Abstract:
The present disclosure relates to the field of pulse compression in signal processing, and more particularly, to systems and methods for the synthesis of waveforms for suppressing sidelobes and sidebands using a combination of time and spectral control. More specifically, the present disclosure relates to a set of waveform symbols which can be used to maximize use of disaggregated grey-space spectrum, adapt to changing spectral condition, and maintain or enhance data rates relative to standard binary phase-shift keying (BPSK) under normal conditions.

Description:
RELATED APPLICATION(S) 
     This application is a continuation-in-part of application Ser. No. 12/940,599 entitled, “Systems and Methods for Suppressing Radar Sidelobes Using Time and Spectral Control,” filed Nov. 5, 2010, claims the benefit of U.S. provisional Application No. 61/258,950, filed Nov. 6, 2009, both of which are incorporated by reference herein in their entireties. 
    
    
     FIELD OF THE INVENTION 
     The present disclosure relates to the field of pulse compression in signal processing and communications, and more particularly, to systems and methods for the synthesis of waveforms for suppressing radar and communications sidelobes and spectral sidebands using a combination of time and spectral control. More specifically, the present disclosure relates to a set of waveform symbols which can be used to maximize use of disaggregated grey-space spectrum, adapt to changing spectral conditions, and/or maintain or enhance data rates relative to standard binary phase-shift keying (BPSK) under normal conditions. 
     BACKGROUND OF THE DISCLOSURE 
     Radio frequency spectrum is becoming increasingly disaggregated. This is due to a number of factors, including the historical pattern in which various bands of spectrum have been allocated to various commercial and/or military interests. The growth and proliferation of various digital communications have placed increasing pressure on the efficient use of spectrum. The tremendous growth of this industry sector has made the robustness and reliability of wideband communications an increasing challenge. Further, the proliferation of communications devices and the relative increase in data rates has generated an increasing amount of data that needs to be transmitted over a reduced amount of spectrum. 
     These problems are exacerbated by multi-path complications, as well as various terrain features. In addition, the number of competing systems has increased. The physical range between systems has decreased, and the susceptibility of various communications to interference from other systems has increased. Further complicating this scenario is the tremendous proliferation of mobile devices and communication systems, both in the civilian and military sectors. This has produced a number of technical challenges. 
     In the field of radar signal processing, a technique known as “pulse compression” has long been used to improve the range resolution of radars. In general, pulse compression involves modulating a transmitted radar pulse (i.e., with a “code” or “waveform”) and then correlating the received signal with an appropriate “filter” function, based on the known modulation. This technique can also be applied to communication systems to create waveforms that effectively make use of fragmented spectrum, maintain or increase data rates, and change in accordance to varying spectral conditions. 
     One reason for implementing a pulse compression system is to obtain the high range resolution of a short pulse, while realizing the higher signal-to-noise ratio (SNR) of a longer uncoded pulse. This is accomplished by increasing the bandwidth of the longer pulse by introducing signal modulation. This technique can extend the maximum detectable range, improve the probability of detection (P D ), and affect a lower probability of intercept (LPI), by lowering the peak power requirements for the same SNR. In a communication system, however, pulse compression is used to control the data rate by manipulating the time and spectral properties of the signal. 
     A major principle in radar is the coherent combination of signals in order to affect what signals are seen and what signals are suppressed in order to separate targets from clutter. One manifestation of this coherent combination is in spatial domain processing, which takes the form of azimuth and elevation transmit/receive antenna beams. Another manifestation is in the time domain, in which targets in a range cell of limited size are enhanced while targets outside this range cell are suppressed. In this second form, there is a correspondence between the range cell size (resolution) and the signal bandwidth in the frequency domain. In communications applications the broader bandwidth translates into a higher data rate. In all three domains (i.e. spatial, temporal, and frequency), there will be unwanted “sidelobes” that can be sources of interference and false-targets, or data rate reductions. 
     The set of performance measures that determine the design of the code and the associated filters used in radar pulse compression include SNR loss, code amplitude, peak response broadening, and sidelobe behavior. The design of such codes and filters constitutes a tradeoff among the various performance measures. Optimized pulse compression search techniques have been developed that can compute many codes and filter combinations in response to each set of performance requirements. The corresponding filters can be “matched” to the codes in length/time and in amplitude and phase, so as to improve SNR gain and resolution, or “mismatched” in length/time, amplitude and phase, so as to reduce the correlation of sidelobes. Similar tradeoffs can be achieved in the design of communications signals. 
     One prior method of minimizing sidelobes by optimizing matched filter codes is described in “Multi-parameter Local Optimization for the Design of Superior Matched Filter Polyphase Pulse Compression Codes,” by Nunn and Welch. One benefit of using a matched filter is that it maximizes the gain in the SNR (i.e., the processing gain). Hence, the matched filter has no SNR loss because its filter characteristics are precisely matched to the received waveform. Conversely, when implementing a mismatched filter to reduce correlation side lobes, some of this SNR processing gain and/or target resolution may be lost. 
     Synthetic aperture radar (SAR) is a particular type of radar that uses a plurality of small, low-directivity, stationary antennas scattered near or around the target area, or an antenna moving over stationary targets. Echo waveforms received by the moving or plurality of antennas can be processed to resolve the target. In some cases, SAR radar may be improved by combining many radar pulses to form a synthetic aperture, using additional antennas or significant additional processing. SAR operation typically involves transmitting signals that cover a broad spectrum, or frequency bandwidth, to obtain desirable resolution (e.g., 200 MHz to 2 GHz). For example, in SAR applications, the bandwidth occupied by the radar is so large that it overlaps with large swaths of heavily utilized and important spectral regions. These applications tend to cause heavy in-band, and sometimes out-of-band, spectral interference. Similarly high data-rate communications involves transmitting signals that cover a broad spectrum which can overlap with bands reserved for other purposes. 
     The demands and prevalence of modern electronic communications, navigation, and other systems make it difficult to obtain large swaths of contiguous bandwidth. In the real world, spectrum is a precious commodity that is carefully managed. For example, it may be necessary to satisfy spectrum managers that restrict transmission frequency bandwidth. Although radar designers have developed various methods to reduce time sidelobes, they have been less successful at minimizing spectral sidelobes and in band spectral properties. As a result, many of these pulse compression codes have not been heavily utilized in real world radar systems because of their spectral shortcomings. For general pulse compression applications, these out-of-band spectral emissions can interfere with other communications or radar devices at nearby frequencies. In broadband applications, the problem is even more serious. These spectral interference problems are both time and location dependent. In the communications realm much emphasis has been placed on reducing spectral impact, but at a high cost to peak-to-average-power ratio (PAPR) and intersymbol interference. 
     Thus, for many applications, most notably SAR applications and high data rate communications, it is important to have available methods to rapidly and simultaneously control both the time side lobes and spectral characteristics of these transmit waveforms. Nunn has developed methods to achieve fine control over time side lobe code characteristics, such as peak sidelobe levels (PSLs) or integrated sidelobe levels (ISLs) of discrete, constant amplitude pulse compression codes using constrained optimization techniques. Nunn has also used the same methodology to create mismatched filters with excellent ISL, PSL and loss characteristics. Prior to the current effort, these methods have not been used to address the spectral issues. 
     The proliferation of radio-frequency devices has had two principal effects. First, it has resulted in the fragmentation of available communications spectrum due to interference sub-bands. Second, it has resulted in the imposition of varying and often extremely difficult spectral mask requirements. Prior known communications approaches have involved the use of multiple carriers that make use of disaggregated spectrum. These approaches, however, lead to very high peak-to-average power ratios (PAPR), limiting their utility in certain environments. Other techniques involve spectral filtering which imposes a cost in the form of intersymbol interference. 
     Demands for bandwidth continue to grow and available bandwidth is becoming increasingly disaggregated. There remains a need to maximize use of disaggregated spectrum. Specifically, this requires maximum use of spectrum under spectral mask conditions imposed at fragment edges. There is also a continuing need to be able to adapt to changing spectral conditions. Further, there remains a need to maintain or enhance data rates relative to standard BPSK under normal conditions. 
     SUMMARY OF DISCLOSURE 
     In accordance with an embodiment of the present disclosure, a computer-implemented method is disclosed for achieving spectral compliance while enhancing the data rate. Embodiments of the present disclosure narrow the bands of the individual signals. This lengthens signal duration while increasing signal-to-noise ratio. Lengthening the signal duration provides additional degrees of freedom which makes it easier to meet sideband requirements with minimal PAPR penalties. Embodiments of the present disclosure increase the data rate while using lengthened symbols through the use of low-order orthogonal signaling techniques. 
     The signal-to-noise ratio requirement increases with the number of members in an orthogonal signaling alphabet. In embodiments of the present disclosure, increased signal duration, however, provides additional gain, and in some instance more gain is acquired through coherent addition than is lost to this increasing SNR requirement. Embodiments of the present disclosure use varying numbers of symbols per spectral fragment. In an embodiment of the present disclosure, four symbols represent a unique code for each spectral fragment derived from the primary relative location of the fragment within the spectral band. In an alternative embodiment, eight symbols represent a unique code for each spectral fragment derived from the primary relative location of the fragment within the spectral band. The sidebands of the individual symbols are constrained outside of the spectral fragment and are not constrained within the fragment. This enables sideband hiding among the symbols within a fragment. 
     The members of each group of symbols have very low synchronized cross-correlations while independently meeting spectral compliance objectives. These notches provide clean, low-error, separation of auto-correlation functions, with peak-at-center, from the synchronized cross-correlations. The width of this cross-correlation notch can be an optimization criterion on the waveform set. A larger notch may be required in severe multipath conditions. 
     In alternative embodiments of the present disclosure, the increase in SNR provides the capability to implement M-ary phase coding to each symbol, up to 4-bits, or 16 phases. This provides improvements over BPSK signaling using the same sub-band while achieving sub-80 dB sideband with near-vertical spectral fall-off, and without substantial inter-symbol interference. The increase in SNR from increased symbol length provides an additional SNR advantage. Increased correlation widths and appropriate mis-matched filtering may further provide resistance to multi-path effects. The SNR improvement may be substantial in situations when the available bandwidth is marginal, including situations in which the bandwidth is overused, being jammed, or in severe multipath situations. 
     Embodiments of the present disclosure may increase the data rate logarithmically. This may be accomplished by sequential hopping through sub-bands of the overall spectrally fragmented band corresponding to the expanding alphabet. This may be backward-compatible with older or lower cost communications systems. Embodiments of the present disclosure employing multiple-input-multiple-output (MIMO) techniques may achieve a linear increase in data rates through multiple parallel high power amplifiers with a linear increase in power. 
     Embodiments of the present disclosure achieve high data rate expansion through simultaneous use of a number of fragments. This provides linear, and larger, data rate expansion. Embodiments of the present disclosure employ multiple amplifiers operating in parallel: one amplifier per spectral fragment. Use of multiple amplifiers would result in a linear increase in transmitted power using low peak-to-average-power-ratio PAPR waveforms along with a linear increase in data rate. This may avoid the very large peak-to-average power penalties seen, for example, in Orthogonal Frequency division Multiplexing (OFDM), which can be as high as 15 dB. 
     In embodiments of the present disclosure, individual symbols are computed and optimized. They may be selected in real-time as spectral sideband requirements are determined, scaled, and frequency shifted. They can be back-fit by the incorporation of appropriate modulators/demodulators (MODEMS) rendering the front-end operations transparent to existing systems. 
     In an embodiment, a computer-implemented method is disclosed for suppressing sidelobes in pulse compression signal processing. The method includes creating an objective function that quantifies signal characteristics as a sum of integrated sidelobe levels of a pulse; defining spectral and time constraints to further constrain the objective function; converting the objective function into an unconstrained optimization problem using a penalty method; and using a processor to obtain gradients of the unconstrained optimization problem in the spectral domain. 
     In an embodiment, a system is disclosed for suppressing sidelobes in pulse compression signal processing. The system includes a memory configured to store instructions for generating an optimized waveform; and a processor configured to: receive an objective function that quantifies signal characteristics; define spectral and time constraints to further constrain the objective function; convert the objective function into an unconstrained optimization problem; and use a gradient descent method to solve the unconstrained optimization problem and generate an optimized waveform. 
     In an embodiment, a computer implemented method is disclosed for generating an optimized mismatched filter for a pulse compression system. The method includes receiving spectral and time constraints that define a pulse; creating an objective function that quantifies signal characteristics as a sum of integrated sidelobe levels of a pulse, based on the spectral and time constraints; converting the objective function into an unconstrained optimization problem using a penalty method; and using a processor to obtain gradients of the unconstrained optimization problem in the spectral domain. 
     In an embodiment, a computer implemented method is disclosed for suppressing sidelobes in pulse compression signal processing. The method includes generating a function that quantifies signal characteristics; and using a processor to calculate time-domain derivatives of the function in the frequency domain. 
     In an embodiment, a computer implemented method is disclosed for suppressing sidelobes in pulse compression signal processing. The method includes generating a function that quantifies signal characteristics; and using a processor to optimize time and spectral characteristics of the signal characteristics using frequency domain calculations. 
     In an embodiment, a computer implemented method is disclosed for suppressing sidelobes in pulse compression signal processing. The method includes generating a function that quantifies signal characteristics as a sum of functions of sidelobe levels; and using a processor to minimize the function by obtaining gradients of the function in the spectral domain. 
     In an embodiment, a computer implemented method is disclosed for suppressing sidelobes in pulse compression signal processing. The method includes generating a function that quantifies signal characteristics as a sum of functions of sidelobe levels; and using a processor to control the sidelobe levels by obtaining gradients of the function in the spectral domain. 
     In an embodiment, a method for enhancing the use of disaggregated spectrum comprising generating one or more waveform symbol sets; where said waveform symbol set forms a unique coding scheme comprising 2 to 16 symbols; said waveform symbol set correlates to a relative location within the spectral band; and applying said waveform symbol set to narrow the bands of each individual signal. 
     It will be apparent to those skilled in the art that various modifications and variations can be made in the system and method for reception in communication networks. It is intended that the standard and examples be considered as exemplary only, with a true scope of the disclosed embodiments being indicated by the following claims and their equivalents. 
     In this respect, before explaining at least one embodiment of the disclosure in detail, it is to be understood that the disclosure is not limited in its application to the details of construction and to the arrangements of the components set forth in the following description or illustrated in the drawings. The disclosure is capable of embodiments in addition to those described and of being practiced and carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein, as well as the abstract, are for the purpose of description and should not be regarded as limiting. 
     The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate certain embodiments of the disclosure, and together with the description, serve to explain the principles of the disclosure. 
     As such, those skilled in the art will appreciate that the conception upon which this disclosure is based may readily be utilized as a basis for designing other structures, methods, and systems for carrying out the several purposes of the present disclosure. It is important, therefore, to recognize that the claims should be regarded as including such equivalent constructions insofar as they do not depart from the spirit and scope of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts an exemplary pulse compression system consistent with embodiments of the present disclosure; 
         FIGS. 2A and 2B  depict flowcharts of exemplary methods for quantifying signal characteristics and obtaining gradients in the spectral domain; 
         FIG. 3  depicts a flowchart of an exemplary method of generating an optimized waveform and an exemplary method of operating a pulse compression system using an optimized waveform; 
         FIG. 4  depicts a flowchart of another exemplary embodiment of a method of generating an optimized waveform for a pulse compression system; and 
         FIG. 5A  depicts a flowchart of an exemplary embodiment of a transmit MODEM design.  FIG. 5B  is a flowchart of an exemplary embodiment of a receive MODEM design. 
         FIG. 6  is a schematic diagram of an embodiment of the present disclosure. 
         FIG. 7  is a graph of power as a function of frequency of a BPSK symbol. 
         FIG. 8  is a graph of power as a function of frequency of a short waveform symbol set of an embodiment of the present disclosure. 
         FIG. 9  is a graph of power as a function of frequency of an intermediate waveform symbol set of an embodiment of the present disclosure. 
         FIG. 10  is a graph of power as a function of frequency of a long waveform symbol set of an embodiment of the present disclosure. 
         FIG. 11  is a graph of power as a function of frequency of a set of two short symbols (80 bit/μs). 
         FIG. 12  is a graph of power as a function of frequency of a set of two longer symbols (72 bit/μs). 
         FIG. 13  is a graph of power as a function of frequency of a set of four short symbols (133 bit/μs). 
         FIG. 14  is a graph of power as a function of frequency of a set of four longer symbols (120 bit/μs). 
         FIG. 15  is a graph of power as a function of frequency of a set of eight short symbols (107 bit/μs). 
         FIG. 16  is a graph of power as a function of frequency of a set of eight longer symbols (106 bit/μs). 
     
    
    
     DETAILED DESCRIPTION 
     Reference will now be made in detail to the present embodiments of the disclosure, certain examples of which are illustrated in the accompanying drawings. 
       FIG. 1  depicts an exemplary pulse compression system  100  consistent with embodiments of the present disclosure. Pulse compression system  100  may include a waveform generator  103 , which is configured to generate a low power waveform signal. Waveform generator  103  may be provided in communication with a power amplifier transmitter  104 , which is configured to increase the power to the desired peak transmit power. Power amplifier transmitter  104  may be provided in communication with a duplexer  106 , which is provided in communication with an antenna  108  and configured to switch between transmission and receiving modes. Duplexer  106  may also be provided in communication with a low-noise amplifier  110 , which is configured to amplify the received echo signal. Low-noise amplifier  110  may be provided in communication with a mixer  112 , which is configured to mix the received signal to an intermediate frequency (IF). Mixer  112  may be provided in communication with an IF amplifier  114 , which is configured to amplify the IF signal. IF amplifier  114  may be provided in communication with a pulse compression filter  116 , which is configured to process the received echo to yield a narrow compressed pulse response, with a main lobe width that does not depend on the duration of the transmitted pulse. Pulse compression filter  114  may include a matched filter  118  and/or a mismatched/weighting filter  120 . The output signal from pulse compression filter  116  may then be processed through an envelope detector  122 , a video amplifier  124 , and/or any other desired signal processing apparatus, before being directed to a display  126 . A synchronizer  128  may be provided in communication with display  126 , waveform generator  103 , and/or power amplifier transmitter  104 . 
     In one embodiment, radar pulse compression system  100  may also include a computer  102  provided in communication with waveform generator  103 , pulse compression filter  116 , and/or a network  101 , such as a local-area network (LAN), or a wide-area network (WAN), such as the Internet. Computer  102  may include any type of memory configured to store electronic data, and any type of processor configured to execute instructions and/or perform data operations. In one embodiment, computer  102  may be configured to generate an optimized waveform/filter combination useful in pulse compression. Thus, waveform generator  103  may be configured to receive one or more optimized waveforms from computer  102  and/or from other computers connected to network  101 . Additionally, or alternatively, waveform generator  103  may include a memory and a processor configured to generate one or more optimized waveforms, consistent with the exemplary methods described herein. 
     Thus, pulse compression system  100  may be configured to generate an optimized waveform/filter pair and to perform radar pulse compression based on the optimized waveform.  FIGS. 2-4  depict exemplary embodiments of methods for defining a function of radar signal characteristics, obtaining time-domain derivatives of the function in the frequency domain, generating an optimized waveform using the time-domain derivatives, and performing pulse compression based on the optimized waveform. 
     In general, constrained optimization techniques may be used to control spectral emissions while at the same time maintaining adequate time sidelobe control. Because SAR and high data rate communications applications require waveforms with large time-bandwidth (BT) products and suitable time sidelobes, efficient methods are now disclosed for creating the derivatives needed by the constrained optimization process to control the spectrum. These efficient computation methods are simultaneously applied to the associated time sidelobe and spectral characteristics. In these new methods, many needed calculations may be performed in the frequency domain using fast Fourier transforms (FFTs). Many of these calculations, if done in the time domain, would result in processing on the order of N^2, whereas doing these calculations in the frequency domain results in a process that has calculations on the order of log — 2(N)*N, where N is the number of degrees of freedom used to describe the waveform or filter. The resulting efficiency allows for the simultaneous optimization of spectral characteristics and time side lobe characteristics for applications requiring large time-bandwidth products. 
     Accordingly, systems and methods are disclosed in which time and spectral characteristics of a constant (or near constant) or otherwise defined amplitude pulse compression code or waveform are optimized using frequency domain calculations instead of time domain calculations. Specifically, a method is disclosed for minimizing an objective function that can be used to control ISLs and PSLs and other time domain characteristics in the time domain and spectral levels in the frequency domain, by calculating the time-domain derivatives of this objective function in the frequency domain. One exemplary embodiment of this technique may include the simultaneous optimization of waveform ISL and spectral characteristics using a gradient descent method, such as a “steepest descent method” or a “conjugate gradient method,” as will be described in more detail below. 
       FIGS. 2A and 2B  depict flowcharts of exemplary methods consistent with the present disclosure. Method  150  may include generating a function that quantifies signal characteristics (step  152 ), calculating time-domain derivatives of the function in the frequency domain (step  154 ), and then using the time domain derivatives to iterate or optimize the function (step  156 ). 
     Method  160  may include generating a function that quantifies signal characteristics as a sum of sidelobe levels (step  162 ), minimizing the function by obtaining gradients of the function in the spectral domain (step  164 ), and using the minimized function to generate a waveform that controls sidelobe levels (step  166 ). 
       FIG. 3  depicts a flowchart of an exemplary method  202  for generating an optimized waveform, which may include the step of creating an objective function that quantifies signal characteristics (step  204 ). For example, a given arbitrary set of waveform characteristics may be defined and optimized together to generate a transmitted waveform that is reasonably smooth looking and has desired pulse compression properties. In one embodiment, the initial created objective function may be the sum of the autocorrelation function&#39;s ISL of a filtered code. The value of the objective function may be lower for better signals and higher for worse performing signals, such that minimization of the objective function leads to desired waveforms. 
     Method  202  may also include defining spectral and time constraints to create a constrained optimization problem (step  206 ). The objective function may be defined to take the absolute value of each constraint, determine whether the value is larger than a constraint, and if so, take the square of the difference between the absolute value and the constraint. Thus, the objective function is higher valued in places where the constraint is not met, and quantifies how close the constraints are to being satisfied. Various defined constraints may include: (1) individual and composite time sidelobes, (2) individual and composite spectral sidelobes, (3) broadening characteristics, (4) mismatch filter losses, (5) waveform amplitude, etc. For example, given a spectrum of 235-450 MHz, it may be desired to create a notch in the spectrum between 250-260 MHz. It may also be desired to control time characteristics, such as the main beam peak, ISL and PSL levels, and the length of the waveform. In some embodiments, it may also be desirable to constrain the waveform to have relatively constant amplitude. Constant amplitude waveforms allow transmitters to run at saturation (full power) and prevent potential max power from being wasted during transmission. The objective function may also include antenna constraints if it is desirable to prevent an antenna from transmitting in a certain direction, e.g., due to clutter in that direction. 
     Thus, the objective function may be a function that maps vectors corresponding to a discretized version of the waveform or filter into a real value. For each type of constraint, the necessary objective functions may be described in the spectral domain using the fast convolution technique. The necessary gradients may then be created in the spectral domain and mapped back to the time domain, as will be described below in more detail. 
     Method  202  may further include converting the constrained optimization problem into an unconstrained optimization problem (step  208 ). For example, the constrained optimization problem described above may be transformed into an unconstrained optimization problem by, for example, using a penalty method (described above), an augmented Lagrange method, etc. 
     Method  202  may further include using a gradient descent method to solve the unconstrained optimization problem (step  210 ). Any suitable optimization technique may be used to solve the unconstrained optimization problem. However, in certain embodiments, a gradient descent method, such as the steepest descent method, or conjugate gradient method, may be desirable. Such methods may involve taking the gradient of the objective function with respect to each one of the individual components, and performing a 1-D line search in the negative gradient or similar direction to find a minimum. In one exemplary embodiment, a conjugate gradient method (e.g., Polak-Ribiere method) may be used, as will be described in more detail below. 
     Method  202  may also include outputting the optimized waveform obtained by solving the unconstrained optimization problem (step  212 ). For example, as described above, computer  102  may store, display, and/or transmit the generated optimized waveform to waveform generator  103 . Waveform generator  103  may then send the optimized waveform to power amplifier transmitter  104 , where it is directed to antenna  108  through duplexer  106 . 
     Method  214  for performing pulse compression, may include transmitting a modulated radar pulse based on an optimized waveform generated by method  202  (step  216 ). In particular, antenna  108  may be configured to transmit a modulated radar pulse based on the optimized waveform (step  216 ). For example, power amplifier  104 , duplexer  106 , and antenna  108  may be used to transmit a waveform similar to a linear frequency modulated radar pulse, by inhabiting each of the desired frequencies, and skipping bands that are not desired. Antenna  108  may also be configured to receive a reflected modulated echo waveform (step  218 ). The received echo waveform may be passed through low-noise amplifier  110 , mixer  112 , and IF amplifier  114  to pulse compression filter  116 . The pulse compression filter  116  may be configured to compress the reflected, modulated echo waveform based on the optimized waveform (step  220 ). For example, pulse compression filter  116  may apply any type or combination of autocorrelation or cross-correlation matched or mismatched filters  118 . 
     Method  214  may also include applying one or more matched or mismatched filters and/or weighting filters to minimize time sidelobes and other characteristics (step  220 ). For example, pulse compression filter  116  may include any number or type of mismatched filters and/or weighting filters  120  designed to control time sidelobes or filter unwanted spectral components of the received signal. For example, a secondary algorithm may be used to control time sidelobes and improve range resolution to efficiently extract desired information from returned signal. Inputs to a mismatched filter may include constraints on broadening, length, and total and individual sidelobe levels. Method  214  may also include performing further signal processing, as desired. 
       FIG. 4  depicts a particular exemplary embodiment of a method  302  for generating an ISL-optimized waveform, consistent with the more general methods  150 ,  160 , and  202 . For example, in one embodiment, method  302  may include creating an objective function that quantifies radar signal characteristics as a sum of the integrated sidelobe levels (ISL) (step  304 ). One exemplary constrained optimization problem corresponds to minimizing the integrated sidelobes of a pulse compression waveform subject to waveform spectral and amplitude considerations. For example, an exemplary function that would control frequency domain content as well as time sidelobe behavior along with a degree of waveform amplitude control would be to solve the constrained optimization problem:
 
Minimized Σ i   |R   i ( {right arrow over (x)},{right arrow over (x)} )| 2  
 
Subject to  P   i ( {right arrow over (x)} )≦ c   i   , |x   i | 3   ≦d   i  and | x   i | 2   ≧d   i  
 
where {right arrow over (x)} is a vector comprising a discretized version of the waveform; where R i  is the i th  correlation sidelobe; and the summation of the absolute values squared of the R i  defines the integrated sidelobes. Moreover, P i  is the l&#39;th element of the discretized spectral power of the waveform; c i  is some constant that describes the max spectral power levels in the frequency domain; and |x i | is the amplitude component desired to be a constant d i .
 
     A penalty method may be used to convert the above constrained optimization problem into an unconstrained optimization problem via the construction of an appropriate objective function, which can be solved using the conjugate gradient method, as described below. The solution of this problem may yield a near constant amplitude waveform with suitable integrated sidelobe level (ISL) behavior and user controlled frequency domain behavior. 
     Thus, method  302  may include defining spectral and time constraints as discussed above to further constrain the optimization problem (step  306 ), and then using a penalty method to convert the constrained optimization problem into an unconstrained optimization problem (step  308 ). In general, a penalty method may include converting formal constraints into terms of the objective function whose minimization achieves the desired results. For example, if it were desired to minimize ISL, the objective function would be the ISL (i.e., sum of squares of time sidelobes) and penalty terms would be added for additional sidelobe characteristics desired to be suppressed. If one region of spectrum is desired to be suppressed, the spectrum can be discretized, and a penalty term added to the objective function in the form of the square of the difference between the discretized spectral power and the goal when the power is greater than the goal, (i.e. (P i ({right arrow over (x)})−c i ) 2  where P i  ({right arrow over (x)}) is the power at the l&#39;th discretized component of the spectrum and c i  is the goal at that position. Thus, the penalty method can be utilized to supplement the initial objective function in order to formulate an unconstrained optimization problem suitable for use with the conjugate gradient method. 
     As described above, a useful function may be constructed to implement the above information using a penalty method, which transforms a constrained optimization problem of the form:
 
Minimize  f ( {right arrow over (x)} )
 
Subject to  g   i ( {right arrow over (x)} )≦ k   i  for  I= 1 , . . . , L.  
 
into an unconstrained optimization. To accomplish this, a sequence of functions g p  ({right arrow over (x)}) may be formed for which:
 
 g   p ( {right arrow over (x)} )= f ( {right arrow over (x)} )+Σ i=1   L   p   i ( g   i ( {right arrow over (x)} )− k   i ) 2 ,
 
where p i =0 if g i ({right arrow over (x)})&lt;=k i , and equals a large positive constant, p, otherwise.
 
     Thus, a sequence of functions g p ({right arrow over (x)}) may be minimized with ever increasing values of p until the constraints are sufficiently well satisfied. For each new and larger value of p, the optimization can be initialized with optimal value of {right arrow over (x)} for the previous value of p, until the ith element of {right arrow over (g)} reaches a given level. If g i ; is desired to be less than k i , the term “p i (g i ({right arrow over (x)})−k i ) 2 ” can be added it to the rest of the objective function when g i ({right arrow over (x)})&gt;k i i(i) to achieve the same result that the formal constraint “g i ({right arrow over (x)})&lt;=k i i(i)” would have. 
     When this penalty process is applied to the sidelobe minimization function described above, the following result for gp(i) may be obtained.
 
 g   p ( {right arrow over (x)} )= h   1 ( {right arrow over (R)} ( {right arrow over (x)},{right arrow over (x)} ))+ h   2 ( {right arrow over (P)} ( {right arrow over (x)} ))
 
     Once the penalty method has been used to convert the constrained optimization problem into an unconstrained optimization problem, the method can further include obtaining gradients in the spectral domain (step  310 ). In particular, it may be desirable to define various parameters in the spectral domain instead of the time domain, by using the Fourier transform. In one exemplary embodiment, pulse compression may be performed in the frequency domain by (1) taking the Fourier transform of the signal to be compressed, (2) taking the Fourier transform of the filter to be compressed against, (3) conjugating one of the transformed values, (4) multiplying the two values, and (5) taking the inverse Fourier transform of the product. Performing pulse compression in the frequency domain may speed up pulse compression calculations by performing them at a speed on the order of log — 2(N)*N where N is the number of components of the vectorized signal. 
     However, optimization in the spectral domain may be limited by the gradients that can be obtained in the spectral domain (i.e., the gradients that can be calculated using FFT techniques). In particular, it can be desirable to obtain the gradient of functions involving the circular correlation of two vectors, and the convolution of a vector with itself. Equations (1) and (2) below may be utilized to calculate the gradients desired in using FFTs to ensure that large scale problems can be solved with modest hardware requirements, by working in the spectral domain. 
     For purposes of illustration, if {right arrow over (x)} and {right arrow over (y)} are two complex vectors of length N (representing the waveform code {right arrow over (x)} and filter {right arrow over (y)} in the time domain), the circular correlation {right arrow over (R)}({right arrow over (x)},{right arrow over (y)}) of the two vectors may be found by point-wise multiplication of the conjugate of the discrete Fourier transform of one of the vectors with the discrete Fourier transform of the other vector, and then applying an inverse Fourier transform to the result. In practice, the process of calculating the Fourier, and inverse Fourier transforms may be accomplished using FFT methods, but for simplicity of notation, may be expressed using an N×N discrete Fourier matrix denoted by the symbol F. In this way, the discrete Fourier transform  X  and  Y  are given by  X =F{right arrow over (x)},  Y =F{right arrow over (y)}. Similarly {right arrow over (x)}=F −1   X  and {right arrow over (y)}=F −1   Y . 
     Using this notation, the correlation 
                 R   ⇀     ⁡     (       x   ⇀     ,     y   ⇀       )       =         F     -   1       ⁡     (           F   ⁢     x   ⇀       _     ∘   F     ⁢     y   ⇀       )       =       F     -   1       ⁡     (         X   ⇀     _     ∘     Y   ⇀       )               
where ◯ denotes the Hadamard or Schur product, which is defined by point wise multiplication of two equal sized matrices or vectors.
 
     In order to minimize the objective function, as described above, it may be desirable to minimize real valued functions of the form 
               h   ⁡     (       R   ⇀     ⁡     (       x   ⇀     ,     y   ⇀       )       )       =     Σ   ⁢           ⁢         h   i     ⁡     (       R   ⇀     ⁡     (       x   ⇀     ,     y   ⇀       )       )       .             
In order to do this, it may be useful to calculate the function ∇ {right arrow over (y)} h efficiently. This can be done using the function:
 
∇ {right arrow over (y)}   h=F   −1 (conj(conj( F∇   {right arrow over (R)}   h )◯ F{right arrow over (x)} ).
 
where ∇ {right arrow over (p)} h is the gradient of h with respect to R (the correlated sidelobes), and F{right arrow over (x)} is the FFT, which maps the time domain {right arrow over (x)} into the frequency domain. Thus, equation (1) can be used to obtain the gradient of h({right arrow over (R)}({right arrow over (x)},{right arrow over (y)})), the advantage being that the inverse FFT has taken the gradient back into the time domain.
 
     Similarly, if the interest is in manipulating the spectral component of the vector {right arrow over (x)} via optimization of a function of the spectral power {right arrow over (P)}({right arrow over (x)}), given by h({right arrow over (P)}({right arrow over (x)}))=Σh i ({right arrow over (P)}({right arrow over (x)})) where {right arrow over (P)}({right arrow over (x)})={right arrow over (X)}◯{right arrow over (  X  (i.e. the convolution of x with itself) then the following gradient can be used:
 
∇ {right arrow over (x)}   {right arrow over (h)}=F   −1 (2 N∇   {right arrow over (p)}   {right arrow over (h)}◯{right arrow over (X)} ).
 
where ∇ {right arrow over (p)} {right arrow over (h)} is the gradient of {right arrow over (h)} with respect to {right arrow over (P)} (i.e., the vectorized power of the waveform in the spectral domain) and 2N∇ p h◯{right arrow over (X)} is moved from the frequency domain back into time domain by the inverse FFT. Thus, the function ({right arrow over (h)}({right arrow over (P)}({right arrow over (x)})) can then be minimized using a conjugate gradient algorithm, which implements the gradient described in equation (2). As described above, because the penalty method may be used to obtain g p ({right arrow over (x)}) in the form: g p ({right arrow over (x)})=h 1 ({right arrow over (R)}({right arrow over (x)},{right arrow over (x)}))+h 2 ({right arrow over (P)}({right arrow over (x)})), the gradients ∇ {right arrow over (x)} g p  may be obtained using equations (1) and (2) above.
 
     In one embodiment, the Polak-Ribiere conjugate gradient method can then utilize the above gradients to find optima of the respective functions (step  312 ). For instance, the conjugate gradient algorithm may start with an initial guess to a local minimum {right arrow over (x)} 0 , and proceed with a line search to find a minimum in the negative gradient direction to find a better estimate {right arrow over (x)} 1 . From that point forward, a sequence of line searches may be used which would, in the case of a purely quadratic minimization problem, force the algorithm to converge in a number of steps Jess than or equal to the number of DOFs. 
     The algorithm may include starting with an initial estimate {right arrow over (x)} 0 :
         1. Let {right arrow over (g)} 0 =∇f({right arrow over (x)} 0 ) and let {right arrow over (d)} 0 =−{right arrow over (g)} 0      2. For k=0 . . . n−1
           a. Let {right arrow over (x)} k+1 ={right arrow over (x)} k +α k {right arrow over (d)} k  where α k  minimizes f({right arrow over (x)} k +α{right arrow over (d)} k )   b. Compute {right arrow over (g)} k+1 =∇f({right arrow over (x)} k+1 )   c. If k=n−1 got to step 3, otherwise let {right arrow over (d)} k+1 =−{right arrow over (g)} k+1 +β k {right arrow over (d)} k , where   
               

     
       
         
           
             
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             3. Replace {right arrow over (x)} k  by {right arrow over (x)} 0  and return to step 1
 
until a suitable solution for {right arrow over (x)} is obtained. As described above, any obtained optimized waveform {right arrow over (x)} can then be saved, stored, and/or transmitted to waveform generator  103  for use in radar pulse compression system  100  (step  314 ).
 
           
         
       
    
       FIGS. 5A and 5B  are flowcharts depicting exemplary methods of an embodiments of the present disclosure. Method  450  may include reading symbol bits and phase bits (step  452 ), using the symbol bits to pick symbols in a digital lookup table (step  454 ), modulating the symbol by the appropriate M-ary phase (step  456 ), performing amplitude modulation compatible with indicated sideband requirements (step  458 ), and performing digital to analog conversion (step  460 ). 
     Method  660  may include a quadrature conversion (step  662 ), filtering the symbols through a filter bank (step  664 ), passing the symbols through a symbol detector (step  666 ), combining the header phase reference and symbol detector (step  680 ), passing the resulting combination to a phase detector (step  690 ). 
       FIG. 6  is a schematic diagram of an embodiment of the present disclosure. As depicted in  FIG. 6 , a digital T x  message is received by a Receive Modem of the type depicted in  FIG. 5 . The modem converts the input digital T x  message to coded symbols. The coded symbol stream is then transferred to a signal generator where it is processed by a Digital-to-Analog converter. The analog symbol stream is then transferred to a software defined radio platform and output as a T x  message. 
     Similarly, as shown in  FIG. 6 , a R x  message may be received by the software defined radio platform. The R x  message is converted to an analog symbol stream and supplied to Signal Generator. Analog to-Digital-converter converts analog symbol stream into digital symbol stream and delivers the digital symbol stream to Transmit Modem of the type depicted in  FIG. 5 . Transmit modem converts the digital symbol stream to uncoded series of message bits and outputs a digital R x  message. 
     Embodiments of the present disclosure also include features to identify changing spectral conditions and adapt the method of the present disclosure to provide effective communications in spite of changing spectral conditions. As shown in  FIG. 6 , RF signals are detected. Spectral conditions are identified and analyzed to identify features and characteristics of the spectral conditions that are significant to maintain high quality communications. A digital stream indicating RF interference and a description of the interference detected are delivered to modem of the type depicted in  FIG. 5  to enhance processing of the communications. 
     Embodiments of the present disclosure address the challenges that increasingly disaggregated bandwidth pose for wideband communications in a number of ways. Embodiments of the present invention make maximum use of available disaggregated spectrum through the use of deep spectral notches. These help prevent interference with other systems, even at close spatial range. The use of matched and mismatch filters enhances protection against interference such as multipath. Embodiments of the present disclosure maintain and enhance data rates under changing spectral conditions. Spectral notches are configured so that they do not impair data throughput. 
     Embodiments of the present invention also maximize signal fidelity. They do this through enhancement of the signal-to-noise ratio by using moderate increases in signal length. Further, signal fidelity is enhanced through reduced inter-symbol interference. 
     Further, embodiments of the present invention are able to offset multi-path effects. They do this by controlled, time-domain cross-correlation notches in the matched and missed-matched filters. The method of the present disclosure has particular application to communications systems. 
     Embodiments of the present disclosure maximize use of each individual spectral fragment in a disaggregated spectrum. By the use of spectral mask conditions imposed at the fragment edges, sidebands of the individual symbols are constrained outside the fragment and are not constrained within the spectral fragment. This enables a degree of sideband hiding among the symbols in the fragment. Further, data rate capabilities are expanded by simultaneous utilization of spectrum across multiple fragments in environments with severe interference. Traditional multi-carrier approaches, in contrast, make the use of this disaggregated spectrum prohibitive due to very high peak-to-average power ratios. 
     Embodiments of the present invention achieve the required spectral compliance while maximizing data rate by narrowing the bands of the individual signals this lengthens symbol durations while increasing their available signal-to-noise ratio (SNR). This lengthening provides additional degrees of freedom, making it easier to meet sideband requirements with minimal peak-to-average power penalties. Certain embodiments of the present disclosure increase the data rate through the use of these lengthened symbols by using low-order orthogonal signaling techniques. 
     As the number of members in the orthogonal signaling alphabet increases, the signal-to-noise ratio requirement also increases. Increased signal duration, however, can provide enough, and sometimes substantially more signal-to-noise ratio gain through coherent addition to make up the required signal-to-noise ratio. Groups of 2 to 16 symbols per spectral fragment are effective in various embodiments of the present disclosure each of the symbol sets in the fragment represents a unique coding scheme. The coding scheme is derived from its primary relative location within the spectral band. 
     These symbol sets possess very low synchronized cross-correlations. All independently meet the spectral compliance objectives of the communication system. These notches provide clean, low-error, separation of the auto-correlation functions with peak-at-center, from the synchronized cross correlations. The width of this cross-correlation notch can be part of the optimization criterion imposed on the waveform set. Larger notches may be required in severe multipath conditions. 
     The increase in signal-to-noise ratio provides the capability to implement M-ary phase coding to each symbol. This can add to the 3 bits provided through the alphabet of 8 orthogonal signals improvements were achieved over data rate obtained from BPSK signaling using the same sub-band while achieving sub −80 dB sideband with near vertical spectral fall off and without inter-symbol interference. In certain embodiments, the signal-to-noise ratio increases due to increased symbol length provide an additional signal-to-noise ratio advantage. Increased SNR may provide resistance to multipath effects. This signal-to-noise ratio advantage may be critical in marginal situations, such as where spectrum is oversubscribed, jammed, multipath effects occur, or changing spectral conditions would otherwise adversely affect the quality of communications. 
       FIGS. 7 ,  8 ,  9 , and  10 , illustrate the bit-rate of trade-space using a BPSK symbol and 3 different spectrally-controlled symbols of embodiments of the present disclosure. A chip rate of 0.01 μs was used. This translates into a 200 MHz null-to-null bandwidth. Three different symbols of embodiments of the present disclosure were optimized to three different side-band levels. Instead of using traditional weighting functions, such as a root-raised cosine (RRC) function, the symbols are optimized to contain the weights within the symbols. For each of the symbols depicted in  FIGS. 8 ,  9 , and  10 , the length of symbol needed to recover the signal-to-noise ratio lost by using a peak-power limited transmitter with the measured the PAPR of the symbol was determined. The symbol length necessary for the symbol null-to-null bandwidth to remain in the 200 MHz null-to-null bandwidth of the BPSK symbol was also calculated. Using the more restrictive of these two measures, the bit rate of the three symbols was calculated. 
     The bit rates were 41, 35, and 31 and BPS for symbols shown in  FIGS. 8 ,  9 , and  10 , respectively. The longest symbol length, corresponding to the 31 bit rate showed the highest suppression of side lobe signal. 
       FIGS. 9-14 , illustrate the trade-off between spectral sideband levels, M-ary orthogonal symbol set size and M-ary phase set size. The power spectrum of the longer symbol sets illustrates how a larger symbol size allows for a lower sideband level via a longer time-bandwidth product. It is also illustrates the ability to hide sidebands of the orthogonal symbols within their common sidebands. The bit error rate for a given spectral fragment can be recovered, by higher use of M-ary phase coding. This is due to the increase in SNR gain, as a result of symbol lengthening. 
     Embodiments of the present disclosure achieved higher data rate expansion through simultaneous use of a number of fragments this provides linear and higher data rate expansion. Simultaneous use may be enabled through the use of multiple amplifiers operating in parallel, one per spectral fragment. The use of multiple transmitters may result in a linear increase in total transmitted power along with a linear increase in data rate while avoiding the very large peak-to-average power penalties seen in, for example, OFDM. 
     Embodiments of the present disclosure may compute waveform symbol sets either off-line or in real time. In real time, the waveform symbol sets may be selected based on spectral sideband requirements, scaling, and frequency shifting. Moreover, the waveform symbol sets may be back fitted to existing systems through the use of appropriate modulators/demodulators (MODEMS). This may render the front and operations of the present disclosure transparent to existing systems. 
     Accordingly, the presently-disclosed systems and methods may be used to perform optimization of both time and spectral characteristics of constant amplitude, near constant amplitude or otherwise defined amplitude pulse compression code or waveform, by using frequency domain calculations instead of time domain calculations. Specifically, the above objective function may be minimized to control integrated-sidelobe-levels (ISL), and peak sidelobe levels (PSL) in the time domain and spectral levels in the frequency domain, by calculating the time-domain derivatives of this objective function with relevant calculations in the frequency domain. Thus, the waveform ISL and spectral characteristics may be quickly and simultaneously optimized using a suitable gradient descent method. Because the calculations are performed in the frequency domain with the number of calculation of the order of log — 2base N, they can be performed more quickly. 
     Although described herein in relation to radar pulse compression, the exemplary methods and systems described herein may be applied to any type of electromagnetic waveforms, such as communications waveforms. Thus, the presently disclosed systems and methods may be used for pulse compression processing of communications waveforms, so as to control and reduce unwanted spectral sidebands that may interfere with communications in adjacent bands. 
     It will be apparent to those skilled in the art that various modifications and variations can be made in the system and method for reception in communication networks. It is intended that the standard and examples be considered as exemplary only, with a true scope of the disclosed embodiments being indicated by the following claims and their equivalents.