Abstract:
A partial response QAM Modem is provided wherein a transmit signal is generated by modulating band limited, multilevel, partial response precoded data with inphase and quadrature carriers in inphase and quadrature phase channels, respectively, and summing the modulated outputs. The received signal is processed by a passband automatic complex equalizer. The equalized signal is delayed and then demodulated with recovered inphase and quadrature carriers to recover the inphase and quadrature phase baseband signals. The recovered baseband signals are sampled at appropriate instants to produce estimates of the transmitted symbols. These symbols are partial response decoded and processed by error correctors to improve the overall system performance. The error corrector outputs are decoded to convert the multilevel estimated data to binary data and the binary data of the two channels is summed to obtain estimates of the binary data transmitted.

Description:
This is a continuation of application Ser. No. 736,581, filed Oct. 28, 1976, now abandoned. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to communication systems utilizing quadrature amplitude modulation schemes and, more particularly, to an improved system using quadrature amplitude modulation in combination with partial response techniques. 
     BACKGROUND OF THE INVENTION 
     As is set forth hereinbelow, the present invention combines quadrature amplitude modulation techniques with partial response coding techniques resulting in modems which provide unusually high quality digital transmission at specific data rates. Before considering the invention itself, quadrature amplitude modulation and partial response coding techniques will be briefly considered by way of background. 
     Quadrature Amplitude Modulation (QAM) techniques have been used by data communication engineers for some time. In accordance with this general technique, the data to be transmitted is split into inphase (I) and quadrature phase (Q) channels. After suitable shaping (band-limiting), the signal in each channel is modulated with inphase and quadrature carriers, respectively. The modulated signals in the two channels are then summed to produce a band-limited QAM signal. In general, the error rate performance of a data communication using QAM techniques is less susceptible to small amounts of phase jitter in the demodulating carrier than systems using vestigial sideband (VSB) or single sideband (SSB) modulation techniques. In a high performance VSB or SSB system, a portion of the carrier and pilot tone is transmitted with the information bearing portion of the signal in order to permit recovery of the carrier. In a QAM system, all of the transmitted power is allotted to the information bearing portion of the signal, and the carrier can be recovered from the quadrature amplitude modulated signal itself (see Stiffler, &#34;Theory of Synchronous Communications&#34;, Prentice Hall, 1971). Reference is also made to, e.g., Lucky, Salz and Weldon, &#34;Principles of Data Communication&#34;, McGraw-Hill, 1968, for a further description of QAM systems. 
     Partial response techniques, wherein a controlled amount of intersymbol interference is employed in order to increase the transmission rate, are a more recent development and have been used with an AM-SSB scheme to achieve certain spectral shaping effects in data transmission. Reference is made to, e.g., Kretzmer, &#34;Generalization of a Technique for Binary Data Communication&#34; IEEE Transactions on Communication Technology, February 1966, for a description of various classes of partial response schemes and precoding techniques, such precoding being used to avoid error propagation in the decoded data. Reference is also made to U.S. Pat. No. 3,388,330 (Kretzmer). For a binary input signal, the partial response baseband signal includes at least three levels. A partial response system requires a higher signal-to-noise ratio than an ideal binary system for the same error rate. However, since the number of levels in a partial response system is always higher than the number of input levels, certain sequences of the estimated data can be eliminated as being impossible and thus a system using a three-level (ternary) partial response baseband signal can perform substantially as well as an ideal binary system. Reference is made to U.S. Pat. Nos.  3,492,578 (Gerrish et al) and 3,679,977 (Howson) for descriptions of multilevel precoded partial response data transmission systems. 
     SUMMARY OF THE INVENTION 
     According to the invention, a data communication system is provided which combines QAM and partial response techniques. The system of the invention is specifically adapted to provide satisfactory quality digital transmission through a telephone voice channel at specific data rates. In particular, an optimum combination of multilevel encoding and QAM and partial response techniques is employed to provide the high quality performance referred to, at bit rates of 8,000 bits per second (BPS) and 16,000 BPS. 
     A number of factors must be considered in attempting to transmit an 8000 BPS or 16000 BPS signal through a telephone voice channel. The telephone channel has a bandwidth of approximately 300 to 3000 Hz (2700 Hz bandwidth), has various envelope delay characteristics and is perturbed by noise, phase jitter and frequency translation. In accordance with the present invention, for an 8000 BPS signal, the incoming data is converted into a 5×5 partial response coded QAM signal having a total bandwidth (at the natural zero of the spectrum) of ± 1,333.3 Hz, (2,666.6 Hz) which fits the 2700 Hz bandwidth of the channel. For a 16000 BPs signal, an 11×11 partial response QAM system is provided which has a total bandwidth of ± 1,600.00 Hz (3,200 Hz). While this total bandwidth somewhat exceeds the channel bandwidth, there are no signal components at ± 1,600 Hz and the components at ± 1,400 Hz and ± 1,500 Hz are so low in level that loss thereof is allowable. 
     In accordance with a preferred embodiment of the invention, the incoming serial binary data is split into inphase and quadrature phase channels, after being scrambled with a pseudo-random sequence. The binary data in each channel is converted to a multilevel signal thereby increasing the data rate per unit bandwidth. For example, for the 8000 BPS system referred to above, the binary data is converted to three levels while for the 16000 BPS system, sets of 5 bits are converted into pairs of six-level signals. The multilevel encoded data is partial response precoded and, using a shaping filter, converted to band-limited analog waveforms in each channel. The outputs of the shaping filters are multiplied (modulated) by inphase and quadrature phase carriers and the resultant outputs are summed to produce a band-limited partial response QAM signal. 
     At the receiver, the received partial response QAM signal is processed by a passband automatic complex equalizer to compensate for the amplitude and delay characteristics of the line. The inphase and quadrature phase carriers are recovered and applied to first and second demodulators along with delayed outputs from the equalizer to recover the inphase and quadrature phase baseband signals. These latter signals are sampled at appropriate instants to generate estimated symbols as well as error signals which are fed back to the passband automatic complex equalizer to provide appropriate adjustment of the tap voltages (coefficients). The estimated symbols in each channel are partial response decoded and corrected to generate multilevel estimated data. Multilevel decoders (e.g., the three-level to two-level decoders) convert the multilevel data into a sequence of estimated binary data. The sequences from each channel are combined and pseudo-random descrambling is used to recover the original transmitted signal. 
     Other features and advantages of the invention will be set forth in, or apparent from, the detailed description of a preferred embodiment found hereinbelow. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block circuit diagram of a partial response QAM transmitter in accordance with a preferred embodiment of the invention; 
     FIG. 2 is a block circuit diagram of a partial response QAM receiver in accordance with a preferred embodiment of the invention; 
     FIG. 3 is a signal space diagram for a 5×5 partial response QAM system; 
     FIG. 4 is a signal space diagram for an 11×11 partial response QAM system; 
     FIG. 5 is a representative diagram of the waveform at the output of the two-level to three-level encoders of FIG. 1; 
     FIG. 6 is a diagram of the waveform at the output of the partial response coders of FIG. 1; 
     FIG. 7a, 7b, and 7c are diagrams providing an example of two-level to three-level class I partial response encoding; 
     FIGS. 8a and 8b are diagrams providing an example of three-level to five-level class I partial response encoding; and 
     FIG. 9 is a diagram of the filtered waveform of the five-level eye pattern of a class I partial response system at the output of the low pass filters of FIG. 1. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring to FIG. 1, a block diagram of the transmitter of the partial response QAM system of the invention is shown. Serial binary data at a rate R bits per second (BPS) is applied to scrambler 12. The scrambling is achieved with a pseudo-random sequence with a long pattern. The scrambler 12 thus converts the serial binary data into a totally random distate data. This random data is fed serially into a pair of shift registers 16 and 18. Shift registers 16 and 18 are clocked at a rate of R Hz per second, as controlled from a clock input line 20. Thus, the random data is shifted R times per second. The output of registers 16 and 18 are connected to the inputs of a pair of registers 22 and 24, respectively. Registers 22 and 24 are clocked at a frequency of R/2 Hz per second, obtained by dividing the output of clock line 20 by two. The outputs of registers 22 and 24 are therefore parallel pairs of data bits conventionally termed &#34;dibits&#34;. Hence, by the arrangement described, serial data bits are converted to parallel dibits, and two channels of dibit streams are obtained at the outputs of registers 22 and 24. These outputs form the inputs to the inphase (I) channel and quadrature (Q) channel. The outputs of registers 22 and 24 are connected to first and second two-level to three-level encoders 30 and 28, respectively. 
     Encoders 28 and 30 serve to code the two streams of random dibits into two streams of three-level tribits. Details of a suitable two-level to three-level encoder are described in U.S. Pat. No. 3,883,727 (Stuart et al). The outputs of the encoders 28 and 30 are three-level semi-square eye patterns, as shown in FIG. 5. These outputs are connected to first and second partial response coders 32 and 34. 
     The coders 32 and 34 are class I partial response type coders. Such coders provide both precoding and encoding. Parital response coding or signaling, which is also referred to as correlative level coding and duobinary coding, imposes certain restraints on the three-level tribits in changing from one level to another. In accomplishing this, the regenerated streams of tribits data are encoded into five level streams of tribits as shown in FIG. 6. This results is retaining the same tribit capacity per channel while at the same time compressing the baseband frequency spectrum to one half of that prior to partial response coding. In class I partial response coding the transition from one extreme level to another extreme level in two successive tribits or keying intervals is prohibited, the keying interval being defined as the shortest real time interval between two successive level changes. Therefore, the Nyquist frequency in the frequency spectrum at the output of the two coders 32, 34 is one half of that prior to partial response coding. Consequently, the first natural null of the baseband spectrum in displaced by the same proportion, i.e., to the Nyquist frequency of the input frequency spectrum prior to partial response coding. 
     In class I partial response coding, the output will, in general, have 2L-1 levels, where L is the number of levels of the signal at the input of the coder. Before considering generalized L level partial response coding, two-level to three-level partial response coding will be described to aid in explaining, in real time, the process of two-to-one bandwidth compression. Thus, referring to the two-level binary random pattern shown in FIG. 7(a), this sequence of two-level bits is to be transformed to a new sequence of three-level partial response class I signalling. This process is implemented by time delaying the sequence by one bit interval, i.e., 1/R seconds where R is the bit rate in bits per seconds, and algebraically adding the delayed output to the input as shown in Table 1 below. 
     
                       TABLE 1______________________________________a.sub.i     a.sub.i-1     a.sub.i + a.sub.i-1______________________________________+1/2        --            ---1/2        +1/2          0+1/2        -1/2          0+1/2        +1/2          +1-1/2        +1/2          0-1/2        -1/2          -1+1/2        -1/2          0-1/2        +1/2          0-1/2        -1/2          -1-1/2        -1/2          -1--          -1/2          --;______________________________________ 
    
     In Table 1, the first column represents the pattern shown in FIG. 7(a), but referenced so as to be bipolar. The second column represents the same pattern as the first column but delayed by one bit. The third column represents the algebraic sum of the first and second columns. The time function corresponding to the third column is shown in FIG. 7(c). As illustrated in FIG. 7(c), there is no transition between one extreme level to another extreme level. 
     Turning now to the precoding process, an error in the partial response encoded signal at the receiver, once made, will tend to propagate throughout the rest of the data bits. This is due to the fact that if the bit a i-1  is in error then the bit a 1  is also likely to be in error. To avoid this, some kind of compensation at the receiver or some technique of precoding at the transmitter is necessary. The latter is preferred from the standpoint of simplicity and ease of implementation. 
     For an n level signal the precoding rule is 
     
         bi = a.sub.i -.sub.n b.sub.i-1 
    
     where a i  is the input to the precoder; b i  is the output of the precoder; and b i-1  is the output of the precoder as delayed by one baud, (a baud being a bit, dibit, tribit and so on, depending on the number of level n). 
     The symbol - n  denotes modulo-n subtracted and indicates that a i  and b i-1  are added algebrically and the result, considering the sign, is interpreted using the rules of modulo-n arithmetic. This means that a newly precoded sequence of bauds b i  is selected such that the modulo sum of a queue or series of i bauds, ending with the ith baud, is equal to the ith bit of the original sequence of bauds a i . 
     Considering the general rules for N modulo-n arithmetic, for positive N, the modulo-n number is the remainder of N/n. For negative N, the modulo-n number is the alegrabic sum N+n, carried to the lowest negative number, N being any number. As an example, for n = 3, and N arbitrarily selected to be ± 6, the rule operates as shown in Table 2 below. 
     
                       TABLE 2______________________________________N          modulo-3______________________________________6          05          24          13          02          21          10          0-1         2-2         1-3         0-4         2               (-1=2)-5         1               (-2=1)-6         0               (-3=0)______________________________________ 
    
     Thus, considering a three-level random sequence of bauds, the results of the modulo-3 subtraction b i  =a i  - 3  b i-1  is listed in Table 3 below. 
     
                       TABLE 3______________________________________b.sub.i = a.sub.i ⊖ .sub.3 b.sub.i-1a.sub.i  b.sub.i-1          b.sub.i______________________________________0      (0)0      0         ←   01      0         ←   02      1         ←   10      1         ←   11      2         ←   21      2         ←   22      2         ←   2 = 1-2 = -1 = 20      0         ←   02      0         ←   00      2         ←   21      1         ←   10      0         ←   01      0         ←   02      1         ←   11      1         ←   12      0         ←   0                     2.______________________________________ 
    
     The first level under b i-1  is arbitrarily selected as zero. Then b i  is calculated and is then delayed by one baud to obtain the second b i-1  from which the second b i  is obtained. The process is repeated for the rest of the pattern. The third column headed b i , represents the new precoded sequence of bauds. This sequence is still a three-level sequence of bauds and the waveform still looks like that of FIG. 5. The significance of the precoding provided will be evident from the discussions hereinbelow with respect to the decoding, or &#34;undoing&#34;, of the precoding at the receiver. 
     Considering the encoding (in contrast to the precoding) process, the precoded sequence of bauds is encoded into a class I partial response signal before modulation. The encoding rule can be represented by the equation: 
     
         c.sub.i = b.sub.i + b.sub.i-1. 
    
     In encoding, the precoded sequence. b i  is applied to a delay line providing one baud of real time delay. The output and the input of the delay line are added algebraically to obtain a multilevel class I partial response signal. In this class, a three-level signal will be encoded into a five-level signal and by virtue of the process, the spectrum bandwidth is reduced to one half of that of the three-level spectrum. 
     As an aid to understanding this process, consider the exemplary three-level random sequence of bauds listed in the first column in Table 4 below. The waveform corresponding to this sequence is shown in FIG. 8(a). The sequence is delayed by one baud which corresponds to shifting the first column by one baud to obtain the second, b i-1  column, and the two columns are added algebraically. 
     
                       TABLE 4______________________________________c.sub.i = b.sub.i + b.sub.i-1b.sub.i     b.sub.i-1      c.sub.i______________________________________-1          --             --+1          -1              0-1          +1              0 0          -1             -1 0           0              0+1           0             +1 0          +1             +1-1           0             -1 0          -1             -1 0           0              0+1           0             +1+1          +1             +2-1          +1              0-1          -1             -2 0          -1             -1--           0             --;______________________________________ 
    
     The resultant waveform of c i  is shown in FIG. 8(b) and is a five-level signal. Note that there are no transitions from one extreme level to the other extreme level in two consecutive transitions. The eye pattern at this stage will look like that shown in FIG. 6. 
     Partial response multilevel data systems are discussed in patent and non-patent literature and, for example, reference is made to U.S. Pat. No. 3,388,330 (Kretzmer) for a further discussion of such systems. 
     Referring again to FIG. 1, the five-level signals at the output of 32 and 34 are applied to the low pass shaping filters 36 and 3 respectively. Shaping filters reject higher harmonics of the semi-square eye pattern, shown in FIG. 6. The output of each shaping filter 36, 38 is a smooth eye pattern like that shown in FIG. 9. The shaping filters 36, 38 provide a cutoff frequency of R/6 Hz. 
     The outputs of filters 36 and 38 are connected to modulators 40 and 42, respectively. Modulator 40 is modulated with the inphase carrier cosw c  t, while modulator 42 is modulated with quadrature carrier sinw c  t. The output of the two modulators are added algebraically by the summer 44 to produce a band limited partial response QAM line signal which constitutes the output of the transmitter. For systems employing two-level to three-level encoders, such as described above, the line signal is 5×5 constellation and can be represented by the signal space diagram shown in FIG. 3. For systems employing two-level to six-level encoders, the line signal is 11×11 constellation and can be represented by the signal space diagram shown in FIG. 4. 
     Referring to FIG. 2, a block diagram of the receiver is shown. The received partial response QAM signal is applied to a passband conjugate complex automatic equalizer 50. A detailed description of this type of equalizer can be found in Gitlin et al, &#34;Passband Equalization of Differentially Phase-Modulated Data Signals&#34;, Bell System Technical Journal, (February, 1973). 
     The output of the equalizer 50 is connected to a carrier recovery circuit 52 which is preferably of the type disclosed in A. Bhopale, U.S. Pat. No. 3,984,778 entitled &#34;Carrier Recovery Scheme for a SSB-SC Signal&#34;. 
     The output of the equalizer 50 is connected also to a pair of identical time delay networks 54 and 56. The outputs of the delay networks 54 and 56 are applied to amplitude modulators or multipliers 58 and 60, respectively. The output of the carrier recovery circuit 52, i.e., the recovered inphase carrier, is connected to modulator 58, the inphase carrier being used in demodulating the received line signal. The output of the carrier recovery circuit 52 is also connected to a 90° phase shift network 62. The output of phase shift network 62, i.e., the recovered quadrature carrier is connected to modulator 60, the recovered quadrature carrier being used in demodulating the line signal. The delay networks 54 and 56 are designed to introduce a real time delay into the equalizer output equal to that of the real time used in recovering and processing the carrier by carrier recovery circuit 52. Thus, the recovered inphase carrier cos W c  t, and the delayed line signal at the output of 54, are in real time synchronism. Further, the 90° phase shifted carrier sin W c  t, at the output of phase shifter 62, is in phase quadrature with the carrier of the received and delayed line signal. i.e., the signal at the output of delay network 56. 
     The outputs of the demodulators 58 and 60 are passed through post detection low pass filters 64 and 66 to select the baseband spectrum. The cutoff frequency of the post detection filters is R/6 Hz, which is the Nyquist frequency of the two-to-three level precoded and the three-to-five level partial response encoded signal. The outputs of the post detection filters 64 and 66 are smooth five-level eye patterns of the form shown in FIG. 9. 
     The two recovered five-level eye patterns in both channels are sampled by the decision and error generator circuits 68 and 70. The sampling clock, or timing information, is obtained from clock recovery circuit 72. The clock recovery circuit 72 obtains the timing information from one of the recovered five-level eye patterns. A method of doing this is disclosed in U.S. Pat. No. 3,746,800, issued on July 17, 1973. Error signals are obtained by comparing the levels of the recovered five-level eye patterns with five preset d.c. voltage levels. The two error signals are used to adjust the tap coefficients of the automatic equalizer (not shown) to provide convergence to a point where a minimum error signal and the best eye opening is achieved. The outputs of the decision circuits 68 and 70 are the recovered and equalized eye patterns, and are the five-level partial response encoded and three-level precoded data. These outputs are applied to partial response decoders and error corrector circuits 74 and 76, respectively. 
     Considering the decoding process, decoding is simply the reverse of encoding. The decoding process is used to convert the five-level eye patterns into three-level eye patterns, i.e., to obtain a three-level precoded signal from the five-level partial response encoded signal. Encoding being an algebraic process, decoding is also an algebraic process. The decoding rule employed is b i  = C i  - b i-1 . 
     Refer now to, as an example, the precoded and encoded sequence of data shown in Table 5. 
     
                       TABLE 5______________________________________b.sub.i              C.sub.ia.sub.i= a.sub.i ⊖ 3 b.sub.i-1             b.sub.i-1                     = b.sub.i + b.sub.i-1 b.sub.i______________________________________0                    00    0            0       00 - 0 = 0 0 1 2 0 0 1 1 2      0 0 1 1                      ##STR1##1    2            2       44 - 2 = 21    2            2       44 - 2 =  22    0            2       22 - 2 = 00    0            0       00 - 0 = 02    2            0       22 - 0 = 20    1            2       33 - 2 = 11    0            1       11 - 1 = 00    0            0       00 - 0 = 01    1            0       11 - 0 = 12    1            1       22 - 1 = 11    0            1       11 - 1 = 02    2            0       22 - 0 = 2______________________________________ 
    
     The first column in Table 5 is the original sequence of three-level of tribits a i  copied from the first column in Table 3 above. The second column in Table 5 is also a three-level sequence of tribits b i , obtained by precoding a i  according to the modulo-3 subtraction rule, which is copied from the second column in Table 3. The third column is the second column delayed by one baud. The fourth column is the partial response three-level to five-level encoding of precoded sequence of tribits b i . The fifth column is the decoded sequence of precoded tribits obtained from the fourth column according to the decoding rule set forth above. The first value of b i  is assumed arbitrarily to be zero. This value is subtracted from c i  to obtain the second b i , and the process is repeated to obtain the rest of the precoded tribits b i . At this point the three-level precoded sequence of tribits b i  is recovered. It should be noted that the columns headed a i , b i  and b i-1  are not needed for decoding process and are listed merely for reference and comparison. 
     Considering the postcoding process, postcoding is simply the reverse of precoding and is used to convert the precoded sequence of three-level tribits to the original sequence of tribits. The postcoding rule employed is: a i  = b i  +  n  b i-l , where the + n  symbol denotes modulo-n summation, b i  is the precoded sequence of bauds, b i-1  is b i  sequence of bauds delayed by one baud time interval, and a i  is the original sequence of data bauds. For the three-level precoded data sequence of bauds consider the sequence listed in the fifth column in Table 5. This sequence is copied in the first column in Table 6. The second column in Table 6 is the b i  sequence of the first column as delayed by one tribit. The third column is the modulo-3 sum of the first and second columns. The symbol = 3 indicates modulo-3 rule restraints. The third column of Table 6 is the original tribit sequence at the transmitter shown in the first column in Table 5 and Table 3 above. 
     
                       TABLE 6______________________________________                  a.sub.ib.sub.i   b.sub.i-1    = b.sub.i ⊕ 3 b.sub.i-1______________________________________0         0            0 + 0 = 0 = 00         0            0 + 0 = 0 = 01         0            1 + 0 = 1 = 11         1            1 + 1 = 2 = 22         1            2 + 1 = 3 = 02         2            2 + 2 = 4 = 12         2            2 + 2 = 4 = 10         2            0 + 2 = 2 = 20         0            0 + 0 = 0 = 02         0            2 + 0 = 2 = 21         2            1 + 2 = 3 = 00         1            0 + 1 = 1 = 10         0            0 + 0 = 0 = 01         0            1 + 0 = 1 = 11         1            1 + 1 = 2 = 20         1            0 + 1 = 1 = 12         0            2 + 0 = 2 = 2     2______________________________________ 
    
     The outputs of circuits 74 and 76 are connected to first and second 3-level to 2-level decoders 78 and 80, respectively. The corrected outputs of circuits 74 and 76 constitute the three-level estimated data and this data is read in pairs of tribits and converted into three serial bits in each channel using decoders 78 and 80. The outputs of decoders 78 and 80 are combined and converted into serial estimates of the transmitted data, by a parallel-to-serial converter 90. 
     The output of converter 90 is descrambled using a descrambler 92 which receives, as the other input thereto, the descrambling PN sequence, thereby providing recovery of the originally transmitter serial binary data. 
     The system described hereinabove is particularly adapted for use in transmitting an 8000 BPS through a telephone voice channel. The 8000 BPS signal is divided by two-thirds in the two-level to three-level encoding and the data stream is split using the QAM techniques to provide a partial response encoded signal having a bit rate of 2666.6 BPS and a Nyquist Frequency of 1333.3 Hz so that, as stated hereinbefore, the total bandwidth at the natural zeros of the spectrum is ± 1,333.3 Hz or 2,666.6 Hz. The space signal diagram, showing a plurality of points each representing the resultant vector sum of first and second modulated waves in quadrature, for the five by five (5×5) partial response QAM signal is shown in FIG. 3. The 25-state system makes efficient use of transmitted power and the signal power requirements are only slightly inferior to prior art 16-state systems. 
     As also was discussed, where it is desired to transmit 16000 BPS through a telephone voice channel, the binary signals are converted, in groups of five serial bits, to six-state characters and alternate six-level states are partial response encoded to provide two 11-level signals. The 11-level signals are applied to respective modulators which receive inphase and quadrature carriers and whose outputs are summed to produce a QAM signal in the manner described hereinabove. Demodulation of the received signal is also accomplished in a manner similar to that already described in connection with FIG. 2. The partial response precoding provides for multiplication of the Nyquist frequency by two-fifth and the two channel splitting provided by QAM modulation results in two signals having a baud rate of 3200 BPS and a Nyquist frequency of 1600 Hz. Thus, as stated, the total bandwidth at the natural zeros of the spectrum is ± 1600 Hz (or 3200 Hz). As explained previously, while this bandwidth is somewhat in excess of the channel bandwidth there are no signal components at ± 1600 Hz and those at ± 1500 Hz and 1400 Hz are of such a low level that loss thereof is without consequence. The signal space diagram for the 11×11 partial response QAM system is, as noted, shown in FIG. 4. More generally, this technique, i.e., encoding sets of five bits into pairs of six-level signals and coding these signals using Class I partial response techniques, can be used to reduce the bandwidth requirements for data transmission at a given rate of the preferred response QAM system. 
     Although the invention has been described relative to examplary embodiments thereof, it will be understood that other variations and modifications can be effected in these embodiments without departing from the scope and spirit of the invention.