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Application of ARMA models to automatic channel equalization
I N F O R M A T I O N S C I E N C E S 22, 1 0 7 - 1 2 9 (1980)
107
Application of A R M A Models to Automatic C h a n n e l E q u a l i z a t i o n * G. C. GOODWIN H. B, DOAN and A. CANTONI
Department of Electrical and Computer Engineering, University of Newcastle, New South Wales, Australia, 2308 Communicated by Harold W. Sorenson
ABSTRACT This paper describes a novel approach to automatic channel equalization m digital transmission systems. The approach is based on the use of a finite-dimensional rational approximation to the channel characteristics. This class of channel approximation has the following advantages: it allows a finite parametrization of the channel impulse response which may be of infinite duration, it aLlows for the possibility of the noise being colored, it applies to either single- or multiple-channel systems, and it has the pedigogical advantage that many other algorithms in current use are based on models which are special cases of this model. The rational approximation to the channel characteristics is used in the paper to develop a new receiver structure using fixed-lag smoothing ideas. Simulation studies are presented showing that the receiver offers advantages over other algorithms for mitigating the effects of intersymbol and interchannel interference including those arising from carder phase errors.
1.
INTRODUCTION
I n t e r s y m b o l i n t e r f e r e n c e ( I S I ) arises in h i g h - s p e e d p u l s e a m p l i t u d e m o d u l a t e d ( P A M ) d a t a t r a n s m i s s i o n d u e t o the d i s p e r s i o n of t h e p u l s e s h a p e b y i m p e r f e c t c h a n n e l c h a r a c t e r i s t i c s . T h e p e r f o r m a n c e of m a n y c h a n n e l s u s e d f o r digital t r a n s m i s s i o n is a f f e c t e d b y this k i n d of s i g n a l d i s t o r t i o n a n d this l i m i t s t h e r a t e of t r a n s m i s s i o n . *This work was supported by the Australian Research Grants Committee and the Radio Research Board. ©Elsevier North HoLland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017
0020-0255/80/08107-23501.75
108
G . C . GOODWIN, H. B. DOAN, A N D A. CANTONI
Over the past decade efforts to mitigate the effect of ISI and noise have been centered around the development of linear [1-4] and nonlinear [5-9] receivers. Linear receivers based on peak-distortion and mean-square-error performance indices have been extensively studied (e.g. [1,2,3,4]). These receivers are relatively simple to implement. However, they do not perform optimally, because they tend to enhance the noise, especially when the channel possesses nulls in the amplitude characteristics [2]. Nonlinear receivers are capable of improved performance compared with linear receivers, but usually at the cost of significantly greater complexity. For this reason, the usual approach, in practice, has been to use either linear receivers or simple nonlinear receivers such as those incorporating decision feedback and threshold detection [5, 10]. More complicated receivers have also been studied in an attempt to achieve a better compromise between receiver complexity and performance. For example, Qureshi and Newhall [11]. and Falconer and Magee [12] have used a linear prefilter plus Viterbi detector. More recently, Cantoni and Doan [13,28] have given an interpretation to this receiver structure based on a least-squares identification algorithm. In the case of multichannel systems, it is possible for the signals to experience interchannel interference as well as intersymbol interference. For example in QAM systems [29], phase errors in the reference carriers used for demodulation change the allocation of the two portions of the received signal that contribute to the two demodulated waveforms. Other examples of interchannel interference arise from atmospheric propagation anomalies in dual polarized radio communication [30] and from electromagnetic coupling between cable pairs in multicore cable systems. The characteristics of interchannel interference can be constant or randomly time varying. Moreover, they are frequently unknown a p r i o r i at the receiver. Hence adaptive equalizer receivers are often required to adapt the detection scheme. In a recent paper [31] adaptive algorithms for mitigating interchannel interference in dual-channel systems having no intersymbol interference were studied. In a more general case, several maximum-likelihood receivers for multiple-channel transmission systems with multidimensional interference were developed [32, 33]. In the current paper we use an autoregressive-moving-average (ARMA) description of the channel characteristics. This model has a finite number of parameters but allows for an infinite channel impulse response and for colored noise. The ARMA model is a very compact way of describing the channel characteristics, since it incorporates both poles (the autoregressive component) and zeros (the moving-average component). The ARMA description never contains more parameters than a standard finite-impulse-response (FIR) model since the latter is simply a special case of the A R M A model corresponding to the unlikely occurrence that the channel has no poles and the noise is white.
AUTOMATIC C H A N N E L EQUALIZATION
109
A new receiver structure is developed using the ARMA channel model together with Kalman filtering and fixed-lag smoothing. The receiver is designed to minimize the intersymbol and interchannel interference. Previous applications of Kalman filtering and fixed-lag smoothing to equalizer design have assumed an FIR model for the channel [35-41]. As mentioned above, an adequate FIR approximation of a channel usually requires a large number of terms in the FIR description. Thus the resulting Kalman equalizer of fixed-lag smoother is generally complex, since it requires at least as many states as are in the FIR model. This makes the equalizer complex to implement and difficult to make adaptive [41, 40, 35]. By contrast, the fixed-lagged smoothing algorithm that we shall develop in this paper has few states, since it will be based on the more compact ARMA description for the channel. This means that it is easier to implement and potentially simpler to make adaptive. The resulting fixed-lag smoother receiver is essentially a linear filter incorporating both poles and zeros. In order to obtain an estimate of the transmitted signal, we propose that the linear filter be followed by a simple slicer. Since the filter is linear, it could perhaps be argued that its characteristics could be adequately approximated by a somewhat longer FIR filter. The advantage of the latter approach would be that simple least-squares techniques could be used to determine the filter coefficients. On the other hand, an F I R filter will generally require many more coefficients than the fixed-lagged smoother, and this in turn implies increased complexity in the receiver and computation difficulties if the receiver is made adaptive [13, 27, 21]. This indicates that the compact fixed-lag smoother based on the ARMA description may offer advantages in practice and is therefore worthy of consideration as an alternative receiver structure. The layout of the paper is as follows: in Sec. 2 we describe the A R M A model and relate it to other commonly used channel models; in Sec. 3 we describe recursive algorithms that can be used to adaptively fit the A R M A model, in Sec. 4 we describe the linear fixed-lag smoothing algorithm for input recovery, and in Sec. 5 we describe single-channel simulation studies comparing the new algorithm with other algorithms. Finally in Sec. 6 we present some simulation results for QAM systems having both interchannel and intersymbol interference. The simulation results were obtained for various receiver structures which have approximately the same computational complexity for the equalizer implementation. 2.
THE C H A N N E L MODEL
In this paper we shall use a discrete-time channel model which allows for both severe ISI and colored noise. The equivalent baseband signal model that
110
G . C . G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
we propose is a mixed A R M A model having the following form in the single-channel case [14]: "
y,= - • k-- I
'
akyt_k +
bkU,_k + ~_~ c~,et_k, k-- I
(2.1)
k--O
where {u,} denotes the input data sequence, and {e,} denotes a white noise sequence. In this equation a = ( a j ..... an) r is the vector of autoregressive parameters, fl=(bl ..... bin)r is the vector of moving average parameters and y = ( c I ..... c=) is the vector of noise parameters (c o can be normalized to l without loss of generality). The variance of e, is taken to be or. The channel model given in (2.I) allows for a rational input-output transter function of the form B ( z ) / A ( z ) and an output noise power density spectrum of the form
~P~(z)- C ( z ) C ( z - I ) o,2, A(z)A(z-')
z=e/~',
where
B(~-')=b~z-' + . . . +b,,z-", A(z-m)= l + a l z - ' + ''- +a,,z-", C ( z - l ) = 1+ c l z - t + " .
+clz-t.
Note that the impulse response of the above channel model can be infinitely long, since the roots of A act as system poles. The roots of B give the transmission zeros. The channel model given in Eq. (2.1) contains, as special cases, other commonly used models, for example: (1) The usual finite-impulse-response model [15] with white output noise can be obtained by setting = 0 (no autoregressive component), /3 = h (the channel impulse response), ~t=0. (2) A transfer-function model with white noise is obtained by setting y = a . Moreover, many standard equalization procedures can be given a new interpretation based on the use of (2.1) as a channel model. For example, if we use j S = ( b l , 0 , 0 .... 0) and y = 0 , then it can be seen from (2.1) that if we operate
AUTOMATIC CHANNEL EQUALIZATION
111
on the received sequence (y,) as follows:
zt=Yt+ ~ akyt-k, k--l then u , _ l = l / b l z , + l / b : , , a n d i / b l z , is a suitable estimate of the input signal u,_ 1. This can be seen to be equivalent to the standard linear equalizer structure [3]. The model of Eq. (2.1) can be readily extended to multichannel systems [14]. The essential changes are to interpret (Yt), {u,), {e,} as vectors, to replace ( a k, b k, Ck) by square matrices with c o = I , and to define the covariance of {et) as a positive definite matrix Y.. 3.
ON-LINE ESTIMATION ALGORITHMS
W e propose to use a recursive algorithm for fitting models of the type described in Sec. 2 to channel data. F o r simplicity, we consider the scalarchannel case, for which the general algorithm [14] m a y be described by
(3.1)
~t+l ~ t + Kt+ ltgt+ l ,
e,~,÷,
(3.2)
Kt+ 1 - I + , # r , P:p,÷, '
p,+,
= ?,
P,~,+,'C,P, T
,
(3.3)
1 + ~,+ iPt~t+ l t)t + i = Y t + 1
-I-y,
..... - y , _ , + ~ , u . . . . . . u , _ , . + , , ~,_, . . . . . ~,_,+~] 0 ( t ) ,
(3.4)
where 0 = ( a r, fir, y r ) r is the p a r a m e t e r vector to be estimated and the quantities i,, %, @, may be different for different algorithms. The results described in this p a p e r were o b t a i n e d by use of the extendedmatrix method [14]. In this method, i ( t ) , ~p(t), and @(t) are as follows:
T
el+l = v t + l =Yt+l --~,+ l~t , ~b, = % = [ - y , _ I . . . . . - y . . . . u,_ 1. . . . . u . . . . v , - I . . . . . v,-t] r.
(3.5)
In the above algorithm, knowledge of the transmitted sequence {u,} is required. This difficulty can be overcome by using two modes of operation for
112
G . C . GOODWIN, H. B. DOAN, A N D A. C A N T O N I
the adjustment of the coefficients. In the initial training mode the input is set to a known sequence. Then during normal operation the coefficients can be adapted by replacing the unknown inputs by the final output of the detector. If the computational constraints are very severe, then it may not be possible to use the time varying gain computation given in (3.2) and (3.3). In these cases, a simple stochastic-approximation variant can be used in which K, is computed from 1
,36,
and rt is a scalar given by r,
1
r
r, , + 7 ( q , t # t - - r , _ , ) .
(3.7)
This simplified algorithm requires less computation but is slower to converge. Ljung [16, 19] and Solo [42] have recently presented comprehensive analyses of the convergence properties of algorithms of the general form of (3.1). This analysis shows that the extended matrix method will converge under the weak is positive real, i.e. Re[l/C(e-iW)-½]>O, wE condition that I / C( z - i ) _ [-~r, cr]. (The stochastic-approximation form requires only C(z -I) positive real.) A closely related algorithm, called approximate maximum likelihood or R M L 2 [16, 19], can be shown to always converge to a local minimum. Our experience, gained from simulations, is that the extended-matrix method works well for fitting A R M A models provided the roots of C are projected into the unit circle. If required, the algorithms may be modified to track slowly time varying channels by a number of standard techniques, e.g. by the introduction of an exponential forgetting factor [14]. In fact, initial convergence can often be improved by introducing a time varying forgetting factor which converges exponentially to 1 [34, 42]. The algorithm can be applied, virtually unaltered, to the multichannel case, the only problem being that additional constraints are required on the matrices (a k, b k, ck} to ensure that the model is canonical [14] and hence to guarantee that the parameters are uniquely determinable. A simple solution is to make each of the ck matrices diagonal. Moreover, this approach has the added advantage [24] that the parameters can be estimated by a series of scalar algorithms in which 0 denotes the parameters in the ith rows of C(z-I), A(z-i), and B(z-1), and q~, is replaced by qS, = [ -y,'_ t ..... _ y , _ . , _y2_l ..... _y,2_ . . . . . . _y,q_., b l 't _ l ~ . . . ~
- - I d t '_ m ~
~,-~ . . . . . ~ , - t ] r
uL,
.....
-
~,~_.,
,...,
- ~q_.., (3.8)
AUTOMATIC CHANNEL EQUALIZATION
113
It is important to note that the recursive algorithm given in (3.1) to (3.5) is no more complicated than similar recursive schemes that are used with the more conventional F I R models [27, 28]. 4.
INPUT RECOVERY VIA FIXED-LAG SMOOTHING
Using the A R M A signal model described in Sec. 2, there are a n u m b e r of possible algorithms that could be used to recover the input sitrnal. W e shall describe an algorithm based on fixed-lag smoothing. As noted in the introduction, K a l m a n filter and fixed-lag smoothers have traditionally been used with F I R models. In the case of A R M A models, there is an a d d e d complication in that the states are not delayed versions of the input. Thus a completely different fixed-lag smoother structure is required. F o r simplicity, we describe the scalar-channel case, but the extension to multiple channels is i m m e d i a t e provided scalars are changed to vectors and matrices where necessary. W e first prefilter (Yt} to form (z,) as follows: n
z,-~y,+ ~
Sky,_ k
(4.1)
k--I
where (d I . . . . . 5 , ) denotes the estimates of ( a I . . . . . a , ) . It follows from (2.1) that if 5 k ' ~ ' a k, then (z,) can be m o d e l e d as follows: m
1
Z, = Z bkUt--k -~ Z Ck~'t-k" k--l k--O
(4.2)
Since the filter used in (4.1) is stably invertible, {z,) carries the same information content as (Yt). A detection technique is now derived to recover the transmitted d a t a (u,} from {z,}. F o r simplicity we assume that m = l. The following state space model is then equivalent to (4.2): x,+ I = F x t + B u , + C e , , (4.3) z t ~ Hxt'~'et, where
F=
ioo... 1
0
-.-
0
0
1
--'
0
0
0
-••
1
(4.4) .1×.,
114
G . C . G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
B = [ bm C=[
b._,
Cm
H=[0
""
Cm-- 1
---
"'"
0
bl ] r , el]
r ,
1] r.
(4.5)
(4.6) (4.7)
We seek to determine a fixed-lag smoothed estimate of u,_u, viz. a , _ ~ l , = E [ u , - N l Z o , Z l . . . . . z,]
(4.8)
for all .t and some fixed-lag N. Consider the following augmented-signal model: xl+ i = Fxc+ But + Cet, gt-~ nxt'k-et,
Ul+ 1 =Ig t ,
(4.9)
hi2+ 1 - - Igt - - U t - I
/d3+ l = l/t2 = /,/I_ 1 = / ' t _ 2 ,
N+ I ~/dtN~ Igt+ I
,, "
"~ l g t - N "
Introducing a new state vector
X t
x;=
I
(4.10)
I
u U+l
we can express (4.9) in the form x;+ ~ = A x ; + Gut + J e t , z, =Lx~ +et,
(4.1 l)
AUTOMATIC CHANNEL EQUALIZATION
!,0
115
where A, G, J, L have the following structure:
-b-t
A=
o .
0
G=
i]I:----°-] =/o:
1
=
1
L:
J
el-- ,
(4.13)
(4.14)
J=
,~=[,
(4.12)
.o
...
o]
(4.15)
N o w the optimal linear filter for {x;} is well known [17] to have the following form:
~;+,~,=A~;._, +( AP,~,_,L~+ S)( LP,j._,L T+ R)-'v,.
(4.16)
P,+II,=AP,I,_IAr+Q -(Ap.._,LT+S)(Lp,._
,L ~ +R)
-,
(APot_,Ar+S) r, (4.17)
where ~;+tl, denotes the best estimate of x,+~ given (z o, z~,. • •, z,), a n d v, = z, -
L~:;I ,_ l,
(4.18)
Q=GGrod +JJrod,
(4.19)
S=Jo~.
(4.20)
116
G . C . G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
Due to the special structure of Eqs. (4.12) to (4.15), the above equations can be partitioned and simplified as follows:
[ Tt ] =( AP, I,_,L r + S)( LP, It_,L r + R )- ',
Pt+ ,It --
p/+l
lit L ll'] • p/2
(4.21)
(4.22)
21 ' 22 Pt+ l[t P t + lIt
Using the above partitioned matrices, the filter can be rewritten as follows: £,+ tit----F'~,I,-.
+ 7,[ z,- H.~,lt_ I ],
"Yt--(FP, I~_,HT+Co,2)(HP~I~_,HF+o,2)-I, Pt+II,--" - FPtI',-IFT +°dBBT
/dlJt- 1
t~]+ ,It ..[[21
li,21,- I
AN+I [d/+ lit
I~N+ 1 l
+,,[
(4.25)
z,- HX,f,_ l ],
(4.26)
tit-
11 T rk,--~Pt~)_,H T ( HPtl,_,H +o,2) - I , 21 et+l[,--
(4.24)
+°, 2CCT
- y,( FP,I',_ ,Hr+ Co,2) r, t~J+ ii t
(4.23)
[ 2 p 2tit1 IFT+ou2
e,B r -¢k,(Fe, IIl,_tHr+Co~) r.
(4.27) (4.28)
Finally, the required linear fixed-lag smoothed estimate of the input is given by I^d t - N I t - - b l^N+ t + llIt •
(4.29)
REMARK 4.1. The linear fixed-lag algorithm described above does net take account of the fact that {u,} is drawn from a finite alphabet. The quantized estimate of {u,} can be obtained by simply selecting the nearest level to fi,-Np,, i.e. in the binary (__+1) case u, _ N (quantized) = sign a,_ N"
AUTOMATIC CHANNEL EQUALIZATION
117
REMARK 4.2. Note that Eq. (4.23) to (4.28) are not the standard fixed-lag smoothing equations for state estimation. The equations we have presented are simpler because we have to sort only estimates of the input, not the complete state vectors. Thus Pt+H, 21 has dimension ( N + I ) × n only, where n is the order of the model and N is the lag length. The structure of the smoother is depicted in Fig. I. It can be seen from the figure that t~,_Ni, can be recovered from the scalar innovations sequence ( z , - H ~ , I ,_ l) by ( N ) multiplications using the scalar quantities ¢2 ..... eta+ t. We have found that n = 2 and N = 6 give good performance. (Increasing n and
~ <
z +
t
[Data]
I
[ADDITIONAL SIGNAL PROCE~q S ING REQUIRED TO RECOVER THE INPUT]
u2,
tit-1
u~+llt~ i
^N+I
ut+llt ^N+I
~
~'
Ut+llt = Ut_N] t
<
[THE RECOVERED UNIT]
Fig. 1. Structure of the fixed-lag smoother.
118
G . C . GOODWIN, H. B. DOAN, AND A. CANTONI
N beyond these values gives marginal improvement, but most of the gain can be obtained with the small values quoted above.) REMARK 4.3. The closed-loop system matrix for the fixed-lag smoother is given by A-,=A - K , L _ [ F-y,H - ¢k, H
0] ~2
"
(4.30)
It is clear from this matrix that the stability properties of the smoother are determined by those of the nonlagged Kalman filter for the state. It is readily verified from Eqs. (4.3) to (4.7) that the state-space model for the channel is completely observable. Thus it is known [17] that the Kalman filter is asymptotically stable and hence this also applies to the fixed-lag smoother for recovering (u,). I1 REMARK 4.4. It should be noted that the quantities "t,, P,+~I,, ¢,, P , +21~ I , given by Eqs. (4.24), (4.25), (4.27), (4.28) tend to steady-state constant values as t becomes large. Thus, in subsequent calculations the steady-state values can be used in Eqs. (4.23) and (4.26) without further need to use Eq. (4.24), (4.25), (4.27), (4.28). In steady state, the quantities y,, q~,:..... ¢~ in Fig. 1 can be replaced by the constant values Yc, '#f ..... ¢~. This represents a significant computational saving, since for the examples described later, steady state is achieved in less than 10 iterations of Eqs. (4.24), (4.25), (4.27), (4.28).
5.
SINGLE-CHANNEL SIMULATION RESULTS
In this section applications of the ARMA model and of the associated detection scheme are demonstrated for single channels by simulation examples. A number of different baseband channels have been tested [25]. Here we show the results for two channels. Channel 1 (see Fig. 2) is taken from Proakis and Miller [4], and Channel 2 (see Fig. 3) is a typical telephone channel [43]. The simulation experiments were carried out as follows: (1) A pseudorandom binary (PRBN) training sequence was applied to the channel, and the channel output was corrupted by white (or colored) Gaussian noise. The noise level was set to give a specified signal-to-noise ratio at the receiver input (channel output). (2) The training data (input-output pairs) were used in the recursive algorithm of Sec. 3 to fit an ARMA model to the channel and noise characteristics. (3) When the model parameters converged, these parameters were used to determine the steady-state coefficients in the fixed-lag smoother by solving the ~equations given in Sec. 4.
119
AUTOMATIC CHANNEL EQUALIZATION
,
|
*
(a)
IH(w) t
J
i
Co) CHANNEL
/
Fig. 2. Channel I: (a) channel impulse response; Co) frequency response.
[
I
I. t
,
o~2~
(a)
la(~,) ]
\ w
(b)
CHANNEL 2 Fig. 3. Channel 2: (a) impulse response; ~)'frequency rcsponsc.
120
G . C . GOODWIN, H. B. DOAN, A N D A. C A N T O N I
(4) An unknown PRBN signal was applied to the channel, and noise was again added at the output. The received data were passed through the steady state filter (3), and the output of the filter was passed to a slicer to give the recovered input. (5) Multiple runs of the type described in (4) were performed to estimate the probability of error. Up to l06 binary digits were used to give a good estimate of the probability of error depending on the signal-to-noise ratio. Before we discuss the results, some preliminary observations are important: (l) Effect of model time delay. In practice, the received signal message possesses an inherent delay property [26] and this delay property plays an important role in modeling the channel as in Eq. (2.1). The time delay must be appropriately adjusted to achieve accurate channel modeling. Our experience with a number of simulation examples indicates that the optimal value for the time delay is approximately equal to the first major peak on the impulse response. (2) Effect of model order. It is of course necessary to determine appropriate orders for the polynomials A ( z - l ) , B ( z - i ) , and C ( z - l ) . Moreover, for a fixed total number of parameters, the problem arises how best to apportion these parameters between A ( z - i ) , B ( z - i ) , and C ( z - i). This problem is neither new nor peculiar to the A R M A model. For example, in compromised Viterbi receivers a choice must be made between the length of any prefilter and the truncated impulse response. Fortunately, for the fixed-lag smoothing algorithm described here, we have found that the choice of the relative degrees in the A R M A model is not critical. We obtained good results by fixing the orders of A, B, C at 2. Increasing the orders beyond these values gave insignificant improvement. (3) Effect of lag length. It is also important to set the lag length N of the smoothing algorithm to achieve reasonable probability of error without excessive computing. In our simulations a lag length of 5 or 6 gave the best performance. Increasing the lag length further actually deteriorated the performance because the modeling errors become more significant at high lags. Simulations were done for following channel configurations: (a) Addititm white noise at the channel output. This is the usual equivalentchannel model where the input sequence is passed through a linear transfer function and the observed output has a white noise sequence rl, =e, added to it. Co) Additive colored noise at the channel output. The situation is similar to that described in (a) except that the additive noise is colored. The noise sequence at the output was generated by the following model:
Tlt=et+O.8e,_ I + 0.6et_ 2 with {e,} white noise.
(5.1)
AUTOMATIC CHANNEL EQUALIZATION
121
Figures 4, 5, and 6 show typical results from the simulation studies. The figures show results for the following algorithms: (a) Smoother. This is the fixed-lag smoothing algorithm described in Sec. 4. In all cases the order was 2, the lag length 3 to 6. (b) RLS-VA. This is the least-squares Viterbi algorithm described in [13], [28]. The number of autoregressive prefilter coefficients was set to 7, and the number of points on the desired impulse response (DIR) was set to 3. This particular algorithm was chosen for comparison with the smoothing algorithm described in this paper because it has been shown in [13] that the RLS-VA algorithm has approximately equivalent performance to the optimal-eigenvalue algorithm described in [12] and superior performance to all other algorithms of equivalent complexity. Moreover it is shown in [13] that the RLS-VA algorithms can be considered as a recursive form of the prefiltcr algorithms proposed by Falconer and Magee [12] and also by Cantoni and Kwong [21]. (c) FIR algorithm. This is a finite-impulse-response equalizer as in [25] with I I coefficients.
Pr(e)
I0 °
1o-]
\
\
! I0-5~
m
i0
15
SNR(dB)
Fig. 4. Error performance for channel 1 with white output noise: A, smoother; B, RLS-VA; C, FIR equalizer; D, one-shot bound.
122
G . C . GOODWIN, H. B. DOAN, A N D A. C A N T O N I Pr{e)
i0 °
I0
-i
10 - 2
10 -3
10 -4 .
B
10
-5 ,
,
10
15
SNR(dB)
Fig. 5. Error performance for channel 2 with white output noise: A, smoother; B, RLS-VA; D, one shot bound. (d) One-shot bound. It can be shown [25] that for any detection scheme, the one-shot error probability is bounded below by Q(½ v ~ ), where Q gives the tail probability of the N(0, 1) Gaussian distribution, and ¢¢
Here Am is the minimum difference in input signal level (i.e. 2 for the binary _1 case), H ( z - I ) = B ( z - ' ) / A ( z -I) is the channel impulse response, and • ,(e jw) is the channel output noise power density spectrum. It can be seen from the results in Figures 4 to 6 that the A R M A model plus fixed-lag smoothing algorithm gives good probability-of-error performance in the presence of severe intersymbol interference. The RLS-VA algorithm actually performs better than the smoother, but the computing time per recovered sample using RLS-VA is approximately twice that of the smoothing algorithm [25].
AUTOMATIC CHANNEL EQUALIZATION
123
Pr(a)
i0 -]
5
i0
15
SNR (dB)
Fig. 6. Error performance of channel 2 with colored output noise: A, smoother; B, RLS-VA; D, one-shot bound.
6.
MULTICHANNEL SIMULATION RESULTS
A Q A M system was simulated having a phase error 8 between the received signal and the local oscillator. The situation is described as follows: Consider the block diagram of the coherent demodulator as shown in Figure 7. Suppose there is a phase error 8 between the received signal and the local oscillator. As a result of this error the received signal will be demodulated using cos(60ct+8 ) and sin(o~ct+8 ). The output of the demodulator for the in-phase and quadrature channels after low-pass filtering are
yl(t) =
½sl(t) cos 6 - ½s2(t)sin 6,
(6.1)
y 2 ( t ) - ' i s 2 ( t ) c o s 6+ ½sl(t)sin6.
(6.2)
-
124
G . C . G O O D W I N , H. B. D O A N , A N D A.
CANTONI
S(t)=s1(t)c°swe~~/ Low Pass Filter
1 Yt
~ coS(~ct+6)
~ ~_~
+ s2(t)sinWct ~
90 °
t
Low pass ] Filter
2
Yt
Fig. 7. Block diagram of coherent demodulation system.
It can be seen that the effect of the phase error is to introduce a gain factor which causes a reduction in noise immunity; in addition it introduces crosschannel interference. In general, the equivalent b a s e b a n d signals of the Q A M system with channel impulse responses h2(t), h'2(t) will be of the form cos 8
sin 8
~- ~ , u n h ' l ( t - n T ) - T ~,v~h'2(t-nT)+nl(t ),
yl(t,8)-
tl
y2(t,(~)-
rl
cos 8
sin 6 ~ - ~ v , h2(t-nT)+--~-~u~h't(t-nT)+n2(t);
rt
/1
(6.3)
combining the gain factor (cos 8 ) / 2 into the impulse response, this becomes:
yl(t, 8) = ~ u~hl(t-nT)-
(tanS) ~
n
vnh2(t-nT) +nl(t ),
n
y2(t,8)=~,v, h2(t-nT)+(tanS)~u,hl(t-nT)+n2(t). n
(6.4)
rl
A number of different b a s e b a n d channels have been used in the simulation tests [25]. In Table 1 we show the results for a channel having two identical paths of the form used by Proakis and Miller [12]. TABLE I Channel h~ hi
Sampled impulse response 0.06 0.06
-0.07 -0.07
0.1 0.1
-0.3 -0.3
-0.7 -0.7
1.0 0.5 0.0 0.3 0.05 1.0 0.5 0.0 0.3 0.05
0.1 0.1
AUTOMATIC CHANNEL EQUALIZATION
125
In the simulation tests the input sequences (Uk} and (Vk} tO the in-phase and quadrature channels were pseudorandom binary (_+ 1) sequences having zero mean and unit variance. The norm of each channel impulse response was normalized to 1 so that no signal gain or attenuation was introduced by any channel. The additive noise sequence to each path was taken to be zero-mean white Gaussian. Simulation tests were run at signal-to-noise ratios between 7 and 20 dB. The extent of interchannel interference was varied by adjusting tan 8 in the range 0.0 to 1.0. The following algorithms were tested: (a) Data from the Q A M system {uk, v,, y , i y,2) were separated into 2 pairs { u , , y ~ ) and {vk,Y~}. From each pair of real data the parameters in a single-input single-output A R M A model for the in-phase and quadrature paths were estimated separately. Two single-channel smoothers of the type described in Secs, 4 and 5 were then applied to the corresponding paths to obtain the probability of error for each path. The total probability of error was taken to be the sum of those for the two paths.
Pr(e)
100
10-1 a (~0)..
1o_2
__- " / "~(~o)
lo -~
,2'
~(io)
,'/
~(2o~.../ c(20) b(20)
10-4
10-5 0
! 0.2
! 0.5
! 0.7
! 1.0
tan 5 (rad)
Fig. 8. Probability of error versus tan 8 for channel of Table I. (a) 2 separate single-channel smoothers; (b) 2 separate channel least-squares Viterbi algorithms; (c) multichannel smoother. The numbers (10) and (20) denote 10- and 20-dB SNR, respectively.
126
G. C. G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
(b) A recursive least-squares algorithm was applied to each path of the system to obtain the characteristics of the paths separately. The received signals of in-phase and quadrature paths were equalized to {z)) and {z~}, and two separate Viterbi detectors were then used to process these samples to recover the two streams of data input {Uk} and {Vk) respectively. This algorithm is precisely as described in [13]. The interchannel interference is simply included in the noise term for each channel. (c) The data {Uk, Vk,y2,y~} from the Q A M system were used to fit a multivariate A R M A model of the form described in Sec. 2. The fixed-lag smoothing algorithm derived in Sec. 4, suitably modified for the multichannel case, was then applied to the equalizer samples (z~, z~) to obtain the estimate of the input sequence (u,, v,). Figures 8 and 9 show the performance of the above schemes when applied to the channel of Table 1. From this and other simulation studies [25], the following conclusions can be drawn: (i) For tan 8 = 0 , algorithms (a) and (c) have the same probability-of-error performance. This is expected, since for tan 8 = 0 there is no interchannel
Pr (e~
10 0
i0 - I
b
10 -2
10 -3
10 -4
10 -5 10
,
,
12
15
.
17
•
20
]~
SNR(dB)
Fig. 9. Probability of error versus SNR for channel of Table l at tan 8 = 0.5. (b) single-channel least-squares Viterbi algorithm; (c) rnultichannel smoother incorporating full C(z-I) polynomial; (d) multichannel smoother with C(z-J)=I.
A U T O M A T I C C H A N N EL E Q U A L I Z A T I O N
127
interference, and then the multiple-channel smoother given in (c) reduces to two separate single-channel smoothers as described in (a). (ii) The performance of the multiple-channel smoother is virtually independent of tan 8. However, the performance of algorithms (a) and (b) deteriorates rapidly as the degree of interchannel interference is increased. (iii) For small interchannel interference, algorithm (b) gives slightly better peHormance than (a) or (c) at the expense of additional computational effort (approximately a factor of 2). (iv) Figure 8 also shows results tor the case where the polynomial C ( z - i ) in the model (3.1) is assumed to be I. In this case ordinary least squares [14] can be used to estimate the channel parameters [the parameters in A ( z - i) and B(z-~)]. It can be seen from the figure that the smoother-detector using me C(z -~) polynomial performs slightly better than the smoother-detector designed with C ( z - ~ ) = l . However, in many practical applications it may be desirable to put C ( z - ~ ) = I, since this leads to a faster algorithm for estimating the parameters, even though there will be a slight deterioration in the probability-of-error performance. 7.
CONCLUSIONS
This paper has explored the use of rational A R M A models for automatic channel equalization in digital transmission systems. These models allow for a finite parametrization of the channel impulse response, which can have infinite duration, and also allow for the noise being colored. A new detection scheme based on the transfer-function models has been developed using fixed-lag smoothing. The algorithm appears to have particular merit in the case of multiple-channel systems, where it yields probability-of-error performance which is virtually independent of phase errors. Since the smoother is a linear filter with few states, it is relatively easy to implement. Also it can be made adaptive if required. Therefore the A R M A approach may be worthy of consideration as an alternative to existing equalizer design methods. REFERENCES 1. D. W. Tufts, Nyquist's problem--the joint optirniTation of transmitter and receiver in pulse amplitude modulation, Proc. IEEE 53:248-259 (Mar. 1965). 2. M. R. Aaron and D. W. Tufts, Intersymbol interference and error probability, IEEE Trans. Information Theory IT-12:26-34 (Jan. 1964). 3. R. W. Lucky, Techniques for adaptive equalization of digital communication systems, Bell System Tech. J. 45:255-256 (Feb. 1966). 4. J. G. Proakis and J. H. Miller, An adaptive receiver for digital signalling through channels with intersymbol interference, IEEE Trans. Information Theory IT-15:484-497 (July 1969).
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GOODWIN,
H. B. D O A N ,
A N D A. C A N T O N I
5. M. E. Austin, Decision feedback equaliTation for digital communication over dispersive channels, Sc. D. dissertation, Dept. of Electrical Engineering, Mass. Inst. of Tech, Cambridge, 1967. 6. R. W. Chang and J. C. Hancock, On receiver structures for channels having memory, IEEE Tram. Information Theory 1T-12:463-468 (Oct. 1966). 7. A. J. Viterbi, Error bounds for convolution codes and an asymptotical opttmum decoding algorithm, IEEE Trans. Information Theory IT-13:260-269 (Apr. 1967). 8. D. G. Forney, Jr., Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference, IEEE Trans. Information Theory IT-18: (May 1972). 9. G. Ungerboeck, Nonlinear equalization of binary signals in Gaussian noise, IEEE Tram. Comm. Tech. COM-19, No. 6 (Dec. 1971). 10. P. Monsen, Feedback equalization for fading dispersive channels, 1EEE Tram. Information Theory IT-17:56-64. (Jan. 1971). 11. S. Qureshi and E. Newhall, An adaptive receiver for data transmission over time-dispersive channels, Trans. Information Theory IT-19:448-457 (July 1973). 12. D. D. Falconer and F. R. Magee, Jr., Adaptive channel memory truncation for maximum likelihood sequence estimation, Bell System Tech. J. 52:1541 - 1562 (Nov. 1973). 13. A. Cantoni and Hoang B. Doan, Fast adaptive equalization via an identification approach in Proceedings of the International Communications Conference, Toronto, June 1978, pp. 30.4. l - 30.4.4. 14. G. C. Goodwin and R. L. Payne, Dynamic System Identification: Experiment Design and Data Anatysis, Academic, 1977. 15. P. Buffer and A. Cantoni, Non-iterative automatic equalization, I E E E Tram. Comm. Tech. COM-23 (6):621-633 (June 1975). 16. L. Ljung, Analysis of recursive stochastic algorithm.% I E E E Tram. Automatic Control AC-22 (4):551-575 (Aug. 1977). 17. B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, 1979. 18. J. B. Moore, Discrete-time fixed-lag smoothing algorithms, Automatica 9(2):163-174 (Mar. 1973). 19. L. Ljung, On positive real transfer functions and the convergence of some recursive schemes, IEEE Trans. Automatic Control AC-22 (4):539-551 (Aug. 19T7). 20. F. R. Magee and J. G. Proakis, Adaptive maximum likelihood sequence estimation for digital signalling in the presence of intersymbol interference, IEEE Trans. Information Theory IT-19:129-124 (Jan. 1973). 21. A. Cantoni and K. Kwong, Further results on the Viterbi algorithm equalizer, 1EEE Tram. Information Theory IT-20 (6):764-767 (Nov. 1974). 22. N. Levinsson, The Wiener R.MS (root mean square) error criterion in filter design and prediction, J. Mathematical Phys. 25:261-278 (1947). 23. D. Falconer and L. Ljung, Application of fast estimation to adaptive equaliT~tion, Report LITH-ISY-I-0158, Department of Electrical Engineering, Linkoping Univ., June 1977. 24. R. L. Kashyap and R. E. Nasbury, Parameter estimation in multivariate stochastic difference equation, IEEE Trans. Automatic Control AC-19, No. 6 (Dec. 1974). 25. H. B. Doan, Ph.D. Thesis, Univ. of Newcastle, Australia, 1978. 26. P. Hedelin, Can the zero-lag filter be a good smoother?, I E E E Trans. Information Theory IT-23:490-499 (July 1977). 27. D. Goddard, Channel equaliT~tion using a Kalman filter for fast data transmission, IBM. J. Rea. Develop. 18 (3):267-273. 28. H. B. Doan and A. Cantoni, Fast adaptation of equalizer and desired impulse response, IEEE J. Circuits and Systems 2 (5): 159-166 (Sept. 1978). 29. A. P. Clark and J. D. Harvey, Detection of distorted Q.A.M. signal.% Electronic Circuit~ and Systems I, No. 3 (Apr. 1977).
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30. T. S. Chu, Restoring the orthogonality of two polarizations m radio communication systems, I Bell System Tech. J. 50:3063-3069 (Nov. 1971). 31. H. E. Nichols, A. A. Giordano, and J. G. Proakis, MLD and MSE algorithml for adaptive detection of digital signals in the presence of interchannel interference, I E E E Trans. Information Theory IT-23, No. 5 (1977). 32. W. Van Etten, Maximum likelihood receiver for multiple channel transmission systems, I E E E Trans. Comm. Tech. COM-24:276-283 (Feb. 1976). 33. W. Van Etten, An optimum linear receiver for multiple channel digital transmission system, I E E E Trans. Comm. Tech. 23:828-834 (Aug. 1975). 34. T. Soderstrom, L. Ljung, and 1. Gustavsson, A theoretical analysis of recursive identification methods, Automatica--J. I F A C 14:231-244 (1978). 35. R. E. Lawrence and H. Kaufman, The Kalman filter for the equaliT~ation of a digital communications channel, I E E E Trans. Comm. Tech. COM-19 (6):1137-1141 (Dec. 1971). 36. T. W. Mark, A note on the modified Kalman filter, Proc. I E E E 61 (7):481-482 (Apr. 1973). 37. S. Benedetto and E. Biglieri, On hnear receivers for digital transmission systems, I E E E Trans. Comm. Tech. COM-22 (9): 1205-1215 (Sept. 1974). 38. S. B. Kleibnov, V. B. Prival'skii, and I. V. Tmse, Kalman filter for equaliT~tion of digital communications channel, Automat. Remote Control 35 (Part 1): 1094-1102 (Dec. 1974). 39. A. Cantoni and P. Bulter, Linear minimum square error estimators applied to channel equalization, I E E E Trans. Comm. Teeh. COM-25 (4):441-445 (Apr. 1977). 40. P. Butler, Adaptive equalizer--structures and algorithms, Ph.D. Dissertation, Univ. Of Newcastle, Australia, Dec. 1975. 41. A. Guardinceri, A. Luvison, and G. Pirceni, Synchronous Kalman receiver for carrier modulation systems, C S E L T Rapputi Tecnic, No. 1, pp. 59-63 (Mar. 1976). 42. V. Solo, Time series recursions and stochastic approximation Ph.D. dissertation, The Australian National University, Sept. 1978. 43. F. R. Magee, private communication. Received November 1979
107
Application of A R M A Models to Automatic C h a n n e l E q u a l i z a t i o n * G. C. GOODWIN H. B, DOAN and A. CANTONI
Department of Electrical and Computer Engineering, University of Newcastle, New South Wales, Australia, 2308 Communicated by Harold W. Sorenson
ABSTRACT This paper describes a novel approach to automatic channel equalization m digital transmission systems. The approach is based on the use of a finite-dimensional rational approximation to the channel characteristics. This class of channel approximation has the following advantages: it allows a finite parametrization of the channel impulse response which may be of infinite duration, it aLlows for the possibility of the noise being colored, it applies to either single- or multiple-channel systems, and it has the pedigogical advantage that many other algorithms in current use are based on models which are special cases of this model. The rational approximation to the channel characteristics is used in the paper to develop a new receiver structure using fixed-lag smoothing ideas. Simulation studies are presented showing that the receiver offers advantages over other algorithms for mitigating the effects of intersymbol and interchannel interference including those arising from carder phase errors.
1.
INTRODUCTION
I n t e r s y m b o l i n t e r f e r e n c e ( I S I ) arises in h i g h - s p e e d p u l s e a m p l i t u d e m o d u l a t e d ( P A M ) d a t a t r a n s m i s s i o n d u e t o the d i s p e r s i o n of t h e p u l s e s h a p e b y i m p e r f e c t c h a n n e l c h a r a c t e r i s t i c s . T h e p e r f o r m a n c e of m a n y c h a n n e l s u s e d f o r digital t r a n s m i s s i o n is a f f e c t e d b y this k i n d of s i g n a l d i s t o r t i o n a n d this l i m i t s t h e r a t e of t r a n s m i s s i o n . *This work was supported by the Australian Research Grants Committee and the Radio Research Board. ©Elsevier North HoLland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017
0020-0255/80/08107-23501.75
108
G . C . GOODWIN, H. B. DOAN, A N D A. CANTONI
Over the past decade efforts to mitigate the effect of ISI and noise have been centered around the development of linear [1-4] and nonlinear [5-9] receivers. Linear receivers based on peak-distortion and mean-square-error performance indices have been extensively studied (e.g. [1,2,3,4]). These receivers are relatively simple to implement. However, they do not perform optimally, because they tend to enhance the noise, especially when the channel possesses nulls in the amplitude characteristics [2]. Nonlinear receivers are capable of improved performance compared with linear receivers, but usually at the cost of significantly greater complexity. For this reason, the usual approach, in practice, has been to use either linear receivers or simple nonlinear receivers such as those incorporating decision feedback and threshold detection [5, 10]. More complicated receivers have also been studied in an attempt to achieve a better compromise between receiver complexity and performance. For example, Qureshi and Newhall [11]. and Falconer and Magee [12] have used a linear prefilter plus Viterbi detector. More recently, Cantoni and Doan [13,28] have given an interpretation to this receiver structure based on a least-squares identification algorithm. In the case of multichannel systems, it is possible for the signals to experience interchannel interference as well as intersymbol interference. For example in QAM systems [29], phase errors in the reference carriers used for demodulation change the allocation of the two portions of the received signal that contribute to the two demodulated waveforms. Other examples of interchannel interference arise from atmospheric propagation anomalies in dual polarized radio communication [30] and from electromagnetic coupling between cable pairs in multicore cable systems. The characteristics of interchannel interference can be constant or randomly time varying. Moreover, they are frequently unknown a p r i o r i at the receiver. Hence adaptive equalizer receivers are often required to adapt the detection scheme. In a recent paper [31] adaptive algorithms for mitigating interchannel interference in dual-channel systems having no intersymbol interference were studied. In a more general case, several maximum-likelihood receivers for multiple-channel transmission systems with multidimensional interference were developed [32, 33]. In the current paper we use an autoregressive-moving-average (ARMA) description of the channel characteristics. This model has a finite number of parameters but allows for an infinite channel impulse response and for colored noise. The ARMA model is a very compact way of describing the channel characteristics, since it incorporates both poles (the autoregressive component) and zeros (the moving-average component). The ARMA description never contains more parameters than a standard finite-impulse-response (FIR) model since the latter is simply a special case of the A R M A model corresponding to the unlikely occurrence that the channel has no poles and the noise is white.
AUTOMATIC C H A N N E L EQUALIZATION
109
A new receiver structure is developed using the ARMA channel model together with Kalman filtering and fixed-lag smoothing. The receiver is designed to minimize the intersymbol and interchannel interference. Previous applications of Kalman filtering and fixed-lag smoothing to equalizer design have assumed an FIR model for the channel [35-41]. As mentioned above, an adequate FIR approximation of a channel usually requires a large number of terms in the FIR description. Thus the resulting Kalman equalizer of fixed-lag smoother is generally complex, since it requires at least as many states as are in the FIR model. This makes the equalizer complex to implement and difficult to make adaptive [41, 40, 35]. By contrast, the fixed-lagged smoothing algorithm that we shall develop in this paper has few states, since it will be based on the more compact ARMA description for the channel. This means that it is easier to implement and potentially simpler to make adaptive. The resulting fixed-lag smoother receiver is essentially a linear filter incorporating both poles and zeros. In order to obtain an estimate of the transmitted signal, we propose that the linear filter be followed by a simple slicer. Since the filter is linear, it could perhaps be argued that its characteristics could be adequately approximated by a somewhat longer FIR filter. The advantage of the latter approach would be that simple least-squares techniques could be used to determine the filter coefficients. On the other hand, an F I R filter will generally require many more coefficients than the fixed-lagged smoother, and this in turn implies increased complexity in the receiver and computation difficulties if the receiver is made adaptive [13, 27, 21]. This indicates that the compact fixed-lag smoother based on the ARMA description may offer advantages in practice and is therefore worthy of consideration as an alternative receiver structure. The layout of the paper is as follows: in Sec. 2 we describe the A R M A model and relate it to other commonly used channel models; in Sec. 3 we describe recursive algorithms that can be used to adaptively fit the A R M A model, in Sec. 4 we describe the linear fixed-lag smoothing algorithm for input recovery, and in Sec. 5 we describe single-channel simulation studies comparing the new algorithm with other algorithms. Finally in Sec. 6 we present some simulation results for QAM systems having both interchannel and intersymbol interference. The simulation results were obtained for various receiver structures which have approximately the same computational complexity for the equalizer implementation. 2.
THE C H A N N E L MODEL
In this paper we shall use a discrete-time channel model which allows for both severe ISI and colored noise. The equivalent baseband signal model that
110
G . C . G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
we propose is a mixed A R M A model having the following form in the single-channel case [14]: "
y,= - • k-- I
'
akyt_k +
bkU,_k + ~_~ c~,et_k, k-- I
(2.1)
k--O
where {u,} denotes the input data sequence, and {e,} denotes a white noise sequence. In this equation a = ( a j ..... an) r is the vector of autoregressive parameters, fl=(bl ..... bin)r is the vector of moving average parameters and y = ( c I ..... c=) is the vector of noise parameters (c o can be normalized to l without loss of generality). The variance of e, is taken to be or. The channel model given in (2.I) allows for a rational input-output transter function of the form B ( z ) / A ( z ) and an output noise power density spectrum of the form
~P~(z)- C ( z ) C ( z - I ) o,2, A(z)A(z-')
z=e/~',
where
B(~-')=b~z-' + . . . +b,,z-", A(z-m)= l + a l z - ' + ''- +a,,z-", C ( z - l ) = 1+ c l z - t + " .
+clz-t.
Note that the impulse response of the above channel model can be infinitely long, since the roots of A act as system poles. The roots of B give the transmission zeros. The channel model given in Eq. (2.1) contains, as special cases, other commonly used models, for example: (1) The usual finite-impulse-response model [15] with white output noise can be obtained by setting = 0 (no autoregressive component), /3 = h (the channel impulse response), ~t=0. (2) A transfer-function model with white noise is obtained by setting y = a . Moreover, many standard equalization procedures can be given a new interpretation based on the use of (2.1) as a channel model. For example, if we use j S = ( b l , 0 , 0 .... 0) and y = 0 , then it can be seen from (2.1) that if we operate
AUTOMATIC CHANNEL EQUALIZATION
111
on the received sequence (y,) as follows:
zt=Yt+ ~ akyt-k, k--l then u , _ l = l / b l z , + l / b : , , a n d i / b l z , is a suitable estimate of the input signal u,_ 1. This can be seen to be equivalent to the standard linear equalizer structure [3]. The model of Eq. (2.1) can be readily extended to multichannel systems [14]. The essential changes are to interpret (Yt), {u,), {e,} as vectors, to replace ( a k, b k, Ck) by square matrices with c o = I , and to define the covariance of {et) as a positive definite matrix Y.. 3.
ON-LINE ESTIMATION ALGORITHMS
W e propose to use a recursive algorithm for fitting models of the type described in Sec. 2 to channel data. F o r simplicity, we consider the scalarchannel case, for which the general algorithm [14] m a y be described by
(3.1)
~t+l ~ t + Kt+ ltgt+ l ,
e,~,÷,
(3.2)
Kt+ 1 - I + , # r , P:p,÷, '
p,+,
= ?,
P,~,+,'C,P, T
,
(3.3)
1 + ~,+ iPt~t+ l t)t + i = Y t + 1
-I-y,
..... - y , _ , + ~ , u . . . . . . u , _ , . + , , ~,_, . . . . . ~,_,+~] 0 ( t ) ,
(3.4)
where 0 = ( a r, fir, y r ) r is the p a r a m e t e r vector to be estimated and the quantities i,, %, @, may be different for different algorithms. The results described in this p a p e r were o b t a i n e d by use of the extendedmatrix method [14]. In this method, i ( t ) , ~p(t), and @(t) are as follows:
T
el+l = v t + l =Yt+l --~,+ l~t , ~b, = % = [ - y , _ I . . . . . - y . . . . u,_ 1. . . . . u . . . . v , - I . . . . . v,-t] r.
(3.5)
In the above algorithm, knowledge of the transmitted sequence {u,} is required. This difficulty can be overcome by using two modes of operation for
112
G . C . GOODWIN, H. B. DOAN, A N D A. C A N T O N I
the adjustment of the coefficients. In the initial training mode the input is set to a known sequence. Then during normal operation the coefficients can be adapted by replacing the unknown inputs by the final output of the detector. If the computational constraints are very severe, then it may not be possible to use the time varying gain computation given in (3.2) and (3.3). In these cases, a simple stochastic-approximation variant can be used in which K, is computed from 1
,36,
and rt is a scalar given by r,
1
r
r, , + 7 ( q , t # t - - r , _ , ) .
(3.7)
This simplified algorithm requires less computation but is slower to converge. Ljung [16, 19] and Solo [42] have recently presented comprehensive analyses of the convergence properties of algorithms of the general form of (3.1). This analysis shows that the extended matrix method will converge under the weak is positive real, i.e. Re[l/C(e-iW)-½]>O, wE condition that I / C( z - i ) _ [-~r, cr]. (The stochastic-approximation form requires only C(z -I) positive real.) A closely related algorithm, called approximate maximum likelihood or R M L 2 [16, 19], can be shown to always converge to a local minimum. Our experience, gained from simulations, is that the extended-matrix method works well for fitting A R M A models provided the roots of C are projected into the unit circle. If required, the algorithms may be modified to track slowly time varying channels by a number of standard techniques, e.g. by the introduction of an exponential forgetting factor [14]. In fact, initial convergence can often be improved by introducing a time varying forgetting factor which converges exponentially to 1 [34, 42]. The algorithm can be applied, virtually unaltered, to the multichannel case, the only problem being that additional constraints are required on the matrices (a k, b k, ck} to ensure that the model is canonical [14] and hence to guarantee that the parameters are uniquely determinable. A simple solution is to make each of the ck matrices diagonal. Moreover, this approach has the added advantage [24] that the parameters can be estimated by a series of scalar algorithms in which 0 denotes the parameters in the ith rows of C(z-I), A(z-i), and B(z-1), and q~, is replaced by qS, = [ -y,'_ t ..... _ y , _ . , _y2_l ..... _y,2_ . . . . . . _y,q_., b l 't _ l ~ . . . ~
- - I d t '_ m ~
~,-~ . . . . . ~ , - t ] r
uL,
.....
-
~,~_.,
,...,
- ~q_.., (3.8)
AUTOMATIC CHANNEL EQUALIZATION
113
It is important to note that the recursive algorithm given in (3.1) to (3.5) is no more complicated than similar recursive schemes that are used with the more conventional F I R models [27, 28]. 4.
INPUT RECOVERY VIA FIXED-LAG SMOOTHING
Using the A R M A signal model described in Sec. 2, there are a n u m b e r of possible algorithms that could be used to recover the input sitrnal. W e shall describe an algorithm based on fixed-lag smoothing. As noted in the introduction, K a l m a n filter and fixed-lag smoothers have traditionally been used with F I R models. In the case of A R M A models, there is an a d d e d complication in that the states are not delayed versions of the input. Thus a completely different fixed-lag smoother structure is required. F o r simplicity, we describe the scalar-channel case, but the extension to multiple channels is i m m e d i a t e provided scalars are changed to vectors and matrices where necessary. W e first prefilter (Yt} to form (z,) as follows: n
z,-~y,+ ~
Sky,_ k
(4.1)
k--I
where (d I . . . . . 5 , ) denotes the estimates of ( a I . . . . . a , ) . It follows from (2.1) that if 5 k ' ~ ' a k, then (z,) can be m o d e l e d as follows: m
1
Z, = Z bkUt--k -~ Z Ck~'t-k" k--l k--O
(4.2)
Since the filter used in (4.1) is stably invertible, {z,) carries the same information content as (Yt). A detection technique is now derived to recover the transmitted d a t a (u,} from {z,}. F o r simplicity we assume that m = l. The following state space model is then equivalent to (4.2): x,+ I = F x t + B u , + C e , , (4.3) z t ~ Hxt'~'et, where
F=
ioo... 1
0
-.-
0
0
1
--'
0
0
0
-••
1
(4.4) .1×.,
114
G . C . G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
B = [ bm C=[
b._,
Cm
H=[0
""
Cm-- 1
---
"'"
0
bl ] r , el]
r ,
1] r.
(4.5)
(4.6) (4.7)
We seek to determine a fixed-lag smoothed estimate of u,_u, viz. a , _ ~ l , = E [ u , - N l Z o , Z l . . . . . z,]
(4.8)
for all .t and some fixed-lag N. Consider the following augmented-signal model: xl+ i = Fxc+ But + Cet, gt-~ nxt'k-et,
Ul+ 1 =Ig t ,
(4.9)
hi2+ 1 - - Igt - - U t - I
/d3+ l = l/t2 = /,/I_ 1 = / ' t _ 2 ,
N+ I ~/dtN~ Igt+ I
,, "
"~ l g t - N "
Introducing a new state vector
X t
x;=
I
(4.10)
I
u U+l
we can express (4.9) in the form x;+ ~ = A x ; + Gut + J e t , z, =Lx~ +et,
(4.1 l)
AUTOMATIC CHANNEL EQUALIZATION
!,0
115
where A, G, J, L have the following structure:
-b-t
A=
o .
0
G=
i]I:----°-] =/o:
1
=
1
L:
J
el-- ,
(4.13)
(4.14)
J=
,~=[,
(4.12)
.o
...
o]
(4.15)
N o w the optimal linear filter for {x;} is well known [17] to have the following form:
~;+,~,=A~;._, +( AP,~,_,L~+ S)( LP,j._,L T+ R)-'v,.
(4.16)
P,+II,=AP,I,_IAr+Q -(Ap.._,LT+S)(Lp,._
,L ~ +R)
-,
(APot_,Ar+S) r, (4.17)
where ~;+tl, denotes the best estimate of x,+~ given (z o, z~,. • •, z,), a n d v, = z, -
L~:;I ,_ l,
(4.18)
Q=GGrod +JJrod,
(4.19)
S=Jo~.
(4.20)
116
G . C . G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
Due to the special structure of Eqs. (4.12) to (4.15), the above equations can be partitioned and simplified as follows:
[ Tt ] =( AP, I,_,L r + S)( LP, It_,L r + R )- ',
Pt+ ,It --
p/+l
lit L ll'] • p/2
(4.21)
(4.22)
21 ' 22 Pt+ l[t P t + lIt
Using the above partitioned matrices, the filter can be rewritten as follows: £,+ tit----F'~,I,-.
+ 7,[ z,- H.~,lt_ I ],
"Yt--(FP, I~_,HT+Co,2)(HP~I~_,HF+o,2)-I, Pt+II,--" - FPtI',-IFT +°dBBT
/dlJt- 1
t~]+ ,It ..[[21
li,21,- I
AN+I [d/+ lit
I~N+ 1 l
+,,[
(4.25)
z,- HX,f,_ l ],
(4.26)
tit-
11 T rk,--~Pt~)_,H T ( HPtl,_,H +o,2) - I , 21 et+l[,--
(4.24)
+°, 2CCT
- y,( FP,I',_ ,Hr+ Co,2) r, t~J+ ii t
(4.23)
[ 2 p 2tit1 IFT+ou2
e,B r -¢k,(Fe, IIl,_tHr+Co~) r.
(4.27) (4.28)
Finally, the required linear fixed-lag smoothed estimate of the input is given by I^d t - N I t - - b l^N+ t + llIt •
(4.29)
REMARK 4.1. The linear fixed-lag algorithm described above does net take account of the fact that {u,} is drawn from a finite alphabet. The quantized estimate of {u,} can be obtained by simply selecting the nearest level to fi,-Np,, i.e. in the binary (__+1) case u, _ N (quantized) = sign a,_ N"
AUTOMATIC CHANNEL EQUALIZATION
117
REMARK 4.2. Note that Eq. (4.23) to (4.28) are not the standard fixed-lag smoothing equations for state estimation. The equations we have presented are simpler because we have to sort only estimates of the input, not the complete state vectors. Thus Pt+H, 21 has dimension ( N + I ) × n only, where n is the order of the model and N is the lag length. The structure of the smoother is depicted in Fig. I. It can be seen from the figure that t~,_Ni, can be recovered from the scalar innovations sequence ( z , - H ~ , I ,_ l) by ( N ) multiplications using the scalar quantities ¢2 ..... eta+ t. We have found that n = 2 and N = 6 give good performance. (Increasing n and
~ <
z +
t
[Data]
I
[ADDITIONAL SIGNAL PROCE~q S ING REQUIRED TO RECOVER THE INPUT]
u2,
tit-1
u~+llt~ i
^N+I
ut+llt ^N+I
~
~'
Ut+llt = Ut_N] t
<
[THE RECOVERED UNIT]
Fig. 1. Structure of the fixed-lag smoother.
118
G . C . GOODWIN, H. B. DOAN, AND A. CANTONI
N beyond these values gives marginal improvement, but most of the gain can be obtained with the small values quoted above.) REMARK 4.3. The closed-loop system matrix for the fixed-lag smoother is given by A-,=A - K , L _ [ F-y,H - ¢k, H
0] ~2
"
(4.30)
It is clear from this matrix that the stability properties of the smoother are determined by those of the nonlagged Kalman filter for the state. It is readily verified from Eqs. (4.3) to (4.7) that the state-space model for the channel is completely observable. Thus it is known [17] that the Kalman filter is asymptotically stable and hence this also applies to the fixed-lag smoother for recovering (u,). I1 REMARK 4.4. It should be noted that the quantities "t,, P,+~I,, ¢,, P , +21~ I , given by Eqs. (4.24), (4.25), (4.27), (4.28) tend to steady-state constant values as t becomes large. Thus, in subsequent calculations the steady-state values can be used in Eqs. (4.23) and (4.26) without further need to use Eq. (4.24), (4.25), (4.27), (4.28). In steady state, the quantities y,, q~,:..... ¢~ in Fig. 1 can be replaced by the constant values Yc, '#f ..... ¢~. This represents a significant computational saving, since for the examples described later, steady state is achieved in less than 10 iterations of Eqs. (4.24), (4.25), (4.27), (4.28).
5.
SINGLE-CHANNEL SIMULATION RESULTS
In this section applications of the ARMA model and of the associated detection scheme are demonstrated for single channels by simulation examples. A number of different baseband channels have been tested [25]. Here we show the results for two channels. Channel 1 (see Fig. 2) is taken from Proakis and Miller [4], and Channel 2 (see Fig. 3) is a typical telephone channel [43]. The simulation experiments were carried out as follows: (1) A pseudorandom binary (PRBN) training sequence was applied to the channel, and the channel output was corrupted by white (or colored) Gaussian noise. The noise level was set to give a specified signal-to-noise ratio at the receiver input (channel output). (2) The training data (input-output pairs) were used in the recursive algorithm of Sec. 3 to fit an ARMA model to the channel and noise characteristics. (3) When the model parameters converged, these parameters were used to determine the steady-state coefficients in the fixed-lag smoother by solving the ~equations given in Sec. 4.
119
AUTOMATIC CHANNEL EQUALIZATION
,
|
*
(a)
IH(w) t
J
i
Co) CHANNEL
/
Fig. 2. Channel I: (a) channel impulse response; Co) frequency response.
[
I
I. t
,
o~2~
(a)
la(~,) ]
\ w
(b)
CHANNEL 2 Fig. 3. Channel 2: (a) impulse response; ~)'frequency rcsponsc.
120
G . C . GOODWIN, H. B. DOAN, A N D A. C A N T O N I
(4) An unknown PRBN signal was applied to the channel, and noise was again added at the output. The received data were passed through the steady state filter (3), and the output of the filter was passed to a slicer to give the recovered input. (5) Multiple runs of the type described in (4) were performed to estimate the probability of error. Up to l06 binary digits were used to give a good estimate of the probability of error depending on the signal-to-noise ratio. Before we discuss the results, some preliminary observations are important: (l) Effect of model time delay. In practice, the received signal message possesses an inherent delay property [26] and this delay property plays an important role in modeling the channel as in Eq. (2.1). The time delay must be appropriately adjusted to achieve accurate channel modeling. Our experience with a number of simulation examples indicates that the optimal value for the time delay is approximately equal to the first major peak on the impulse response. (2) Effect of model order. It is of course necessary to determine appropriate orders for the polynomials A ( z - l ) , B ( z - i ) , and C ( z - l ) . Moreover, for a fixed total number of parameters, the problem arises how best to apportion these parameters between A ( z - i ) , B ( z - i ) , and C ( z - i). This problem is neither new nor peculiar to the A R M A model. For example, in compromised Viterbi receivers a choice must be made between the length of any prefilter and the truncated impulse response. Fortunately, for the fixed-lag smoothing algorithm described here, we have found that the choice of the relative degrees in the A R M A model is not critical. We obtained good results by fixing the orders of A, B, C at 2. Increasing the orders beyond these values gave insignificant improvement. (3) Effect of lag length. It is also important to set the lag length N of the smoothing algorithm to achieve reasonable probability of error without excessive computing. In our simulations a lag length of 5 or 6 gave the best performance. Increasing the lag length further actually deteriorated the performance because the modeling errors become more significant at high lags. Simulations were done for following channel configurations: (a) Addititm white noise at the channel output. This is the usual equivalentchannel model where the input sequence is passed through a linear transfer function and the observed output has a white noise sequence rl, =e, added to it. Co) Additive colored noise at the channel output. The situation is similar to that described in (a) except that the additive noise is colored. The noise sequence at the output was generated by the following model:
Tlt=et+O.8e,_ I + 0.6et_ 2 with {e,} white noise.
(5.1)
AUTOMATIC CHANNEL EQUALIZATION
121
Figures 4, 5, and 6 show typical results from the simulation studies. The figures show results for the following algorithms: (a) Smoother. This is the fixed-lag smoothing algorithm described in Sec. 4. In all cases the order was 2, the lag length 3 to 6. (b) RLS-VA. This is the least-squares Viterbi algorithm described in [13], [28]. The number of autoregressive prefilter coefficients was set to 7, and the number of points on the desired impulse response (DIR) was set to 3. This particular algorithm was chosen for comparison with the smoothing algorithm described in this paper because it has been shown in [13] that the RLS-VA algorithm has approximately equivalent performance to the optimal-eigenvalue algorithm described in [12] and superior performance to all other algorithms of equivalent complexity. Moreover it is shown in [13] that the RLS-VA algorithms can be considered as a recursive form of the prefiltcr algorithms proposed by Falconer and Magee [12] and also by Cantoni and Kwong [21]. (c) FIR algorithm. This is a finite-impulse-response equalizer as in [25] with I I coefficients.
Pr(e)
I0 °
1o-]
\
\
! I0-5~
m
i0
15
SNR(dB)
Fig. 4. Error performance for channel 1 with white output noise: A, smoother; B, RLS-VA; C, FIR equalizer; D, one-shot bound.
122
G . C . GOODWIN, H. B. DOAN, A N D A. C A N T O N I Pr{e)
i0 °
I0
-i
10 - 2
10 -3
10 -4 .
B
10
-5 ,
,
10
15
SNR(dB)
Fig. 5. Error performance for channel 2 with white output noise: A, smoother; B, RLS-VA; D, one shot bound. (d) One-shot bound. It can be shown [25] that for any detection scheme, the one-shot error probability is bounded below by Q(½ v ~ ), where Q gives the tail probability of the N(0, 1) Gaussian distribution, and ¢¢
Here Am is the minimum difference in input signal level (i.e. 2 for the binary _1 case), H ( z - I ) = B ( z - ' ) / A ( z -I) is the channel impulse response, and • ,(e jw) is the channel output noise power density spectrum. It can be seen from the results in Figures 4 to 6 that the A R M A model plus fixed-lag smoothing algorithm gives good probability-of-error performance in the presence of severe intersymbol interference. The RLS-VA algorithm actually performs better than the smoother, but the computing time per recovered sample using RLS-VA is approximately twice that of the smoothing algorithm [25].
AUTOMATIC CHANNEL EQUALIZATION
123
Pr(a)
i0 -]
5
i0
15
SNR (dB)
Fig. 6. Error performance of channel 2 with colored output noise: A, smoother; B, RLS-VA; D, one-shot bound.
6.
MULTICHANNEL SIMULATION RESULTS
A Q A M system was simulated having a phase error 8 between the received signal and the local oscillator. The situation is described as follows: Consider the block diagram of the coherent demodulator as shown in Figure 7. Suppose there is a phase error 8 between the received signal and the local oscillator. As a result of this error the received signal will be demodulated using cos(60ct+8 ) and sin(o~ct+8 ). The output of the demodulator for the in-phase and quadrature channels after low-pass filtering are
yl(t) =
½sl(t) cos 6 - ½s2(t)sin 6,
(6.1)
y 2 ( t ) - ' i s 2 ( t ) c o s 6+ ½sl(t)sin6.
(6.2)
-
124
G . C . G O O D W I N , H. B. D O A N , A N D A.
CANTONI
S(t)=s1(t)c°swe~~/ Low Pass Filter
1 Yt
~ coS(~ct+6)
~ ~_~
+ s2(t)sinWct ~
90 °
t
Low pass ] Filter
2
Yt
Fig. 7. Block diagram of coherent demodulation system.
It can be seen that the effect of the phase error is to introduce a gain factor which causes a reduction in noise immunity; in addition it introduces crosschannel interference. In general, the equivalent b a s e b a n d signals of the Q A M system with channel impulse responses h2(t), h'2(t) will be of the form cos 8
sin 8
~- ~ , u n h ' l ( t - n T ) - T ~,v~h'2(t-nT)+nl(t ),
yl(t,8)-
tl
y2(t,(~)-
rl
cos 8
sin 6 ~ - ~ v , h2(t-nT)+--~-~u~h't(t-nT)+n2(t);
rt
/1
(6.3)
combining the gain factor (cos 8 ) / 2 into the impulse response, this becomes:
yl(t, 8) = ~ u~hl(t-nT)-
(tanS) ~
n
vnh2(t-nT) +nl(t ),
n
y2(t,8)=~,v, h2(t-nT)+(tanS)~u,hl(t-nT)+n2(t). n
(6.4)
rl
A number of different b a s e b a n d channels have been used in the simulation tests [25]. In Table 1 we show the results for a channel having two identical paths of the form used by Proakis and Miller [12]. TABLE I Channel h~ hi
Sampled impulse response 0.06 0.06
-0.07 -0.07
0.1 0.1
-0.3 -0.3
-0.7 -0.7
1.0 0.5 0.0 0.3 0.05 1.0 0.5 0.0 0.3 0.05
0.1 0.1
AUTOMATIC CHANNEL EQUALIZATION
125
In the simulation tests the input sequences (Uk} and (Vk} tO the in-phase and quadrature channels were pseudorandom binary (_+ 1) sequences having zero mean and unit variance. The norm of each channel impulse response was normalized to 1 so that no signal gain or attenuation was introduced by any channel. The additive noise sequence to each path was taken to be zero-mean white Gaussian. Simulation tests were run at signal-to-noise ratios between 7 and 20 dB. The extent of interchannel interference was varied by adjusting tan 8 in the range 0.0 to 1.0. The following algorithms were tested: (a) Data from the Q A M system {uk, v,, y , i y,2) were separated into 2 pairs { u , , y ~ ) and {vk,Y~}. From each pair of real data the parameters in a single-input single-output A R M A model for the in-phase and quadrature paths were estimated separately. Two single-channel smoothers of the type described in Secs, 4 and 5 were then applied to the corresponding paths to obtain the probability of error for each path. The total probability of error was taken to be the sum of those for the two paths.
Pr(e)
100
10-1 a (~0)..
1o_2
__- " / "~(~o)
lo -~
,2'
~(io)
,'/
~(2o~.../ c(20) b(20)
10-4
10-5 0
! 0.2
! 0.5
! 0.7
! 1.0
tan 5 (rad)
Fig. 8. Probability of error versus tan 8 for channel of Table I. (a) 2 separate single-channel smoothers; (b) 2 separate channel least-squares Viterbi algorithms; (c) multichannel smoother. The numbers (10) and (20) denote 10- and 20-dB SNR, respectively.
126
G. C. G O O D W I N , H. B. D O A N , A N D A. C A N T O N I
(b) A recursive least-squares algorithm was applied to each path of the system to obtain the characteristics of the paths separately. The received signals of in-phase and quadrature paths were equalized to {z)) and {z~}, and two separate Viterbi detectors were then used to process these samples to recover the two streams of data input {Uk} and {Vk) respectively. This algorithm is precisely as described in [13]. The interchannel interference is simply included in the noise term for each channel. (c) The data {Uk, Vk,y2,y~} from the Q A M system were used to fit a multivariate A R M A model of the form described in Sec. 2. The fixed-lag smoothing algorithm derived in Sec. 4, suitably modified for the multichannel case, was then applied to the equalizer samples (z~, z~) to obtain the estimate of the input sequence (u,, v,). Figures 8 and 9 show the performance of the above schemes when applied to the channel of Table 1. From this and other simulation studies [25], the following conclusions can be drawn: (i) For tan 8 = 0 , algorithms (a) and (c) have the same probability-of-error performance. This is expected, since for tan 8 = 0 there is no interchannel
Pr (e~
10 0
i0 - I
b
10 -2
10 -3
10 -4
10 -5 10
,
,
12
15
.
17
•
20
]~
SNR(dB)
Fig. 9. Probability of error versus SNR for channel of Table l at tan 8 = 0.5. (b) single-channel least-squares Viterbi algorithm; (c) rnultichannel smoother incorporating full C(z-I) polynomial; (d) multichannel smoother with C(z-J)=I.
A U T O M A T I C C H A N N EL E Q U A L I Z A T I O N
127
interference, and then the multiple-channel smoother given in (c) reduces to two separate single-channel smoothers as described in (a). (ii) The performance of the multiple-channel smoother is virtually independent of tan 8. However, the performance of algorithms (a) and (b) deteriorates rapidly as the degree of interchannel interference is increased. (iii) For small interchannel interference, algorithm (b) gives slightly better peHormance than (a) or (c) at the expense of additional computational effort (approximately a factor of 2). (iv) Figure 8 also shows results tor the case where the polynomial C ( z - i ) in the model (3.1) is assumed to be I. In this case ordinary least squares [14] can be used to estimate the channel parameters [the parameters in A ( z - i) and B(z-~)]. It can be seen from the figure that the smoother-detector using me C(z -~) polynomial performs slightly better than the smoother-detector designed with C ( z - ~ ) = l . However, in many practical applications it may be desirable to put C ( z - ~ ) = I, since this leads to a faster algorithm for estimating the parameters, even though there will be a slight deterioration in the probability-of-error performance. 7.
CONCLUSIONS
This paper has explored the use of rational A R M A models for automatic channel equalization in digital transmission systems. These models allow for a finite parametrization of the channel impulse response, which can have infinite duration, and also allow for the noise being colored. A new detection scheme based on the transfer-function models has been developed using fixed-lag smoothing. The algorithm appears to have particular merit in the case of multiple-channel systems, where it yields probability-of-error performance which is virtually independent of phase errors. Since the smoother is a linear filter with few states, it is relatively easy to implement. Also it can be made adaptive if required. Therefore the A R M A approach may be worthy of consideration as an alternative to existing equalizer design methods. REFERENCES 1. D. W. Tufts, Nyquist's problem--the joint optirniTation of transmitter and receiver in pulse amplitude modulation, Proc. IEEE 53:248-259 (Mar. 1965). 2. M. R. Aaron and D. W. Tufts, Intersymbol interference and error probability, IEEE Trans. Information Theory IT-12:26-34 (Jan. 1964). 3. R. W. Lucky, Techniques for adaptive equalization of digital communication systems, Bell System Tech. J. 45:255-256 (Feb. 1966). 4. J. G. Proakis and J. H. Miller, An adaptive receiver for digital signalling through channels with intersymbol interference, IEEE Trans. Information Theory IT-15:484-497 (July 1969).
128
G.C.
GOODWIN,
H. B. D O A N ,
A N D A. C A N T O N I
5. M. E. Austin, Decision feedback equaliTation for digital communication over dispersive channels, Sc. D. dissertation, Dept. of Electrical Engineering, Mass. Inst. of Tech, Cambridge, 1967. 6. R. W. Chang and J. C. Hancock, On receiver structures for channels having memory, IEEE Tram. Information Theory 1T-12:463-468 (Oct. 1966). 7. A. J. Viterbi, Error bounds for convolution codes and an asymptotical opttmum decoding algorithm, IEEE Trans. Information Theory IT-13:260-269 (Apr. 1967). 8. D. G. Forney, Jr., Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference, IEEE Trans. Information Theory IT-18: (May 1972). 9. G. Ungerboeck, Nonlinear equalization of binary signals in Gaussian noise, IEEE Tram. Comm. Tech. COM-19, No. 6 (Dec. 1971). 10. P. Monsen, Feedback equalization for fading dispersive channels, 1EEE Tram. Information Theory IT-17:56-64. (Jan. 1971). 11. S. Qureshi and E. Newhall, An adaptive receiver for data transmission over time-dispersive channels, Trans. Information Theory IT-19:448-457 (July 1973). 12. D. D. Falconer and F. R. Magee, Jr., Adaptive channel memory truncation for maximum likelihood sequence estimation, Bell System Tech. J. 52:1541 - 1562 (Nov. 1973). 13. A. Cantoni and Hoang B. Doan, Fast adaptive equalization via an identification approach in Proceedings of the International Communications Conference, Toronto, June 1978, pp. 30.4. l - 30.4.4. 14. G. C. Goodwin and R. L. Payne, Dynamic System Identification: Experiment Design and Data Anatysis, Academic, 1977. 15. P. Buffer and A. Cantoni, Non-iterative automatic equalization, I E E E Tram. Comm. Tech. COM-23 (6):621-633 (June 1975). 16. L. Ljung, Analysis of recursive stochastic algorithm.% I E E E Tram. Automatic Control AC-22 (4):551-575 (Aug. 1977). 17. B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, 1979. 18. J. B. Moore, Discrete-time fixed-lag smoothing algorithms, Automatica 9(2):163-174 (Mar. 1973). 19. L. Ljung, On positive real transfer functions and the convergence of some recursive schemes, IEEE Trans. Automatic Control AC-22 (4):539-551 (Aug. 19T7). 20. F. R. Magee and J. G. Proakis, Adaptive maximum likelihood sequence estimation for digital signalling in the presence of intersymbol interference, IEEE Trans. Information Theory IT-19:129-124 (Jan. 1973). 21. A. Cantoni and K. Kwong, Further results on the Viterbi algorithm equalizer, 1EEE Tram. Information Theory IT-20 (6):764-767 (Nov. 1974). 22. N. Levinsson, The Wiener R.MS (root mean square) error criterion in filter design and prediction, J. Mathematical Phys. 25:261-278 (1947). 23. D. Falconer and L. Ljung, Application of fast estimation to adaptive equaliT~tion, Report LITH-ISY-I-0158, Department of Electrical Engineering, Linkoping Univ., June 1977. 24. R. L. Kashyap and R. E. Nasbury, Parameter estimation in multivariate stochastic difference equation, IEEE Trans. Automatic Control AC-19, No. 6 (Dec. 1974). 25. H. B. Doan, Ph.D. Thesis, Univ. of Newcastle, Australia, 1978. 26. P. Hedelin, Can the zero-lag filter be a good smoother?, I E E E Trans. Information Theory IT-23:490-499 (July 1977). 27. D. Goddard, Channel equaliT~tion using a Kalman filter for fast data transmission, IBM. J. Rea. Develop. 18 (3):267-273. 28. H. B. Doan and A. Cantoni, Fast adaptation of equalizer and desired impulse response, IEEE J. Circuits and Systems 2 (5): 159-166 (Sept. 1978). 29. A. P. Clark and J. D. Harvey, Detection of distorted Q.A.M. signal.% Electronic Circuit~ and Systems I, No. 3 (Apr. 1977).
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30. T. S. Chu, Restoring the orthogonality of two polarizations m radio communication systems, I Bell System Tech. J. 50:3063-3069 (Nov. 1971). 31. H. E. Nichols, A. A. Giordano, and J. G. Proakis, MLD and MSE algorithml for adaptive detection of digital signals in the presence of interchannel interference, I E E E Trans. Information Theory IT-23, No. 5 (1977). 32. W. Van Etten, Maximum likelihood receiver for multiple channel transmission systems, I E E E Trans. Comm. Tech. COM-24:276-283 (Feb. 1976). 33. W. Van Etten, An optimum linear receiver for multiple channel digital transmission system, I E E E Trans. Comm. Tech. 23:828-834 (Aug. 1975). 34. T. Soderstrom, L. Ljung, and 1. Gustavsson, A theoretical analysis of recursive identification methods, Automatica--J. I F A C 14:231-244 (1978). 35. R. E. Lawrence and H. Kaufman, The Kalman filter for the equaliT~ation of a digital communications channel, I E E E Trans. Comm. Tech. COM-19 (6):1137-1141 (Dec. 1971). 36. T. W. Mark, A note on the modified Kalman filter, Proc. I E E E 61 (7):481-482 (Apr. 1973). 37. S. Benedetto and E. Biglieri, On hnear receivers for digital transmission systems, I E E E Trans. Comm. Tech. COM-22 (9): 1205-1215 (Sept. 1974). 38. S. B. Kleibnov, V. B. Prival'skii, and I. V. Tmse, Kalman filter for equaliT~tion of digital communications channel, Automat. Remote Control 35 (Part 1): 1094-1102 (Dec. 1974). 39. A. Cantoni and P. Bulter, Linear minimum square error estimators applied to channel equalization, I E E E Trans. Comm. Teeh. COM-25 (4):441-445 (Apr. 1977). 40. P. Butler, Adaptive equalizer--structures and algorithms, Ph.D. Dissertation, Univ. Of Newcastle, Australia, Dec. 1975. 41. A. Guardinceri, A. Luvison, and G. Pirceni, Synchronous Kalman receiver for carrier modulation systems, C S E L T Rapputi Tecnic, No. 1, pp. 59-63 (Mar. 1976). 42. V. Solo, Time series recursions and stochastic approximation Ph.D. dissertation, The Australian National University, Sept. 1978. 43. F. R. Magee, private communication. Received November 1979