi or wireless local area network (“WLAN”), a wireless wide area network (“WWAN”), and/or a cellular network. The communication network  106  can be, or can include a wireless network and/or wireless network implemented using, for example, gateway devices, bridges, switches, and/or so forth. The network  106  can include one or more segments and/or can have portions based on various protocols such as Internet Protocol (IP) and/or a proprietary protocol. The communication network  106  can include at least a portion of the Internet. In some instances, the communication network  106  can include multiple networks or subnetworks operatively coupled to one another by, for example, network bridges, routers, switches, gateways and/or the like (not shown). 
     Fast Unitary Transformations 
     One or more methods and systems described above usually involve a matrix operation on a vector. If the length of the vector is N and the size of the matrix is N×N (e.g., when the matrix is a unitary matrix), then the matrix operation on the vector involves O(N 2 ) multiplications. Accordingly, as N increases, the computational burden on the telecommunication system can be prohibitive. 
     In some embodiments, some fast unitary transformations can be employed to reduce the calculation complexity. For example, the matrix operation on the vector can be achieved using Fourier matrix, Walsh-Hadamard matrix, Haar matrix, slant matrix, certain types of Toeplitz matrix, and certain types of circulant matrices that can be operated on a vector in a fast complexity class. These types of matrices, however, only form a limited class of transformations and therefore the resulting level of security may not be satisfactory. 
     To address the complexity issues while maintaining the security of the communication, one or more systems and methods described herein employ an approach to build an arbitrary unitary matrix up from smaller matrices. In this approach, unitary matrices are built up in layers. Each layer includes two operations. The first operation is a permutation and the second operation is a direct sum of U(2) matrices. Permutation matrices are unitary matrices that do not require any floating point operations and therefore are computationally free, i.e., with O(1) complexity. U(2) matrices are matrices where most of the values are 0, except the 2×2 blocks along the diagonal (also referred to as block-U(2) matrices). These block-U(2) matrices involve only 4×N/2=2×N multiplications. As a result, a layer including a block-U(2) involves 2×N multiplications for the block-U(2) and no multiplications for the permutation. In other words, one layer during construction of a unitary matrix has complexity O(N). 
     The total complexity of constructing a unitary matrix is the product of the number of layers and O(N) that is the complexity of each layer. In some embodiments, the total number of layers can be log(N), and the total complexity of all of the layers is therefore O(N×log(N)), which is equivalent to the complexity of a standard OFDM. In addition, log(N) layers of block-U(2) and permutation matrices can produce a dense unitary. While the space of fast unitary matrices is not as large as the full space of unitary matrices, it can still be sufficiently large to make an attack by an eavesdropper prohibitive. 
     In some embodiments, the approach described herein can employ block-U(m) matrices to build unitary matrices, where m is a positive integer (e.g., m=3, 4, 5, etc.). In some embodiments, matrices having different sizes can also be used within a single layer when constructing a unitary matrix. In some embodiments, different layers can use matrices having different sizes, e.g., a first layer uses block-U(m) matrices and a second layer uses block-U(l) matrices, where m is different from l. For example, if N=8, a set of four 2×2 block-U(2) matrices can be used in the first layer, followed by a permutation. Then two U(3) matrices and a single U(2) matrix can be used in the second layer, followed by another permutation. The third layer can include a block-U(2) matrix, a block-U(4) matrix, and then another block-U(2) matrix, followed by a third permutation. 
     In some embodiments, certain types of fast unitary matrices can also be written in terms of layers, each of which includes a permutation and a direct sum of blocks of smaller matrices. These types of matrices include, for example, Fourier matrices, Walsh-Hadamard matrices, Haar matrices, slant matrices, and Toeplitz matrices. In some embodiments, the unitary matrix that can be constructed using the layered approach includes any matrix that is not a direct sum of discrete Fourier matrices. 
     The layered approach described herein can be used in any situation that involves the construction of a unitary matrix. For example, the layered approach can be used by the initial vector generation manager  130  in the system  100  illustrated in  FIG. 1  and described above. 
       FIG. 2  is a flowchart illustrating a method  200  of communication including a layered approach to build unitary matrices, according to an embodiment. The method  200  includes, at  210 , generating, via a first processor of a first compute device, a plurality of symbols based on an incoming data. At  220 , a unitary matrix of size N×N is decomposed (where N is a positive integer). The decomposition includes: 1) applying a permutation to each symbol from the plurality of symbols using a permutation matrix, to produce a permuted plurality of symbols, and 2) transforming each symbol from the permuted plurality of symbols using at least one primitive transformation matrix of size M×M, where M is a positive integer having a value smaller than or equal to N. The result of step 2) is to produce a plurality of transformed symbols. In some embodiments, each primitive transformation matrix can include a block-U(M) matrix as described above. 
     The method  200  also includes, at  230 , sending a signal representing the plurality of transformed symbols to a plurality of transmitters. The transmitters then transmit a signal representing the plurality of transformed symbols from the plurality of transmitters to a plurality of receivers. At  240 , a signal representing the unitary matrix is sent to a second compute device for transmission of the unitary matrix to the plurality of receivers. In some embodiments, the unitary matrix can be transmitted to the receivers before the transmission of the signal representing the transformed symbols. The receivers can use the received unitary matrix for recovery of the symbols (i.e., symbols generated at  210 ). 
     In some embodiments, the decomposition of the unitary matrix at  220  can be achieved by multiple layers, each of which includes a permutation and a primitive transformation. For example, the first layer uses a first permutation matrix and a first primitive transformation matrix, and the second layer uses a second permutation matrix and a second primitive transformation matrix. In some embodiments, the total number of layers can be comparable to log(N), where N is the number of symbols generated at  210 . 
     In some embodiments, the unitary matrix decomposed at  220  includes one of a Fourier matrix, a Walsh matrix, a Haar matrix, a slant matrix, or a Toeplitz matrix. In some embodiments, during the decomposition of the unitary matrix at  220 , applying the permutation is not immediately followed by another permutation. 
     In some embodiments, the primitive transformation matrix has a dimension (e.g., a length) with a magnitude of 2, and the constructing the unitary matrix includes an iterative process that occurs log 2  N times. In some embodiments, other lengths can also be used for the primitive transformation matrix. For example, the primitive transformation matrix can have a length greater than 2 (e.g., 3, 4, 5, etc.). In some embodiments, the primitive transformation matrix includes a plurality of smaller matrices having diverse dimensions. For example, the primitive transformation matrix can include block-U(m) matrices, where m can be different values within a single layer or between different layers. 
     In some embodiments, the receiver used in the method  200  includes a plurality of antenna arrays. The plurality of receivers and the plurality of transmitters are configured to perform Multiple Input Multiple Output (MIMO) operations. 
     In some embodiments, a system for communication using layered construction of unitary matrices, according to an embodiment. The system includes a signal transmitter or a plurality of signal transmitters (e.g., numbered 1 to i) and a signal receiver or a plurality of signal receivers (e.g., numbered 1 to j), where i and j are both positive integers. In some embodiments, i and j can equal. In some other embodiments, i can be different from j. In some embodiments, the transmitters and the receivers are configured to perform Multiple Input Multiple Output (MIMO) operations. 
     In some embodiments, the transmitters can be substantially identical to the signal transmitter  301  illustrated in  FIG. 3  and described below. In some embodiments, the receivers can be substantially identical to the signal receiver  401  illustrated in  FIG. 4  and described below. In some embodiments, each transmitter includes an antenna and the antennas of multiple transmitters can form an antenna array. In some embodiments, each receiver includes an antenna and the antennas of multiple receivers can also form an antenna array. 
     The system also includes a processor operably coupled to the signal transmitters. In some embodiments, the processor includes a single processor. In some embodiments, the processor includes a group of processors. In some embodiments, the processor can be included in one or more of the transmitters. In some embodiments, the processor can be separate from the transmitters. For example, the processor can be included in a compute device configured to process the incoming data and then direct the transmitters to transmit signals representing the incoming data. 
     The processor is configured to generate a plurality of symbols based on an incoming data and decompose a unitary transformation matrix of size N×N into a set of layers, where N is a positive integer. Each layer includes a permutation and at least one primitive transformation matrix of size M×M, where M is a positive integer smaller than or equal to N. 
     The processor is also configured to encode each symbol from the plurality of symbols using at least one layer from the set of layers to produce a plurality of transformed symbols. A signal representing the plurality of transformed symbols is then sent to the plurality of transmitters for transmission to the plurality of signal receivers. In some embodiments, each transmitter in the transmitters can communicate with any receiver in the receivers. 
     In some embodiments, the processor is further configured to send a signal representing one of: (1) the unitary transformation matrix, or (2) an inverse of the unitary transformation matrix, to the receivers, prior to transmission of the signal representing the transformed symbols to the signal receivers. This signal can be used to by the signal receivers to recover the symbols generated from the input data. In some embodiments, the unitary transformation matrix can be used for symbol recovery. In some embodiments, the recovery can be achieved by using the inverse of the unitary transformation matrix. 
     In some embodiments, the fast unitary transformation matrix includes one of a Fourier matrix, a Walsh matrix, a Haar matrix, a slant matrix, or a Toeplitz matrix. In some embodiments, the primitive transformation matrix has a dimension (e.g., a length) with a magnitude of 2 and the set of layers includes log 2  N layers. In some embodiments, any other length can be used as described above. In some embodiments, the signal receivers are configured to transmit a signal representing the plurality of transformed symbols to a target device. 
       FIG. 3  is a schematic block diagram of an example signal transmitter  301  that can be a part of an gUBDM system such as the gUBDM system  100  described above with reference to  FIG. 1 , according to an embodiment. The signal transmitter  301  can be structurally and functionally similar to the signal transmitters  101 , 102  of the system  100  illustrated in  FIG. 1 . In some embodiments, the signal transmitter  301  can be, or can include, processors configured to process instructions stored in a memory The signal transmitter  301  can be a hardware-based computing device and/or a multimedia device, such as, for example, a server, a desktop compute device, a smartphone, a tablet, a wearable device, a laptop and/or the like. The signal transmitter  301  includes a processor  311 , a memory  312  (e.g., including data storage), and a communicator  313 . 
     The processor  311  can be, for example, a hardware based integrated circuit (IC) or any other suitable processing device configured to run and/or execute a set of instructions or code. For example, the processor  311  can be a general purpose processor, a central processing unit (CPU), an accelerated processing unit (APU), an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA), a programmable logic array (PLA), a complex programmable logic device (CPLD), a programmable logic controller (PLC) and/or the like. The processor  311  can be operatively coupled to the memory  312  through a system bus (for example, address bus, data bus and/or control bus). 
     The processor  311  can be configured to receive a signal to be transmitted and to perform processing to transform the signal into a transformed signal by applying an arbitrary transformation. In some implementations, the processor  311  can apply an arbitrary transformation that is defined to be a unitary transformation such that the transformed signal can be transmitted in a secure and efficient manner using the gUBDM system. 
     The processor  311  can include a set of components including a converter  314 , an arbitrary transform selector  315 , and an arbitrary transform applier  316 . The processor  311  can receive a set of signals  321 A,  321 B, perform a set of arbitrary transformations  331 A,  331 B, and send a set of transformed signals  341 A,  341 B. 
     In some embodiments, each of the converter  314 , an arbitrary transform selector  315 , and an arbitrary transform applier  316  can be software stored in the memory  312  and executed by processor  311 . For example, each of the above mentioned portions of the processor  311  can be code to cause the processor  311  to execute the converter  314 , the arbitrary transform selector  315 , and the arbitrary transform applier  316 . The code can be stored in the memory  312  and/or a hardware-based device such as, for example, an ASIC, an FPGA, a CPLD, a PLA, a PLC and/or the like. In other embodiments, each of the converter  314 , the arbitrary transform selector  315 , and the arbitrary transform applier  316  can be hardware configured to perform the respective functions. In some embodiments, each of the components can a combination of software and hardware based. In some embodiments one or more of the components (e.g., converter  314 , the arbitrary transform selector  315 , the arbitrary transform applier  316 ) of the processor  311  can be configured to operate based on one or more platforms (e.g., one or more similar or different platforms) that can include one or more types of hardware, software, firmware, operating systems, runtime libraries, and/or so forth. In some implementations, the components of the signal transmitter can be configured to operate within a cluster of devices (e.g., a server farm). In such an implementation, the functionality and processing of the components of the signal transmitter  301  can be distributed to several devices of the cluster of devices. The components of the signal transmitter  301  and signal receiver  401  can be, or can include, any type of hardware and/or software configured to process attributes. 
     The converter  314  can be configured to receive a signal to be transmitted and prepare the signal in a form that can be transformed by the processor  311  using an arbitrary transformation. For example, in some embodiments, the processor  311  can receive a signal in the form of a serial set of symbols b n . The converter  314  can be configured to perform a serial-to-parallel computation on the set of symbols b n  to convert the serial set of symbols b n  to a parallel set of symbols. In some embodiments, the converter  314  can generate a plurality of vectors (e.g., vectors  321 A and  321 B) based on the set of symbols. In some implementations, the converter  314  can receive a signal in the form of a plurality of input bits. The converter  314  can be configured to generate a plurality of symbols based on the plurality of input bits. The converter  314  can be further configured to generate a plurality of blocks based on the plurality of symbols where each block from the plurality of blocks represents a vector from a plurality of vectors (e.g., vectors  321 A,  321 B). Alternatively, the converter  314  can be further configured to generate multiple pluralities of blocks based on the plurality of symbols where each plurality of blocks from the multiple pluralities of blocks represents a vector from a plurality of vectors (e.g., vectors  321 A,  321 B). 
     The arbitrary transform selector  315  can be configured to select, based at least partly on the signal to be transmitted or the plurality of vectors generated by the converter  314 , an arbitrary transformation (e.g., arbitrary transformation  331 A,  331 B) to be applied on the plurality of vectors (e.g., vectors  321 A,  321 B) to securely and efficiently transmit the vectors from the signal transmitter  201  to one or more receivers associated with the gUBDM system. The arbitrary transformation (e.g., arbitrary transformation  331 A,  331 B) can include one of, or a combination of any of, a non-linear transformation, a unitary transformation, an ETF transformation, or a NETF transformation. In some embodiments, the arbitrary transform selector  315  can have access to a library of arbitrary transformations that are unitary by design (e.g., arbitrary transformation  331 A,  331 B) from which one can be selected for transmitting a signal. The arbitrary transform selector  315  can select the arbitrary transformation based, for example, on a transformation type and/or a criteria negotiated between two communicants via a telecommunications handshake or otherwise input by a participant in the communications system. The criteria can include, for example, one or more of: a desired security level, a latency threshold, an error rate threshold, a minimum data rate, a maximum data rate, etc. Notably, unitary transformation is the largest class of transformations that can be performed on a vector of symbols that leaves the total power of the signal unchanged. If a non-unitary transformation is used, then the inverse transformation at the receiver will necessarily amplify noise in some of the received symbols, whereas this is not the case of unitary transformations. 
     In some instances, the arbitrary transformation selector  315  can be configured to select a transformation that is not an identity matrix, a discrete Fourier matrix, or is any other direct sum of Fourier matrices. For example in some implementations the arbitrary transformations selector  315  can have a library of unitary transformations and based on a set of guidelines select one unitary transformation U and perform computations to check if U is an identity matrix, or a discrete Fourier matrix, or is any other direct sum of a set of Fourier matrices. If U is one of the three above categories, in some embodiments the arbitrary transform selector  315  can discard U and select another transformation that can meet the guideline of not being any of the above three categories. If the arbitrary transformation selector  315  picks a transformation U that is not an identity matrix, a discrete Fourier matrix, or is any other direct sum of Fourier matrices it can then assign U as the arbitrary transformation A to be used for an instance of transforming a signal to be transmitted using a gUBDM system according to that embodiment. 
     In some implementations, the arbitrary transform selector  315  can perform the selection based on a set of inputs received by the processor  311 . In some implementations, the arbitrary transform selector  315  can perform the selection based on a set of parameters associated with the signal, the plurality of vectors, the nature of signal transmission (e.g., a security requirement, sensitivity of information content in the signal, path of signal transmission, etc.). In some implementations, the arbitrary transform selector  315  can be configured to define and generate an arbitrary transformation according a set of inputs received by the processor  311  (e.g., a set of user inputs received by the processor  311 ). 
     The arbitrary transform applier  316  can apply the selected arbitrary transformation on the plurality of vectors (e.g., vectors  321 A,  321 B) to generate a plurality of transformed vectors (e.g., transformed vectors  341 A,  341 B). In some implementations, the plurality of transformed vectors can have a total magnitude that substantially equals a total magnitude of the plurality of vectors. 
     The transformed vectors can then be sent to the signal transmitter antennas  317  and  318  included in the communicator  313  to be sent to one or more signal receivers associated with a signal receiver. In some implementations, for example, the arbitrary transform applier  316  can be configured to perform matrix operations to apply a transformation matrix A on a set of vectors to generate transformed vectors. In some implementations, the arbitrary transform applier  316  can be configured to perform any suitable number of procedures (e.g. signal processing procedures, suitable matrix operations) on a set of vectors before applying an arbitrary transformation. 
     While illustrated to include two signal transmitter antennas  317  and  318 , as described above, a similar signal transmitter could include a single transmitter antenna according to some embodiments. A similar signal transmitter could include any suitable higher number of signal transmitter antennas (i.e., more than two transmitter antennas) according to still other embodiments. In some embodiments the signal transmitter  301  can include a plurality of antenna arrays configured to perform Multiple Input Multiple Output (MIMO) operations. 
     The memory  312  of the signal transmitter  301  can be, for example, a random access memory (RAM), a memory buffer, a hard drive, a read-only memory (ROM), an erasable programmable read-only memory (EPROM), and/or the like. The memory  312  can store, for example, one or more software modules and/or code that can include instructions to cause the processor  311  to perform one or more processes, functions, and/or the like (e.g., functions associated with the converter  314 , the arbitrary transform selector  315 , the arbitrary transform applier  316 ). In some embodiments, the memory  312  can include extendable storage units that can be added and used incrementally. In some implementations, the memory  312  can be a portable memory (for example, a flash drive, a portable hard disk, and/or the like) that can be operatively coupled to the processor  311 . In other instances, a memory can be remotely operatively coupled with the signal transmitter  301 . For example, a remote database server can serve as a memory and be operatively coupled to the signal transmitter  301 . 
     The communicator  313  can be a hardware device operatively coupled to the processor  311  and memory  312  and/or software stored in the memory  312  executed by the processor  311 . The communicator  313  can include a signal transmitter antenna  317  and optionally a signal transmitter antenna  318 . While a second transmitter antenna  318  in addition to the transmitter  317  is shown in  FIG. 3 , a signal transmitter similar to the signal transmitter  301  can have any number of transmitter antennas, according to some embodiments, or just a single signal transmitter antenna, according to some other embodiments. The communicator  313  can be, for example, a network interface card (NIC), a Wi-Fi™ module, a Bluetooth® module and/or any other suitable wired and/or wireless communication device. Furthermore the communicator  313  can include a switch, a router, a hub and/or any other network device. The communicator  313  can be configured to connect the signal transmitter  301  to a communication network (such as the communication network  106  shown in  FIG. 1 ). In some instances, the communicator  313  can be configured to connect, via one or more communication channels, to a communication network such as, for example, the Internet, an intranet, a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), a worldwide interoperability for microwave access network (WiMAX®), an optical fiber (or fiber optic)-based network, a Bluetooth® network, a virtual network, and/or any combination thereof. 
     In some instances, the communicator  313  can facilitate receiving and/or transmitting a file and/or a set of files via one or more communication channels through a communication network (e.g., the communication network  106  in the gUBDM system  100  of  FIG. 1 ). In some instances, a received file can be processed by the processor  211  and/or stored in the memory  312  as described in further detail herein. In some instances, as described previously, the communicator  313  can be configured to send a plurality of transformed vectors, via the signal transmitter antennas  317  and  318 , to one or more signal receiver antennas associated with one or more signal receivers connected to a communication network as part of a gUBDM system. The communicator  313  can also be configured to send and/or receive data associated with a library of arbitrary transformation systems. 
     Returning to  FIG. 1 , the signal transmitters  101 , 102  that are connected to gUBDM system  100  can be configured to communicate with and transmit signals to signal receivers  103 ,  104  via one or more communication channels defined in the communication network  106 .  FIG. 4  is a schematic representation of a signal receiver  401  that is part of gUBDM system. The signal receiver  401  can be structurally and functionally similar to the signal receivers  103 ,  104  of the system  100  illustrated in  FIG. 1 . The signal receiver  401  includes a processor  411 , a memory  412 , and a communicator  413 . 
     The processor  411  can be, for example, a hardware based integrated circuit (IC) or any other suitable processing device configured to run and/or execute a set of instructions or code. For example, the processor  311  can be a general purpose processor, a central processing unit (CPU), an accelerated processing unit (APU), an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA), a programmable logic array (PLA), a complex programmable logic device (CPLD), a programmable logic controller (PLC) and/or the like. The processor  411  can be operatively coupled to the memory  412  through a system bus (for example, address bus, data bus and/or control bus). 
     The processor  411  can be configured to receive a transformed signal that is securely transmitted via one or more communication channels defined in a communication network (e.g., network  106  of  FIG. 1 ), obtain information associated with an arbitrary transformation that was used to generate the transformed signal, and based on the information process the transformed signal to recover an original signal (e.g., by applying an inverse of the arbitrary transformation) such that the original signal can be received by a destination in a secure and efficient manner using the gUBDM system, according to an embodiment. 
     The processor  411  can include a set of components including a converter  414 , an arbitrary transform identifier  415 , and an arbitrary transform reverser  416 . The processor  411  can include, or access from memory  412 , a plurality of transformed vectors  441 A,  441 B, representing transformed signals, received from one or more transmitter antennas of a signal transmitter (e.g., transmitter antennas  317  and  318  of signal transmitter  301 ) that is part of the gUBDM system that the signal receiver  401  is part of. The processor  411  can include or access in memory  412  a set of arbitrary transformations  431 A and  431 B, identified based on information associated with a signal received from a signal transmitter, and a set of reverse transformations  451 A,  451 B, computed based on the identified arbitrary transformations, and a plurality of vectors  421 A,  421 B representing a set of original signals. 
     The arbitrary transform identifier  415  can be configured to receive information associated with a transformed signal (e.g., transformed signal represented by transformed vectors  441 A,  441 B) received via the signal receivers  417  and  418 , the information including an indication of the identity of an arbitrary transformation that was used in generating the transformed signals. The arbitrary transform identifier  415  is configured to, based on the information, identify the arbitrary transformation that can be used to recover an original signal (e.g., original signal represented by plurality of vectors  421 A,  421 B) from the transformed signal (e.g., transformed signals  441 A,  441 B). 
     The arbitrary transform reverser  416  generates, based on the identity of the arbitrary transformation, an inverse of the identified arbitrary transformation, also referred to as a reverse transformation (e.g., reverse transformations  451 A,  451 B) configured to reverse the effects of the identified arbitrary transformation to recover the original signal from a transformed signal. For example, in some embodiments, the arbitrary transform reverser  416  generates a reverse transformation (A′)  451 A configured to be applied on a plurality of transformed vectors  441 A and  441 B, representing a transformed signal, and received by the signal receiver  401 , so that the reverse transformation (A′)  451 A can reverse the effects of an arbitrary transformation (A)  431 A and recover a plurality of vectors  421 A and  421 B representing an original signal. 
     The converter  414  can be configured to receive a recovered plurality of vectors (e.g.,  421 A and  421 B) representing an original signal and regenerate the original signal from the recovered plurality of vectors. For example, in some embodiments, the processor can receive a parallel set of symbols b n . The converter  414  can be configured to perform a parallel-to-serial computation on the set of symbols b n  to convert the parallel set of symbols b n  to a serial set of symbols that can be similar to the original signal. In some embodiments, the converter  414  can receive a plurality of recovered vectors (e.g., vectors  421 A and  421 B) and generate, based on the vectors, an original signal including a set of symbols. In some embodiments, the converter  414  can receive a plurality of recovered vectors (e.g., vectors  421 A and  421 B) and generate, based on the recovered vectors pluralities of blocks each plurality of blocks representing a vector of the plurality of vectors. The converter  414  can then regenerate, based on the pluralities of blocks, a plurality of input bits from which it can recover an original signal. 
     The memory  412  of the signal receiver  401  can be similar in structure and/or function to the memory  312  of the signal transmitter  301 . For example, the memory  412  can be a random access memory (RAM), a memory buffer, a hard drive, a read-only memory (ROM), an erasable programmable read-only memory (EPROM), and/or the like. The memory  412  can store, for example, one or more software modules and/or code that can include instructions to cause the processor  411  to perform one or more processes, functions, and/or the like (e.g., functions associated with the converter  414 , the arbitrary transform identifier  415 , the arbitrary transform reverser  416 ). In some embodiments, the memory  412  can include extendable storage units that can be added and used incrementally. In some implementations, the memory  412  can be a portable memory (for example, a flash drive, a portable hard disk, and/or the like) that can be operatively coupled to the processor  411 . In other instances, the memory can be remotely operatively coupled with the signal receiver  401 . For example, a remote database server can serve as a memory and be operatively coupled to the signal receiver  401 . 
     The communicator  413  can be a hardware device operatively coupled to the processor  411  and memory  412  and/or software stored in the memory  412  executed by the processor  411 . The communicator  413  can include a signal receiver antenna  417  and optionally a signal receiver antenna  418 . While a second receiver  418  in addition to the receiver  417  is shown in  FIG. 4 , a signal receiver similar to the signal receiver  401  can have any number of receivers, according to some embodiments, or just a single signal receiver, according to some other embodiments. The communicator  413  can be, for example, a network interface card (NIC), a Wi-Fi™ module, a Bluetooth® module and/or any other suitable wired and/or wireless communication device. Furthermore the communicator  413  can include a switch, a router, a hub and/or any other network device. The communicator  413  can be configured to connect the signal receiver  401  to a communication network (such as the communication network  106  shown in  FIG. 1 ). In some instances, the communicator  413  can be configured to connect to a communication network such as, for example, the Internet, an intranet, a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), a worldwide interoperability for microwave access network (WiMAX®), an optical fiber (or fiber optic)-based network, a Bluetooth® network, a virtual network, and/or any combination thereof. 
     In some instances, the communicator  413  can facilitate receiving and/or transmitting a file and/or a set of files via one or more communication channels defined in a communication network (e.g., the communication network  106  in the gUBDM system  100  of  FIG. 1 ). In some instances, a received file can be processed by the processor  411  and/or stored in the memory  412  as described in further detail herein. In some instances, as described previously, the communicator  413  can be configured such that the signal receivers  417  and  418  include one or more antennas tuned to receive transformed signals of a particular predetermined center frequency within a predetermined bandwidth, to receive transformed signals securely and efficiently transmitted by one or more signal transmitter antennas associated with one or more signal transmitters connected to a communication network as part of a gUBDM system. The communicator  413  can also be configured to send and/or receive data associated with a library of arbitrary transformation systems. In some embodiments the signal receiver  401  can include a plurality of antenna arrays configured to perform Multiple Input Multiple Output (MIMO) operations. 
     In some embodiments, the gUBDM system (e.g., gUBDM system  100 ) can be in some aspects partly similar in stru-cture and/or function to an Orthogonal Frequency Divisional Multiplexing (OFDM) system. For example, an example pipeline for an OFDM system  500 ′ can include a set of operations as presented in  FIG. 5A , where vector b can be a set of symbols b n . 
     In the example OFDM system  500 ′, the symbols b n  enter an OFDM transmitter and are first put through a “serial-to-parallel” (labeled “S/P” above) computation, and then they are run through an inverse FFT (labeled “iFFT” above). In some embodiments, they may be given a cyclic prefix, and undergo a pulse shaping procedure. An OFDM receiver can be configured to perform the above operations in a reverse order, except an FFT replaces the iFFT. 
     Compared to the above described OFDM system  500 ′, operations carried out by a gUBDM system  500  described herein (e.g., gUBDM system  100 ) are illustrated in  FIG. 4B . The gUBDM  500  can include an extra operator (e.g., a linear operator) A between the S/P block  514  and the iFFT block, as shown in  FIG. 5B . In use, according to the example embodiment associated with  FIG. 5B , the gUBDM  400  operates such that symbols b n  are received by the signal transmitter and are first put through a serial-to-parallel block (e.g., converter similar to converter  314  of the signal transmitter  301 ) to generate a converted set of vectors. The converted set of vectors then undergo the linear transformation A to generate a set of transformed vectors. For example, the transformation can be carried out by an arbitrary transformation applier  515  similar to arbitrary transformation applier  316  and the linear transformation A being selected by arbitrary transformation selector similar to the arbitrary transformation selector  315 . In some embodiments, the transformed vectors are then put through an iFFT block to generate a second transformed vectors and the resulting second transformed vectors can be transmitted to one or more receivers in the gUBDM system. In some other embodiments, the iFFT block can be skipped and the transformed vectors generated by the arbitrary transformation applier can be transmitted to one or more receivers in the gUBDM system. Expressed in another way,
 
   b   → A     b   →   s   =   A     b   .
 
(where   is the discrete Fourier matrix). In some embodiments, A can be unitary by design, as described herein, and F is known to be unitary. By property of unitary matrices as a group, the product FA will also be unitary. Therefore, because A can be any unitary, including the iFFT matrix is unnecessary, and according to some embodiments a gUBDM system can be configured by replacing the iFFT block with an arbitrary unitary A, as illustrated in  FIG. 5C  showing the operations in a gUBDM system  500 , including an arbitrary transform applier  515 , according to an embodiment.
 
     Following the above description a signal transmitter and a signal receiver operable with an OFDM system (e.g., OFDM system  500  of  FIG. 5A ) can be easily adapted to be used with a gUBDM system described herein (e.g., gUBDM systems  500  in  FIGS. 5B and 5C ) with the only changes being a replacement of an iFFT operation with an arbitrary transformation operation using A at the transmitter and the FFT with A′ at the signal receiver to reverse the transformation. Other details of an OFDM system can remain. 
     The above described gUBDM system, in use, can be used to transmit signal in a highly secure and efficient manner as described in detail below. Given a signal transmission system, where one or more signals are transmitted from a source associated with a user Alice to a destination associated with a user Bob, such a system may be vulnerable to eavesdropping by a third party user Eve who may have access to the transmitted signal or transmitted vectors. Given that a gUBDM system is being used for the signal transmission, where an arbitrary transformation A is used to generate the transformed signal or transformed vectors that are being transmitted, if Eve doesn&#39;t know the matrix A and is only able to base her attack on knowing cipher, the amount of work she has to do to recover the data can be prohibitively large. In some other embodiments, the arbitrary transformation can be non-linear in nature, making it even more complicated and infeasible for Eve to find the non-linear transformation to recover signals even if she has access to plaintext/ciphertext pairs. 
       FIG. 6  illustrates a flowchart describing an example method  600  of preparing a signal and transmitting a signal in a secure and efficient manner using a gUBDM system, according to an embodiment. At  671 , according to the method  600 , a signal transmitter of a gUBDM system (e.g., the signal transmitter  201  described above) receives data including a plurality of input bits. The plurality of input bits can represent an original signal that is to be transmitted in a secure and efficient manner. The data can further include other attributes associated with the signal represented by the input bits. For example the data can include information related to the nature of the signal, the nature of the input bits, the size, sensitivity of the information contained, security requirement, etc. 
     At  672 , the signal transmitter generates a plurality of symbols based on the plurality of input bits. In some instances, the signal transmitter can generate a plurality of symbols with a symbol being described as a pulse in a digital complex baseband signal. In some implementations, a symbol can be a waveform, or a state that, when transmitted through a communication channel defined in a communications network, can change/alter and/or maintain a state or a significant condition of the communication channel such that the state or condition persists, for a fixed period of time. In some instances, a signal transmitter can break up a plurality of input bits associated with a serial signal into a plurality of symbols that can be modified and/or transmitted in parallel using a Multiple Input and Multiple Output system of transmission as described further below. In some instances, a signal transmitter can use a converter (e.g., converter  314 ) to convert a serial plurality of input bits into a parallel plurality of symbols. In some implementations, the generating a plurality of symbols based on a plurality of input bits can be via using a bit-to-symbol map. 
     At  673 , the signal transmitter generates pluralities of blocks based on the plurality of symbols, each plurality of blocks from the pluralities of blocks representing a vector from a plurality of vectors. In some instances, a signal transmitter can receive a serial plurality of symbols associated with a serial signal and break it up into pluralities of blocks each plurality of block representing a vector from a plurality of vectors, the vectors being configured to be transformed and/or transmitted in parallel using a Multiple Input and Multiple Output system of transmission as described herein. In some instances, a signal transmitter can use a converter (e.g., converter  314 ) to convert the serial plurality of symbols into the pluralities of blocks. 
     At  674 , the signal transmitter select, based at least partially on the plurality of vectors, an arbitrary transformation configured to be applied to the vectors to generate a plurality of transformed vectors. For example, the signal transmitter can have access to a library of arbitrary Transformations including unitary transformations, equiangular tight frame (ETF) transformations, and a nearly equiangular tight frame (NETF) transformations. The signal transmitter can use an arbitrary transformation selector (e.g., arbitrary transformation selector  315 ) to select arbitrary transformation, for example a unitary transformation, to be applied on the plurality of vectors to generate a plurality of transformed vectors. In some instances, the arbitrary transformation can select an equiangular tight frame (ETF) transformation, or in some other instances the arbitrary transformation selector can select a nearly equiangular tight frame (NETF) transformation. In some implementations, the arbitrary transformation selector can be configured such that the arbitrary transformation selected is based on a matrix that is not an identity matrix or a discrete Fourier matrix. In some implementations, the arbitrary transformation selector can be configured such that the arbitrary transformation selected is based on a matrix that is not a direct sum of discrete Fourier matrices. 
     At  675 , the signal transmitter applies the arbitrary transformation to each vector of the plurality of vectors to produce the plurality of transformed vectors. In some instances, the applying the arbitrary transformation can be such that the plurality of transformed vectors has a total magnitude that substantially equals a total magnitude of the plurality of vectors. 
     At  676 , the signal transmitter sends a signal representing the plurality of transformed vectors to a plurality of transmitter antennas for transmission of a signal representing the plurality of transformed vectors from the plurality of transmitter antennas to a plurality of signal receivers. In some instances, the plurality of transformed vectors can be configured to be sent in parallel via multiple transmitter antennas associated with the signal transmitter antenna device (e.g., transmitter antennas  317  and  318  associated with the signal transmitter  301 ) and through multiple communication channels using a Multiple Input and Multiple Output system of transmission such that the transformed vectors sent in parallel can be received by a plurality of receivers associated with one or more signal receivers associated with the gUBDM system being used. For example, the plurality of signal receivers can include a plurality of antenna arrays, and the plurality of signal receivers be associated with signal receivers such as the signal receiver  401  and the plurality of signal transmitter antennas can be associated with signal transmitters such as the signal transmitter  301 , where in the signal transmitter and the signal receiver are configured to perform Multiple Input Multiple Output (MIMO) operations. 
     In some implementations, the signal can include a set of transformed symbols associated with the plurality of transformed vectors and a signal transmitter (e.g., signal transmitter  301 ) can place a set of transformed symbols on the communication channel (s) (e.g., via a transmitter  317 ) at a fixed and known symbol rate. A signal receiver can perform the task of detecting the sequence of transformed symbols to reconstruct the transformed vectors. In some instances, there may be a direct correspondence between a transformed symbol and a small unit of data. For example, each transformed symbol may encode one or several binary digits or ‘bits’. The data may also be represented by the transitions between transformed symbols, or even by a sequence of many transformed symbols. 
     In some implementations, the signal transmitter can be configured to send the signal representing the plurality of transformed vectors to the plurality of transmitters via a physical layer associated with an open system interconnection model (OSI). The OSI model is a conceptual model that characterizes and standardizes the communication functions of a telecommunication or computing system without regard to its underlying internal structure and technology with the goal of achieving interoperability of diverse communication systems using standard communication protocols. The OSI model uses partitioning of information exchanged via communication channels of a communication network into abstraction layers (e.g., seven layers) with each layer including information of a specific type. 
     For example, a layer 1 can include a physical layer used for the transmission and reception of unstructured raw data between a signal transmitter and a physical transmission medium (e.g., a wireless communication channel in a communication network such as network  106 ). It is configured to convert digital bits included in the signals transmitted into electrical, radio, or optical signals. Layer specifications define characteristics such as voltage levels, the timing of voltage changes, physical data rates, maximum transmission distances, modulation scheme, channel access method and physical connectors. This includes the layout of pins, voltages, line impedance, cable specifications, signal timing and frequency for wireless devices. Bit rate control is done at the physical layer and may define transmission mode as simplex, half duplex, and full duplex. The components of a physical layer can be described in terms of a network topology. The communications channel used to transmit the signal can have specifications for a physical layer. 
     At  677 , the signal transmitter provides the arbitrary transformation to the plurality of signal receivers, the providing being in association with the sending of the plurality of transformed vectors, the providing further being configured for a recovery of the plurality of vectors at the plurality of signal receivers. In some implementations, the plurality of signal receivers is further configured to transmit a signal representing the plurality of transformed vectors to a target device. For example the plurality of signal receivers can be associated with one or more signal receivers that can be configured to transmit a signal representing the plurality of transformed vectors to a target device. 
     In some instances, the signal transmitter can send a signal that, in addition to representing the plurality of transformed vectors, can also be representing one of: (1) the arbitrary transformation, or (2) an inverse of the arbitrary transformation to the plurality of signal receivers. In some instances, the signal transmitter can send a first signal representing the plurality of transformed vectors and send a second signal representing the arbitrary transformation or an inverse of the arbitrary transformation. In some implementations the signal transmitter can send the second signal at a time point prior to the sending of the first signal. That is, said in another way, the signal transmitter can send the signal representing the arbitrary transformation or an inverse of the arbitrary Transformation prior to transmission of the signal representing the plurality of transformed vectors to the plurality of signal receivers, such that the plurality of signal receivers recovers the plurality of vectors from the plurality of transformed vectors based on the arbitrary transformation or an inverse of the arbitrary transformation. 
       FIG. 7  illustrates an example method  600  of transmitting a signal in a secure and efficient manner, using a gUBDM system according to an embodiment. The method  700  can be implemented by a processor for example a processor associated with a signal transmitter of a gUBDM system (e.g., the signal transmitter  201  described above). At  771 , an arbitrary transformation is applied to a plurality of vectors to produce a plurality of transformed vectors. The arbitrary transformation can include a unitary transformation, an equiangular tight frame (ETF) transformation, or a nearly equiangular tight frame (NETF) transformation. In some implementations, more than one arbitrary transformations can be applied. For example in some instances, the signal transmitter implementing the method  700  can be configured such that a first arbitrary transformation is applied to the plurality of vectors to produce a first plurality of transformed vectors and a second arbitrary transformation is applied to the plurality of vectors to produce a second plurality of transformed vectors. 
     At  772 , the method includes producing, using the arbitrary transformation, a first transformed signal based on at least a first transformed vector from the plurality of transformed vectors. In some instances the first transformed signal can include a first complex baseband signal. At  773 , the method includes producing, using the arbitrary transformation, a second transformed signal based on at least a second transformed vector from the plurality of transformed vectors. In some instances, the second transformed signal can include a second complex baseband signal. 
     As described above, in some implementations the second transformed signal can be based on a second transformed vector the second plurality of transformed vectors generated using the second arbitrary transformation. 
     At  774 , the method  700  includes transmitting the first transformed signal, via a communications channel, to a first signal receiver that is configured to detect the first transformed signal. At  775 , the method includes transmitting the second transformed signal, via the communications channel, to a second signal receiver that is configured to detect the second complex baseband signal. In some instances, the transmitting the second transformed signal is via a second communications channel different from the first communications channel. 
     At  776 , the method includes providing a signal representing the arbitrary transformation to the first signal receiver and the second signal receiver in association with the transmitting the first transformed signal and the transmitting the second transformed signal, for recovery of the plurality of vectors at the first signal receiver and the second signal receiver based on the arbitrary transformation. In some instances, the providing the signal representing the arbitrary transformation is done prior to transmitting the first transformed signal and the transmitting the second transformed signal. In some other instances, the providing the signal representing the arbitrary transformation can be done after the transmitting the first transformed signal and the transmitting the second transformed signal, in which case the signal receivers can store the transformed signal(s) received and recover the original signals at a later point in time after receiving the signal representing the arbitrary transformation. In some instances, the signal receivers can be configured to transmit a transformed signal to a target device. For example, the signal receivers can be configured to transmit a signal representing the plurality of transformed vectors to a designated target device. 
     As described above, in some instances where a first arbitrary transformation is used to produce the first plurality of transformed vectors and a second arbitrary transformation is used to the second plurality of transformed vectors, the providing a signal representing the arbitrary transformation can include providing a first signal representing the first arbitrary transformation and providing a second signal representing the second arbitrary transformation. In some implementations, the transmitting the first transformed signal and the providing the first signal representing the first arbitrary transformation can be to a first receiver associated with a first receiver, and the transmitting the second transformed signal produced using the second arbitrary transformation and the providing the second signal representing the second arbitrary transformation can be to a second receiver antenna associated with a second receiver different from the first receiver. In some instances, the first and second signals representing the first and second arbitrary transformations can be broadcast together to a wide audience including the first and second signal receivers. In some instances the first signal representing the arbitrary transformation can be broadcast widely but not the second signal representing the arbitrary transformation, such that the first signal receiver is able to recover the first plurality of vectors but the second receiver is unable to recover the second plurality of transformed vectors until the second signal representing the second arbitrary transformation is provided or broadcast. 
     While described as a variation of an OFDM system, some embodiments of a gUBDM system operate as a variation of a DSSS system wherein a “code map” is used and is bandwidth limited. The explicit form, as given in the &#39;839 patent referred to above, is
 
   c   :   N →   M  
 
   v         c   (   v   ),  (33)
 
where the m th  component of  c ( v )∈   M  is given by
 
     
       
         
           
             
               
                 
                   
                     
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     Here, v n  is the n th  component of  v , the κs are a set of N distinct numbers satisfying
 
κ n −κ m   ∈     ∀m,n,   (35)
 
     and M is an integer chosen so that M&gt;2 max n |κ n |. This map has the properties discussed above (band-limited and dot-product preserving). Typically, M≈N if the κ are sequential integers centered around 0. 
     So, to create a maximal set of mutually orthogonal spreading codes, a unitary matrix A∈U(N) is chosen. If the n th  column is denoted (or row, doesn&#39;t matter which as long as there is consistency) of A as Ā n , then the N codes are  c (Ā n ) for n∈[1, . . . , N]. 
     If one device is to transmit data on all N codes, then it will be able to take the N symbols b n , multiply each one by every component of its spreading code, and then add the resulting vectors together. So the transmitted vector  s  is 
                       s   _     =       ∑     n   =   1     N     ⁢       b   n     ⁢       c   _     ⁡     (       A   _     n     )             ,           (   36   )               
where b n  are the symbols.
 
     But to do this, the transmitter multiplies a symbol b_n∈C which is typically a complex number (a float, double, etc), times all M≈N components of  c (Ā_n). This is repeated for all N symbols b_n. So, there are N symbols, each being multiplied by N components of the code. This makes the complexity O (N ∧2), which is prohibitive for wide-band applications. (Compare to OFDM, which is O(N log N).) 
     Notably for multiple access applications, where each user is given a subset of the codes, they only have to do O(N) work, which is better than OFDM. That makes the DSSS implementation very good for multiple access applications. 
     To obtain a UBDM that is O(N log N), to match OFDM reinterpret (0.0.4). The transmitted baud is 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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       FIG. 8  is a flowchart describing an example method for a reception of a plurality of transformed vectors and recovery of a plurality of vectors, using a gUBDM system according to an embodiment. The method  800  can be implemented by a processor associated with a signal receiver (e.g., signal receiver  401 ) described herein. 
     At  871 , the method  800  includes receiving from a plurality of signal transmitters and via a plurality of signal receivers, a signal representing a plurality of transformed vectors. 
     At  872 , the method includes receiving an indication of an arbitrary transformation configured to be used to recover a plurality of vectors based on the plurality of transformed vectors. In some implementations, the receiving the indication of the arbitrary transformation can be from the plurality of signal transmitters and via the plurality of signal receivers. In some instances the receiving the indication of the arbitrary transformation can be prior to the receiving the signal representing a plurality of transformed vectors. In some instances the indication can include an inverse of the arbitrary transformation. 
     At  873 , the method includes applying the arbitrary transformation to each transformed vector of the plurality of transformed vectors to produce a plurality of vectors. At  874 , the method includes recovering, based on the plurality of vectors, an original signal. In some instances for example the recovering the original signal can be performed by a converter (e.g., converter  414 ) associated with a signal receiver. In some instances the method  800  can skip the recovering the original signal at  873  and instead store or send the plurality of vectors to another device to perform the recovering of the original signal. 
     Another advantage of the above described gUBDM system is that it is designed to take full advantage of the richness and structure of the unitary groups. One opportunity the gUBDM system described affords is the ability to incorporate ETF/NETFs into an adopted and modified OFDM system variation—this is something that is impossible in an OFDM system otherwise unmodified. 
     The gUBDM system also affords a signal transmission source the ability to include code division multiplexing into an OFDM system upon modification into a gUBDM system. This means that in addition to time division, frequency division, and spatial multiplexing, we can do code division multiplexing can be performed. This adds an enormous degree of freedom for system engineers. 
     It should be noted that an iFFT will still be likely performed after applying a general unitary A, in some implementations, which can make equalization easier. So, take a data vector b and send it through the steps b→Ab→FAb, where F is a Fourier transformation. However, because of the group structure of U(N), it is known that if F and A are both elements of U(N) are used, then their product will be as well. Because we are using the entire group U(N), there is no difference between claiming a single matrix A and claiming a single matrix A followed by a Fourier matrix. No matter how many unitary matrices we multiplied together, the result is still just another element of U(N). 
     In other words, a key advantage of this approach is the security. If the act of modulating the data is able, by itself, to fully secure the content to an eavesdropper on that channel, denying her access to the bits (or anything above OSI layer 1), then the attack surface for the eavesdropper has changed radically. All possibilities of traffic analysis attacks, protocol weakness attacks, control data leakage attacks, etc. are completely eliminated. Furthermore, in networks where the security provided by traditional encryption causes delay/latency that adversely impacts the network, the encryption (usually at OSI layer 3 or higher) can be optionally completely removed. This eliminates the space, power, heat, or time to include the encryption, as well as the overhead usually associated with encryption. Furthermore, the delays/latency associated with encryption (everything from simply having to pass the information up and down the OSI stack to the latency associated with simply having to run those bits through the cryptologic) can be eliminated. All the system needs to do is transmit. The modulation itself takes care of the security. 
     The signal receiver is open to any computation upon receiving the transformed signal. In some implementations, the signal receiver can simply demodulate the signal and recover the symbols and bits. In some implementations, the signal receiver may also want to store the digitized I and Q, or pass the digitized I and Q on to some other system without applying the inverse of the unitary matrix. 
       FIG. 9  is an example constellation diagram. As shown in  FIG. 9 , the constellation diagram is a two-dimensional xy-plane scatter diagram, representing a signal modulated by a digital modulation scheme. As such, each instance of a constellation diagram can have a particular modulation scheme with which it is associated.  FIG. 10  is a flow diagram illustrating a first method for modulation-agnostic UBDM signal transformation (e.g., employing gUBDM), according to an embodiment. As shown in  FIG. 10 , the method  1000  include selecting, at  1071  and via a processor of a communications system, a block size. The communications system can be configured to perform at least one of Quadrature Amplitude Modulation (QAM), Amplitude Phase Shift Keying (APSK) modulation, or Orthogonal Frequency-Division Multiplexing. In addition, the communications system can include one or more of a wireless communication system, a wired communication system, or a fiber optic communication system. At  1072 , the processor identifies a set of constellation points of a constellation diagram, based on a received plurality of bits. The processor can identify the constellations points based, for example, on the constellation diagram itself, data associated with the constellation diagram, a representation of a constellation diagram or portion thereof, etc. Identifying the set of constellation points can include mapping the received plurality of bits to the constellation diagram (e.g., using a gray code). The constellation diagram is associated with a modulation scheme. At  1073 , the processor generates a plurality of symbol blocks based on the set of constellation points. Each symbol block from the plurality of symbol blocks has a size equal to the block size and includes a subset of constellation points from the set of constellation points. At  1074 , the processor applies a unitary braid division multiplexing (UBDM) transformation to each symbol block from the plurality of symbol blocks to produce a plurality of complex numbers, and at  1075 , the processor sends the plurality of complex numbers. Optionally, the method  1000  also includes causing transmission of at least one signal representing the plurality of complex numbers using the modulation scheme (not shown). 
       FIG. 11  is a flow diagram illustrating a second method for modulation-agnostic UBDM signal transformation, according to an embodiment. As shown in  FIG. 11 , the method  1100  includes receiving, via a processor of a communications system, a plurality of input bits at  1171 . The communications system can include one or more of: a wireless communication system, a wired communication system, or a fiber optic communication system. The method  1100  also includes converting, at  1172 , the plurality of input bits into a plurality of complex values. Converting the plurality of input bits into a plurality of complex values can include performing bit-to-symbol mapping (e.g., using a gray code) based on the plurality of input bits, and applying a unitary braid division multiplexing (UBDM) transformation (e.g., including a plurality of nonlinear layers and a plurality of linear layers). The bit-to-symbol mapping can be based, for example, on the constellation diagram itself, data associated with the constellation diagram, a representation of a constellation diagram or portion thereof, etc. At  1173 , the plurality of complex values is sent, via the processor and using a predetermined modulation technique, for subsequent processing (e.g., pulse shaping and/or the application of at least one filter). The predetermined modulation technique can include one or more of Quadrature Amplitude Modulation (QAM), Amplitude Phase Shift Keying (APSK) modulation, or Orthogonal Frequency-Division Multiplexing. Optionally, the method  1100  also includes causing transmission of at least one signal representing the plurality of complex numbers using the modulation scheme (not shown). 
       FIG. 12  is a flow diagram illustrating a third method for modulation-agnostic UBDM signal transformation, according to an embodiment. As shown in  FIG. 12 , the method  1200  includes receiving a plurality of input bits, at  1271 , and mapping each input bit from the plurality of input bits (e.g., to a constellation), at  1272 , using a bit-to-symbol map to identify a plurality of symbols. The constellation can be a constellation of a constellation diagram for a signal to be transmitted wirelessly, or through wired or fiber optic communication. Subsets of symbols from the plurality of symbols are grouped at  1273  into a plurality of blocks, each block from the plurality of blocks having a size N. At  1274 , a UBDM transformation is applied to each block from the plurality of blocks to produce a plurality of complex numbers, and the resulting complex numbers are sent at  1275 , for example, to a downstream portion of the communication system (e.g., to one or more filters of, or operably coupled to, the processor) for at least one of pulse shaping or filter application. In some such implementations, the method does not include the application of an inverse Fourier transform prior to sending the transmitted signal. Alternatively or in addition, the method does not include the generation of spreading codes. 
     EXAMPLE EMBODIMENTS 
     In a first example embodiment, a method includes applying an arbitrary transformation to a plurality of vectors to produce a plurality of transformed vectors, the arbitrary transformation including one of a unitary transformation, an equiangular tight frame (ETF) transformation, or a nearly equiangular tight frame (NETF) transformation. For example, the arbitrary transformation can include a matrix having rows that form one of an equiangular tight frame or a nearly equiangular tight frame. The method also includes producing a first transformed signal, using the arbitrary transformation, based on at least a first transformed vector from the plurality of transformed vectors. The method also includes producing a second transformed signal, using the arbitrary transformation, based on at least a second transformed vector from the plurality of transformed vectors. The method also includes transmitting the first transformed signal (e.g., via a first transmitter), via a first communications channel, to a first signal receiver that is configured to detect the first transformed signal, and transmitting the second transformed signal (e.g., via a second transmitter different from the first transmitter), via a second communications channel, to a second signal receiver that is configured to detect the second transformed signal. The method also includes causing transmission of a signal representing the arbitrary transformation, for subsequent recovery of the plurality of vectors based on the arbitrary transformation. The method optionally also includes performing an inverse fast Fourier transform (iFFT) on at least one of the first transformed signal and the second transformed signal prior to transmitting the first transformed signal and the second transformed signal. 
     The first communications channel can be different from the second communications channel. Alternatively or in addition, the plurality of transformed vectors can have a total magnitude that substantially equals a total magnitude of the plurality of vectors. Alternatively or in addition, producing the first transformed signal and producing the second transformed signal do not include the use of a spreading code vector. Alternatively or in addition, transmitting the first transformed signal and transmitting the second transformed signal can be performed using multiple access communication. 
     The first example embodiment can also include generating the plurality of vectors. The generating can include generating a plurality of symbols based on a plurality of input bits using a bit-to-symbol map, and generating a plurality of blocks based on the plurality of symbols, each block from the plurality of blocks representing a vector from the plurality of vectors. 
     In some implementations of the first example embodiment, the arbitrary transformation is based on a matrix that is not an identity matrix or a discrete Fourier matrix. Alternatively or in addition, the arbitrary transformation is based on a matrix that is not a direct sum of discrete Fourier matrices. 
     In a second example embodiment, a system includes a plurality of signal receivers, a plurality of signal transmitters, and at least one processor operably coupled to the plurality of signal transmitters. The at least one processor is configured to generate a plurality of vectors. The at least one processor can be configured to generate the plurality of vectors, for example, by generating a plurality of symbols based on a plurality of input bits using a bit-to-symbol map, and generating pluralities of blocks based on the plurality of symbols, each plurality of blocks from the pluralities of blocks representing a vector from the plurality of vectors. The at least one processor is also configured to apply an arbitrary transformation to each vector from the plurality of vectors to produce a plurality of transformed vectors. The arbitrary transformation includes one of a unitary transformation, an equiangular tight frame (ETF) transformation, or a nearly equiangular tight frame (NETF) transformation. 
     The arbitrary transformation may be based on a matrix that is not an identity matrix or a discrete Fourier matrix. Alternatively or in addition, the arbitrary transformation may be based on a matrix that is not a direct sum of discrete Fourier matrices. The plurality of transformed vectors can have a total magnitude that substantially equals a total magnitude of the plurality of vectors. The at least one processor is also configured to send a signal representing the plurality of transformed vectors to the plurality of transmitters (e.g., via the physical layer of the open system interconnection model) for transmission of the plurality of transformed vectors to the plurality of signal receivers. Optionally, the at least one processor is also configured to send a signal representing one of: (1) the arbitrary transformation, or (2) an inverse of the arbitrary transformation to the plurality of signal receivers prior to transmission of the signal representing the plurality of transformed vectors to the plurality of signal receivers, such that the plurality of signal receivers recovers the plurality of vectors from the plurality of transformed vectors based on the arbitrary transformation or an inverse of the arbitrary transformation. The plurality of signal receivers can also be configured to transmit a signal representing the plurality of transformed vectors to a target device. 
     In some implementations of the second example embodiment, the plurality of signal receivers includes a plurality of antenna arrays, and the plurality of signal receivers and the plurality of signal transmitters are configured to perform Multiple Input Multiple Output (MIMO) operations. 
     In a third example embodiment, a method includes generating a plurality of vectors, and applying an arbitrary transformation to each vector from the plurality of vectors to produce a plurality of transformed vectors. The arbitrary transformation can include one of a unitary transformation, an equiangular tight frame (ETF) transformation, or a nearly equiangular tight frame (NETF) transformation. The plurality of transformed vectors can have a total magnitude that substantially equals a total magnitude of the plurality of vectors. The method also includes sending a signal representing the plurality of transformed vectors to a plurality of transmitters for transmission of a signal representing the plurality of transformed vectors from the plurality of transmitters to a plurality of signal receivers. The method also includes providing the arbitrary transformation to the first signal receiver and the second signal receiver, for recovery of the plurality of vectors at the first signal receiver and at the second signal receiver. Optionally, the method also includes sending the signal representing the plurality of transformed vectors to the plurality of transmitters via the physical layer of the open system interconnection model. 
     In some implementations of the third example embodiment, the plurality of signal receivers includes a plurality of antenna arrays, and the plurality of signal receivers and the plurality of signal transmitters are configured to perform Multiple Input Multiple Output (MIMO) operations. 
     The arbitrary transformation can be based on a matrix that is not an identity matrix or a discrete Fourier matrix. Alternatively or in addition, the arbitrary transformation can be based on a matrix that is not a direct sum of discrete Fourier matrices. 
     While various embodiments have been described above, it should be understood that they have been presented by way of example only, and not limitation. Where methods and/or schematics described above indicate certain events and/or flow patterns occurring in certain order, the ordering of certain events and/or flow patterns may be modified. While the embodiments have been particularly shown and described, it will be understood that various changes in form and details may be made. 
     Although various embodiments have been described as having particular features and/or combinations of components, other embodiments are possible having a combination of any features and/or components from any of embodiments as discussed above. 
     Some embodiments described herein relate to a computer storage product with a non-transitory computer-readable medium (also can be referred to as a non-transitory processor-readable medium) having instructions or computer code thereon for performing various computer-implemented operations. The computer-readable medium (or processor-readable medium) is non-transitory in the sense that it does not include transitory propagating signals per se (e.g., a propagating electromagnetic wave carrying information on a transmission medium such as space or a cable). The media and computer code (also can be referred to as code) may be those designed and constructed for the specific purpose or purposes. Examples of non-transitory computer-readable media include, but are not limited to, magnetic storage media such as hard disks, floppy disks, and magnetic tape; optical storage media such as Compact Disc/Digital Video Discs (CD/DVDs), Compact Disc-Read Only Memories (CD-ROMs), and holographic devices; magneto-optical storage media such as optical disks; carrier wave signal processing modules; and hardware devices that are specially configured to store and execute program code, such as Application-Specific Integrated Circuits (ASICs), Programmable Logic Devices (PLDs), Read-Only Memory (ROM) and Random-Access Memory (RAM) devices. Other embodiments described herein relate to a computer program product, which can include, for example, the instructions and/or computer code discussed herein. 
     In this disclosure, references to items in the singular should be understood to include items in the plural, and vice versa, unless explicitly stated otherwise or clear from the context. Grammatical conjunctions are intended to express any and all disjunctive and conjunctive combinations of conjoined clauses, sentences, words, and the like, unless otherwise stated or clear from the context. Thus, the term “or” should generally be understood to mean “and/or” and so forth. The use of any and all examples, or exemplary language (“e.g.,” “such as,” “including,” or the like) provided herein, is intended merely to better illuminate the embodiments and does not pose a limitation on the scope of the embodiments or the claims. 
     Some embodiments and/or methods described herein can be performed by software (executed on hardware), hardware, or a combination thereof. Hardware modules may include, for example, a general-purpose processor, a field programmable gate array (FPGA), and/or an application specific integrated circuit (ASIC). Software modules (executed on hardware) can be expressed in a variety of software languages (e.g., computer code), including C, C++, Java™, Ruby, Visual Basic™, and/or other object-oriented, procedural, or other programming language and development tools. 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.