Abstract:
A communication system and method for continuous phase modulation providing for transmission of a phase-modulated carrier having a phaseform representative of concurrently transmitted symbols. The phaseform of the phase-modulated signal is a sum of shift bi-orthogonal functions, each term in that sum being weighted by one of the overlapping symbols. The communication system and method provide full-response demodulation for the recovery of a particular symbol from among the concurrently transmitted symbols by selecting a receiving filter function shift bi-orthogonal to the transmitter filter function corresponding to the particular symbol. The communication system and method then provide for nulling, by integration over a time interval during which the particular symbol is transmitted, those transmitter filter functions that do not correspond to the particular symbol. This results in the separation of the particular symbol from the other concurrently transmitted symbols.

Description:
RELATED APPLICATIONS 
     This application is a division of application Ser. No. 09/469,581 filed Dec. 22, 1999, which claims priority from U.S. Provisional Application No. 60/114,274 filed on Dec. 30, 1998 which is hereby incorporated by reference. 
    
    
     TECHNICAL FIELD OF THE INVENTION 
     This invention relates to general wireless communication systems, and in particular to continuous phase modulation methods and systems. 
     BACKGROUND 
     When transmitting a collection of symbols across a noisy channel, there exists an inherent tradeoff between the power used to transmit each symbol and the length of time or the bandwidth required to transmit the symbol. This tradeoff is shown graphically in the communication efficiency plane of FIG. 1, in which the horizontal axis is the ratio of energy-per-bit to noise power spectral density, E b /N o , and the vertical axis is the ratio of the communication rate to bandwidth (bits/sec/Hz). The Shannon limit represents the theoretical maximum capacity of the channel to transfer data as, E b /N o , varies. It is apparent from inspection of FIG. 1 that as more power is used to transmit information, information can be communicated across a channel more rapidly. 
     The somewhat abstract tradeoff illustrated in FIG. 1 can immediately be understood by considering the case of a speaker attempting to communicate information, for example a telephone number, to a listener across a noisy room. One approach in such a case is for the speaker to shout, thereby rendering the numbers clearly audible to the listener over the noise in the room. Using this method, the speaker can communicate the entire telephone number in a very short time. 
     Another, far more subtle, approach is for the speaker to speak in a whisper but to elongate the delivery of each syllable. When the speaker transmits the message using this method, the listener can generally pick out the underlying drone of the speaker&#39;s message from the background noise, in effect filtering out the speaker&#39;s message by integrating the received signal and noise over a sufficiently long integration interval. The disadvantage of this second approach is that it requires considerably more time to transmit the entire telephone number across the room. 
     In the context of FIG. 1, the speaker who shouts the phone number across the room operates toward the right-hand side of the communication efficiency plane. As a result, that speaker will be able to operate at a high baud rate and to therefore transmit the entire number very quickly. In contrast, the speaker who whispers the phone number operates toward the left-hand side of the communication efficiency plane and is therefore constrained by the Shannon limit to operate at a low baud rate and to therefore transmit the entire number slowly. 
     Every communication system can be characterized by an operating point in the communication efficiency plane of FIG.  1 . One goal of communication systems design is to place that operating point as close as possible to the Shannon limit. Where bandwidth is a limited resource, one can approach the Shannon limit using a high power system. Such a system does not require large bandwidth for near-error free transmission. Conversely, where power is a limited resource, one can approach the Shannon limit by using a low power system operating over a large bandwidth. 
     In a wireless communication system, the availability of power directed toward the receiver is limited by the transmitter&#39;s power rating and antenna gain. The availability of bandwidth is constrained by the desirability of sharing the available spectrum with as many channels as possible and by FCC governmental and international regulations. Consequently, in a wireless communication system, both power and bandwidth are limited resources. 
     In addition, a communication system in which different symbols are modulated onto the carrier as different power levels will have certain symbols represented by the lowest power level. Because these symbols are transmitted at lower than average power levels, they will inevitably be more prone to corruption by noise than symbols represented by higher than average power levels. As a result, the error rate associated with the transmission of a message will depend on the content of that message. A higher average power will be required for these symbols to reduce the error rate to a desired level. 
     Constant power signals are preferred for many wireless systems such as satellite communication systems which typically use more efficient class C operating amplifiers. Because these are non-linear amplifiers that operate at or beyond saturation, a waveform having other than constant amplitude can experience profound distortion as it passes through such an amplifier. 
     Constant power transmission can be achieved by modulating either the frequency or the phase of a carrier wave. Of these two modulation alternatives, phase modulation is far preferable for satellite communication systems because of its greater bandwidth efficiency. The reason for this greater bandwidth efficiency can be readily understood by considering the operation of a frequency modulation system. 
     In a frequency modulation system such as frequency shift keying (FSK), each symbol corresponds to a particular frequency. When two symbols correspond to two frequencies that are very close together, the probability of a demodulation error due to noise in the communication channel is high. In order to reduce the probability of such an error, the difference of the two frequencies must exceed a certain fixed amount that is proportional to the channel bandwidth. An increase in the signaling rate entails an increase in the number of frequencies employed. This, of course, consumes bandwidth. 
     In a conventional phase modulation system, each symbol corresponds to a particular phase angle associated with a carrier having a single frequency. As a result, a large number of symbols can be transmitted without requiring multiple frequencies. Phase modulation systems are thus particularly desirable in applications such as satellite communication systems in which bandwidth is at a premium. 
     In its simplest form, a phase modulation system operates by transmitting a carrier having a fixed phase angle which is representative of a first symbol during a first time interval. In the context of this application, the “phase angle” of a carrier refers to the principal branch of the arc tangent of the ratio of the imaginary part of the carrier to the real part of the carrier. During a second time interval, the system transmits the carrier but with a fixed phase angle corresponding to a second symbol (not necessarily different from the first symbol). The system then operates at the second phase angle for the duration of the second time interval. The system continues to operate in the foregoing manner until all the symbols that make up the message have been sent. The phase considered as a function of time (hereinafter referred to as the “phaseform”) for this system thus traces a discontinuous path in time as shown in the phase cylinder in FIG.  2 . 
     A disadvantage of the foregoing method of operating a phase modulation system is that it is not possible to efficiently shape the phaseform so that spectral energy will remain concentrated within the allocated bandwidth while maintaining signaling rate. For systems that employ non-overlapping phase symbols, the symbols are typically phases that are held constant during each symbol transmission interval. For instance, PSK employs constant phase symbols. 
     Unfortunately, phaseform symbols that have constant values introduce discontinuities whenever a symbol is followed by a different one, and the discontinuity causes energy to spill into adjacent channels, causing adjacent channel interference. If the duration of each of the successive non-overlapping symbols is increased to allow shaping for improved energy concentration and decreased adjacent channel interference, then the signaling rate is decreased. 
     Another approach to overcoming the foregoing disadvantage is to increase the time interval required to transmit a particular symbol but to also allow portions of two or more symbols to be transmitted during each time interval. This type of phase modulation is best understood with reference to FIG. 3, which shows the transmission of six symbols, each represented by a particular phase angle on the vertical axis, and each of which requires three time intervals, ΔT, for transmission. 
     As shown in FIG. 3, transmission of the first symbol begins at t=0. At the end of the first clock period, t=ΔT (ΔT=1 clock period) before the first symbol can be completely transmitted, transmission of the second symbol begins. At the end of the second clock period t=2ΔT, before either the first or second symbol can be completely transmitted, transmission of the third symbol begins. Thus, during the third clock period (between t=2ΔT and t=3ΔT), three symbols are concurrently being transmitted. Since no more than three symbols are transmitted concurrently, a phase modulation system operating as shown in FIG. 3 is said to have a “modulation latency” of three. 
     It is apparent from FIG. 3 that a phase modulation system having a high latency can, after an initial latency interval of duration equal to the duration of the first symbol, transmit a set of symbols at the same rate as a phase modulation system having a latency of one, notwithstanding the fact that each individual symbol may take longer to transmit. What is less apparent is that the transmission of overlapping symbols in the manner shown in FIG. 3 can be made to eliminate the discontinuities shown in the phase cylinder of FIG. 2 by selecting the phasefom symbols to vary continuously with time. If the individual symbols are continuous, and if the boundaries between adjacent symbols are also continuous, then every linear combination of such symbols overlapped arbitrarily in time will also be continuous. Since it is the discontinuities in the phaseform that result in the spray of high-frequency energy into adjacent channels, phase modulation as shown in FIG. 3 (often referred to as “continuous phase modulation”) tends to better confine the signal energy into its allocated bandwidth. 
     Because energy from two or more overlapping phaseform symbols is commingled, known demodulators cannot readily separate the energy associated with any particular phaseform symbol. In these types of systems, referred to as “partial response” systems, there is less energy per symbol available to the demodulator. In order to compensate for this reduced symbol energy, sequence demodulators, such as Viterbi decoders, are used to recover the symbols. Unfortunately, the use of such sequence demodulators imposes computational costs that increase exponentially with the number of symbols concurrently processed by the demodulator to retrieve the current symbol. 
     It is known that, in theory, one can separate the individual overlapping symbols that constitute a phaseform if the phaseform is a linear combination of time-shifted sinc functions (sinc (x)=sin (x)/x) if the coefficients of the shifted sinc functions are known functions of the information symbols. Because the temporally shifted sinc functions are orthogonal to one another, it is theoretically possible to recover all the energy associated with a particular symbol, and hence achieve “full response” demodulation, by demodulating the phaseform using a matched filter corresponding to the sinc function associated with that symbol. However, this procedure is physically unrealizable because the sinc function has infinite duration. 
     It is therefore desirable to provide systems and methods for achieving physically-realizable full-response continuous phase modulation. 
     SUMMARY OF THE INVENTION 
     In a communication system and method according to the invention, a sequence of information symbols is encoded as a phase trajectory (herein referred to as a phaseform) of a phase-modulated carrier, thereby generating a phase-modulated signal. This phase-modulated signal is representative of a selected symbol transmitted during a selected-symbol transmission interval, and at least two symbols concurrently being transmitted during that interval. The phaseform of the phase-modulated signal is generated by evaluating a combination of transmitting-filter responses having finite duration. The foregoing combination of transmitting-filter responses includes a selected transmitting-filter response weighted by the selected symbol. 
     Recovery of the selected symbol is accomplished at a receiver by first demodulating the phase-modulated signal to obtain the phaseform. The receiver then filters the phaseform with a receiving-filter response corresponding to the selected transmitting-filter response. This results in a weighted output representative of the combination of transmitting-filter responses weighted by the receiving-filter response. Finally, the receiver accumulates the values of the weighted output over the transmitting interval. This accumulation process can be implemented by an integrator, for an analog system implementation, or alternatively, by a summer, for a digital system implementation. The result of the accumulation process is a signal representative of the selected symbol separated from the at least two overlapping symbols transmitted concurrently. 
     The receiving filter-responses may be selected to be shift-biorthogonal to the transmitting-filter responses. In the case in which the receiving-filter response is the same as the corresponding transmitter-filter response, the shift-biorthogonal receiving-filter responses will be shift-orthogonal and the transmitter-filter and corresponding receiver-filter will be a matched filter pair. The result in either case is that the accumulation of the values of the weighted output over the transmitting interval nulls or cancels constituents of the weighted output that correspond to symbols other than the selected symbol. The result of the accumulation process is thus a signal representative of the selected symbol alone. In this way, the communication system and method of the invention separates the selected symbol from among the commingled symbols present in the phaseform. 
     The operation of the communication system and method of the invention does not rely on the nature of the signal and can be implemented in conjunction with analog or digital carriers and with analog or digital phaseforms. What is important is that the transmitting-filter functions and the receiving-filter functions be selected so that phaseform constituents corresponding to symbols other than the symbol of interest can be eliminated by an accumulation process. 
     In a second embodiment of the communication system and method of the invention, a symbol to be recovered and at least two other symbols are transmitted concurrently. This second embodiment is identical to the first embodiment with the exception that the receiving-filter function is selected to have the additional property of being matched, or shift-orthogonal, to the transmitting-filter function. 
    
    
     The foregoing and other objects, features and advantages of the invention will be apparent from the following description and the accompanying drawings, in which: 
     BRIEF DESCRIPTION OF THE FIGURES 
     FIG. 1 shows the Shannon limit on the bandwidth efficiency plane; 
     FIG. 2 shows a complex baseband representation of a phase cylinder on which is traced a discontinuous phaseform from a conventional phase modulation system; 
     FIG. 3 illustrates the concept of modulation latency by showing the transmission of six symbols, each of which requires three clock periods for complete transmission; 
     FIG. 4 shows a general communication system incorporating the principles of the invention; 
     FIG. 5 shows the components of the transmitter shown in FIG. 4; 
     FIG. 6 shows the signals present at various points within the transmitter of FIG. 5 after m clock periods have elapsed; 
     FIG. 7 shows representative filter functions for the communication system of FIG. 1 in which the modulation latency is 3; 
     FIG. 8 shows the architecture of the receiver shown in FIG. 4; 
     FIG. 9 shows the architecture of one symbol decoder shown in FIG. 8; and 
     FIG. 10 shows the architecture of a second symbol decoder shown in FIG.  8 . 
    
    
     DETAILED DESCRIPTION 
     With reference to FIG. 4, a communication system  10  incorporating the principles of the invention includes a transmitter  12  and a receiver  14  separated by a communication channel, represented in FIG. 4 by a noise source  15 . 
     In a communication system  10  according to the invention, the transmitter  12  forms and transmits a phase-modulated wave having a phaseform in which are encoded a sequence of symbols. At successive instants of time separated by a clock period having a duration of ΔT seconds, the transmitter begins sending a new information symbol. Each symbol requires g clock periods for transmission. Hence g is referred to as the “modulation latency,” or simply the “latency” of the system. At any instant, the phaseform can represent as many symbols as the modulation latency of the system. The receiver  14  then demodulates the phase-modulated wave and recovers the symbols encoded in the phaseform. 
     To facilitate exposition and understanding of the structure and operation of the communication system  10 , the structure and operation of the transmitter  12  and receiver  14  are discussed with reference to separate sets of figures. The structure and operation of the transmitter  16  is discussed with reference to FIGS. 5-7. The structure and operation of the receiver  14  is discussed with reference to FIG. 8-10. 
     Transmitter 
     Referring to FIG. 5, the transmitter  12  incorporating the subject matter of the invention includes a plurality of transmitting shift-biorthogonal function generators  14   a-g  and a plurality of transmitting mixers  16   a-g . The number of such transmitting shift-biorthogonal function generators  14   a-g  and the number of such transmitting mixers  16   a-g  is at least equal to the modulation latency, g. An input binary digital information stream is transformed into a stream of numerical information symbols τ 0 , τ 1 , τ 2 , . . . τ L−1 . These numerical values are transmitted by the transmitter  12 , and will be recovered by a receiver, to be described below in connection with FIG. 8. A post-processing operation reconverts the numerical symbols into a binary digital information stream. 
     The input symbol stream τ 0 , τ 1 , τ 2 , . . . τ L−1  is demultiplexed into a plurality of demultiplexed streams  18   a-g , each of which is directed to a corresponding transmitting mixer  16   a-g . The first symbol, τ 0  from the first demultiplexed stream  18   a  is presented to the first mixer  16   a . The resulting output  20   a  from the first mixer  14   a  is τ 0 L 0 . One symbol start period ΔT later, the second symbol τ 1 , from the second demultiplexed stream  18   b , is presented to the second mixer  16   b . This procedure continues until the g th  symbol, τ g−1  is presented to the g th  transmitting mixer  16   g . The process repeats with the (g+1) th  symbol, τ g  from the first demultiplexed stream  18   a  being presented to the input of the first mixer  16   a . The resultant mixer outputs  20   a-g  of these transmitting mixers  16   a-g  are (g) signals that are representative of a filtered symbol. The transmitting-mixer outputs  20   a-g  are connected to a corresponding number of inputs  21   a-g  of a transmitting accumulator  22 . The transmitting accumulator  22  has a transmitting-accumulator output  24  on which it provides a signal representative of the sum of the overlapping filtered symbols. A phase modulator  26  then encodes the signal present on the transmitting-accumulator output  24  onto the phase of a carrier wave. This carrier wave, with its phase now modulated by the signal present on the accumulator output  24 , then enters the communication channel  15 . 
     FIG. 6 shows the signals present on the various inputs and outputs of the transmitter  12  of FIG. 5 after m clock periods have elapsed. The g symbols τ k  for k=m, m+1, . . . , m+g−1, which are a known numerical function of the information bits in the message to be transmitted, being transmitted in clock period m+1, namely symbols τ m , τ m+1 , . . . , τ m+g−1 , are presented at the symbol inputs  19   a-g  of the transmitting mixers  16   a-g . The transmitting-filter generators  14   a-g  then generate corresponding transmitting-filter functions L m (x), L m+1 (x), . . . , L m+g−1 (x). These transmitting-filter functions are presented at the filter inputs  18   a-g  of the corresponding transmitting mixers  16   a-g . These transmitting-filter functions, representative examples of which are shown in FIG. 7, are preferably chosen to be temporally-shifted shift-biorthogonal wavelet basis functions having compact support. Each mixer  16   a-g  mixes a transmitting-filter function with a corresponding symbol. The output of each transmitting mixer  16   a-g  is thus a shift-biorthogonal wavelet function having an amplitude weighted by a corresponding symbol being transmitted: τ i L i (x). 
     Those skilled in the art will recognize that the transmitting-filter generators  14   a-g  and the transmitting mixers can be implemented directly in hardware or by suitable software instructions provided to a computer system. 
     For simplicity of notation, hereinafter let x=t/ΔT. In FIG. 7, each transmitting-filter function L i (x) is preferably shifted in time (x) so that it is non-zero only during clock periods during which its corresponding symbol τ i  is being transmitted. Since it takes g clock periods to transmit a given symbol, each transmitting-filter function is non-zero for only g clock periods. For the filter functions shown in FIG. 7, the modulation latency is three. Hence, L m (x) is non-zero only during clock periods between t=mΔT and t=(m+3)ΔT. Similarly, L m+1 (x) is non-zero only during those clock periods between t=(m+1)ΔT and t=(m+4)ΔT. 
     As shown in FIG. 6, the transmitting-accumulator  22  provides, at the transmitting-accumulator output  24 , a phaseform representative of a weighted sum of shifted wavelet basis functions:          φ        (     x   ,   m     )       =       ∑     i   =   m       m   +   g   -   1              τ   i            L   i          (   x   )                                  
     The phase modulator  26  then encodes the signal from the transmitting-accumulator output  24  onto a carrier exp(iωx). The signal present at the output  32  of the phase modulator  26  is thus a wave having a time-varying phase modulated by a sum of weighted and temporally-shifted wavelet basis functions corresponding to those symbols that are currently being transmitted: 
     
       
           exp ( i (ω x +φ( x,m )))  
       
     
     Consideration of FIGS. 5-7 reveals that a transmitter  12  operating according to the principles of the invention begins transmitting a new symbol with each clock interval. Since the transmission of each symbol requires several clock intervals, during any clock interval, there can be several symbols being transmitted, each at a different stage of transmission. The transmitter  12  weights each symbol by mixing it with a transmitting-filter function which is non-zero only during those clock periods in which the symbol is being transmitted. The transmitter  12  then evaluates a sum over the clock periods during which the transmitting-filter functions are non-zero. This sum, or integral, is then used to modulate the phase of a carrier wave. 
     Receiver 
     Referring now to FIG. 8, the receiver  14  incorporating the principles of the invention includes a phase demodulator  32  for recovering the phaseform          φ        (     x   ,   m     )       =       ∑     i   =   m       m   +   g   -   1              τ   i            L   i          (   x   )                                  
     from the phase-modulated signal. The process of recovering the phaseform can include conventional signal processing techniques for noise-suppression. These functions are accomplished by known subsystems within the phase demodulator  32 . 
     The phase demodulator  32  is connected to phaseform inputs  33   a-g  of each of a plurality of symbol decoders  34   a-g . The phaseform provided to each symbol decoder  34   a-g  is thus a finite sum of phase terms, each of which is a temporally-shifted wavelet basis function weighted by one of the symbols being transmitted. The purpose of each symbol decoder  34   a-g  is to separate one of these constituent phase terms from the phaseform. The output of each symbol decoder  34   a-g  is thus the symbol associated with one of the constituent phase terms in the above phaseform. Since the number of phase terms, and hence the number of symbols, present in the phaseform at any instant is no greater than the modulation latency, the number of symbol decoders  34   a-g  need be no greater than the modulation latency. 
     The observant reader will note that in FIG. 8, only the first symbol decoder  34   a  is shown as having an output. This is meant to suggest that not all the symbol decoders have their outputs ready at the same time. The reason this is so is that each symbol takes several clock intervals to transmit. Thus, if the transmitter were to begin transmitting symbol τ i  before beginning the transmitting symbol τ i+1 , one would expect that the symbol decoder charged with extracting symbol τ i  would complete its extraction process before the symbol decoder charged with extracting symbol τ i+1  could complete its extraction process. 
     Because the symbol decoders  34   a-g  all operate in fundamentally the same manner, it is instructive to consider in detail the operation of a first symbol decoder  34   a , shown in detail in FIG.  9 . 
     Those skilled in the art will recognize that the illustrated demodulator  32  and symbol decoders  34   a-g  can be implemented directly in hardware or by suitable software instructions provided to a computer system. 
     Referring now to FIG. 9, the first symbol decoder  34   a  includes a receiving mixer  35   a  and a receiving-filter generator  36   a . The receiving mixer  35   a  has a phaseform terminal  33   a  connected to the phase demodulator  32 , a receiving-filter input terminal  38   a  connected to the receiving-filter generator  36   a , and a receiving-mixer output terminal  40   a.    
     In operation, the receiving-filter generator  36   a  generates and places a receiving-filter function R m (x) at the receiving-filter input terminal  38   a  of the receiving mixer  35   a . The signal leaving the receiving mixer  35   a  is thus                    R   m          (   x   )              ∑     i   =   m       m   +   g   -   1              τ   i            L   i          (   x   )             =                    τ   m            R   m          (   x   )              L   m          (   x   )         +       τ     m   +   1              R   m          (   x   )              L     m   +   1            (   x   )         +   …              +                                τ     m   +   g   -   1              R   m          (   x   )              L     m   +   g   -   1            (   x   )                                      
     where x=t/ΔT is as described in connection with the discussion of the transmitter. The receiving-filter function R m (x) is selected such that the integral of the product R m (x)L n (x) is non-zero only when n=m. Consequently, in order to extract the symbol τ m  from the signal present at the receiving-filter input terminal  33   a , one need only integrate over an appropriate time interval. Since each of the L i (x) have compact support, it follows that each term in the above sum, and hence the sum itself, is non-zero over a finite interval. It is this finite interval that is the appropriate time interval for integration. 
     Filter pairs having the foregoing property can be shift bi-orthogonal filter pairs for which                       ∫       R        (   x   )            L        (     x   -   n     )               x         =     {         0           if                 n     ≠   0               c   ,     c   ≠   0               if                 n     =   0                                      
     or shift-orthonormal filters for which          ∫       R        (   x   )            L        (     x   -   n     )               x         =     {         0           if                 n     ≠   0             1           if                 n     =   0                                    
     Alternatively, filter pairs suitable for use in the method and system of the invention can be bi-orthogonal filters which are defined as having the additional properties: 
     
       
         ∫ R ( x ) dx= 1  
       
     
     and 
     
       
         ∫ L ( x ) dx= 1  
       
     
     In the case in which R(x)=L(x), the transmitter and receiver filters are matched filters. 
     The signal present leaving the receiving mixer  35   a  is made available to a signal input terminal of a receiving accumulator  44   a . In addition to a signal input terminal, the receiving accumulator  44   a  includes: a receiving-accumulator output terminal  46   a  on which is placed an accumulated quantity; and an associated controller  48   a  for providing control signals to the receiving accumulator  44   a . These control signals specify when to begin accumulation, how long accumulation is to occur, and when to place the accumulated quantity on the receiving-accumulator output terminal  46   a.    
     The receiving accumulator  44   a  can act as a summer or integrator, depending on whether the communication system  10  is an analog or digital communication system. The associated controller  48   a  selects the appropriate time interval for integrating or summing the signal present at the signal input terminal. The receiving accumulator  44   a  and its associated controller  48   a  thus cooperate to place on the receiving accumulator output terminal  46   a  a signal representative of a sum or integral over a finite and selected time interval. 
     In order to recover a particular symbol, the receiving accumulator  44   a  begins the integration process when the phase term corresponding to that symbol first becomes one of the constituents of the phaseform. The integration process carried out by the receiving accumulator  44   a  continues throughout the interval during which the phase term corresponding to that symbol remains one of the constituents of the phaseform. Once the phaseform no longer has, as one of its constituents, the phase term corresponding to the particular symbol, the integration process ceases and the phase term corresponding that symbol is made available at the receiving accumulator output terminal  46   a.    
     Depending on the choice of receiving filter function and transmitting-filter function, the signal at the receiving-accumulator output terminal  46   a  may need to be scaled. This is readily accomplished by feeding that signal into a multiplier  47   a  and scaling it by a suitable constant available in a scaling register  49   a . The value stored in the scaling register  49   a  can be pre-selected or can be obtained from a look-up table. 
     With the exception of differences in the filter functions generated by the receiving-filter generator  36   a  and in the start and stop times for accumulation as provided by the receiving accumulator controller  48   a , the remaining symbol decoders  34   b-g  operate in the same manner as the first symbol decoder  34   a.    
     As shown in FIG. 10, the second symbol decoder  34   b , like the first symbol decoder  34   a , includes a receiving mixer  35   b  and a receiving-filter generator  36   b . The receiving mixer  35   b  has a phaseform terminal  33   b  connected to the phase demodulator  32 , a receiving-filter input terminal  38   b  connected to the receiving-filter generator  36   b , and a receiving-mixer output terminal  40   b.    
     In operation, the receiving-filter generator  36   b  places a different receiving-filter function R m+1 (x) at the receiving-filter input terminal  38   b  of the receiving mixer  34   b . The signal present at the output terminal is thus                    R     m   +   1            (   x   )              ∑     i   =     m   +   1         m   +   g              τ   i            L   i          (   x   )             =                    τ     m   +   1              R     m   +   1            (   x   )              L     m   +   1            (   x   )         +                                  τ     m   +   2              R     m   +   1            (   x   )              L     m   +   2            (   x   )         +   …              +                                τ     m   +   g   +   1              R     m   +   1            (   x   )              L     m   +   g            (   x   )                                      
     The receiving-filter function R m+1 (x) for the second symbol decoder  34   b  is likewise selected such that the integral of the product R m+1 (x)L n (x) is non-zero only when n=m+1. The receiving-filter function can be a time shifted version of R m (x), however any function satisfying the foregoing condition is suitable. Consequently, in order to extract the symbol τ m+1  from the signal present at the receiving-filter input terminal  33   b , one need only integrate over an appropriate time interval. Since each of the L i (x) have compact support, it follows that each term in the above sum, and hence the sum itself, is non-zero over a finite interval. It is again this finite time interval that is the appropriate interval for integration. However, the start time and stop time for this finite time interval are different from those used in the first symbol decoder  34   a.    
     As was the case in the first symbol decoder  34   a , the signal leaving the receiving mixer  35   b  is made available to a signal input terminal of a receiving accumulator  44   b . In addition to a signal input terminal, the receiving accumulator  44   b  includes: a receiving-accumulator output terminal  46   b  on which is placed an accumulated quantity; and an associated controller  48   b  for providing control signals to the receiving accumulator  44   b . These control signals specify when to begin accumulation, how long accumulation is to occur and when to place the accumulated quantity on the receiving-accumulator output terminal  46   b.    
     The receiving accumulator  44   b  can act as a summer or integrator, depending on whether the communication system  10  is an analog or digital communication system. The associated controller  48   b  selects the appropriate time interval for integrating or summing the signal present at the signal input terminal. This time interval is typically delayed by one time interval relative to the time interval used in the first symbol decoder  34   a . The receiving accumulator  44   b  and its associated controller  48   b  thus cooperate to place on the receiving accumulator output terminal  46   b  a signal representative of a sum or integral over a finite and selected time interval. 
     In order to recover the symbol τ m+1 , the receiving accumulator  44   b  begins the integration process when the phase term corresponding to τ m+1  first becomes one of the constituents of the phaseform. The integration process carried out by the receiving accumulator  44   b  continues throughout the interval during which the phase term corresponding to τ m+1  remains one of the constituents of the phaseform. Once the phaseform no longer has, as one of its constituents, the phase term corresponding to τ m+1 , the integration process ceases and the phase term corresponding τ m+1  is made available at the receiving-accumulator output terminal  46   b.    
     It will be apparent to those of ordinary skill in the art that the operation of the invention relies upon the judicious choice of filter functions and not on whether the underlying communication system is a digital or analog communication system. Hence, it is intended that the scope of the following claims cover both digital and analog communication systems.