Patent Publication Number: US-7596183-B2

Title: Joint optimization of transmitter and receiver pulse-shaping filters

Description:
FIELD OF THE INVENTION 
   The present invention relates generally to communication systems, and particularly to methods and tools for designing and optimizing transmitter and receiver pulse-shaping filters. 
   BACKGROUND OF THE INVENTION 
   Several methods are known in the art for designing transmitter and receiver pulse-shaping filters. Such methods are described, for example, by Zeng in “Pulse Shaping Filter Design and Interference Analysis in UWB Communication Systems,” PhD Thesis submitted to the Virginia Polytechnic Institute and State University, Falls Church, Va., Jul. 20, 2005, which is incorporated herein by reference. Chapter 4, pages 62-80 describes a two-stage method in which the transmitter pulse-shaping filter of an ultra wideband (UWB) system is first designed to meet a given spectral mask. Based on the optimized transmitter filter, the receiver pulse-shaping filter is designed to suppress multiple-access interference and best recover the transmitted signal. 
   Mir and Roy describe a method for transmitter and receiver filter optimization in a digital subscriber line (DSL) system in “Optimum Transmitter/Receiver Design for a Narrowband Overlay in Noncoordinated Subscriber Lines,” IEEE Transactions on Communications, (52:6), June 2004, pages 992-998, which is incorporated herein by reference. The optimization method attempts to reduce cross-talk interference between DSL systems occupying the same frequency band. 
   Another transmitter/receiver optimization method is described by Ho Cho in “Joint Transmitter and Receiver Optimization in Additive Cyclostationary Noise,” IEEE Transactions on Information Theory, (50:12), December 2004, pages 3396-3405, which is incorporated herein by reference. The method considers strictly band-limited linear modulations in additive cyclostationary noise. Optimum transmitter and receiver waveforms that jointly minimize the mean-squared error at the output of the receiver are derived. 
   Jung and Wunder describe a mathematical framework for joint transmitter and receiver pulse shape optimization with respect to the scattering function of a wide-sense stationary uncorrelated scattering (WSSUS) channel in “The WSSUS Pulse Design Problem in Multicarrier Transmission,” e-printed in arXiv.org operated by Cornell University (Ithaca, N.Y.), Sep. 27, 2005, which is incorporated herein by reference. This paper is also available at www.arxiv.org/abs/cs.IT/0509079. 
   SUMMARY OF THE INVENTION 
   Embodiments of the present invention provide improved methods for jointly designing transmitter and receiver pulse-shaping filters. Unlike some known filter design methods, the disclosed methods enable a system designer to trade off a wide variety of different performance characteristics and figures-of-merit of the transmitter and receiver when designing the filters. For example, allowing a certain amount of inter-symbol interference (ISI) by the filters often enables significant reduction in the peak-to-average ratio (PAR) of the transmitted signal. As will be demonstrated below, the design methods described herein enable the designer to define constraints to be met by the transmitter and/or receiver, such as a specified transmitter spectral mask. Other requirements may be relaxed in order to optimize the transmitter and receiver filters to suit a desired performance trade-off. 
   In a typical design process, the designer defines one or more performance-related variables formulated in terms of at least some of the coefficients of the transmitter and receiver pulse-shaping filters. Based on at least some of the filter coefficients and performance-related variables, the designer may set one or more optimization constraints. The designer defines a cost function over the variables. Optimized coefficient values of the transmitter and receiver pulse-shaping filters are then jointly calculated by applying an optimization process to the cost function while meeting the one or more constraints. 
   There is therefore provided, in accordance with an embodiment of the present invention, a method for jointly designing a transmitter pulse-shaping filter and a receiver pulse-shaping filter having respective filter coefficients, including: 
   defining one or more performance-related variables based on at least some of the filter coefficients of the transmitter and receiver pulse-shaping filters; 
   setting one or more constraints applicable to one or more of the filter coefficients and variables; 
   evaluating a cost function defined over the variables; and 
   jointly calculating optimized filter coefficient values of the transmitter and receiver pulse-shaping filters by applying an optimization process to the cost function while meeting the one or more constraints. 
   In an embodiment, the one or more performance-related variables include a peak to average ratio (PAR) at an output of a transmitter including the transmitter pulse-shaping filter. In another embodiment, the one or more performance-related variables include an adjacent channel interference (ACI) at an output of the receiver pulse-shaping filter. Additionally or alternatively, the one or more performance-related variables may include a residual self inter-symbol interference (SISI) at the output of the receiver pulse-shaping filter. Further additionally or alternatively, the one or more performance-related variables may include a signal to noise ratio (SNR) degradation caused by a deviation from a matched filter response between the transmitter and receiver pulse-shaping filters. 
   In another embodiment, the one or more performance-related variables include a distance from a spectral mask defined for a transmitter including the transmitter pulse-shaping filter. In yet another embodiment, the one or more performance-related variables include a normalized mean square error (NMSE) caused by a non-linear noise floor in at least one of a transmitter including the transmitter pulse-shaping filter and a receiver including the receiver pulse-shaping filter. 
   In still another embodiment, the one or more performance-related variables include a symbol rate of a transmitter including the transmitter pulse-shaping filter and a receiver including the receiver pulse-shaping filter. Additionally or alternatively, the one or more performance-related variables include a variable indicative of hardware-related distortion in at least one of a transmitter including the transmitter pulse-shaping filter and a receiver including the receiver pulse-shaping filter. 
   In an embodiment, setting the one or more constraints includes at least one of setting respective limits on one or more of the variables and setting a limit on a function defined over the one or more of the variables. Additionally or alternatively, setting the one or more constraints includes defining a spectral mask to be met by a signal at an output of a transmitter including the transmitter pulse-shaping filter. Further additionally or alternatively, setting the one or more constraints includes defining an upper limit on a residual self inter-symbol interference (SISI) at the output of the receiver pulse-shaping filter. In another embodiment, setting the one or more constraints includes defining an upper limit on an adjacent channel interference (ACI) at an output of the receiver pulse-shaping filter. In yet another embodiment, setting the one or more constraints includes defining an upper limit on a peak to average ratio (PAR) at the output of a transmitter including the transmitter pulse-shaping filter. In still another embodiment, setting the one or more constraints includes defining an upper limit on a normalized mean square error (NMSE) caused by a non linear noise floor in at least one of a transmitter including the transmitter pulse-shaping filter and a receiver including the receiver pulse-shaping filter. 
   In a disclosed embodiment, jointly calculating the optimized filter coefficient values includes relaxing a requirement from a first performance-related variable imposed on the transmitter and receiver pulse-shaping filters while improving a performance of a second performance-related variable imposed on at least one of the transmitter and receiver pulse-shaping filters. The first performance-related variable may include a residual self inter-symbol interference (SISI) at the output of the receiver pulse-shaping filter, and the second performance-related variable may include a peak to average ratio (PAR) at an output of a transmitter including the transmitter pulse-shaping filter. 
   In an embodiment, applying the optimization process to the cost function includes minimizing a peak to average ratio (PAR) at an output of a transmitter including the transmitter pulse-shaping filter. Applying the optimization process to the cost function may include maximizing a symbol rate of a transmitter including the transmitter pulse-shaping filter and a receiver including the receiver pulse-shaping filter while meeting a spectral mask defined for the transmitter. 
   In another embodiment, jointly calculating the optimized filter coefficient values includes optimizing at least one of a tap configuration, a number of coefficients and a coefficient quantization of at least one of the transmitter and receiver pulse-shaping filters. Additionally or alternatively, jointly calculating the optimized filter coefficient values may include applying at least one of a gradient-based and an exhaustive search-based optimization method. 
   There is also provided, in accordance with an embodiment of the present invention, apparatus for jointly designing a transmitter pulse-shaping filter and a receiver pulse-shaping filter having respective filter coefficients, including: 
   a user interface, which is arranged to accept definitions of one or more performance-related variables based on at least some of the filter coefficients of the transmitter and receiver pulse-shaping filters, of one or more constraints applicable to one or more of the filter coefficients and variables, and of a cost function defined over the variables; and 
   a processor, which is arranged to jointly calculate optimized filter coefficient values of the transmitter and receiver pulse-shaping filters by applying an optimization process to the cost function while meeting the one or more constraints. 
   There is additionally provided, in accordance with an embodiment of the present invention, a computer software product for jointly designing a transmitter pulse-shaping filter and a receiver pulse-shaping filter having respective filter coefficients, the product including a computer-readable medium, in which program instructions are stored, which instructions, when read by a computer, cause the computer to accept definitions of one or more performance-related variables based on at least some of the filter coefficients of the transmitter and receiver pulse-shaping filters, to accept definitions of one or more constraints applicable to one or more of the filter coefficients and variables, to accept a cost function defined over the variables, and to jointly calculate optimized filter coefficient values of the transmitter and receiver pulse-shaping filters by applying an optimization process to the cost function while meeting the one or more constraints. 
   There is further provided, in accordance with an embodiment of the present invention, a communication system, including: 
   a transmitter, which includes a transmitter pulse shaping filter; and 
   a receiver, which includes a receiver pulse shaping filter, 
   wherein the transmitter and receiver pulse shaping filters have respective filter coefficients, which are determined by defining one or more performance-related variables based on at least some of the filter coefficients of the transmitter and receiver pulse-shaping filters, setting one or more constraints applicable to one or more of the filter coefficients and variables, evaluating a cost function defined over the variables, and jointly optimizing the filter coefficient values of the transmitter and receiver pulse-shaping filters by applying an optimization process to the cost function while meeting the one or more constraints. 
   The present invention will be more fully understood from the following detailed description of the embodiments thereof, taken together with the drawings in which: 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a block diagram that schematically illustrates a wireless communication system, in accordance with an embodiment of the present invention; 
       FIG. 2  is a block diagram that schematically illustrates a filter design tool, in accordance with an embodiment of the present invention; 
       FIG. 3  is a flow chart that schematically illustrates a method for joint optimization of transmitter and receiver pulse-shaping filters, in accordance with an embodiment of the present invention; and 
       FIGS. 4A ,  4 B,  5 A and  5 B are graphs that show frequency-domain responses of jointly-optimized transmitter and receiver pulse-shaping filters, in accordance with an embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF EMBODIMENTS 
   System Description 
     FIG. 1  is a block diagram that schematically illustrates a wireless communication system  20 , in accordance with an embodiment of the present invention. System  20  comprises a transmitter  24 , which accepts digital data, converts it to a modulated radio frequency (RF) signal and transmits the signal to a receiver  28 . Receiver  28  receives the RF signal and demodulates it so as to reproduce the data. 
   The digital data provided to transmitter  24  is modulated by a modulator  32  to produce a sequence of modulated samples that represent symbols, in accordance with a certain modulation format. The modulated samples are filtered by a transmitter pulse-shaping filter  36  to produce a sequence of filtered samples. Filter  36  comprises a digital filter, which may be implemented as a finite impulse response (FIR) filter, an infinite impulse response (IIR) filter, or using any other suitable configuration known in the art. In some embodiments, a transmitter front end  38  processes the output of the transmitter pulse-shaping filter. Transmitter front end  38  may perform functions such as interpolation, predistortion and/or other filtering operations. 
   The filtered samples are provided to a digital-to-analog converter (DAC)  40 , which produces an analog signal representing the modulated and filtered sample sequence at its input. The analog signal is then upconverted to a suitable radio frequency by an upconverter  44  and amplified by a power amplifier (PA)  48 . The high power RF signal is transmitted via a transmit antenna  52 . 
   A receive antenna  56  receives the transmitted signal and provides it to a downconverter  60 , which downconverts the signal from RF to a suitable intermediate frequency (IF) or baseband analog signal. The analog signal is digitized by an analog-to-digital converter (ADC)  64 , which produces a sequence of digitized samples. In some embodiments, the digitized samples are processed by a receiver front end  66 , which performs functions such as decimation, automatic gain control (AGC) and/or other adaptive filtering operations. The samples are then filtered by a receiver pulse-shaping filter  68 . Filter  68  comprises a digital filter, which may be implemented as a finite impulse response (FIR) filter, an infinite impulse response (IIR) filter, or using any other suitable configuration known in the art. The samples at the output of filter  68  are demodulated by a demodulator  72 , which reproduces and outputs the data provided to transmitter  24 . 
   The embodiments described below are mainly concerned with methods for jointly designing and optimizing pulse-shaping filters  36  and  68 . The system configuration of  FIG. 1  is a simplified configuration, chosen purely for the sake of conceptual clarity in order to show the functions of pulse shaping filters  36  and  68  in communication system  20 . The design methods described herein can be used to produce transmitter and receiver pulse-shaping filters for use in any suitable communication system, such as microwave links and other point-to-point wireless links, cellular communication systems and satellite communication systems. The design methods described herein can also be used to produce transmitter and receiver pulse-shaping filters for use in wired communication systems such as cable modems and DSL systems. 
   The order of functional blocks in the transmitter and receiver shown in  FIG. 1  is an exemplary order. In alternative embodiments, the functional blocks may be arranged and applied in any other suitable order. 
     FIG. 2  is a block diagram that schematically illustrates a filter design tool  74 , in accordance with an embodiment of the present invention. Tool  74  comprises a user interface  76  for interacting with a user, such as a system designer. The designer inputs definitions such as implementation-specific parameters, variables, constraints and cost functions that define the desired optimization. Interface  76  may comprise any suitable interface for accepting the user definitions and for providing the optimization results, such as a suitable graphic user interface (GUI) or a file transfer interface. An optimization processor  78  accepts the user definitions and applies an optimization process, to produce optimized coefficient values of filters  36  and  68 . The resulting filter coefficient values are provided to the designer via interface  76 . 
   Typically, optimization processor  78  comprises a general-purpose computer, which is programmed in software to carry out the functions described herein. The software may be downloaded to the computer in electronic form, over a network, for example, or it may alternatively be supplied to the computer on tangible media, such as CD-ROM. 
   Joint Optimization of Transmitter and Receiver Pulse-Shaping Filters 
   The spectral responses of filters  36  and  68  often have significant influence on the performance of communication system  20 . For example, filter  36  limits the spectral bandwidth of the transmitted signal, often determining the level of adjacent channel interference and other spurious emissions generated by transmitter  24 . When transmitter  24  conforms to a particular communication standard, the bandwidth of the transmitted signal is typically specified in the standard, often using a spectral mask, which should not be exceeded. In some cases, filter  36  may increase the peak-to-average ratio of the transmitted signal, which may limit the output power of power amplifier  48  or cause non-linear distortion in the transmitted signal. 
   The filtering operation of filters  36  and  68  often introduces a certain amount of inter-symbol interference (ISI) into the filtered samples, which may degrade the performance of demodulator  72 . The response of filter  68  may determine the selectivity of receiver  28 , i.e., its ability to suppress out-of-band interference. Other effects of filters  36  and  68  on the performance of system  20  will be apparent to those skilled in the art. 
   In order to minimize ISI in the signal demodulated by demodulator  72 , many known communication methods and systems implement filters  36  and  68  such that their combined spectral response produces a raised cosine (RC) spectral response. In many cases, the RC response is split equally between the transmitter and receiver filters, so that each of filters  36  and  68  has a root raised cosine (RRC) response. In such configurations, the transmitter and receiver pulse-shaping filters are matched to one another. 
   Although optimized for ISI minimization, the RC response, and in particular the splitting of this response into two RRC responses in the transmitter and receiver, may be non-optimal for other performance characteristics, such as transmitted signal peak-to-average ratio, adjacent channel interference and/or compliance with a particular spectral mask. In some practical cases, however, a different pulse-shaping filter design may relax the ISI requirement (i.e., introduce a tolerable amount of ISI) while enabling significant improvement in other performance characteristics. 
   Embodiments of the present invention provide design methods for jointly optimizing the transmitter and receiver pulse-shaping filters to specified performance characteristics and constraints. These methods provide greater flexibility to a system designer in determining the performance trade-offs of the system. 
     FIG. 3  is a flow chart that schematically illustrates a method for joint optimization of transmitter and receiver pulse-shaping filters  36  and  68 , in accordance with an embodiment of the present invention. Two optimization examples are shown in  FIGS. 4A ,  4 B,  5 A and  5 B further below. In the description that follows, transmitter filter  36  comprises a FIR filter having n coefficients denoted g[1] . . . g[n]. Receiver filter  68  comprises a FIR filter having m coefficients denoted h[1] . . . h[m]. 
   The method begins with a designer specifying implementation-specific parameters, at an implementation definition step  80 . Implementation-specific parameters may comprise, for example, the configuration (FIR, IIR) of filters  36  and  68 , the number of coefficients (taps) in each of the filters and the number of bits allocated to each filter coefficient (i.e., tap quantization). The designer provides the implementation-specific parameters to tool  74  using interface  76 . 
   The designer then defines one or more performance-related variables, at a variable definition step  84 . The variables typically comprise performance figures-of-merit of system  20 , expressed in terms of the coefficients of filters  36  and  68 . The designer provides the variables to tool  74  using interface  76 . 
   For example, the designer may define a variable denoted Y 1  expressing the peak-to-average ratio (PAR) of the signal produced by transmitter pulse-shaping filter  36 : 
                   Y   ⁢           ⁢   1     =         (       ∑     i   =   1     n     ⁢          g   ⁡     [   i   ]              )     2         1   N     ·       ∑     i   =   1     n     ⁢            g   ⁡     [   i   ]            2                   [   1   ]               
wherein N denotes the samples per symbol ratio of filter  36  (i.e., the number of samples used to express each symbol in the response g[1] . . . g[n]). In some cases, when the transmitted signal is represented using N samples per symbol, the maximum PAR value should be selected at a particular sampling offset. In such cases, Y 1  can be expressed as
 
   
     
       
         
           
             
               
                 
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   Equation [2] chooses the sampling time offset (denoted k) that provides the maximum PAR value. 
   As another example, the designer may define a variable denoted Y 2 , which specifies the amount of adjacent channel interference (ACI) in the receiver. Let G=DTFT(g) and H=DTFT(h) denote the Discrete-Time Fourier Transform (DTFT), or frequency-domain transfer functions, of filters  36  and  68 , respectively. The lengths of the two DTFTs are assumed to be the same. If the two filters are not the same length, one of them can be zero padded to fulfill this condition. Let T 1  denote the combined frequency-domain transfer function of filters  36  and  68 , when the transmitter frequency is set to the adjacent channel. T 1  can be written as T 1 =H(f)·G(f−ΔCH), (assuming the Nyquist frequency used is sufficiently high to avoid aliasing in G(f−ΔCH)). ΔCH denotes the adjacent channel spacing, or the frequency offset between two adjacent systems. Let T 2  denote a system frequency-domain transfer function used as a reference. T 2  is given by T 2 =H(f)·G(f). The adjacent channel interference can be written as 
                   Y   ⁢           ⁢   2     =     10   ⁢           ⁢       log   10     [         ∑   k     ⁢            T   ⁢           ⁢     1   ⁡     [   k   ]              2           ∑   k     ⁢            T   ⁢           ⁢     2   ⁡     [   k   ]              2         ]               [   3   ]               
wherein k sums over the frequency bins of a suitable frequency range that covers both T 1  and T 2 . Although equation [2] does not explicitly show the dependence of Y 2  on the time-domain filter coefficient values g[ ] and h[ ], Y 2  does depend on these values according to the definitions of T 1 , T 2 , G and H.
 
   As yet another example, the designer may specify a variable denoted Y 3 , which defines the residual ISI in the received signal. Let SISI denote the mean square error due to ISI, or self ISI, in the received signal. The normalized Self ISI is given by 
                 SISI   =           E   ⁡     (       z   k     -     x   k       )       2         E   ⁡     (     x   k     )       2       =         ∑     n   ≠   ctrcoeff       ⁢            L   ⁡     [     n   ·   N     ]            2       +            1   -   ctrcoeff          2     +       ∑   n     ⁢            s   ⁡     [     n   ·   N     ]            2                   [   4   ]               
wherein x k  denotes a symbol and z k  denotes a received sample. The signal is assumed to be synchronized, i.e., sampled at the correct timing. L[n] denotes a convolution of the two pulse-shaping filters, i.e.,
 
             L   ⁡     [   n   ]       =       g   *   h     =       ∑   k     ⁢       g   ⁡     [   k   ]       ⁢       h   ⁡     [     n   -   k     ]       .                 
ctrcoeff denotes the value of the center coefficient of L[n]. The term
 
             ∑     n   ≠   ctrcoeff       ⁢     L   ⁡     [   n   ]             
in equation [4] above excludes the center coefficient, and the samples of L[·] should be chosen at the correct sampling instant which minimizes ISI. s[n] is defined as
 
             s   ⁡     [   n   ]       =       ∑   k     ⁢       g   ⁡     [   k   ]       ⁢     ⅇ       -   j     ⁢     ∏   k         ⁢         h   *     ⁡     [     n   -   k     ]       .               
The term S[n] accounts for aliasing effects encountered when the receiver uses direct IF sampling, as is known in the art. When the receiver uses other downconversion methods, this term can be omitted.
 
   As another example, the designer may define a variable denoted Y 4 , which specifies the degradation in signal to noise ratio (SNR) caused when filter  68  (response h) deviates from an optimal matched filter for the transmitted signal produced by filter  36  (response g). The optimal SNR can be written as 
                   SNR   opt     =         ∑   k     ⁢            g   ⁡     [   k   ]            2         N   0               [   5   ]               
wherein N 0  denotes the noise power at the output of filter  68 . The SNR degradation variable is given by
 
   
     
       
         
           
             
               
                 
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   In some embodiments, variables can be defined for quantifying certain hardware-related distortion mechanisms. For example, if a non-linear model of PA 48 is available, the designer can define a variable specifying the non-linear distortion generated by the PA. 
   Additionally or alternatively, the designer may define any other suitable variable that specifies a performance figure-of-merit of system  20  in terms of coefficients g[1] . . . g[n] and h[1] . . . h[m] of filters  36  and  68 . Such variables may comprise, for example, the system symbol rate (baud rate), the distance from a spectral mask defined for the transmitted signal and the normalized mean square error (NMSE) caused by a non-linear noise floor (e.g. a non-linear noise floor caused by the transmitter power amplifier). 
   The designer then sets one or more constraints, at a constraint definition step  88 . Constraints are typically derived from requirements in the system specification or requirements imposed by applicable communication standards or regulations. For example, a constraint may define a regulatory spectral mask that should be met by the transmitted signal. Other exemplary constraints may require that the residual ISI (Y 3 ) and/or ACI (Y 2 ) be smaller than certain predetermined values. In general, constraints may depend on the variables defined in step  84  above, on the coefficients of filters  36  and  68 , or both. The constraints are provided to tool  74  via interface  76 . 
   The designer defines a cost function, at a cost definition step  92 . The cost function, denoted COST, is defined over one or more of the variables specified in step  84  above and defines the desired joint optimization of filters  36  and  68 . For example, COST=MIN(Y 2 ) defines that the coefficients of filters  36  and  68  are to be optimized for minimum ACI. COST=MIN(Y 1 ) defines that the filters are to be optimized for minimum peak-to-average ratio of the transmitted signal. The cost function may be defined over several variables. For example, COST=MIN{f(Y 1 , Y 2 )} defines that the filter coefficients should be optimized for a function of both peak-to-average ratio and minimum ACI. The function f defines the relative weight, or significance, given to each of the variables in the optimization process. The designer provides the cost function to tool  74  using interface  76 . 
   Optimization processor  78  performs an optimization process to determine the optimized filter coefficients, at an optimization step  96 . The processor jointly determines the values of coefficients g[1] . . . g[n] and h[1] . . . h[m] of filters  36  and  68 , respectively, which optimize the cost function COST under the defined constraints. Processor  78  may use any suitable optimization method known in the art for this purpose, such as various gradient methods, various exhaustive search and other search methods that explore the space of coefficient values. 
   For example, in some embodiments, processor  78  uses the Optimization Toolbox of the MATLAB® computational software tool produced by The Mathworks, Inc. (Natick, Mass.) for performing the optimization process. Specifically, processor  78  may use the fmincon optimization function defined in the optimization toolbox. The fmincon function finds a minimum of a constrained non-linear multivariable function. Documentation of this optimization function can be found at www.mathworks.com/access/helpdesk/help/toolbox/optim/ug/fmincon.html. Information regarding the optimization toolbox can be found at www.mathworks.com/access/helpdesk/help/toolbox/optim/ug/index.html. 
   Processor  78  outputs the optimized coefficient values via interface  76 , at an output step  100 , and the method terminates. In some embodiments, the optimization process may optimize some or all of the implementation-specific parameters, as well. For example, the coefficient quantization and/or the length (i.e., number of coefficients) of one or both pulse-shaping filters can be optimized. 
   Exemplary Optimization Results 
     FIGS. 4A ,  4 B,  5 A and  5 B are graphs that show frequency-domain responses of jointly-optimized transmitter and receiver pulse-shaping filters, in accordance with an embodiment of the present invention. 
   In order to evaluate the effectiveness of the optimization method of  FIG. 3  above, pulse-shaping filters optimized using the disclosed method were compared with RRC filters having the same implementation-specific parameters (i.e., same number of coefficients and quantization). In the examples that follow, both the reference (RRC-based) and the optimized communication system use 16-signal Quadrature amplitude (16-QAM) modulation. The transmitter pulse-shaping filter comprises a 65 coefficient FIR filter. The receiver pulse-shaping filter comprises a 33 coefficient FIR filter. The coefficients of both filters are quantized non-uniformly using a rounding function. The center coefficients of both filters are represented using 12 bits. The quantization level gradually decreases, with edge coefficients represented using 4 bits. The adjacent channel spacing (ΔCH) is 14 MHz. (As noted above, the symbol rate may also be defined as a variable or as a cost function.) In the reference system, both transmitter and receiver RRC filters have a roll-off factor of 0.25. 
     FIG. 4A  shows a comparison between the optimized and RRC transmitter pulse-shaping filters. The corresponding comparison between the receiver pulse-shaping filters is shown in  FIG. 4B  below. The transmitter and receiver pulse-shaping filters were jointly optimized using the method of  FIG. 3  above. In  FIG. 4A , a curve  120  shows a spectral mask defined for the transmitted signal. In the present example, curve  120  shows the spectral mask defined in the European Telecommunications Standards Institute (ETSI) EN 302 217-2-2 standard entitled “Fixed Radio Systems; Characteristics and Requirements for Point-to-Point Equipment and Antennas.” This standard is available at www.etsi.org. 
   A curve  124  shows the spectral response of the reference RRC pulse-shaping filter. A curve  128  shows the spectral response of an optimized transmitter pulse-shaping filter. In  FIG. 4B , a curve  132  shows the spectral response of the reference RRC receiver pulse-shaping filter. A curve  134  shows the spectral response of the optimized receiver pulse-shaping filter. 
   The optimization that produced the optimized transmitter and receiver filters of curves  128  and  134 , respectively, attempted to minimize the peak-to-average ratio of the transmitted signals (variable Y 1  as defined in equation [1] above). The constraints defined in the optimization were (1) self ISI (Y 3 )≦−38 dB, (2) SNR degradation (Y 4 )≦0.2 dB, (3) ACI (Y 2 )≦−17 dB and (4) compliance with the spectral mask of curve  120 . 
   The symbol rate used in both reference and optimized systems is 12.5 MHz. As can be seen in  FIG. 4A , curve  124  violates the mask of curve  120  by approximately 1.5 dB at this symbol rate. Although the example shows a RRC filter having a rolloff factor of 0.25, it was not possible to reach a symbol rate of 12.5 MHz using an RRC transmitter filter while meeting the spectral mask. Curve  128 , however, shows that the optimized transmitter filter does meet the spectral mask constraint at a symbol rate of 12.5 MHz. 
   The following table shows a comparison between the figures-of-merit of the reference and optimized systems, following completion of the optimization process: 
   
     
       
         
             
             
             
             
             
           
             
                 
             
             
                 
               Peak-to- 
                 
                 
                 
             
             
                 
               average 
                 
                 
               SNR 
             
             
                 
               ratio 
                 
               Residual 
               degradation 
             
             
               Parameter 
               (Y1) 
               ACI (Y2) 
               ISI (Y3) 
               (Y4) 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
             
             
             
             
             
             
          
             
               Reference 
               5.4 dB 
               −29.6 
               dB 
               −45.4 
               dB 
               0.02 
               dB 
             
             
               system 
             
             
               Constraint 
               — 
               ≦−17.0 
               dB 
               ≦−38 
               dB 
               ≦0.2 
               dB 
             
             
               Optimized 
               2.5 dB 
               −17.0 
               dB 
               −39.7 
               dB 
               0.2 
               dB 
             
             
               system 
             
             
                 
             
          
         
       
     
   
   As can be seen from the table, the optimization significantly reduced signal peak-to-average ratio of the transmitted signal from 5.4 to 2.5 dB, while meeting the defined constraints. This example shows that a pair of jointly-designed transmitter and receiver pulse-shaping filters allowing a certain amount of signal distortion enables significant peak-to-average ratio reduction. These filters also enable achieving a higher symbol rate, which in turn enables better performance due to reduced coding rate. 
     FIGS. 5A and 5B  show a similar comparison at a slightly lower symbol rate of 12 MHz.  FIG. 5A  shows a comparison of transmitter pulse-shaping filters. A curve  138  shows the spectral response of the reference RRC transmitter filter. A curve  142  shows the spectral response of the optimized transmitter filter. At a symbol rate of 12 MHz, the RRC transmitter filter complies with the spectral mask of curve  120 .  FIG. 5B  shows the corresponding comparison of receiver pulse-shaping filters. A curve  146  shows the spectral response of the reference RRC receiver filter. A curve  150  shows the spectral response of the optimized receiver filter. 
   The following table shows the figure-of-merit comparison between the reference and optimized systems at a symbol rate of 12 MHz: 
   
     
       
         
             
             
             
             
             
           
             
                 
             
             
                 
               Peak-to- 
                 
                 
                 
             
             
                 
               average 
                 
                 
               SNR 
             
             
                 
               ratio 
                 
               Residual 
               degradation 
             
             
               Parameter 
               (Y1) 
               ACI (Y2) 
               ISI (Y3) 
               (Y4) 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
             
             
             
             
             
             
          
             
               Reference 
               5.4 dB 
               −43.3 
               dB 
               −45.4 
               dB 
               0.02 
               dB 
             
             
               system 
             
             
               Constraint 
               — 
               ≦−17.0 
               dB 
               ≦−38 
               dB 
               ≦0.2 
               dB 
             
             
               Optimized 
               2.5 dB 
               −20.1 
               dB 
               −39.7 
               dB 
               0.2 
               dB 
             
             
               system 
             
             
                 
             
          
         
       
     
   
   It can be seen that the optimization process reduced the peak-to-average ratio of the transmitted signal from 5.4 to 2.5 dB while meeting the defined constraints. 
   It will be appreciated that the embodiments described above are cited by way of example, and that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and sub-combinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art.