CA1329642C - Arrangement for combating intersymbol interference and noise - Google Patents

Abstract

ABSTRACT:
"Arrangement for combating intersymbol interference and noise."

An arrangement for combating intersymbol interference and noise, which are introduced into a data signal (ak) transmitted at a symbol rate 1/T by a dispersive transmission channel (CH), comprises an adaptive equalizer (EQ) with a symbol detector (ID) for forming tentative symbol decisions (âk) as well as a post-detector (PD) for forming final symbol decisions (âk-m) using an auxiliary signal which is derived from the transmitted data signal (rk) at the input of the arrangement. By using a priori knowledge of the main character of the transfer characteristic of the transmission channel (CH) and by using as an auxiliary signal for the post-detector (PD) the input signal (ak) of the symbol detector (ID) in lieu of the transmitted data signal (rk), this post-detector (PD) can be arranged in a non-adaptive and thus simpler way without a resulting noticeable loss of transmission quality. (Fig. 4).

Description

c~
1329~2 PHN 12.039 ~Arrangement for combating intersymbol interference and noise.~

The invention relates to an arrangement for combating intersymbol interference and noise which are introduced into a data signal transmitted at a symbol rate 1/T by a dispersive transmission channel, said arrangement comprising an adaptive equalizer having a fixed symbol detector for forming tentative symbol decisions, as well as a post-detector for forming final symbol decisions from the tentative ~ymbol decisions utilizing an auxiliary signal derived from the transmitted data signal at the input of the arrangement.
Such arrangements are known for the case in which the equalizer is a linear equalizer and for the case in which the equalizer is of the decision feedback type. The former option is described in an article ~Adaptive Cancellation of Nonlinear Intersymbol Interference for Voiceband Data Transmission~ by E.Biglieri et al., published in IEEE J.
Select. Areas Co~mun., Vol. SAC-2, No. 5, pp. 765-777, September 1984, more specifically, Figure 7 and the associated description, the latter option being known from an article A Maximum-Likelihood Sequence Estimator with Decision-Feedback Equalization~ by W.U.Lee and F.S.Hill Jr., published in IEEE Trans. Commun., Vol. COM-25, No. 9, pp.971-979, September 1977, more specifically, Figure 2 and the associated description. As already shown by the two titles, the post-detectors considered in these articles are of different types. More specifically, in the article by Biglieri et al. an intersYmbol interference canceller is utilized, whereas in the article by Lee and Hill a post-detector is utilized which is arranged for estimating the maximum-likelihood sequence of transmitted data symbols. In both cases the post-detector comprises a final symbol detector for forming final symbol decisions on the basis o~ an input signal with a well defined correlation structure, and the transmitted data signal at the input of the arrangement constitutes the auxiliary signal which is utilized in the post-detector for improving the quality of the symbol decisions. For transmission channels whose transfer characteristics are not accurately known a priori, the correlation structure of this auxiliary signal is also 132gS~2 uncertain. In such cases the post-detector should be arranged in an adaptive way so as to realize the predetermined correlation structure at the input of the final symbol detector.
The invention now has for its object to provide an arrangement of the type mentioned in the preamble in which a priori knowledge is used of the main character of the transfer characteristic of the transmission channel for simplifying the post-detector.
Thereto, the system according to the invention is characterized in that the post-detector ls arranged for forming the final symbol decisions in a non-adaptive way and in that the auxiliary signal is constituted by the input signal of the symbol detector.
To state the invention another way, there is provid_d arrangement for combating intersymbol interference and noise which are introduced into a data signal transmitted at a symbol rate 1/T
by a dispersive transmission channel with a transfer characteristic only the main character of which is known a priori, said arrangement comprising an adaptlve equalizer having at least one adaptive filter for deriving a first detection signal from an input signal of the arrangement and having a fixed symbol detector for taking tentative symbol decisions on the basis of the detection signal, the arrangement further comprlsing combining means for combinlng the detection slgnal and a signal derived from the tentative symbol decisions lnto a detection final signal and a non-adaptive post-detector for taking final symbol decisions on the basis of the final detection signal.
In contradistinction to the situation with the known arrangements, the correlation structure of this auxiliary signal is no longer uncertain because it is tailored to the fixed symbol detector owing to the adaptability of the equalizer.
Consequently, the post-detector need no longer be arranged adaptively so that a simpler implementation is possible. The loss of transmission quality attending this simplification can be minimized, when dimensioning the post-detector, ~y taking account of a priori knowledge of the main character of the transfer .r~ 2 ~f 1 3 ~ 2 characteristic of the transmission channel.
This general notion will now be discussed in more detail for a number of embodiments of the invention with reference to the drawing in which:
Fig. 1 shows a functional discrete-time model of a data transmission system having a dispersive transmission channel and an arrangement known from the above-mentioned prior art having an adaptive equalizer and an adaptive post-detector;
Fig. 2A and Fig. 2B show block diagrams of an adaptive linear equalizer and an adaptlve equallzer of the decision feedback type, respectively;
Flg. 3A shows a block diagram of a known adaptive post-detector ln the form of an intersymbol interference canceller, and Fig. 3B shows a block diagram of a known adaptive post-detector which is arranged for estimatlng the maximum-likelihood sequence of transmitted data symbols;
Fig. 4 shows a block diagram of an arrangement according to the invention having an adaptive linear equalizer and a non-adaptive - 2a 1329~2 PHN 12.039 3 post-detector in which intersymbol interference cancellation is used;
Fig. 5 shows a block diagram of an arrangement according to the invention having an adaptive decision feedback equalizer and a non-adaptive post-detector in which intersymbol interference S cancellation is used;
Fig. 6A and Fig. 6B show a set of graphs for illustrating the quality of tentative and final symbol decisions in the arrangement according to Fig. 5;
Fig. 7 shows a block diagram o$ an arrangement according to the invention having an adaptive decision feedback equalizer and a non-adaptive post-detector which is arranged for estimating the maximum-likelihood sequence of transmitted data symbols.
In all these Figures corresponding elements are always indicated by the same reference symbols.
In the following description a discrete-time modelling of the transmission system and the arrangement is used, as the general notion of the invention can be presented in the most simple way with reference to such a modelling. This does not lead to a loss of generality as the present modelling can be derived unambiguously from the parameters of the continuous-time system, as described, for example, in the book entitled ~Digital Communications~, by J.G.Proakis, McGraw-Hill, New York, 1983, Chapter 6, and especially Section 6.3, pp. 351-357.
Fig. 1 shows a functional discrete-time model of a system for transmitting data symbols ak at a symbol rate 1/T through a discrete-time dispersive channel CH having a causal impulse response fk, which channel introduces also additive white Gaussian noise so that a received data signal rk = (a ~ f)k ~ nk develops, the symbol ~ representing the linear convolution operator.
This received data signal rk is applied to an adaptive equaliser EQ
for obtaining tentative symbol decisions bk which, in the absence of transmission errors, are related in a simple and generally linear manner to the transmitted data symbols ak and from which tentative symbol decisions âk can be derived in a simple ~anner which, in the absence of transmission errors, are equal to ak. When using partial-response techniques this correlation can be described with the aid of a polynomial in a delay operator D representing the 132~6~2 2010g-7795 symbol interval T. Further details about these partial-response polynomials are to be found, for example, in the article "Partial-Response Signaling" by P. Kabal and S. Pasupathy, IEEE Trans.
Commun., Vol. COM-23, No. 9, pp. 921-934, September 1975. In all these cases it is possible to obtain at the output of the equalizer EQ a signal âk which, in absence of transmission errors, is a faithful replica of the transmitted data signal ak. It may well be the case that specific operations whlch are carried out in the equallzer EQ to derive âk from bk are again cancelled in a subsequent adaptive post-detector PD. Naturally, in such cases it i8 wiser in a practical implementation to apply to the post-detector PD such a signal that strictly speaking no redundant operations need to be carried out. For simplicity, it will be assumed hereinafter that the equalizer EQ generates tentative symbol decisions âk which, in the absence of transmission errors, are a faithful replica of the transmitted data symbols ak, so that bk = âk (2) for all k. On the basis of the above considerations it will be evident that this assumption does not impose essential constraints.
The tentative symbol decisions âk at the output of equalizer EQ are applied to the adaptive post-detector PD, which derive~ additional information from an auxlliary signal in the form of the received signal rk and uses this additional information for improving the quality of the tentative symbol decisions âk. The final symbol decisions âk M thus obtained are ideally a version of the transmitted data symbols ak delayed over a specific number of M symbol intervals having length T.
Generally, the adaptive equalizer EQ can be arranged in any way conventionally used for forming final symbol decisions.
More specifically, the equalizer EQ can be a linear or a decision-feedback equalizer, as shown in the Figures 2A and 2B, respectively.
The linear equalizer as shown in Fig. 2A comprises an adaptive feedforward filter FF with an impulse response ck which converts received data signal rk into a real-valued estimate ak f 13296~2 PHN 12.039 5 transmitted data signal ak. From this estimate ak a symbol detector ID subsequently forms tentative symbol decisions âk. On the basis of formula (1) input signal ak can be written as ak = (r 2 c)k = (a ~ f ~ c)k + (n * c)k. (3) As impulse response ck of feedforward filter FF is adjusted adaptively for suppressing intersymbol interference (ISI) in the data component (a ~ f)k of received data signal rk according to formula (1)l the data component (a ~ f ~ c)k of input signal ak should be substantially free from ~SI. Therefore, impulse response ckl apart from an otherwise unimportant scale factorl will satisfy in good approximation (f ~ C)k = ~k where k is the Rronecker delta function with 1I k = O
k =
I k ~ O. (5) When utilizing formulas (4) and (5)l formula (3) can then be simplified to ak = ak ~ (n * c)k (6) so that at the input of symbol detector ID an estimate ak of data signal ak is formed which is practically free from I5I. The filtered noise ~ignal (n ~ c)k which occurs in this expression ~ill generally be an amplified and correlated version of the original noise signal nk, because feedforward filter FF, in addition to an optional phase equalization, normally also performs an amplitude equalization which results in noise components being additionally amplified at frequencies at which transmission channel CH shows a large attenuation.
For completeness, it is observed that the assumption made here that at the input of symbol detector ID a direct estimate ak of ak is formedl leads to a correlation structure of ak at the input of symbol detector ID which substantially corresponds with the generally well-defined correlation structure of ak. Converselyl with the use of partial-response techniques, not discussed in more detail in this Applicationl a linearly transformed version bk of ak is estimatedl which leads to an equally well-defined correlation structure of this estimate bk of b~ at the input of symbol detector ID.
As appears from the abovel a correct adjustment of - 1329~2 PHN 12.039 6 feedforward filter FF always leads to a correlation structure of input signal ak which is required for the proper functioning of symbol detector ID. An adjustment of feedforward filter FF adapted to this desirable correlation structure is achieved under control of an error signal ek which in the case of Fig. 2A is representative of the difference between the input signal ak and output signal âk of symbol detector ID. For simplicity, it is assumed in Fig. 2A that error signal ek is equal to the difference signal ak ~ ~k.
The decision-feedback equalizer EQ shown in Fig. 2~
includes in addition to the elements of Fig. 2A also a feedback filter FB having an impulse response qk which generates a cancelling signal for post-cursive ISI on the basis of tentative symbol decisions âk_i with i 2 1, which decisions have already been made, this cancelling signal being subtracted b7 means of a summator from the output signal of feedforward filter FF for obtaining the input signal ~k of symbol detector ID. Error signal ek, which is representative of the difference between in and output signals ~k and âk f symbol detector ID, is again used for obtaining the desired correlation structure of input signal ak of symbol detector ID. Thereto, it is necessary that at least one of the two filters FF and FB be adjusted adaptively under control of error signal ek.
On the basis of formula (1) input signal ak of symbol detector ID in Fig 2D can be written as ak = ~r ~ c)k - (â ~ q)k (a ~ f ~ c)k ~ (â * q)k * (n ~ c)k ( ) With a correct dimensioning of feedforward filter FF the data component (a ~ f ~ c)k of its output signal will virtually only contain post-cursive ISI, so that in good approximation the following will hold (f ~ C)k = ~ k < -1. (8) For combating post-cursive ISI, feedback filter FB has a causal impulse ~esponse qk for which holds O, k < O
9k = (9) (f ~ C)k' ~ > 1 Due to this causal nature of feedback filter FB its output signal at any instant is only determined by the past tentative symbol decisions âk_i with i > 1. Under normal operating 132~2 PHN 12.039 7 conditions these symbol decisions are correct and in that case, when using formulas ~8) and t9), formula (7) can in good approximation be written as ak = ak + (n ~ c)k.
According to the latter formula, in the absence of erroneous tentative symbol decisions âk, an estimate ak of data signal ak that is practically free from post-cursive ISI is formed at the input of symbol detector ID. For transforming the impulse reponse fk of transmission channel CH into a substantially causal impulse response ~f * c)k according to formula ~8), feedforward filter FF substantially performs a phase equalization resulting in virtually no noise colouring or noise amplification. For simplicity of the next description, the effect of this equalization will be incorporated in impulse response fk f transmission channel CH, so that feedforward filter FF can be o~itted.
It will also be assumed that the gain factor of feedforward filter FF
also incorporated in fk leads to a scaling of fk such that fO = 1. These two assumptions, which do not have a limiting effect on the following considerations, enable to simplify formula ~7) to ak = ~a * f)k - (â t q)k + nk (11) and formula ~9) to 9k = fk ~ k ~12) so that formula ~7) can now be written as = ~a ~ f)k - ~â ~ ~f ~ ))k ~ nk =
ak + ~a - â) * f)k + nk In the absence of erroneous tentative symbol decisions ak the term ((a - ~) * f)k in this expression i~ dropped, so that then the following holds ~k = ak + nk. (14) To illustrate the conventional way of structuring post-detector PD of Fig. 1, Fig. 3A shows a block diagram of a known adaptivepost-detector in the form of an ISI canceller (compare the above article by Biglieri et al.), Fig. 3B showing a block diagram of a known adaptive post-detector which is arranged for estimating the maximum-likelihood sequence of transmitted data symbols (compare the above article by Lee and Hill).
The post-detector shown in Fig. 3A comprises a receive filter RF having impulse response wk. Fro~ received data signal rk 13~96~2 PHN 12.039 8 receive filter RF forms a filtered version Yk in which noise is suppressed in the best way possible. As known (compare the said article by Biglieri et al.), a maximum noise suppression is achieved by arranging receive filter RF as a so-called ~Matched Filterl whose impulse reponse is the time-inverse of the impulse response fk of transmission channel CH. In order to realize a causal and hence physically implementable impulse response wk receive filter RF should also introduce a signal delay MT which corresponds with a number of M
symbol intervals having length T and which is at least egual to the memory span of transmission channel CH, so that ideally for i~pulse response wk it holds that Wk fM-k = f-(k-M) (15) Post-detector PD also includes a final-symbol detector FD for forming the final symbol decisions ~k-M~ and means for cancelling ISI in output signal Yk of receive filter RF. These cancelling means comprise a feedforward section FFS with a causal impulse response p~ for forming a first cancelling signal for pre-cursive ISI in Yk in response to a number of tentative symbol decisions âk, a feedback section FBS with a causal impulse response q for forming a second cancelling signal for post-cursive ISI in Yk in response to a number of final symbol decisions âk_M, and a summator for combining Yk with the two cancelling signals into an input signal ak_M
for final-symbol detector FD.
The feedforward and feedback sections FFS and FBS and possibly also the receive filter RF are adjusted adaptively under control of an error signal ek which is representative of the difference between signal ~k-H and output signal âk_M of final symbol detector FD. As appears from an article On the Performance and Convergence of the Adaptive Canceller of Intersymbol Interference in Data Transmission- by X. Wesolowski, published in IEEE Trans. Commun., Vol. COM-33, No. 5, pp.425-432, May 1985, with the aid of this post-detector an improvement of the transmission quality can often be achieved which corresponds with an improvement of 1-2 d~ in the signal-to-noise ratio. As the correlation structure of received signal rk is not precisely known in advance, the same will apply to output signal Yk of receive filter RF, irrespective of the adaptive or non-adaptive implementation of this receive filter RF. For the sake of the required - 13296~2 PHN 12.039 9 accurate cancellation of pre- and post-cursive ISI it is therefore essential that the feedforward and feedback sections FFS and FBS have an adaptive structure.
On the basis of formula (1) output signal Yk of receive filter RF in Fig. 3A can be described as Yk = (r * w)k = (a * f ~ w)k + (n * W)k. (16) When utilizing a ~Matched Filter~ according to formula (15) for receive filter RF this expression can be rewritten as Yk = (a * (f * f-))k-M + (n ~ f_)k_M
in which the subscribt ~_~ is used to indicate time-inversion, so that, for example, ~f * f_)k represents the autocorrelation function of impulse response fk of transmission channel CH. For input signal ak_M = Yk ~ ~â ~ P')k - ~â * q')k-M (18) of final symbol detector FD it then holds that ak-M = (a ~ (f * f_))k_M - (â * P')k +
- (â ~ g')k-H + (n ~ f_)k_M-As appears from this formula, pre-cursive ISI in the data component (a ~ (f * f_))k_M of Yk can only be cancelled on the basis of the tentative symbol decisions âk not delayed over M symbol intervals, and thus by means of the second term (â ~ P')k on the right-hand side. Conversely, post-cursive ISI in the data component (a * ~f * f ))k-M of Yk can be cancelled both on the basis of tentative symbol decisions âk and on the basis of the qualitatively better final symbol decisions âk_M ~compare also the description of Fig. 1 in the above article by Wesolowski). With cancellation of pre-cursive and post-cursive ISI on the basis of tentative symbol decisions âk feedback section FBS can be omitted, as is shown by a dashed line in Fig. 3A, and feedforward section FFS should have the following impulse response according to formula ~19) O, k < - 1, ~ ~ f_)k_H~ < k < M - 1, Pk ~ k = M (20 (f 2 fJ k-H~ M + 1 < k < 2M - 1 O, k > 2M - 1 When cancelling post-cursive ISI by means of a feedback sectLon FBS
impulse responses Pk and qk of the feedforward and feedback 132~42 PHN 12.039 10 sections FFS and FBS should obviously satisfy 0, k < -1 P = (f ~ f_)k-M~ 0 < k < M - 1, (21) 0, k > M
5 and 0, k < 0 qk = (f ~ f_)k, 1 < k < M - 1, (22) 0. k > M.
Both in the case of formula (20) and the case of formulas (21) and (22) perfect cancellation of pre- and post-cursive ISI is effected in the absence of erroneous tentative and final symbol decisions, in which case formula (19) can be simplified to ak-M = aX-M(f * f_)0 + (n ~ f_)k_M. (23) According to this formula there is formed at the input of final-symbol detector PD an estimate ak_M of a version of data signal ak_M
that is amplified by a factor (f ~ f_)0, in which estimate pre-cursive and post-cursive ISI is fully cancelled and noise is optimally suppressed because receive filter RF has an impulse reponse wk of the ~atched Filter~ type with wk = f_(k_M). As also appears from the above article by R.Wesolowski, this optimum noise suppression leads to a quality improvement of the final symbol decisions âk_M with respect to that of the tentative symbol decisions âk which corresponds with an improvement of 1-2 dB in the signal-to-noise ratio. It is worth mentioning that this improvement is not strictly bound to the use of a ~Matched Filter~ for receive filter RF, as appears, for example from an article ~A Simulation Study of Intersymbol Interference CancellationY by J.W.M.Bergmans and Y.C.Wong, published in AE~, Vol. 41, No. 1, pp.33-37, 1987. More specifically, this article shows that especially deviations from the ~Matched Filter~
characteristic, leading to a smaller amount of pre-cursive ISI, will hardly lead to a loss of transmission quality, with the proviso that feedforward and feedback sections FFS and FBS are dimensioned such that there is perfect cancellation of pre-cursive and post-cursive ISI in the absence of erroneous tentative and final symbol decisions.
The post-detector PD shown in Fig. 3B comprises a feedforward section FFS with a causal impulse response p~ for reducing the span of post-cursive ISI in auxiliary signal rk by means 1329l~42 PHN 12.039 11 of a summator also receiving this auxiliary signal rk. The output signal bk of this summator is applied to a Viterbi detector VD, which determines on the basis of bk the maximum-likelihood sequence âk_M of transmitted data symbols ak. Owing to the reduced span of the ISI in input signal bk a relatively simple Viterbi detector VD will suffice. As also the remaining ISI in input signal bk of Viterbi detector VD has a correlation structure which is not accurately known in advance, both the feedforward section FFS and the Viterbi detector VD will have to be adjusted adaptively under csntrol of, for example, the error signal ek from the Figures 2A and 2B.
According to Fig. 3B input signal bk of Viterbi detector VD can be written as bk = rk ~ (â * P )k- (24) When using formula (1) this formula can be written as bk = (a ~ f)k ~ (â ~ P')k + nk. (25) The reduction of the span of the ISI in auxiliary signal rk which is produced by the second term in the right-hand side of this formula can be explained in a simple way by splitting up the impulse response fk of transmission channel CH according to fk = fk + fr (26) where fk~ < k < L, fk = (27) O,k > L + 1 and O,O < k < L, fk = (28) fk, k > L + 1 According to these formulas fk is split up into a truncated impulse response fk with a suitably chosen and generally small memory span 1T, and a residual impulse response fk. Now it is a task of the feedforward section FFS to cancel the ISI within the span of the residual impulse response fk. Thereto, impulse response Pk f feedforward section FFS i8 chosen such that P~ = fk (29) When using formulas (26) and (29) formula (25) can be written as 132~2 PHN 12.039 12 bk = (a * ft)k + ((a - â) % fr)k + nk. (30) In the absence of erroneous tentative symbol decisions âk the term ((a - â) ~ fr)k in this expression is dropped, so that the following holds bk = (a ~ f )k + nk-which implies on the basis of formula (27) that bk only contains ISI in a reduced span with the length LT, so that a relatively simple Viterbi detector VD will suffice.
In situations in which a priori knowledge about the main character of the transfer characteristic of transmission channel CH is available, the invention now provides a possibility of arranging post-detector PD in a non-adaptive and thus simpler way. The loss of transmission quality attending this simplification remains slight as will be explained hereinafter.
Fig. 4 shows a block diagram of an arrangement according to the invention with an adaptive linear equalizer EQ according to Fig. 2A and a non-adaptive post-detector PD based on the principle of ISI cancellation as also applied in Fig. 3A. In contradistinction to the situation in Fig. 1 the auxiliary signal of post-detector PD is not formed now by received data signal rk, but by input signal ak f symbol detector ID in equalizer EQ of Fig. 4. This signal has a well-defined correlation structure which according to formula ~6) i9 substantially given by the fixed correlation structure of data signal ak. This also holds for the tentative symbol decisions âk and, therefore, a non-adaptive embodiment of post-detector PD will suffice. A proper dimensioning in this respect is to be achieved by using a priori knowledge about the main character of the transfer characteristic of transmission channel CH. This a priori knowledge can be simply represented in terms of a nominal impulse response qk f transmission channel CH, known in advance, which response is causal as is the actual response fk, and which is related to fk according to fk = gk ~ ~k (32) where ~k represents the relatively small difference between the nominal and the actual impulse responses.
According to formula (6), input signal ak of symbol detector ID in the present case contains a noise component (n ~ c)k amplified by feedforward filter FF. Receive filter RF, as in the 13296~2 PHN 12.039 13 conventional combination of Fig. 2A with Fig. 3A, now has to realize as good a suppression of this noise component as possible. If accurate a priori knowledge with respect to impulse response fk of transmission channel CH were available, a maximum noise suppression could be realized by dimensioning receive filter RF such that its output signal Yk has the same noise component (n ~ f_)k_M as in the conventional situation described by formula (17). Because in Fig. 4 it holds that Yk = (a % w)k (33) this means that according to formula (6) impulse response wk of receive filter RF should satisfy (C ~ W)k = f-(k-M)-Owing to the adaptability of equalizer EQ impulse response ck satisfies formula (4), so that formula (34) after convolution with fk can be simplified to Wk = (f ~ f-)k-M
Therefore, impulse response wk of receive filter RF in this case should ideally be a delayed version of autocorrelation function (f ~ f-)k of impulse response fk of transmission channel CH.
~ecause of a priori knowledge of the precise form of fk is actually lacking, receive filter RF can now only be dimensioned in accordance with the nominal impulse response gk of transmission channel CH, so that Wk = (g ~ g-)X-M (36) The noise ~uppression achieved by this choice is substantially equal to the noise suppression achieved in the case of the conventional adaptive equalisation, considering the relatively small difference ~k between the nominal and actual impulse responses gk and fk. With the aid of formulas (6) and (33) it appears that in this case an output signal Yk of receive filter RF develops having the form Yk = (a ~ (g ~ s_))k_M + (n ~ c * (g ~ g_))k_M. (37) This formula distinctly shows that the fixed correlation structure of input signal ak of symbol detector ID in combination with the non-adaptive form of receive filter RF leads to a fixed correlation structure of data component (a ~ (9 ~ g-))k-M of outpu g Yk of receive filter RF. Consequently, it is possible to cancel pre-cursive and post-cursive ISI in this output signal Yk on the basis of non-adaptive feedforward and feedback sections FFS and F~S. Thus, a post-PHN 12.039 14 detector PD which is totally non-adaptive and hence simpler to implement will suffice. The achieved simplification of the implementation of post-detector PD is linked with only a minor impairment of the transmission quality as the noise suppression realized by the receive filter RF still substantially corresponds with the noise suppression in the adaptive situation.
In Fig. 4 post-detector PD is also arranged for producing an error signal ek which is representative of the difference between input signal ak_H and output signal âk_~ of final symbol detector FD. This signal ek is used for adjusting the adaptive filter FF in equalizer EQ of Fig. 4 in lieu of the qualitatively poorer error signal ek from Fig. 2A.
In practice there will be a certain freedom with respect to the exact choice of the nominal impulse response gk of transmission channel CH. This freedom can be used for a further simplification of post-detector PD, as will now be explained for the case in which this impulse response g~ can be characterized by a partial-response polynomial g~D). For, according to the above article by Rabal and Pasupathy, such a polynominal g(D) in the delay operator D generally has a relatively low order and also, apart from an otherwise unimportant scale factor, only integral-valued coefficients. Consequently, the impulse response wk of receive filter RF, according to formula (36) being a delayed version of the autocorrelation function of impulse response gk, will have only a small number of non-zero coefficients which, in addition, are integral. Needless to observe that this will lead to an extremely simple digital implementation. An additional advantage of such a choice is the fact that the small order of a partial-response polynomial in many cases will lead to an output signal Yk of receive filter RF with less pre-cursive ISI than when a ~Matched Filter~ is used. As appears from the above article by Bergmans and Wong this generally leads to a transmission quality which is hardly inferior to the quality when a ~Matched Filter~ is used.
To illustrate this possibility of further simplification of post-detector PD it is observed that the impulse response fk of a much used class of digital magnetic recording systems at low information densities is characterized in good approximation by the simple bipolar polynomial PHN 12.039 15 g(D) = 1 - D ~38) as shown in an article ~Discrete-Time Models for Digital Magnetic Recording~ by J.W.M.Bergmans, Philips J. Res., Vol. 41, No. 6, pp.531-558, 1986. According to formula (36) it then holds that with M = 1 5 ~ -1, k = 0, 1 2, k = 1, Wk = (39) -1, k = 2, 0, k > 3, ~o that receive filter RF can be implemented digitally in an utterly simple manner.
Since the ISI in output signal Yk of receive filter RF
according to formula (33) is completely determined by impulse response Wk of receive filter RF, which in the present example is given by formula (3g), the dimensioning of feedforward and feedback sections FFS
and FBS for cancelling pre-cursive and post-cursive ISI will be completely determined by this impulse response Wk, and in this example for the respective impulse responses Pk and q~ of these two sections it will hold that 20 -1, k = 0 P~ = (40) 0, k > 1, and 0, k = 0 25 qk =-1, k = 1 (41) 0, k > 2.
So, in combination with the sign inversion to be performed in the su~oator, feedforward section FFS degenerates to a direct through-connection, ~hilst feedback section FBS degenerates to an extremely simple delay over one symbol interval T.
Such simplifications as shown in Fig. 4 are also possible in the case when equalizer EQ in the arrangement according to the invention is of the decision feedback type. This will now be further explained with reference to Fig. 5 differing from Fig. 4 in that now decision feedback equalizer EQ according to Fig. 2B is used.
In Fig. 5 adaptive equalizer EQ forms estimates ~k f data signal ak which are given by formula ~13), which formula under PHN 12.039 16 normal operating conditions can be simplified to formula (14).
In the conventional arrangement corresponding with Fig. 5, in which the auxiliary signal for post-detector PD is constituted by received data signal rk (as is shown by a dashed line in Fig. 5), receive filter RF sho~ld be matched in the best way possible to impulse response fk of transmission channel CH, compare for~ula (15), for the benefit of a maximum noise suppression. In the novel configuration of Fig. 5 input signal ~k f symbol detector ID in lieu of received data signal rk is used as an auxiliary signal for post-detector PD. Since ak contains the same noise component nk as received data signal rk, receive filter RF should again be a ~Matched Filter~ according to formula (15) for a maximum suppression of this noise component nk. Since actual knowledge about fk in post-detector PD is lacking, receive filter RF can now only be matched to the nominal impulse response gk of transmission channel CH, so that in this case Wk = gM-k = g-(k-M)- (42) The noise suppression achieved in this way is substantially the same as achieved with the conventional adaptive noise suppression.
For completeness it should be observed that in the conventional adaptive configuration according to Fig. 3~ it is also possible and advisable for reasons of complexity to use a non-adaptively approximated ~Matched Filter~ according to formula (42). In those cases in which actual iDpulse response fk of transmission channel CH is not equal to nominal impulse response gk, this will lead to a data component (a ~ f ~ g-)k-M in output signal Yk of receive filter RF which is not strictly equal to the original data component (a ~ f ~ f_)k_M according to formula (17). When utilizing feedforward section FFS for cancelling pre-cursive and post-cursive ISI, its impulse reponse P will have to be adapted to this modified ISI structure according to 0, k < -1, ~ g_)k_M~ < k < M - 1, P~ 0, k = M ~43) (f ~ s_)k_~ M 1 1 < k < 2M - 1 0. k > 2M - 1, which formula in the nominal case (that is to say, if gk = fh) is naturally the same as the original formula (20). In this case of a non-13296~2 PHN 12.039 17 adaptive receive filter RF according to formula ~42) thus an input signal ak_M of final symbol detector FD of the form k-M = (a * (f * g-))k-H ~ (a ~ P')k +
+ (n ~ g-)k-H (44) will develop.
When using formula (11) output signal Yk = (a * W)X of receive filter RF in the new situation of Fig. 5 can be written as Yk = (a * f * w)k - (t * q * w)k + (n * w)k (45) A comparison of this formula with the corresponding formula (16) for the conventional adaptive situation teaches us that this output signal Yk now has an additional contribution -(â * q * w)k in the form of tentative symbol decisions âk which are filtered by feedback filter FB in equalizer EQ and, subsequently, by receive filter RF in post-detector PD. In order to obtain an input signal ak_M of final symbol detector FD which is equal to the original input signal according to formula (44), this additional contribution -(â * q * w)k will have to be removed by cancelling same with an equally large additional contribution via feedforward section FFS.
Thereto, a contribution (q ~ w)k has to be added to the original impulse response Pk of feedforward section FFS. When cancelling pre-cursive as well as post-cur~ive ISI on the basis of tentative symbol decisions tkl the original impul-~e response Pk being provided by formula (43) owing to the use of an approximated ~Matched Filtern, it will now hold, therefore, that , k < -1, ~f * g-~k-M ~ ~q ~ W)k, 0 < k < M - 1, P~ -~q * W)M, k = M, ~46) ~f ~ g-)k-H ~ ~q * w)k, M + 1 < k < 2M - 1, 0, k > 2M - 1.

According to formulas (12) and (42) it now holds that ~9 * W)k (f ~ g-)k-H 9-(k-H) (47) 13296~2 PHN 12.039 18 so that formula (46) can be simplified to 0, k < -1, g-(k-M)~ 0 < k < M - 1, P = (48) go ~ (f * g_)O~ k = M, 0. k > M + 1, in which for k > M + 1 the a priori knowledge is used that gk is a causal function. According to this formula feedforward section FFS
apparently does not need to cancel post-cursive ISI any longer because this cancellation has already been effected in an adaptive way by the cascade arrangement of adaptive feedback filter F~ in equalizer EQ and non-adaptive receive filter RF in post-detector PD on the basis of tentative symbol decisions âk. It is evident that this will render the implementation of feedforward section FFS simpler. From formula (48) it appears that Pk only for k = M depends on the actual impulse response fk of transmission channel CH, which response is not known a priori. Naturally, the best possible, non-adaptive choice for this value PM is made by substituting the nominal impulse response gk for fk, so that p~ = gO - (g ~ g_~0 (49) To illustrate the dimensioning thus required of feedforward section FFS, the following table summarizes the i~pulse responses p~ of feedforward section FFS that are required when different partial-response polynomials gtD) are used for characterizing nominal impulse response gk of transmission channel CH.
~ ~--P2 ~ ~ ~
~ _1 _ -1 (1 + D)2 1 2 -5 With the exception of the response g(D) ~ D)2 (less 1~296~2 PHN 12.039 19 important in practice) all coefficients P have an absolute value 0 or 1, leading to a highly simplified digital implementation of feedforward section FFS.
As already mentioned hereinbefore, it is basically possible in the conventional adaptive post-detector PD to cancel post-cursive ISI on the basis of qualitatively better final symbol decisions âk_M by means of a feedback section FBS. It is clear that the just described procedure for determining an appropriate non-adaptive dimensioning of feedforward section FFS can also be implemented in this case for both feedforward section FFS and feedback section FBS
of post-detector PD. For brevity, this procedure will not be repeated here, but it will suffice to present the table arranged for this situation.
~s E~ Po _ P2--__ ~31 1 - D-1 -1 1 0 0 _1 20 1 + D1 _1 -1 1 0 l 1 - D2-1 O _1 1 O _l I

~1 + D)2 1 2 _5 -4 _~ 4 1 l The strongly quantized and ever still relatively short impulse responses in this table are easy to realize in practice even at high symbol rates 1IT.
To illustrate the transmission quality that is achievable with the arrangement according to Fig.5, Fig. 6 shows a set of qraphs obtained by simulation, and that for a pair of digital magnetic recording systems of the type described in the above article by Dergmans, which systems have different normalized information densities D. These graphs show the probability of erroneous symbol decisions, in this case the bit error rate (~ER), as a function of the signal-to-noise ratio ~SNR~ at the input of the arrangement. In both cases a partial-13296~2 PHN 12 . 039 20 response polynomial g(D) = 1 - D is used to characterize the nominal impulse response gk of transmission channel CH. For a very low normalized information density D = 0.1 the actual impulse response fk as shown in Fig. 6A is substantially equal to nominal impulse response gk = ~k-~ k-1- For a five times higher density D = 0.5, however, the actual impulse response fk as shown in Fig. 6B already starts deviating from the nominal impulse response gk = ~k ~ k-1~ although gk still forms a reasonable styling of fk. Curve (a) in Fig. 6A and Fig. 6B shows the theoretical optimum that is achievable with a detector arranged for estimating the maximum-likelihood sequence of tran~mitted data symbols ak, which optimum in this case coincides with the so-called ~Matched Filter ~ound~ (MFB).
Curve (b~ in Figs. 6A and 6B shows the bit error rate (BER) for the tentative symbol decisions âk at the output of equalizer EQ in Fig. 5. Curve (c) in Fig. 6A and Fig. 6B shows the bit error rate (BER) for the final symbol decisions ak_M when using a conventional arrangement according to Fig. 1 with an adaptive equalizer EQ as shown in Fig. 2B and an adaptive post-detector PD as shown in Fig. 3A.
Finally, curve (d) in Fig. 6A and Fig. 6B shows the bit error rate 20 (BER) for the final symbol decisions ak_H at the output of post-detector PD as shown in Fig. 5.
8ecause the actual and nominal impulse responses fk and gk in Fig. 6A are virtually identical, the di~ensionings of non-adaptive post-detector PD in Fig. 5 and adaptive post-detector PD as shown in Fig. 3A will be the same in this case, as explained hereinbefore. Therefore, the curves (c) and (d) in this Figure coincide. Comparing curves (c) and (d) on the one hand with curve (b) on the other shows that the quality of the final symbol decisions âk_M in this case is distinctly better than that of the tentative symbol decisions âk. Fig. 6A finally shows that the curves (c) and (d) naturally fall short of the ~Matched Filter Bound~ of curve (a).
In the situation of Fig. 6B nominal impulse response gk = k ~ k-1 forms no more than a reasonable styling of actual impulse response fk. The transmission quality which is achievable with a non-adaptive post-detector PD as shown in Fig. 5 (curve (d)J adjusted to nominal impulse response gk will naturally fall short of that of a PHN 12.039 21 fully adaptive post-detector PD as shown in Fig. 3A (curve (c)), which for that matter is fully adapted to the precise form of actual impulse response fk. Nevertheless, as explained hereinbefore, the differences between curves (c) and (d) remain relatively small. Like in the situation Gf Fig. ~A the quality of the final symbol decisions âk_M according to both curve (c) and curve (d) is distinctly better than that of the tentative symbol decisions âk at the output of equalizer EQ (curve (b)). Needless to observe that the two curve3 (c) and (d) for their part again fall short of the ~Matched Filter Bound~ of curve (a).
A further increase of the information density D leads to an actual impulse response fk deviating ever more from the nominal impulse response gk = k ~ k-1. From a certain density D the nominal impulse response gk will form such a little realistic styling f fk that the quality of the final symbol decisions ak M of post-detector PD in Fig. 5 is not better, but even worse than that of the tentative symbol decisions âk. Simulations have shown that this is the case for densities D from approximately D = 1.2, for which holds fO 1, f1 = ~ 0 33' f2 = ~ 39~ f3 = ~ 15 f4 = - 0.06, f5 = ~ 0 03~ f6 = ~ 0.02, f7 = - 0.01 (50) so that the styling gO = 1, g1 = -1 is clearly unrealistic.
Also in the case when post-detector PD is arranged for estimating the maximum-likelihood sequence of transoitted data symbols a priori knowledge of the main character of the transfer characteristic of transmission channel CH can be used for designing post-detector PD in a non-adaptive and thus simpler way. This will now be further explained with reference to Fig. 7 showing a block diagram of an arrangement according to the invention with a decision feedback equalizer EQ as ~hown in Fig. 2B and a non-adaptive post-detector PD which is arranged for estimating the maximum-likelihood sequence of transmitted data symbols.
In Fig. 7 adaptive equalizer EQ forms estimates ak of data signal ak which are given by formula ~13), which formula under normal operating conditions can be simplified to formula (14).
In the conventional arrangement corresponding with Fig. 7, in which the auxiliary signal for post-detector PD is 1329~42 PHN 12.039 22 constituted by received data signal rk (compare Fig. 3B), feedforward section FFS should have an impulse response P which corre~ponds in the best way possible with residual impulse response fk f transmission channel CH, compare formulas (28) and (29), in order to reduce the span of post-cursive ISI in input signal bk f Viterbi detector VD to the memory span LT of truncated impulse response fk of transmission channel CH, compare formulas (27), (30) and (31). In the novel configuration of Fig. 7, input signal ak of symbol detector ~D in equalizer EQ is used in lieu of received data signal rk as an auxiliary signal for post-detector PD. Compared to rk this auxiliary signal has an additional contribution -(â ~ 9)k = -(â ~ (f- ))k in the form of tentative symbol decisions âk which are filtered by feedback filter FB in equalizer EQ. In order to obtain an input signal bk of Viterbi detector VD which is identical with the original input signal according to formula (30), this additional contribution -(â ~ (f- ~))k will have to be removed by cancelling same by an egually large additional contribution via feedforward section FFS. This means that in case of original impulse response P = frk a contribution (~ - f)k will have to be added. The new impulse response P then becomes P fk ~ ~k ~ fk (51~
which expression, when using formula (26), can be simplified to P~ = k ~ fk- (52) Since a priori knowledge of the precise form of fk and thus also fk is lacking, feedforward section FFS can now only be dimensioned in accordance with the nominal impulse response gk of transmission channel CH, so that P = k ~ gk (53) where gk is the truncated version of the nominal impulse response gk, that is to say, gk~ < k < L, gk = (54 0, k> L ~ 1.
On the basis of this formula and formula (14) it now appears that in the absence of erroneous tentative symbol decisions âk an input signal bk of Viterbi detector VD in post-detector PD will occur that has the form PHN 12.039 23 13 2 ~ ~ 4 2 bk = ak + nk ~ (a 1 ( ~ 9 ))k = (a ~ st)k + nk. (55) This formula clearly shows that the fixed correlation structure of input signal a~ of symbol detector ID in combination with the non-adaptive form of feedforward section FFS leads to a fixed correlationstructure of data component (a ~ gt)k of input signal bk of Viterbi detector VD. Therefore, once again a post-detector PD which is completely non-adaptive and hence simpler to implement will suffice.
Like in the preceding cases, the achieved simplification of the implementation of post-detector PD is accompanied with only a slight impairment of the transmission quality considering the relatively small deviation between the nominal and actual truncated impulse responses gk and fk. In the nooinal case, that is to say, if fkt = gk, non-adaptive post-detector PD according to Fig. 7 is even totally equivalent to conventional adaptive post-detector PD as shown in Fig. 3B.
When using a partial-response polynomial g~D) for characterizing the nominal truncated impulse response gtk an input signal bk of Yiterbi detector VD in post-detector P~ will occur that has an extremely simple correlation structure. In that case it is possible to use a simpler final symbol detector in lieu of Viterbi detector VD for forming final symbol decisions ~k-M. Such detectors are known, for example, from an article ~On Decoding of Correlative Level Coding Systems with Ambiguity Zone Detection~ by J.Xobayashi and D T.Tang, published in IEEE Trans. Commun. Technol., Vol. COH-19, No. 4, pp. 467-477, August 1971. Despite the fact that the detectors described in this article lead to a transmission quality falling slightly below to the quality achieved with the Viterbi detector, their implementation is many times simpler.
Naturally, it is also possible to employ in the post-detectors PD as shown in Fig. 4 and Fig. 5 final symbol detectors of a different kind in lieu of the simple final symbol detector FD employed there, for example, of the type described in the above article by Kobayashi and Tang. In order to achieve the correct correlation structure of the input signal ak_M of such a final symbol detector FD it will generally be necessary to adapt the impulse responses P and qk of feedforward and feedback sections FFS

13296~2 PHN 12.039 24 and FBS derived hereinbefore to this correlation structure. Because it will be evident after the extensive explanations given hereinbefore how such an adaptation has to be effected, this adaptation will not be described any further for brevity.
It will also be clear from the foregoing that the procedure for dimensioning a non-adaptive post-detector on the basis of a priori knowledge of the main character of the transfer characteristic of transmission channel CH is so universal that this procedure can also be implemented for adaptive equalizers EQ and non-adaptive post-detectors PD of a different kind.

Claims (7)

1. Arrangement for combating intersymbol interference and noise which are introduced into a data signal transmitted at a symbol rate 1/T by a dispersive transmission channel with a transfer characteristic only the main character of which is known a priori, said arrangement comprising an adaptive equalizer having at least one adaptive filter for deriving a first detection signal from an input signal of the arrangement and having a fixed symbol detector for taking tentative symbol decisions on the basis of the detection signal, the arrangement further comprising combining means for combining the detection signal and a signal derived from the tentative symbol decisions into a detection final signal and a non-adaptive post-detector for taking final symbol decisions on the basis of the final detection signal.

2. An arrangement as claimed in claim 1, wherein the non-adaptive post-detector comprises means for determining the difference between the final detection signal and an expected value of said final detection signal corresponding to the final symbol decision, and wherein the adaptive equalizer is adapted on the basis of said error signal.

3. An arrangement as claimed in claim 2, characterized in that the adaptive equalizer is a linear equalizer.

4. An arrangement as claimed in claim 1 or 2, characterized in that the adaptive equalizer is a decision feedback equalizer.

5. An arrangement as claimed in claim 1 or 2, characterized in that the post-detector comprises a receive filter connected to the input of the symbol detector, a feedforward section for forming a first cancelling signal for intersymbol interference in response to a number of consecutive tentative symbol decisions, a summator for adding together the output signal of the receive filter and the first cancelling signal to an input signal of a final symbol detector for forming final symbol decisions, the receive filter being matched to the main character of the transfer characteristic of the transmission channel.

6. An arrangement as claimed in claim 5, characterized in that the post-detector also includes a feedback section for forming a second cancelling signal for post-cursive intersymbol interference in response to a number of consecutive final symbol decisions, which second cancelling signal is also supplied to the summator.

7. An arrangement according to claim 1, wherein the post-detector comprises a detector of the Maximum Likelihood Sequence Estimation type.