CA1202726A - Data transmission systems - Google Patents
Data transmission systemsInfo
- Publication number
- CA1202726A CA1202726A CA000388556A CA388556A CA1202726A CA 1202726 A CA1202726 A CA 1202726A CA 000388556 A CA000388556 A CA 000388556A CA 388556 A CA388556 A CA 388556A CA 1202726 A CA1202726 A CA 1202726A
- Authority
- CA
- Canada
- Prior art keywords
- vectors
- signal
- expanded
- expanding
- data signal
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired
Links
Classifications
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/38—Synchronous or start-stop systems, e.g. for Baudot code
- H04L25/40—Transmitting circuits; Receiving circuits
- H04L25/49—Transmitting circuits; Receiving circuits using code conversion at the transmitter; using predistortion; using insertion of idle bits for obtaining a desired frequency spectrum; using three or more amplitude levels ; Baseband coding techniques specific to data transmission systems
- H04L25/4917—Transmitting circuits; Receiving circuits using code conversion at the transmitter; using predistortion; using insertion of idle bits for obtaining a desired frequency spectrum; using three or more amplitude levels ; Baseband coding techniques specific to data transmission systems using multilevel codes
-
- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/37—Decoding methods or techniques, not specific to the particular type of coding provided for in groups H03M13/03 - H03M13/35
- H03M13/39—Sequence estimation, i.e. using statistical methods for the reconstruction of the original codes
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03006—Arrangements for removing intersymbol interference
- H04L25/03178—Arrangements involving sequence estimation techniques
- H04L25/03203—Trellis search techniques
- H04L25/03229—Trellis search techniques with state-reduction using grouping of states
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03006—Arrangements for removing intersymbol interference
- H04L25/03178—Arrangements involving sequence estimation techniques
- H04L25/03248—Arrangements for operating in conjunction with other apparatus
- H04L25/03299—Arrangements for operating in conjunction with other apparatus with noise-whitening circuitry
Abstract
ABSTRACT
Method and Apparatus for decoding received data signals using Viterbi algorithm decoding in which to reduce the computational time an expansion technique is involved in which the m level signal is expanded into nk expanded vectors, where n is smaller than m, and in which k vectors are chosen from the nk expanded vectors for selection of the level of the received signal.
Method and Apparatus for decoding received data signals using Viterbi algorithm decoding in which to reduce the computational time an expansion technique is involved in which the m level signal is expanded into nk expanded vectors, where n is smaller than m, and in which k vectors are chosen from the nk expanded vectors for selection of the level of the received signal.
Description
IMPROVEMENTS IN OR RELATING TO DATA TRANSMISSION SYSTEMS
The present ;nvent;on relates to Data Transmission Systems and more part;cularly to modems for such systemsn The invention is concerned ~ith modems which utilise the Viterbi-alqorithm tForney, GaDo The Viterbi-algorithm, Proc.IEEE, Vol 61, pp268-278, March 1973) as a method of decoding the data signals.
The object of the present invention is to reduce the complexity of a reduced-state Viterbi-algorithm detector operating on a multi level signal, without significantly degrad;ng its performance, and thereby makes ;t possible to ach;eve near maximum-l;kelihood detection at a cost not very much greater than that involved ;n a convent;onal equalizer. The potent;al advantage in tolerance to additive noise, likely to be gained by the system, is up to about 6 dB~
The basic technique of the invention is to double expand either sequentially or simultaneously the incom;n~ s;gnal thus reduc;ng the complexity of the detector.
More part;cularly the ;nvent;ve feature ;nvolves the method that ;s used to select k stored vectors from mk expanded vectors twhere m îs the number of levels of the data s;gnal) ;n the algorithm of a reduced-state V;terb;-algor;thm detector.
~'`' .
0~ 6 On -the receipt of a received sample, each stored vec-tor is here'bxpanded" into m vectors and the "cost" associa-ted wi-th each of these is evaluated.
From -the mk expanded vectors k vec-tors are now selected according -to some criterion. When m and k are both large, mk becomes very large and an excessive number of operations is required per received sample. The basic fea-ture of the invention is to break the expansion process down into two separate operations, which may be performed sequentially or simultaneously and each of which involves expanding the k stored vectors in-to nk vectors (where n~<m~
from which are then selected k s-tored vectors.
When the two expansion processes are performed sequentially, n-~m~ and k vectors are selected after each expansion. When the two expansion processes are performed simul-taneously, n~m~ and the two se-ts of nk expanded vec-tors give an effective total of n2k vectors from which are selected k stored vectors. Use is made of the fact tha-t the cost of any one of -the effec-tive -total of n2k vec-tors is very simply derived from the cos-ts of -the two individual vec-tors, one in each of -the two se-ts of nk expanded vectors and togethergiving the resultant vec-tor7 so that the amount of computa-tion involved corresponds to 2nk expanded vectors rather than n2K vectors.
:~LZ~Z7~
The immedia-te appllca-tion o~ the invention is in a 9600 bits-per-second modem for use o-ver -the public switched telephone ne-twork. 0-ther applications include digital data--transmission systems in which multilevel signals, at ra-tes of up to 100,000 bi-ts-per-second, are sent over channels introducing severe time dispersion of -the transmit-ted signal-elements, an~ where i-t is required to achieve a good tolerance to additive noise.
~mbodiments of -the presen~t i~ention will now be described with reference -to -the accompanying drawings in which:-Figure 1 shows a Data--transmission system according to -the present invention, Figure 2 shows a block diagrammatic circuit for the adaptive adjustmen-t of linear filter and es-timator, Figure 3 shows a) a-ttenuation charac-teristic and b) group-delay charac-teristic of telephone-circuit 1.
Figure ~ shows a) attenuation characteristic and b) Group-delay characteristic of telephone-circuit 2~
Figure 5 shows a) attentua-tion characteristic and b) Group-delay characteristic of telephone-circuit 3O
~Z~ 6 Figure 6 shows a) a-ttenua-tion characteris-tic and b) Group-delay characteristic o~
telephone-circui~t 4.
Figure 7 shows a) attenuation characteristic and b) Group-delay characteris-tic of the combination of transmitter and receiver filters.
Figure 8 shows varia-tion o~ error rate with signal/noise ratio for the different systems opera-ting over the telephone-circuit 1.
Figure 9 shows variation of error rate with signal/noise ratio for the di~fere.nt systems operating over the telephone-circuit 2.
Figure 10 shows variation of error rate with signal/
noise ratio for the different sys-tems operating over the telephone-circuit 3.
Figure 11 shows variation of error rate with signal/noise ratio for the different systems operating over the -telephone-circuit 4.
Figure 12 shows ~ariation of error rate wi-th an inaccuracy in the estimate made by the receiver o~
a) -the level and b) the carrier phase of the received signal, for the System E with m = 6 and n = 7 opera-ting over the telephone-circuits 1-4.
~Z~27~;
1. INTRODUCTION
A telephone c;rcu;t ;n the sw;tched telephone network ~ay have any one of a very w;de range of attenuat;on-frequency, characteristics and any one of a correspondingly wide range of group-delay frequency characteristics, giving an available (or usable) bandwidth that may lie anywhere in the range about 1500 Hz to 3000 Hz.1 It is evident therefore that for the satisfactory transmission of data at a rate as high as 9600 bits/second over the switched telephone network, the receiver must be adaptive in the sense that it takes full account of the distortion introduced into the received data signal7 The conventional approach to this problem is to use an adaptive nonlinear (decision-feedback) equalizer that is adjusted to minimize the mean-square error in the equalized signala2~5 Such systems are known to operate satisfactorily over telephone circuits at rates of up to 4800 bits/second. At higher transmission rates~ however, satisfactory operation is not always obta;ned over the poorer telephone circu;ts.
An alternative approach that has recently been considered is the use of the Yiterbi~algorithm detector.5~7 Th;s selects as the detected message the possible sequence of transmitted data-symbols tsignal-element values) for which there is min;mum mean-square d;fference between the samples of the corresponding ~, ~L2~Z726 received data signal, for the g;ven s;gnal d;stortion but in the absence of no;se, and the samples of the s;gnal actually received~ When the data signal ;s received in the presence of stationary additive white Gauss;an noise, giving statistically independent Gaussian noise samples at the detector input, th;s is a process of maximum-likelihood detection, and when the transmitted data-symbols are statistically independent and equally likely to have any of their possible values~ the detection process min;m;zes the probability of error in the detection of the received message.5 It is assumed here that the received signal is sampled at the Nyquist rate, so that the corresponding samples contain all the information in this signal~
Unfortunately, when the sampled impulse-response of the channel conta;ns a large number of components ~non-zero samples), which ;s the case ;n the part;cular appl;cat;on cons;dered herer the Viterbi-algor;thm involves both an excessive amount of storage and an excess;ve number of operations per received data symbol. One approach for overcoming th;s difficulty is to use a linear feedforward transversal filter at the detector input to reduce the number of components in the channel sampled~impulse-response. The f;lter is adjusted adaptively to give a "desired" sampled impulse-response for the channel and filter, which has a given small number of components and Z7~i may or may not be f;xed.8~11 rhe disadvantage of th;s arrangement is that, for some telephone circuits, the linear filter may equalize some of the ampl;tude distortion ;ntroduced by the telephone circu;t and under these cond;t;ons max;mum-l;kel;hood detect;on ;s no longer ach;eved, lead;ng to an ;nfer;or performance~5 The l;near filter should ;deally perform the function of a "whitened matched-filter",6~14 which ensured that true maximum-likel;hood detec~ion ;s achieved by the V;terb;-algor;thm detector. In the ;deal arrangement and w;th stat;onary add;t;ve wh;te Gauss;an no;se at the receiver ;nput, the data signal ;s transm;tted over the channel at the Nyquist rate and the rece;ver filter has the same bandw;dth as the rece;ved signal, being such that, w;th the sampl;ng of the signal at the output of the rece;ver filter~ once per data sym~ol (at the Wyqu;st rate), the no;se samples at the output of the sampler are stat;st;cally ;ndependent Gaussian random variables w;th a f;xed var;ance. The wh;tened matched-f;lter now becomes the same as the l;near feedforward transversed f;lter that forms the first part of a convent;onal nonl;near tdec;s;on-feedback~ equalizer, where this is fed with the given sampled s;gnal and ;s adjusted to m;nimize the mean-square error tand hence max;m;ze the s;gnal/no;se ratio) in its output signal, subject to the exact equal;zation of ~he 272~i channel.5 The advantage of this is that the linear filter may be held adaptively adjusted for the channel to g;ve approximately the required response, at least at high signal/noise ratios, using basically the same simple S arrangement as that involved in a conventional adaptive nonl;near equal1Yer.5 The signal at the output of the l;near f;lter ;s fed ~o the maximum-l;kel;hood detector, and the feedback f;lter gives an estimate of the sampled impulse-response of the channel and linear filter, which is required by the detector. The details of the arrangement will be described later. When correctly adjusted, the linear filter is such that, ;n the z transform of the sampled impulse-response of the channel and l;near filter, all zeros (roots) of the z transform of the sampled impulse- response of the çhannel, that lie outside the unit circle in the z plane tthat is, have an absolute value greater than unity~, are replaced by the complex conjugates of their rec;procals, all rema;n;ng zeros being left unchanged.5 Thus all zeros of the z transform of the channel and linear filter is ;ns;de or on the un;t c;rcle.
The linear f;lter ;s, in fact, a pure phase equalizer (although not usually for the channel itself~ and so performs an orthogonal transformation on the rece;ved signal~5 In practice~ the linear f;lter may introduce some gain or attenuationf wh;ch however has the same effect ~L2~;27~6 _9_ on the noise as on the da~a s;gnal, and so does not change in any way the statistical relationship between the no;se and data signa~. This means that a Viterbi-algorithm detector operating at the output of the fiLter has the same tolerance to the Gaussian noise as the appropr;ate Viterbi-algorithm detector operating at its input, although generally involving much less complex equipment~
Furthermore, when an allowance has been made for the change in Level, the noise component at the filter output have the same statistical properties as those at its input~ being there~ore statistically ;ndependent and w;th a ~ixed variance. Since the linear filter does not no~ remove any of the amplitude distortion introduced by the channel~ it unfortunately does not always achieve an adequate reduction in the number of components of the resultant sampled impulse-response~
It follows that the Viterbi-algorithm detector must be replaced here by a near-maximum-likelihood detector that uses very much less storage and requires far fewer operations per rece;ved data-symbol, for the given number of components ;n the sampled ;mpulse-response of the channel. Various systems of this type have been developed and tested by computer s;mulation~ with very prom;s;ng results.5,12,13 An alternat;ve approach that has recently been studied il;~f3 ~726 -10~
;s to d;scard t.he l;near f;lter altogether, so that the near-max;mum-l;kelihood detector operates d;rectly on the original sampled signal, at the output of the samplerD14 An estimator is, of course, required here to determine the sampled impulse-response of the channel, but this can be implemented relatively simply, requiring a transversal filter w;th only as many taps as there are components ;n the sampled ;mpulse-response of the channel.15,16 However, somputer-simulation tests (whose results have not been published) have shown that ;n - the presence of very severe phase d;stort;on~ such as can occur over the poorer switched telephone c;rcu;ts, these systems require a considerable amount of storage and a large number of operat;ons per rece;ved data-symbol, in order to ach;eve near-maximum-likelihood detection. The most cost-effective arrangement for the given application is therefore to use an adaptive linear filter of the type previously described~ connected between the sampler and detector and acting as a whitened matched f;lter. The adaptive filter has the important property that it (at least approximately) removes all the phase distortion introduced by the channel and ;n addit;on adjusts the sampled impulse-response of the channel and filter into a form that is ;deally suited to a near-maximum-likelihood detector of the type cons;dered here.5~13 ~2~Z7216 The paper descr;bes the appl;cat;on of the methods invoLved in some recently deYeloped detection processes to the design of a QAM tquadrature amplitude modulated) system operating at 9600 b;ts/second with a 16-point QAM signal.
Details are g;ven here of some novel detection processes that avoid the high complexity involved with a 16-point s;gnal in the prev;ously descr;bed processes. Results of computer-s;mulation tests are then presented~ comparing the performances of the various systems, ~hen operating over different telephone circu;ts, and showing also the effects of inaccuracies in the estimates of the level and carrier phase of the received signalO
The present ;nvent;on relates to Data Transmission Systems and more part;cularly to modems for such systemsn The invention is concerned ~ith modems which utilise the Viterbi-alqorithm tForney, GaDo The Viterbi-algorithm, Proc.IEEE, Vol 61, pp268-278, March 1973) as a method of decoding the data signals.
The object of the present invention is to reduce the complexity of a reduced-state Viterbi-algorithm detector operating on a multi level signal, without significantly degrad;ng its performance, and thereby makes ;t possible to ach;eve near maximum-l;kelihood detection at a cost not very much greater than that involved ;n a convent;onal equalizer. The potent;al advantage in tolerance to additive noise, likely to be gained by the system, is up to about 6 dB~
The basic technique of the invention is to double expand either sequentially or simultaneously the incom;n~ s;gnal thus reduc;ng the complexity of the detector.
More part;cularly the ;nvent;ve feature ;nvolves the method that ;s used to select k stored vectors from mk expanded vectors twhere m îs the number of levels of the data s;gnal) ;n the algorithm of a reduced-state V;terb;-algor;thm detector.
~'`' .
0~ 6 On -the receipt of a received sample, each stored vec-tor is here'bxpanded" into m vectors and the "cost" associa-ted wi-th each of these is evaluated.
From -the mk expanded vectors k vec-tors are now selected according -to some criterion. When m and k are both large, mk becomes very large and an excessive number of operations is required per received sample. The basic fea-ture of the invention is to break the expansion process down into two separate operations, which may be performed sequentially or simultaneously and each of which involves expanding the k stored vectors in-to nk vectors (where n~<m~
from which are then selected k s-tored vectors.
When the two expansion processes are performed sequentially, n-~m~ and k vectors are selected after each expansion. When the two expansion processes are performed simul-taneously, n~m~ and the two se-ts of nk expanded vec-tors give an effective total of n2k vectors from which are selected k stored vectors. Use is made of the fact tha-t the cost of any one of -the effec-tive -total of n2k vec-tors is very simply derived from the cos-ts of -the two individual vec-tors, one in each of -the two se-ts of nk expanded vectors and togethergiving the resultant vec-tor7 so that the amount of computa-tion involved corresponds to 2nk expanded vectors rather than n2K vectors.
:~LZ~Z7~
The immedia-te appllca-tion o~ the invention is in a 9600 bits-per-second modem for use o-ver -the public switched telephone ne-twork. 0-ther applications include digital data--transmission systems in which multilevel signals, at ra-tes of up to 100,000 bi-ts-per-second, are sent over channels introducing severe time dispersion of -the transmit-ted signal-elements, an~ where i-t is required to achieve a good tolerance to additive noise.
~mbodiments of -the presen~t i~ention will now be described with reference -to -the accompanying drawings in which:-Figure 1 shows a Data--transmission system according to -the present invention, Figure 2 shows a block diagrammatic circuit for the adaptive adjustmen-t of linear filter and es-timator, Figure 3 shows a) a-ttenuation charac-teristic and b) group-delay charac-teristic of telephone-circuit 1.
Figure ~ shows a) attenuation characteristic and b) Group-delay characteristic of telephone-circuit 2~
Figure 5 shows a) attentua-tion characteristic and b) Group-delay characteristic of telephone-circuit 3O
~Z~ 6 Figure 6 shows a) a-ttenua-tion characteris-tic and b) Group-delay characteristic o~
telephone-circui~t 4.
Figure 7 shows a) attenuation characteristic and b) Group-delay characteris-tic of the combination of transmitter and receiver filters.
Figure 8 shows varia-tion o~ error rate with signal/noise ratio for the different systems opera-ting over the telephone-circuit 1.
Figure 9 shows variation of error rate with signal/noise ratio for the di~fere.nt systems operating over the telephone-circuit 2.
Figure 10 shows variation of error rate with signal/
noise ratio for the different sys-tems operating over the telephone-circuit 3.
Figure 11 shows variation of error rate with signal/noise ratio for the different systems operating over the -telephone-circuit 4.
Figure 12 shows ~ariation of error rate wi-th an inaccuracy in the estimate made by the receiver o~
a) -the level and b) the carrier phase of the received signal, for the System E with m = 6 and n = 7 opera-ting over the telephone-circuits 1-4.
~Z~27~;
1. INTRODUCTION
A telephone c;rcu;t ;n the sw;tched telephone network ~ay have any one of a very w;de range of attenuat;on-frequency, characteristics and any one of a correspondingly wide range of group-delay frequency characteristics, giving an available (or usable) bandwidth that may lie anywhere in the range about 1500 Hz to 3000 Hz.1 It is evident therefore that for the satisfactory transmission of data at a rate as high as 9600 bits/second over the switched telephone network, the receiver must be adaptive in the sense that it takes full account of the distortion introduced into the received data signal7 The conventional approach to this problem is to use an adaptive nonlinear (decision-feedback) equalizer that is adjusted to minimize the mean-square error in the equalized signala2~5 Such systems are known to operate satisfactorily over telephone circuits at rates of up to 4800 bits/second. At higher transmission rates~ however, satisfactory operation is not always obta;ned over the poorer telephone circu;ts.
An alternative approach that has recently been considered is the use of the Yiterbi~algorithm detector.5~7 Th;s selects as the detected message the possible sequence of transmitted data-symbols tsignal-element values) for which there is min;mum mean-square d;fference between the samples of the corresponding ~, ~L2~Z726 received data signal, for the g;ven s;gnal d;stortion but in the absence of no;se, and the samples of the s;gnal actually received~ When the data signal ;s received in the presence of stationary additive white Gauss;an noise, giving statistically independent Gaussian noise samples at the detector input, th;s is a process of maximum-likelihood detection, and when the transmitted data-symbols are statistically independent and equally likely to have any of their possible values~ the detection process min;m;zes the probability of error in the detection of the received message.5 It is assumed here that the received signal is sampled at the Nyquist rate, so that the corresponding samples contain all the information in this signal~
Unfortunately, when the sampled impulse-response of the channel conta;ns a large number of components ~non-zero samples), which ;s the case ;n the part;cular appl;cat;on cons;dered herer the Viterbi-algor;thm involves both an excessive amount of storage and an excess;ve number of operations per received data symbol. One approach for overcoming th;s difficulty is to use a linear feedforward transversal filter at the detector input to reduce the number of components in the channel sampled~impulse-response. The f;lter is adjusted adaptively to give a "desired" sampled impulse-response for the channel and filter, which has a given small number of components and Z7~i may or may not be f;xed.8~11 rhe disadvantage of th;s arrangement is that, for some telephone circuits, the linear filter may equalize some of the ampl;tude distortion ;ntroduced by the telephone circu;t and under these cond;t;ons max;mum-l;kel;hood detect;on ;s no longer ach;eved, lead;ng to an ;nfer;or performance~5 The l;near filter should ;deally perform the function of a "whitened matched-filter",6~14 which ensured that true maximum-likel;hood detec~ion ;s achieved by the V;terb;-algor;thm detector. In the ;deal arrangement and w;th stat;onary add;t;ve wh;te Gauss;an no;se at the receiver ;nput, the data signal ;s transm;tted over the channel at the Nyquist rate and the rece;ver filter has the same bandw;dth as the rece;ved signal, being such that, w;th the sampl;ng of the signal at the output of the rece;ver filter~ once per data sym~ol (at the Wyqu;st rate), the no;se samples at the output of the sampler are stat;st;cally ;ndependent Gaussian random variables w;th a f;xed var;ance. The wh;tened matched-f;lter now becomes the same as the l;near feedforward transversed f;lter that forms the first part of a convent;onal nonl;near tdec;s;on-feedback~ equalizer, where this is fed with the given sampled s;gnal and ;s adjusted to m;nimize the mean-square error tand hence max;m;ze the s;gnal/no;se ratio) in its output signal, subject to the exact equal;zation of ~he 272~i channel.5 The advantage of this is that the linear filter may be held adaptively adjusted for the channel to g;ve approximately the required response, at least at high signal/noise ratios, using basically the same simple S arrangement as that involved in a conventional adaptive nonl;near equal1Yer.5 The signal at the output of the l;near f;lter ;s fed ~o the maximum-l;kel;hood detector, and the feedback f;lter gives an estimate of the sampled impulse-response of the channel and linear filter, which is required by the detector. The details of the arrangement will be described later. When correctly adjusted, the linear filter is such that, ;n the z transform of the sampled impulse-response of the channel and l;near filter, all zeros (roots) of the z transform of the sampled impulse- response of the çhannel, that lie outside the unit circle in the z plane tthat is, have an absolute value greater than unity~, are replaced by the complex conjugates of their rec;procals, all rema;n;ng zeros being left unchanged.5 Thus all zeros of the z transform of the channel and linear filter is ;ns;de or on the un;t c;rcle.
The linear f;lter ;s, in fact, a pure phase equalizer (although not usually for the channel itself~ and so performs an orthogonal transformation on the rece;ved signal~5 In practice~ the linear f;lter may introduce some gain or attenuationf wh;ch however has the same effect ~L2~;27~6 _9_ on the noise as on the da~a s;gnal, and so does not change in any way the statistical relationship between the no;se and data signa~. This means that a Viterbi-algorithm detector operating at the output of the fiLter has the same tolerance to the Gaussian noise as the appropr;ate Viterbi-algorithm detector operating at its input, although generally involving much less complex equipment~
Furthermore, when an allowance has been made for the change in Level, the noise component at the filter output have the same statistical properties as those at its input~ being there~ore statistically ;ndependent and w;th a ~ixed variance. Since the linear filter does not no~ remove any of the amplitude distortion introduced by the channel~ it unfortunately does not always achieve an adequate reduction in the number of components of the resultant sampled impulse-response~
It follows that the Viterbi-algorithm detector must be replaced here by a near-maximum-likelihood detector that uses very much less storage and requires far fewer operations per rece;ved data-symbol, for the given number of components ;n the sampled ;mpulse-response of the channel. Various systems of this type have been developed and tested by computer s;mulation~ with very prom;s;ng results.5,12,13 An alternat;ve approach that has recently been studied il;~f3 ~726 -10~
;s to d;scard t.he l;near f;lter altogether, so that the near-max;mum-l;kelihood detector operates d;rectly on the original sampled signal, at the output of the samplerD14 An estimator is, of course, required here to determine the sampled impulse-response of the channel, but this can be implemented relatively simply, requiring a transversal filter w;th only as many taps as there are components ;n the sampled ;mpulse-response of the channel.15,16 However, somputer-simulation tests (whose results have not been published) have shown that ;n - the presence of very severe phase d;stort;on~ such as can occur over the poorer switched telephone c;rcu;ts, these systems require a considerable amount of storage and a large number of operat;ons per rece;ved data-symbol, in order to ach;eve near-maximum-likelihood detection. The most cost-effective arrangement for the given application is therefore to use an adaptive linear filter of the type previously described~ connected between the sampler and detector and acting as a whitened matched f;lter. The adaptive filter has the important property that it (at least approximately) removes all the phase distortion introduced by the channel and ;n addit;on adjusts the sampled impulse-response of the channel and filter into a form that is ;deally suited to a near-maximum-likelihood detector of the type cons;dered here.5~13 ~2~Z7216 The paper descr;bes the appl;cat;on of the methods invoLved in some recently deYeloped detection processes to the design of a QAM tquadrature amplitude modulated) system operating at 9600 b;ts/second with a 16-point QAM signal.
Details are g;ven here of some novel detection processes that avoid the high complexity involved with a 16-point s;gnal in the prev;ously descr;bed processes. Results of computer-s;mulation tests are then presented~ comparing the performances of the various systems, ~hen operating over different telephone circu;ts, and showing also the effects of inaccuracies in the estimates of the level and carrier phase of the received signalO
2. MODEL OF DATA TRANSMISSION SYSTEM
The model of the data-transmission system is shown in Figure 1.
This is a synchronous serial 16-point QAM system~ The information to be transmitted is carried by the data~
symbols (s;), where s; = + a + bj (1) __ and j = ~ a = 1 or 3 and b = 1 or 3~ The (s;) are statistically independent and equally likeLy to have any o~
the;r 16 possible values. It is assumed that s; = O for i~ 0, so that s; ~ (t - iT) is the ith signal-element at the input to the transmitter filter. The transmission path is a linear baseband channel that A a~f~Jr~r ;ncludes a telephone circuit together with a linear QAM
modulator at the transmitter and a linear QAM demodulator at the receiver. The basic structure of the modulator and demodulator has been descr;bed elsewhereu17 The signals transmit~ed over the ";n phase" channel of the QAM
system represented by real valued quantities9 and the signals over the "quadrature" channel by imaginary-valued quantities, to give a resultant complex-valued baseband s1gnal at both the input and output of the transmission path in Figure 1~ The carriers of the transmitted signals in the in~phase and quadrature channels are taken to be ~ cos 2~ fct and - ~ sin 2 ~ fct, respectively, where fc = 1800.17 The transmission filter, transmission path and receiver filter together form a linear baseband channel whose impulse response is the complex-valued function q(t), with an effective duration of less than (g~1)T seconds, where g is the appropriate ;nteger. It is assumed for the purpose of this study that qtt) is time invariant over any one transmiss;on. The various types of additive and multiplicatiYe noise normally introduced by telephone circuits are neglected here, and ;t is assumed that the only noise is stationary white Gaussian noise with complex values, zero mean and a flat tfrequency independent) power spectral density, ~hich is added to the data signal at the output of the transmission path~ to give ~2~6 ~ 13 the complex-valued Gaussian noise waveform v(t) at the output of the rece;ver f;lter. Although teLephone c;rcu;ts do not normally introduce signif;cant levels of Gaussian noise, the relative tolerance of different data-transm;ss;on sys~ems ~o wh;te Gaussian no;se ;s a goodmeasure of the;r relat;ve overall tolerance to the addit;ve no;se actually experienced over telephone c;rcu;tsn1 The waveform at the output of the rece;ver f;lter ;s the complex-valued s;gnal p(t) = i~i s;q(t - iT) + v(t) ~1) where Q is some large positive integer~ The waveform pSt) ;s sampled once per data symbol a~ the t;me ;nstants (iT) to g;ve the received samples (p;), wh;ch are fed to the adaptive l;near f;lter (F;gure 1~ This filter ;s taken to have been adjusted exactly as prev;ously descr;bed so that it acts as a whitened matched filter, the only difference being that the filter now operates with complex samples in place of the real-valued samples previously assumed. Thus the sampled impulse-response of the l;near baseband channel and adaptive l;near f;lter (F;gure 1) ;s g;ven by the (9~ component vector Y = Yo Y1 ~ Yg (2) which has complex-valued components and a z ~ransform with all its zeros ;ns;de or on the un;t circle in the z plane.
Furthermore, when an allowance has been made for the change 2~7~
;n level, the no;se components ~w;) at the output of the filter have the same stat;st;~al propert;es as the no;se components (vj) at ;ts input, where vj - v(iT). The delay in transmission over the baseband channel and adaptive fiLter~ other ~han that involved in the time-d;spersion of the received signal, ;s for conven;ence neglected here, so that yo ~ and y; = 0 for i < 0 and ; > 9. Thus the sample value at the f;lter output, at time t = iT, ;~ the complex-valued quant;ty rj = ~ Si-hYh ~ w; (3) where the real and imaginary parts of the (w;) are stat;st;cally ;ndependent Gauss;an random var;ables w;th zero mean and variance = a2.
The adjustment of the adaptive linear filter and the estimat;on of Y is achieved by the arran~ement shown in Figure 2. It can be seen that this is a simple development of a conventional adaptive nonlinear (decision-feedback) equalizer For the case where there is a delay in detection ~of two sampl;ng intervals ;n this particular example~5 Clearly, when ~here is no delay in detection, the arrangement degenerates into a conventional equalizer~ The main tap of the ~-tap adaptive linear feedfor~ard transversal f;lter, ;s close to the last tap, for all channels. A square marked T is a store that holds the corresponding sample value, the s~ored values being sh;fted one place ;n the d;rection shown dur;ng each sampl;ng ;nterval. A square marked ~ is an accumulator that sums the input samples, and * ;ndicates that the corresponding sample value ;s replaced by ;ts complex conjusate. a and b S are small pos;t;ve constants, and ~ rad;ans ;s the carr;er phase correcton necessitated by a pnase error ;n the coherent demodulator. ~ ;s determ;ned by the carr;er phase control circu;t, which ;s not shown ;n F;gures 1 and 2.
The adapt;ve adjustment of ~, together with that of the transversal-f;lter tap-gains in Figure 2, ;s arranged to be such that yo = 1, and the method of adjustment of each transversal filter ;s the same as ~hat ;n a convent;onal equalizer.S Further details of ~his and of the determ;nation of ~ are however beyond the scope of the paper~
Except where otherw;se stated it will be assumed that the adaptive linear f;lter ;s correctly adjusted and that the estimate of Y, which is the sequence yo~ Y1, ..~, yg' stored in the channel estimator, is also correct. The estimate of Y~ together with a prior knowledge of the possible values of s;~ are used by the near maximum-likelihood detector and echo canceller~
The detector (Figure 1) operates on i~s input samples tr~j to g;ve the finally detected data-symbols 9 ;~
~L v~7~
tSi), Si be;ng determined after the rece;pt of ri+n~ where n < 9, so that there ;5 a delay ;n detection of n sampling intervals. The echo canceller removes from the samples (r;) estimates (detected values) of all components involving data symbols (s;) whose final detected values (Si) have already been determ;ned. Thus the echo canceller operates on r; to give h=n+1 When Si_h = sj-h, for all (h)~ as will be assumed for the present, ~ n r; = ~ sj-h Yh + w;
h=o so that the sampled impulse-response of the channel and adaptive filter has been reduced from g + 1 to n ~ 1 components, thus greatly simpl;fying the detection process when n 9.
Let Sk, Rk and Wk be the k-component ro~ vectors whose ith components are s;, ri and w;, respectively, for i = 1, 2~ ~O~ k~ Also let Xk, Zk and Uk be the k-conponent row-vectors whose ith components are x;, z;
and u;~ respectively, for ; - 1~ 2, .~, k, where x;
has one of the 16 possible values of s;, Zi = ~ h Yh t6) h=0 and u; is the possible value of w; satisfy;ng ri = zj + u; (7~
In the k-dimensional complex vector space containing `~:
~Z0~7;:6 the vectors Rk, Zk and Uk, the square o~ the 'lunitary'l distance between the vectors Rk and Zk is kl2 = ~U1¦2 ~ ~U2¦2 + ,,, + lUklZ (~3) where ~ujl is the absolute value (modulus) of u;.
When all real and imag;nary components of the (w;~
are stat;st;cally independent and Jith a fixed var;ance, the maximum-likelihood vector xk is its possible value such that ¦Uk¦2 is minimized. Under the assumed conditions this Xk ;s the poss;ble value of Sk most likely to be correct.
The model of the data-transmission system is shown in Figure 1.
This is a synchronous serial 16-point QAM system~ The information to be transmitted is carried by the data~
symbols (s;), where s; = + a + bj (1) __ and j = ~ a = 1 or 3 and b = 1 or 3~ The (s;) are statistically independent and equally likeLy to have any o~
the;r 16 possible values. It is assumed that s; = O for i~ 0, so that s; ~ (t - iT) is the ith signal-element at the input to the transmitter filter. The transmission path is a linear baseband channel that A a~f~Jr~r ;ncludes a telephone circuit together with a linear QAM
modulator at the transmitter and a linear QAM demodulator at the receiver. The basic structure of the modulator and demodulator has been descr;bed elsewhereu17 The signals transmit~ed over the ";n phase" channel of the QAM
system represented by real valued quantities9 and the signals over the "quadrature" channel by imaginary-valued quantities, to give a resultant complex-valued baseband s1gnal at both the input and output of the transmission path in Figure 1~ The carriers of the transmitted signals in the in~phase and quadrature channels are taken to be ~ cos 2~ fct and - ~ sin 2 ~ fct, respectively, where fc = 1800.17 The transmission filter, transmission path and receiver filter together form a linear baseband channel whose impulse response is the complex-valued function q(t), with an effective duration of less than (g~1)T seconds, where g is the appropriate ;nteger. It is assumed for the purpose of this study that qtt) is time invariant over any one transmiss;on. The various types of additive and multiplicatiYe noise normally introduced by telephone circuits are neglected here, and ;t is assumed that the only noise is stationary white Gaussian noise with complex values, zero mean and a flat tfrequency independent) power spectral density, ~hich is added to the data signal at the output of the transmission path~ to give ~2~6 ~ 13 the complex-valued Gaussian noise waveform v(t) at the output of the rece;ver f;lter. Although teLephone c;rcu;ts do not normally introduce signif;cant levels of Gaussian noise, the relative tolerance of different data-transm;ss;on sys~ems ~o wh;te Gaussian no;se ;s a goodmeasure of the;r relat;ve overall tolerance to the addit;ve no;se actually experienced over telephone c;rcu;tsn1 The waveform at the output of the rece;ver f;lter ;s the complex-valued s;gnal p(t) = i~i s;q(t - iT) + v(t) ~1) where Q is some large positive integer~ The waveform pSt) ;s sampled once per data symbol a~ the t;me ;nstants (iT) to g;ve the received samples (p;), wh;ch are fed to the adaptive l;near f;lter (F;gure 1~ This filter ;s taken to have been adjusted exactly as prev;ously descr;bed so that it acts as a whitened matched filter, the only difference being that the filter now operates with complex samples in place of the real-valued samples previously assumed. Thus the sampled impulse-response of the l;near baseband channel and adaptive l;near f;lter (F;gure 1) ;s g;ven by the (9~ component vector Y = Yo Y1 ~ Yg (2) which has complex-valued components and a z ~ransform with all its zeros ;ns;de or on the un;t circle in the z plane.
Furthermore, when an allowance has been made for the change 2~7~
;n level, the no;se components ~w;) at the output of the filter have the same stat;st;~al propert;es as the no;se components (vj) at ;ts input, where vj - v(iT). The delay in transmission over the baseband channel and adaptive fiLter~ other ~han that involved in the time-d;spersion of the received signal, ;s for conven;ence neglected here, so that yo ~ and y; = 0 for i < 0 and ; > 9. Thus the sample value at the f;lter output, at time t = iT, ;~ the complex-valued quant;ty rj = ~ Si-hYh ~ w; (3) where the real and imaginary parts of the (w;) are stat;st;cally ;ndependent Gauss;an random var;ables w;th zero mean and variance = a2.
The adjustment of the adaptive linear filter and the estimat;on of Y is achieved by the arran~ement shown in Figure 2. It can be seen that this is a simple development of a conventional adaptive nonlinear (decision-feedback) equalizer For the case where there is a delay in detection ~of two sampl;ng intervals ;n this particular example~5 Clearly, when ~here is no delay in detection, the arrangement degenerates into a conventional equalizer~ The main tap of the ~-tap adaptive linear feedfor~ard transversal f;lter, ;s close to the last tap, for all channels. A square marked T is a store that holds the corresponding sample value, the s~ored values being sh;fted one place ;n the d;rection shown dur;ng each sampl;ng ;nterval. A square marked ~ is an accumulator that sums the input samples, and * ;ndicates that the corresponding sample value ;s replaced by ;ts complex conjusate. a and b S are small pos;t;ve constants, and ~ rad;ans ;s the carr;er phase correcton necessitated by a pnase error ;n the coherent demodulator. ~ ;s determ;ned by the carr;er phase control circu;t, which ;s not shown ;n F;gures 1 and 2.
The adapt;ve adjustment of ~, together with that of the transversal-f;lter tap-gains in Figure 2, ;s arranged to be such that yo = 1, and the method of adjustment of each transversal filter ;s the same as ~hat ;n a convent;onal equalizer.S Further details of ~his and of the determ;nation of ~ are however beyond the scope of the paper~
Except where otherw;se stated it will be assumed that the adaptive linear f;lter ;s correctly adjusted and that the estimate of Y, which is the sequence yo~ Y1, ..~, yg' stored in the channel estimator, is also correct. The estimate of Y~ together with a prior knowledge of the possible values of s;~ are used by the near maximum-likelihood detector and echo canceller~
The detector (Figure 1) operates on i~s input samples tr~j to g;ve the finally detected data-symbols 9 ;~
~L v~7~
tSi), Si be;ng determined after the rece;pt of ri+n~ where n < 9, so that there ;5 a delay ;n detection of n sampling intervals. The echo canceller removes from the samples (r;) estimates (detected values) of all components involving data symbols (s;) whose final detected values (Si) have already been determ;ned. Thus the echo canceller operates on r; to give h=n+1 When Si_h = sj-h, for all (h)~ as will be assumed for the present, ~ n r; = ~ sj-h Yh + w;
h=o so that the sampled impulse-response of the channel and adaptive filter has been reduced from g + 1 to n ~ 1 components, thus greatly simpl;fying the detection process when n 9.
Let Sk, Rk and Wk be the k-component ro~ vectors whose ith components are s;, ri and w;, respectively, for i = 1, 2~ ~O~ k~ Also let Xk, Zk and Uk be the k-conponent row-vectors whose ith components are x;, z;
and u;~ respectively, for ; - 1~ 2, .~, k, where x;
has one of the 16 possible values of s;, Zi = ~ h Yh t6) h=0 and u; is the possible value of w; satisfy;ng ri = zj + u; (7~
In the k-dimensional complex vector space containing `~:
~Z0~7;:6 the vectors Rk, Zk and Uk, the square o~ the 'lunitary'l distance between the vectors Rk and Zk is kl2 = ~U1¦2 ~ ~U2¦2 + ,,, + lUklZ (~3) where ~ujl is the absolute value (modulus) of u;.
When all real and imag;nary components of the (w;~
are stat;st;cally independent and Jith a fixed var;ance, the maximum-likelihood vector xk is its possible value such that ¦Uk¦2 is minimized. Under the assumed conditions this Xk ;s the poss;ble value of Sk most likely to be correct.
3~ SYSTEM A
The detection process used here is a mod;fication of an arrangement prev;ously13 referred to as System 2 and ;t operates as follows. Just prior to the receipt of the sample rk a~ the detector input, the detector holds ;n store m n-component vectors ~Qk-1), where m is a multiple of ~, Qk-1 = Xk-n Xk-n~ Xk-1 (9) and x; is as prev;ously def;ned. Each vector Qk-1 ;s assoc;ated w;th the correspond;ng cost ¦Uk_1¦2 (eqnu8), in the evaluat;on of wh;ch ;t is assumed that si = s; for every i. This implies that the echo canceller operates in the ideal manner and means, of course, that there is an inaccuracy in ~Uk_1¦2 whenever one or more of the (si) are incorrect~
~, . .
:~z0~z~
On the receipt of rk, each of the stored vectors (Qk-1) ;s expanded ;nto four (n ~ 13 - component vectors (Pk), where Pk = Xk-n Xk-n~1 . . . Xk (10) The first n components of Pk are as in the or;ginal vector Qk-1 and the last component xk has the four different values + 2 ~ 2j. The cost associated with each vector Pk is evaluatd as Ck = ~Uk-1~2 + ¦rk - h~oXk hl (11) For each of the four possible values of Xko the detector then selects the 4m vectors (Pk) hav;ng the smallest costs (Ck~ to give a total of m selected vectors together w;th the;r associated costs. Each selected vector Pk is next expanded into four vectors tPk) where the first n components are again as in the original vector Qk-1, and to the given value of the last component Xk tnow + 2 + 2j~ are added the four different values 1 + j~ The cost assoc;ated with each expanded vector ;s evaluated as ~Uk~2 = ¦Uk-1¦2 ~ ¦rk -h~oXk-h Yh¦2 (127 The detected data~symbol sk_n ;s then taken as the value of Xk-n ;n the vector Pk with the snallest cost~ and the first symbol xk_n ;n each vector Pk is d;scarded to g;ve the correspond;ng vector Qk, which ;s, of course, assoc;ated w;th the same cost ¦Uk¦2.
Finally, m vectors ~Qk) are selected from the 4m vector (Qk), as foLlows. When m = 32, the detector selects, for each of the 16 possible values of Xk, the two vectors ~Qk) that have the smallest costs and d;ffer in S the value of xk_1. When m = 16, the detector selects, for each of the 16 possible values of Xk, the vector Qk having the smallest cost. The selected vectors are now stored together with their associated costs. These arrangements can be implemented s;mply and ensure that all selected vectors are different, thus preventing any "merging" of the stored vectors.13 The techn;que just described is an arrangement of "double expansion" in which it is assumed that Xk = Xa,k ~ Xb,k (13) where xa,k = ~ 2 + 2j t14) and Xb,k = + 1 + j (15) so that xk is treated as the sum of two separate 4-level data-symbols xa,k and xb,k. In the expansion of the m stored vectors (Qk-1), Xb,k ;s set to zero so that x~ is treated as though it were xa~k.
In the expansion of the m selected vectors (Pk)~ xk is taken as xa~k + xb,k~ ~he value of Xa,k for each vector being, of course~ that determined in the first process of expansion and selection.
7Z~
The arrangement rel;es on the fact that ;f some complex number qk is at a smaller un;lary distance from a g;ven one of the 4 poss;ble values of xa,k than from the 3 rema;n;ng possible values of xa,k, then it is also at a smaller unitary distance from Xa,k ~ Xb,k~ for the g;ven Xa,k and for any of the 4 possible values of Xb,k~ than ;t is from the rema;n;ng 12 poss;ble values of Xa,k ~ Xb,k- Th;s ;mpl;es that the poss;ble value of xk closest to qk may be determ;ned ;n t~o success;ve opera~ions: first the selection of the poss;ble value of xa,k closest to qk, and ~hen, for the selected value of Xa,k~ the determination of ~he possible value of xb,k such that XaOk ~ Xb,k ;s closest to qk.
The detection process used here is a mod;fication of an arrangement prev;ously13 referred to as System 2 and ;t operates as follows. Just prior to the receipt of the sample rk a~ the detector input, the detector holds ;n store m n-component vectors ~Qk-1), where m is a multiple of ~, Qk-1 = Xk-n Xk-n~ Xk-1 (9) and x; is as prev;ously def;ned. Each vector Qk-1 ;s assoc;ated w;th the correspond;ng cost ¦Uk_1¦2 (eqnu8), in the evaluat;on of wh;ch ;t is assumed that si = s; for every i. This implies that the echo canceller operates in the ideal manner and means, of course, that there is an inaccuracy in ~Uk_1¦2 whenever one or more of the (si) are incorrect~
~, . .
:~z0~z~
On the receipt of rk, each of the stored vectors (Qk-1) ;s expanded ;nto four (n ~ 13 - component vectors (Pk), where Pk = Xk-n Xk-n~1 . . . Xk (10) The first n components of Pk are as in the or;ginal vector Qk-1 and the last component xk has the four different values + 2 ~ 2j. The cost associated with each vector Pk is evaluatd as Ck = ~Uk-1~2 + ¦rk - h~oXk hl (11) For each of the four possible values of Xko the detector then selects the 4m vectors (Pk) hav;ng the smallest costs (Ck~ to give a total of m selected vectors together w;th the;r associated costs. Each selected vector Pk is next expanded into four vectors tPk) where the first n components are again as in the original vector Qk-1, and to the given value of the last component Xk tnow + 2 + 2j~ are added the four different values 1 + j~ The cost assoc;ated with each expanded vector ;s evaluated as ~Uk~2 = ¦Uk-1¦2 ~ ¦rk -h~oXk-h Yh¦2 (127 The detected data~symbol sk_n ;s then taken as the value of Xk-n ;n the vector Pk with the snallest cost~ and the first symbol xk_n ;n each vector Pk is d;scarded to g;ve the correspond;ng vector Qk, which ;s, of course, assoc;ated w;th the same cost ¦Uk¦2.
Finally, m vectors ~Qk) are selected from the 4m vector (Qk), as foLlows. When m = 32, the detector selects, for each of the 16 possible values of Xk, the two vectors ~Qk) that have the smallest costs and d;ffer in S the value of xk_1. When m = 16, the detector selects, for each of the 16 possible values of Xk, the vector Qk having the smallest cost. The selected vectors are now stored together with their associated costs. These arrangements can be implemented s;mply and ensure that all selected vectors are different, thus preventing any "merging" of the stored vectors.13 The techn;que just described is an arrangement of "double expansion" in which it is assumed that Xk = Xa,k ~ Xb,k (13) where xa,k = ~ 2 + 2j t14) and Xb,k = + 1 + j (15) so that xk is treated as the sum of two separate 4-level data-symbols xa,k and xb,k. In the expansion of the m stored vectors (Qk-1), Xb,k ;s set to zero so that x~ is treated as though it were xa~k.
In the expansion of the m selected vectors (Pk)~ xk is taken as xa~k + xb,k~ ~he value of Xa,k for each vector being, of course~ that determined in the first process of expansion and selection.
7Z~
The arrangement rel;es on the fact that ;f some complex number qk is at a smaller un;lary distance from a g;ven one of the 4 poss;ble values of xa,k than from the 3 rema;n;ng possible values of xa,k, then it is also at a smaller unitary distance from Xa,k ~ Xb,k~ for the g;ven Xa,k and for any of the 4 possible values of Xb,k~ than ;t is from the rema;n;ng 12 poss;ble values of Xa,k ~ Xb,k- Th;s ;mpl;es that the poss;ble value of xk closest to qk may be determ;ned ;n t~o success;ve opera~ions: first the selection of the poss;ble value of xa,k closest to qk, and ~hen, for the selected value of Xa,k~ the determination of ~he possible value of xb,k such that XaOk ~ Xb,k ;s closest to qk.
4. SYSTEM B
Th;s ;s a development of an arrangement prev;ously13 referred to as System 1 and i~ operates as follows. Just prior to the rece;pt of the sample rk at the detector input~ the detector holds in store m n-component vectors tQk_1) together with the associated costs (IUk~ (eqns~ 9 and 12). On the receipt of rk the detector expands each vector Qk-1 ;nto the correspond;ng 16 vectors (Pk) (eqn. 10) having the 16 poss;ble values of xk ~ c ~given by ~ a ~ bj, for a = 1 or 3 and b = 1 or 3) and ;t evaluates the cost ¦Ukj2 for each of the 16 vectors (Pk). It then selects the vector Pk with the smallest cost and takes the detected data-symbol sk_n to have the value of xk_n in the selected vector Pk~ All ~ectors (Pk) for which Xk-n = Sk-n are now d;scarded, and the f;rst component of each of the rema;n;n~ vectors ~Pk) is omitted to give the corresponding n-component vectors (Qk). The one of these vectors derived from the vector Pk with the smallest cost ¦Uk¦2 is the first selected vector Qk.
The detector then selects from the remaining vectors ~Qk) the m-1 vectors associated with the smallest costs, to give a total of m vectors (Qk) and their associated costs, which are stored. The discard;ng of the given vectors (Pk) prevents the merging of the stored vectors, since it ersures that, if these are all different at the start of transmission, no two or more of them can subsequently become the same~
Th;s ;s a development of an arrangement prev;ously13 referred to as System 1 and i~ operates as follows. Just prior to the rece;pt of the sample rk at the detector input~ the detector holds in store m n-component vectors tQk_1) together with the associated costs (IUk~ (eqns~ 9 and 12). On the receipt of rk the detector expands each vector Qk-1 ;nto the correspond;ng 16 vectors (Pk) (eqn. 10) having the 16 poss;ble values of xk ~ c ~given by ~ a ~ bj, for a = 1 or 3 and b = 1 or 3) and ;t evaluates the cost ¦Ukj2 for each of the 16 vectors (Pk). It then selects the vector Pk with the smallest cost and takes the detected data-symbol sk_n to have the value of xk_n in the selected vector Pk~ All ~ectors (Pk) for which Xk-n = Sk-n are now d;scarded, and the f;rst component of each of the rema;n;n~ vectors ~Pk) is omitted to give the corresponding n-component vectors (Qk). The one of these vectors derived from the vector Pk with the smallest cost ¦Uk¦2 is the first selected vector Qk.
The detector then selects from the remaining vectors ~Qk) the m-1 vectors associated with the smallest costs, to give a total of m vectors (Qk) and their associated costs, which are stored. The discard;ng of the given vectors (Pk) prevents the merging of the stored vectors, since it ersures that, if these are all different at the start of transmission, no two or more of them can subsequently become the same~
5~ SYSTEM C
The technique of double expansion, applied in System A, is here applied to System B~ to give an arran0ement ;nvolving less storage and a smaller number of operations in a detection process than does System B. Following the expansion of the m stored vectors (Qk-1) into 4m ~2~7~6 vectors ~Pk), for which Xk = ~ Z + 2j, and the evaluation of the associated costs (Ck), as in System A, the detector selects the m vec~ors (Pk) w;th the smallest costs, regardless of the values of any of their (x;~. Following the expans;on of the latter vectors ;nto 4m vectors (Pk) and the evaluat;on of the;r costs (¦Uk¦2~, again as in System A, the detector selects the vector Pk with the smallest cost and takes the detected data-symbol Sk~n to have the value of xk_n in the selected vector Pk. All vectors tPk) for which xk_n ~ sk_n are now d;carded, and the detect;on process proceeds as ;n System B~
The technique of double expansion, applied in System A, is here applied to System B~ to give an arran0ement ;nvolving less storage and a smaller number of operations in a detection process than does System B. Following the expansion of the m stored vectors (Qk-1) into 4m ~2~7~6 vectors ~Pk), for which Xk = ~ Z + 2j, and the evaluation of the associated costs (Ck), as in System A, the detector selects the m vec~ors (Pk) w;th the smallest costs, regardless of the values of any of their (x;~. Following the expans;on of the latter vectors ;nto 4m vectors (Pk) and the evaluat;on of the;r costs (¦Uk¦2~, again as in System A, the detector selects the vector Pk with the smallest cost and takes the detected data-symbol Sk~n to have the value of xk_n in the selected vector Pk. All vectors tPk) for which xk_n ~ sk_n are now d;carded, and the detect;on process proceeds as ;n System B~
6. SYSTEM D
Th;s ;s an al~ernat;ve approach to that ;n System C
towards reducing the amount of storage and the number of operations involved in a detect;on process of System B~
On the rece;pt of rk the detector expands each of the m vectors tQk_1) into the correspond;ng four vectors (Pk), where xk has the four values + 1 and + 3, and the detector evaluates ck (eqn. 11) for each of the 4m vectors (Pk). The detector also expands each vector Qk-1 ;nto four vectors ~Pk), where xk has the four values + j and + 3j tj = ~ ~ and again evaluates Ck for each vector Pk~ The detector then discards from each group of four vectors (Pk~ (or;ginating from . ~i:,,,,~
~rJ~
a singLe vector Qk-1 v;a e;ther of the two expans;on processes) the vector Pk with the largest cost Ck, leaving three vectors in each group. Bearing in mind that yo = ~, th;s can be done very s;mply w;thout in fact evaluating ck for any vector Pk, but using just the real and imaginary parts of the quantity dk = rk ~ ~ xk-h Yh (16) h=o ;n eqn. 11. Thus, from the values of dk for the six vectors (Pk), in the two groups of three vectors originating from any single vector Qk 10 the resulting values of dk for the nine vectors (Pk), given by all combinations of the three real and three imaginary values of xk in the six vectors, are very easily evaluated, to give the corresponding costs ~¦Uk¦Z) of the nine vectors, computed accord;ng to eqn. 12. xk is~ of course, complex valued in each of these vectors. The detector now has 9m vectors ~Pk) together with the associated costs. The detector then determines sk n from the value of xk_n in the vector Pk with the smalLest cost, and the detection process proceeds exactly as for System B.
Th;s ;s an al~ernat;ve approach to that ;n System C
towards reducing the amount of storage and the number of operations involved in a detect;on process of System B~
On the rece;pt of rk the detector expands each of the m vectors tQk_1) into the correspond;ng four vectors (Pk), where xk has the four values + 1 and + 3, and the detector evaluates ck (eqn. 11) for each of the 4m vectors (Pk). The detector also expands each vector Qk-1 ;nto four vectors ~Pk), where xk has the four values + j and + 3j tj = ~ ~ and again evaluates Ck for each vector Pk~ The detector then discards from each group of four vectors (Pk~ (or;ginating from . ~i:,,,,~
~rJ~
a singLe vector Qk-1 v;a e;ther of the two expans;on processes) the vector Pk with the largest cost Ck, leaving three vectors in each group. Bearing in mind that yo = ~, th;s can be done very s;mply w;thout in fact evaluating ck for any vector Pk, but using just the real and imaginary parts of the quantity dk = rk ~ ~ xk-h Yh (16) h=o ;n eqn. 11. Thus, from the values of dk for the six vectors (Pk), in the two groups of three vectors originating from any single vector Qk 10 the resulting values of dk for the nine vectors (Pk), given by all combinations of the three real and three imaginary values of xk in the six vectors, are very easily evaluated, to give the corresponding costs ~¦Uk¦Z) of the nine vectors, computed accord;ng to eqn. 12. xk is~ of course, complex valued in each of these vectors. The detector now has 9m vectors ~Pk) together with the associated costs. The detector then determines sk n from the value of xk_n in the vector Pk with the smalLest cost, and the detection process proceeds exactly as for System B.
7. SYSTEM E
This is a simple modification of System D in wh;ch each of the m stored vectors (Qk-1) is expanded into four vectors ~Pk), where xk has the four values oV~ b' 14U
1 and ~ 3, and also into Four vectors (Pk), where Xk has the four values ~ j and ~ 3j, exactly as before, but now the detector selects from each group of four vectors (Pk) the two vectors with the smallest costs (Ck)~ the basic method of selection being that descr;bed for System D. From the four vectors tPk) derived from any single vector Qk~1~ Xk being real valued in two of these and imaginary-valued in the other two, ~he detector forms the four vectors (Pk) having the complex-valued (Xk) given by all comb;nations of the real and ;mag;nary values of Xk ;n the or;ginal four vectors, and evaluates the assoc;ated costs (¦Uk¦2). The detector now has 4m vectors (Pk) toge~her with the associated costs, and proceeds with the detection of sk_n and the selection of m vectors (Qk) exactly as for System B.
This is a simple modification of System D in wh;ch each of the m stored vectors (Qk-1) is expanded into four vectors ~Pk), where xk has the four values oV~ b' 14U
1 and ~ 3, and also into Four vectors (Pk), where Xk has the four values ~ j and ~ 3j, exactly as before, but now the detector selects from each group of four vectors (Pk) the two vectors with the smallest costs (Ck)~ the basic method of selection being that descr;bed for System D. From the four vectors tPk) derived from any single vector Qk~1~ Xk being real valued in two of these and imaginary-valued in the other two, ~he detector forms the four vectors (Pk) having the complex-valued (Xk) given by all comb;nations of the real and ;mag;nary values of Xk ;n the or;ginal four vectors, and evaluates the assoc;ated costs (¦Uk¦2). The detector now has 4m vectors (Pk) toge~her with the associated costs, and proceeds with the detection of sk_n and the selection of m vectors (Qk) exactly as for System B.
8 SYSTEM F
This ;s a simple modif;cat;on of System E which operates exactly as does System E except that the detector derives only three (rather than four) vectors (Pk) w;th complex values for Xk~ from any single vector Qk-1 the vectors ~Pk)being ~hose with ~he smallest costs (IUkl2) ;n the corresponding group of four vectors of System E. Again~ the select;on process for the ~Pk) can be ;mplemented very simply~ without in fact evaluating the costs, wh;ch are determined after the selection process~
Thus the detector generates 3m vectors tPk) together w;th the associated costs ~¦Uk¦2), and then proceeds w;th the detect;on of sk_n and the selection of m vectors (Qk~, exactly as for System 8n
This ;s a simple modif;cat;on of System E which operates exactly as does System E except that the detector derives only three (rather than four) vectors (Pk) w;th complex values for Xk~ from any single vector Qk-1 the vectors ~Pk)being ~hose with ~he smallest costs (IUkl2) ;n the corresponding group of four vectors of System E. Again~ the select;on process for the ~Pk) can be ;mplemented very simply~ without in fact evaluating the costs, wh;ch are determined after the selection process~
Thus the detector generates 3m vectors tPk) together w;th the associated costs ~¦Uk¦2), and then proceeds w;th the detect;on of sk_n and the selection of m vectors (Qk~, exactly as for System 8n
9. COMPUTER SIMULATION TESTS
The tolerance to additive white Gauss;an no;se of the six systems A-F and of a conventional nonlinear (dec1sion-feedback) equalizer have been determ;ned by computer s;mulation over models of four d;fferent telephone c;rcuits, us;ng the arrangements shown in F;gures 1 and Z
and descr;bed ;n Sect;ons 1-8. In every case (;ncluding that of the equalizer) it ;s assumed that the adapt;ve l;near f;lter has an appropr;ately large number of taps an ;s accurately adjusted to perform the ideal linear transformation described ;n Sect;ons 1 and 2.
F;gures 3-6 show the attenuation and group-delay characteristics of the four different telephone circuits uscd in the tests, the telephone circuit forming a part of the transm;ss;on path ;n F;gure 1. Figure 7 shows the resultant attenuat;on and group-delay characterist;cs of the equ;pment f;lters, these be;ng here cons;dered as operating on the transmitted bandpass signal, rather than on the baseband signals in the transmitter and receiver as shown in Figure 1, to demonstrate more clearly the effects of the equ;pment f;lters in Lim;ting some of the ~z~
distortion introduced by the telephone circuit. The equ;pment filters are taken to include the filtering required to convert the sequence of impulses at the transmitter input (Figure 1) into the corresponding rectangular waveform used ;n an actual modem.
TabLe 1 shows the sampled ;mpulse-response of the l;near baseband channel, sample and adaptive linear filter ;n Figure 1, for each of the four different telephone c;rcu;ts testedO This given some ;dea of the resultant distortion introduced by each channel, bearing in mind that the ideal sampled impulse-response has the first component equal to unity and the remainder all ~ero. It can be seen from Figures 3 and 7 that nearly all the signal d;stortion shown in Table 1 for the telephone circuit 1 is in fact introduced by the equ;pment f;lters.
The teLephone circuits 1 and 2 introduce negligible and typical levels of distortion~ respectively, ~hereas the circu;ts 3 and 4 are close to the typical ~orst circuits normally considered for the transmission of data at 96ûO
and 600-1200 bits~second, respectively. The telephone circuit 3, which is close to the Post Office network N69 introcluces severe group-delay distortion, and the teLephone circuit 4, which is close to the Post Off;ce ~ network N3,~ introduces very severe attenuation distortion.
Figures 8-11 sho~ ~he performances of Systems A-F
tlr5~
and of a conventional nonlinear equal;zer, for the telephone circu;ts 1-4. The s;gnal/noise rat;o ;s here taken to be ~dB~ where ~ = 10 log1o t10/2O2) (17) bearing in mind that the mean-square value of the data-symbol s; ;s 10 and 2a2 is the var;ance of the Gauss;an no;se component w;. The 95X confidence lim;ts of the curves ;s better than + z da. System A has been tested w;th both m = 16 and 32 and w;th n = 7O where m ;s the n~mber of stored vectors (Qk) and n is the number of components in Qk). Systems B-F have all been tested with m = 6 and with both n = 3 and 7. For the telephone circuits 3 and 4, System ~ has in addit;on been tested with both m = 4 and 8 and w;th n = 7.
It can be seen from Figures 8-11 that the Systems A-F
all have a much better overall tolerance to add;t;ve wh;te Gauss;an no;se than the convent;onal nonl;near equal;zer.
The System A w;th m = 32 t32 stored vectors) and n = 7 ta delay ;n detect;on of 7 sampling intervals) has the best performance of all systems tested, as would perhaps be expec~ed, but when m = 16 and n - 7 its tolerance to noise at the lower error rates in the tsi) becomes ;nferior to that of the Systems B-F~ with m = 6 and n = 7. The latter arrangements all have a s;m;lar tolerance to no;se at the lower error rates, th;s be;ng signif;cantly better than ~p~v~ o ~v that w;th m = 6 and n = 3, over the telephone c;rcu;ts 3 and 4. No very useful advantage in the tolerance to no;se of the Systems B-F seems l;kely to be achieved by increasing m to a value greater than 6, but when m ;s reduced from 6 to 4 there is a noticeable degradation in performance. At error rates around 1 in 103~ the Systems B-F, with m = 16 and n = 7, have an advantage in tolerance to addit;ve white Gaussian noise, over the nonlinear equalizer, of approximately 2~ 3 and 52 dB, for the telephone circu;ts 1, 2, 3 and 4, respectively.
Figure 17 shows the effect on the error rate in the (si) of ;naccuracies in the estimates made by the receiver of the level and carrier phase of the received signal, when System E with m = 6 and n = 7 operates ove the telephone circuits 1-4. It can be seen that, over the teLephone c;rcuit 3, the system can tolerate an inaccuracy of about 4 d~ in the estimate of the received s;gnal level or else an inaccuracy of about 2 degrees in the estimate of the received signal carrier phase, for an increase of two times in the error rate of the (si), th;s representing a reduction of less than ~d~ in tolerance to add;tive wh;te Gauss;an no;se. It seems l;kely therefore that ;n a well designed modem and w;th the correction of the more ser;ous phase jitter in the rece;ved signal carrier, no very serious degradation in tolerance to ~ 29-additive noise should be experienced due to the inaccurate estimation of the received signal level or carrier phase.
The tolerance to additive white Gauss;an no;se of the six systems A-F and of a conventional nonlinear (dec1sion-feedback) equalizer have been determ;ned by computer s;mulation over models of four d;fferent telephone c;rcuits, us;ng the arrangements shown in F;gures 1 and Z
and descr;bed ;n Sect;ons 1-8. In every case (;ncluding that of the equalizer) it ;s assumed that the adapt;ve l;near f;lter has an appropr;ately large number of taps an ;s accurately adjusted to perform the ideal linear transformation described ;n Sect;ons 1 and 2.
F;gures 3-6 show the attenuation and group-delay characteristics of the four different telephone circuits uscd in the tests, the telephone circuit forming a part of the transm;ss;on path ;n F;gure 1. Figure 7 shows the resultant attenuat;on and group-delay characterist;cs of the equ;pment f;lters, these be;ng here cons;dered as operating on the transmitted bandpass signal, rather than on the baseband signals in the transmitter and receiver as shown in Figure 1, to demonstrate more clearly the effects of the equ;pment f;lters in Lim;ting some of the ~z~
distortion introduced by the telephone circuit. The equ;pment filters are taken to include the filtering required to convert the sequence of impulses at the transmitter input (Figure 1) into the corresponding rectangular waveform used ;n an actual modem.
TabLe 1 shows the sampled ;mpulse-response of the l;near baseband channel, sample and adaptive linear filter ;n Figure 1, for each of the four different telephone c;rcu;ts testedO This given some ;dea of the resultant distortion introduced by each channel, bearing in mind that the ideal sampled impulse-response has the first component equal to unity and the remainder all ~ero. It can be seen from Figures 3 and 7 that nearly all the signal d;stortion shown in Table 1 for the telephone circuit 1 is in fact introduced by the equ;pment f;lters.
The teLephone circuits 1 and 2 introduce negligible and typical levels of distortion~ respectively, ~hereas the circu;ts 3 and 4 are close to the typical ~orst circuits normally considered for the transmission of data at 96ûO
and 600-1200 bits~second, respectively. The telephone circuit 3, which is close to the Post Office network N69 introcluces severe group-delay distortion, and the teLephone circuit 4, which is close to the Post Off;ce ~ network N3,~ introduces very severe attenuation distortion.
Figures 8-11 sho~ ~he performances of Systems A-F
tlr5~
and of a conventional nonlinear equal;zer, for the telephone circu;ts 1-4. The s;gnal/noise rat;o ;s here taken to be ~dB~ where ~ = 10 log1o t10/2O2) (17) bearing in mind that the mean-square value of the data-symbol s; ;s 10 and 2a2 is the var;ance of the Gauss;an no;se component w;. The 95X confidence lim;ts of the curves ;s better than + z da. System A has been tested w;th both m = 16 and 32 and w;th n = 7O where m ;s the n~mber of stored vectors (Qk) and n is the number of components in Qk). Systems B-F have all been tested with m = 6 and with both n = 3 and 7. For the telephone circuits 3 and 4, System ~ has in addit;on been tested with both m = 4 and 8 and w;th n = 7.
It can be seen from Figures 8-11 that the Systems A-F
all have a much better overall tolerance to add;t;ve wh;te Gauss;an no;se than the convent;onal nonl;near equal;zer.
The System A w;th m = 32 t32 stored vectors) and n = 7 ta delay ;n detect;on of 7 sampling intervals) has the best performance of all systems tested, as would perhaps be expec~ed, but when m = 16 and n - 7 its tolerance to noise at the lower error rates in the tsi) becomes ;nferior to that of the Systems B-F~ with m = 6 and n = 7. The latter arrangements all have a s;m;lar tolerance to no;se at the lower error rates, th;s be;ng signif;cantly better than ~p~v~ o ~v that w;th m = 6 and n = 3, over the telephone c;rcu;ts 3 and 4. No very useful advantage in the tolerance to no;se of the Systems B-F seems l;kely to be achieved by increasing m to a value greater than 6, but when m ;s reduced from 6 to 4 there is a noticeable degradation in performance. At error rates around 1 in 103~ the Systems B-F, with m = 16 and n = 7, have an advantage in tolerance to addit;ve white Gaussian noise, over the nonlinear equalizer, of approximately 2~ 3 and 52 dB, for the telephone circu;ts 1, 2, 3 and 4, respectively.
Figure 17 shows the effect on the error rate in the (si) of ;naccuracies in the estimates made by the receiver of the level and carrier phase of the received signal, when System E with m = 6 and n = 7 operates ove the telephone circuits 1-4. It can be seen that, over the teLephone c;rcuit 3, the system can tolerate an inaccuracy of about 4 d~ in the estimate of the received s;gnal level or else an inaccuracy of about 2 degrees in the estimate of the received signal carrier phase, for an increase of two times in the error rate of the (si), th;s representing a reduction of less than ~d~ in tolerance to add;tive wh;te Gauss;an no;se. It seems l;kely therefore that ;n a well designed modem and w;th the correction of the more ser;ous phase jitter in the rece;ved signal carrier, no very serious degradation in tolerance to ~ 29-additive noise should be experienced due to the inaccurate estimation of the received signal level or carrier phase.
10. CONCLUSIONS
Of the various arrangements tested, Systems E and F
~ith m = 6 and n = 7 appear to be the most cost-effective.
They promise to give a subs~antial improvement ;n tolerance to additive noise over the conventional nonlinear equalizer~ without involving a great increase in equipment complexity. Bearing in m1nd that the telephone circuit 4 is basically similar to the typical worst telephone c;rcuit likely to be experienced over the sw;tched telephone network in this country, ;t seems that so long as the accurate convergence of the adaptive linear filter can be achieved and appropriate steps are taken in the design of the modem to combat the effects of carrier-phase jitter, correct operation of the systems should be obta;ned over nearLy all circu;ts on the public switched telephone networkn `~
,~
12. REFERENCES
1. Clark,A.P., "Pr;nciples of D;gital Data Transmission"
(Pentech Press, London, 1976).
2~ Monsen,PO, "Feedback equalization of fading dispersive channels", IEEE Trans. on Information Theory, IT-17, pp.56-64, January 1971~
3. Westcott,.R.J., "An experimental aclapt;vely equalized modem for data transmission over the switched telephone network", The Rad;o and Electronic Engineer, 42, pp.499-507, November 1972.
4. Clark,h.Pc and Tint,U.S., "Linear and non~linear transversal equalizers for baseband channels", The Radio and Electronic Engineer, 45, pp.271-283, June 1975.
15 5. Clark,A.P., "Advanced Data-Transmission Systems"
IPentech Press, London, 1977).
6. Forney,.G.D., "Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference". IEEE Trans., IT-1B, pp.363-378, May 1972.
7~ Forney,G.D., "The Viterbi-algorithm", Proc.IEEE, 61, pp.268-278, llarch 1973.
8. Qureshi,S.ll.H. and Newhall,E.E., "An adaptive receiver for data transmission over t;me dispersive channels", IEEE Trans., IT~19, pp;.448~457, July 1973.
~r3~z~
9~ Falconer~D.D. and Magee,F~R., "Adaptive channel memory truncation for maximum-likelihood sequence est;mat;on", Bell Syst~ Tech. J., 52, pp.1541-1562, November 1973.
10~ Canton;,A. and Kwong,K., "Further results on the V;terb;-algo~;thm equalizer", IEEE Trans., IT-20 pp.764-767~ November 1974.
Of the various arrangements tested, Systems E and F
~ith m = 6 and n = 7 appear to be the most cost-effective.
They promise to give a subs~antial improvement ;n tolerance to additive noise over the conventional nonlinear equalizer~ without involving a great increase in equipment complexity. Bearing in m1nd that the telephone circuit 4 is basically similar to the typical worst telephone c;rcuit likely to be experienced over the sw;tched telephone network in this country, ;t seems that so long as the accurate convergence of the adaptive linear filter can be achieved and appropriate steps are taken in the design of the modem to combat the effects of carrier-phase jitter, correct operation of the systems should be obta;ned over nearLy all circu;ts on the public switched telephone networkn `~
,~
12. REFERENCES
1. Clark,A.P., "Pr;nciples of D;gital Data Transmission"
(Pentech Press, London, 1976).
2~ Monsen,PO, "Feedback equalization of fading dispersive channels", IEEE Trans. on Information Theory, IT-17, pp.56-64, January 1971~
3. Westcott,.R.J., "An experimental aclapt;vely equalized modem for data transmission over the switched telephone network", The Rad;o and Electronic Engineer, 42, pp.499-507, November 1972.
4. Clark,h.Pc and Tint,U.S., "Linear and non~linear transversal equalizers for baseband channels", The Radio and Electronic Engineer, 45, pp.271-283, June 1975.
15 5. Clark,A.P., "Advanced Data-Transmission Systems"
IPentech Press, London, 1977).
6. Forney,.G.D., "Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference". IEEE Trans., IT-1B, pp.363-378, May 1972.
7~ Forney,G.D., "The Viterbi-algorithm", Proc.IEEE, 61, pp.268-278, llarch 1973.
8. Qureshi,S.ll.H. and Newhall,E.E., "An adaptive receiver for data transmission over t;me dispersive channels", IEEE Trans., IT~19, pp;.448~457, July 1973.
~r3~z~
9~ Falconer~D.D. and Magee,F~R., "Adaptive channel memory truncation for maximum-likelihood sequence est;mat;on", Bell Syst~ Tech. J., 52, pp.1541-1562, November 1973.
10~ Canton;,A. and Kwong,K., "Further results on the V;terb;-algo~;thm equalizer", IEEE Trans., IT-20 pp.764-767~ November 1974.
11. Desblache,A.E~, "Optimal short desired impulse response for max;mum-likel;hood sequence estimation", IEEE TRans. on Communications, COM-25, pp. 735-7~8 July 1977.
:
~ !
_ 34 ~LZ~26 , ~
:
~ !
_ 34 ~LZ~26 , ~
12 Fosch1ni,G.J., "~ reduced state variant of maximurn-likelihood seqUence detection attaining optimum performance for high signal-to-noise ratios", IEEE Trans.~ IT-23, pp.605-609, September 1977.
13 Clark,A.P., llarvey,J.D. and Dri~co:Ll,.T.P., "Near-maximuTn-likelihood detection processes for distorted dicJital signals", The Radio and Electronic Engineer, 48, pp.301-309, June 1978.
14 Clark,A.P., Kwong,C.P. and Harvey,J.D., "Detection processes for severely distorted digital signals", Electronic Circuits and Systems, 3, pp.27-37, January ~979.
Magee,F.R. and Proakis,J.G., "Adar~tive max:imurn-likelihoo(i sequence estirnation Eor dirJital signalling in the presence of intersymbol interference", I~EE Trans./ IT-l9~ pp.l20-124, January 1973.
16 Clark,A.P., ~wong,C.P. and McVerry,F., "Estimation of the sampled impulse-response of a channel", Signal Processing, 2, pp.39-53, January 1980.
17 Clark, A.P. and Harvey,J.D., "Detection of distorted q.a.m.
signals", Electronic Circuits and Systems, 1, pp.103-109, April 1977.
~c~ 7~
_ r IF.LEPHONE CIRCUIT TELEP~ONE CIRCUIT TELEPIIONE CIRCUI'r' l'EL~PilOi~ CIRCUI~
2 3 l 4 , REAL IMAGINARY REAL IMAGINARY REAL IMAGIN~RY I ~E~L IMAGINrJ~Y
I P~RTPAR~ i PART P~RT PART ' PART P~RT ¦ p~R7' 1.0000 0.0000 I.0000 10.00-)01.0000 'O.0000, ~ ~.. oOOO I O.OOc)O
0.3412 0.0667 l 0.5091 j 0.19~,0 o.4~3611.09i3i3 0.254~ , 1.99~1 -0. 1298 -0.0358 ~ -0. 1465o~oooo-0. 59800.0703; -1.739~ -0. 2019 o ~ 02630.0051o ~ 0323- o~ ol 71o ~ 1702-o ~ 1938o ~ 6795-o ~ i3086 0.0015 l0.0008 0.0125 0.0200-0.02~5 ~ 0.1000 0.0408 0.5~13 10-0.0019-0.0016 -0.0099 -0.0109 0.01~0 -0.02s~ 1189 -o~ 3 -0.00170.0006 0.0046 0.007~ -~.013~ 0.01]0 o.o343 0.0420 O.OOlL0.0004 -0. 00690.00i3 ~~ .00 s 60 ~ oo4 2- O ~ 01850.03 64 0.00l80.~010 0.0059 0.0076 0.0003 0.0003 0.0139 0.0216 -0.00140.0000 -0.0025 -o~os~ -0.0008 0.00~1 -0.0102 -0.0009 -o.0008-0.0004 -0.0013 o.oo~o o.oooo -0.00~.1 -O.OOl9 -o.oo34 0.0016-0.0001 0. 0024-o~ 002~3o ~ 0007-o ~ 0007o ~ oo37 -o ~ 00~6 ooo6O . OOOl -o ~ ooo9o ~ 00180. 0~37 o ~ 0002 -0. 0028 -0.0006 -0.00060.0003 -0.0006 -0.0006 -0.0019 -0.0025 -O.OOl9 -0.0046 0.0005o . ooolO . OOOl-o ~ 00030.0020 0. ooo8o . oo83 o . oo22 20o.ooooo.ooo4 0.0002 o.ooo8 0.0005 -0.0002 -0.0056 0.0059 0.0002 l-0.0003 o.oooo-0.0006-0.0022 0.0002 -0.0046 -0.0028 -0.000~ 10.0000 -o.0003-O. ooolo ~ 0007-0.00050. oo49-o ~ oo l 9 -O . OOOl 0.0002 -o. ooo2 0.0003-o ~ 0008 o ~ 0002 -o ~ ooo9 o ~ oo37 o ~ 0004 -O . OOOl 0.0003 -0.0002o ~ 0005 o ~ 0005 -o ~ ooo9 o ~ ooo3 O.0000 O.0000 O.0000O.00000.0000 0.0000 0.0000 0.0000 O.0000 ,O.0000 ~O.00000.00000.0000 0.0000 0.0000 0.0000 TABLE 1. Sampled impulse-response of baseband chanllel and adaptive linear fiLter in Fig. 1 , for each oE
the four telephone circuits.
Magee,F.R. and Proakis,J.G., "Adar~tive max:imurn-likelihoo(i sequence estirnation Eor dirJital signalling in the presence of intersymbol interference", I~EE Trans./ IT-l9~ pp.l20-124, January 1973.
16 Clark,A.P., ~wong,C.P. and McVerry,F., "Estimation of the sampled impulse-response of a channel", Signal Processing, 2, pp.39-53, January 1980.
17 Clark, A.P. and Harvey,J.D., "Detection of distorted q.a.m.
signals", Electronic Circuits and Systems, 1, pp.103-109, April 1977.
~c~ 7~
_ r IF.LEPHONE CIRCUIT TELEP~ONE CIRCUIT TELEPIIONE CIRCUI'r' l'EL~PilOi~ CIRCUI~
2 3 l 4 , REAL IMAGINARY REAL IMAGINARY REAL IMAGIN~RY I ~E~L IMAGINrJ~Y
I P~RTPAR~ i PART P~RT PART ' PART P~RT ¦ p~R7' 1.0000 0.0000 I.0000 10.00-)01.0000 'O.0000, ~ ~.. oOOO I O.OOc)O
0.3412 0.0667 l 0.5091 j 0.19~,0 o.4~3611.09i3i3 0.254~ , 1.99~1 -0. 1298 -0.0358 ~ -0. 1465o~oooo-0. 59800.0703; -1.739~ -0. 2019 o ~ 02630.0051o ~ 0323- o~ ol 71o ~ 1702-o ~ 1938o ~ 6795-o ~ i3086 0.0015 l0.0008 0.0125 0.0200-0.02~5 ~ 0.1000 0.0408 0.5~13 10-0.0019-0.0016 -0.0099 -0.0109 0.01~0 -0.02s~ 1189 -o~ 3 -0.00170.0006 0.0046 0.007~ -~.013~ 0.01]0 o.o343 0.0420 O.OOlL0.0004 -0. 00690.00i3 ~~ .00 s 60 ~ oo4 2- O ~ 01850.03 64 0.00l80.~010 0.0059 0.0076 0.0003 0.0003 0.0139 0.0216 -0.00140.0000 -0.0025 -o~os~ -0.0008 0.00~1 -0.0102 -0.0009 -o.0008-0.0004 -0.0013 o.oo~o o.oooo -0.00~.1 -O.OOl9 -o.oo34 0.0016-0.0001 0. 0024-o~ 002~3o ~ 0007-o ~ 0007o ~ oo37 -o ~ 00~6 ooo6O . OOOl -o ~ ooo9o ~ 00180. 0~37 o ~ 0002 -0. 0028 -0.0006 -0.00060.0003 -0.0006 -0.0006 -0.0019 -0.0025 -O.OOl9 -0.0046 0.0005o . ooolO . OOOl-o ~ 00030.0020 0. ooo8o . oo83 o . oo22 20o.ooooo.ooo4 0.0002 o.ooo8 0.0005 -0.0002 -0.0056 0.0059 0.0002 l-0.0003 o.oooo-0.0006-0.0022 0.0002 -0.0046 -0.0028 -0.000~ 10.0000 -o.0003-O. ooolo ~ 0007-0.00050. oo49-o ~ oo l 9 -O . OOOl 0.0002 -o. ooo2 0.0003-o ~ 0008 o ~ 0002 -o ~ ooo9 o ~ oo37 o ~ 0004 -O . OOOl 0.0003 -0.0002o ~ 0005 o ~ 0005 -o ~ ooo9 o ~ ooo3 O.0000 O.0000 O.0000O.00000.0000 0.0000 0.0000 0.0000 O.0000 ,O.0000 ~O.00000.00000.0000 0.0000 0.0000 0.0000 TABLE 1. Sampled impulse-response of baseband chanllel and adaptive linear fiLter in Fig. 1 , for each oE
the four telephone circuits.
Claims (6)
PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:
1. A method of detecting multi-level signals using Viterbi algorithm detection in which the incoming signal has m levels, comprising the steps of:
sampling the signal at regular predetermined intervals, expanding each sample of the received signal so obtained into nk expanded vectors where n is smaller than m, selecting from the nk expanded vectors k vectors with the smallest costs, storing said k vectors together with their associated costs, calculating from said costs the most likely level of the sampled incoming signal at said predetermined intervals, and, outputting said most likely level of said sampled signal as the detected signal.
sampling the signal at regular predetermined intervals, expanding each sample of the received signal so obtained into nk expanded vectors where n is smaller than m, selecting from the nk expanded vectors k vectors with the smallest costs, storing said k vectors together with their associated costs, calculating from said costs the most likely level of the sampled incoming signal at said predetermined intervals, and, outputting said most likely level of said sampled signal as the detected signal.
2. A method of detecting a multi-level input signal as claimed in claim 1 in which the expansion process is performed sequentially in two operations and in which n = m1/2, k vectors being chosen after each expansion process.
3. A method of detecting a multi-level input signal as claimed in claim 1 wherein the expansion process is performed simultaneously in two operations and in which n ? m1/2 such that the two sets of nk expanded vectors give a total of n2k vectors from which are selected the k vectors to be stored.
4. An apparatus for decoding a multi-level data signal employing a Viterbi algorithm wherein the data signal has m levels, said apparatus comprising:
receiving means for receiving said data signal, sampling means for sampling the received data signal at regular predetermined intervals thereby to generate a plurality of signal samples, expanding means for expanding each signal sample of the received data signal into nk expanded vectors wherein n is smaller than m, cost generating means for generating a cost associated with each of said selected k vectors, selecting means for selecting k vectors from said nk expanded vectors which have the smallest cost, storing means for storing said selected k vectors together with a respective cost associated with each of said k vectors, and calculating means for calculating and outputting as the detected signal the most likely level of said received data signal on the basis of said cost.
receiving means for receiving said data signal, sampling means for sampling the received data signal at regular predetermined intervals thereby to generate a plurality of signal samples, expanding means for expanding each signal sample of the received data signal into nk expanded vectors wherein n is smaller than m, cost generating means for generating a cost associated with each of said selected k vectors, selecting means for selecting k vectors from said nk expanded vectors which have the smallest cost, storing means for storing said selected k vectors together with a respective cost associated with each of said k vectors, and calculating means for calculating and outputting as the detected signal the most likely level of said received data signal on the basis of said cost.
5. An apparatus according to claim 4 wherein said expanding means comprises means for sequentially expanding each sample of the received data signal in two sequential operations in which n = m1/2, and means for selecting k vectors after each expansion process.
6. An apparatus as recited in claim 4 wherein said expanding means comprises means for expanding each sample of the received data signal in two simultaneous operations in which n is less than or equal to the square root of m such that the two sets of nk expanded vectors render a total of n2k vectors from which k vectors are selected and stored.
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| GB8034682 | 1980-10-28 | ||
| GB8034682A GB2087198B (en) | 1980-10-28 | 1980-10-28 | Data decoding |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA1202726A true CA1202726A (en) | 1986-04-01 |
Family
ID=10516930
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA000388556A Expired CA1202726A (en) | 1980-10-28 | 1981-10-22 | Data transmission systems |
Country Status (14)
| Country | Link |
|---|---|
| US (1) | US4484338A (en) |
| EP (1) | EP0050930B1 (en) |
| JP (1) | JPS57127349A (en) |
| AT (1) | ATE12866T1 (en) |
| AU (1) | AU541651B2 (en) |
| CA (1) | CA1202726A (en) |
| DE (1) | DE3170013D1 (en) |
| DK (1) | DK476981A (en) |
| ES (1) | ES506607A0 (en) |
| GB (1) | GB2087198B (en) |
| IE (1) | IE52427B1 (en) |
| NO (1) | NO156229C (en) |
| NZ (1) | NZ198717A (en) |
| ZA (1) | ZA817174B (en) |
Families Citing this family (19)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4578800A (en) * | 1982-07-12 | 1986-03-25 | Yutaka Yasuda | Synchronization circuit for a Viterbi decoder |
| US4527279A (en) * | 1982-07-12 | 1985-07-02 | Kokusai Denshin Denwa Co. | Synchronization circuit for a Viterbi decoder |
| US4646305A (en) * | 1983-09-26 | 1987-02-24 | Case Communications, Inc. | High speed data modem using multilevel encoding |
| US4756007A (en) * | 1984-03-08 | 1988-07-05 | Codex Corporation | Adaptive communication rate modem |
| US4631735A (en) * | 1984-12-28 | 1986-12-23 | Codex Corporation | Coded modulation system with feedback |
| GB8609711D0 (en) * | 1986-04-21 | 1986-05-29 | Clark A P | Channel estimation & detection |
| US4945549A (en) * | 1986-11-13 | 1990-07-31 | The United States Of America As Represented By The Administrator Of The National Aeronautics And Space Administration | Trellis coded modulation for transmission over fading mobile satellite channel |
| EP0293620A1 (en) * | 1987-05-25 | 1988-12-07 | BBC Brown Boveri AG | Method of signal transmission |
| US4885757A (en) * | 1987-06-01 | 1989-12-05 | Texas Instruments Incorporated | Digital adaptive receiver employing maximum-likelihood sequence estimation with neural networks |
| US5271042A (en) * | 1989-10-13 | 1993-12-14 | Motorola, Inc. | Soft decision decoding with channel equalization |
| KR930004862B1 (en) * | 1990-12-17 | 1993-06-09 | 삼성전자 주식회사 | Memory instrument of state estimate |
| US5142551A (en) * | 1991-02-28 | 1992-08-25 | Motorola, Inc. | Signal weighting system for digital receiver |
| WO1993008637A1 (en) * | 1991-10-21 | 1993-04-29 | Motorola, Inc. | System and method for calculating a state transition metric in a viterbi equalizer |
| DE4224214C2 (en) * | 1992-07-22 | 1995-02-09 | Deutsche Forsch Luft Raumfahrt | Process for source-controlled channel decoding by expanding the Viterbi algorithm |
| US5363412A (en) * | 1992-12-28 | 1994-11-08 | Motorola, Inc. | Method and apparatus of adaptive maximum likelihood sequence estimation using filtered correlation synchronization |
| JP3275779B2 (en) * | 1997-06-16 | 2002-04-22 | 日本電気株式会社 | Delay decision feedback type sequence estimation receiver |
| FR2783120B1 (en) * | 1998-09-04 | 2000-11-24 | Nortel Matra Cellular | DIGITAL EQUALIZATION METHOD, AND RADIO COMMUNICATION RECEIVER IMPLEMENTING SUCH A METHOD |
| US6347125B1 (en) | 1999-01-11 | 2002-02-12 | Ericsson Inc. | Reduced complexity demodulator for multi-bit symbols |
| US6999510B2 (en) * | 2003-04-18 | 2006-02-14 | Optichron, Inc. | Nonlinear inversion |
Family Cites Families (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4130818A (en) * | 1977-04-21 | 1978-12-19 | Communications Satellite Corporation | Analog threshold decoding |
| US4171516A (en) * | 1977-11-23 | 1979-10-16 | General Electric Company | Tone phase shift detector |
| US4240156A (en) * | 1979-03-29 | 1980-12-16 | Doland George D | Concatenated error correcting system |
| US4293953A (en) * | 1979-12-28 | 1981-10-06 | The United States Of America As Represented By The Secretary Of The Army | Bi-orthogonal PCM communications system employing multiplexed noise codes |
-
1980
- 1980-10-28 GB GB8034682A patent/GB2087198B/en not_active Expired
-
1981
- 1981-10-08 AT AT81304667T patent/ATE12866T1/en not_active IP Right Cessation
- 1981-10-08 DE DE8181304667T patent/DE3170013D1/en not_active Expired
- 1981-10-08 EP EP81304667A patent/EP0050930B1/en not_active Expired
- 1981-10-16 ZA ZA817174A patent/ZA817174B/en unknown
- 1981-10-20 NO NO813533A patent/NO156229C/en unknown
- 1981-10-20 NZ NZ198717A patent/NZ198717A/en unknown
- 1981-10-22 CA CA000388556A patent/CA1202726A/en not_active Expired
- 1981-10-26 US US06/314,974 patent/US4484338A/en not_active Expired - Fee Related
- 1981-10-27 ES ES506607A patent/ES506607A0/en active Granted
- 1981-10-27 JP JP56171958A patent/JPS57127349A/en active Pending
- 1981-10-27 IE IE2507/81A patent/IE52427B1/en unknown
- 1981-10-27 AU AU76856/81A patent/AU541651B2/en not_active Ceased
- 1981-10-28 DK DK476981A patent/DK476981A/en not_active Application Discontinuation
Also Published As
| Publication number | Publication date |
|---|---|
| NO813533L (en) | 1982-04-29 |
| EP0050930A1 (en) | 1982-05-05 |
| IE812507L (en) | 1982-04-28 |
| NO156229C (en) | 1987-08-19 |
| GB2087198A (en) | 1982-05-19 |
| GB2087198B (en) | 1985-02-27 |
| NO156229B (en) | 1987-05-04 |
| ATE12866T1 (en) | 1985-05-15 |
| ES8300232A1 (en) | 1982-10-01 |
| IE52427B1 (en) | 1987-10-28 |
| US4484338A (en) | 1984-11-20 |
| EP0050930B1 (en) | 1985-04-17 |
| ZA817174B (en) | 1982-10-27 |
| DK476981A (en) | 1982-04-29 |
| JPS57127349A (en) | 1982-08-07 |
| DE3170013D1 (en) | 1985-05-23 |
| AU541651B2 (en) | 1985-01-17 |
| NZ198717A (en) | 1985-08-16 |
| ES506607A0 (en) | 1982-10-01 |
| AU7685681A (en) | 1982-05-06 |
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