Maximum Likelihood Sequence Detection for Intersymbol ...

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Maximum Likelihood Sequence Detection for Intersymbol Interference Channels: A New Upper Bound on Error Probability SERGIO VERDir,

MEMBER, IEEE

tight in the low-noise region. The lower bound analysis [2] is based on the error probability of receivers with side information that perform one-shot optimum decisions.The upper bound (see (8) below) was obtained by the proof of the three subevents, amended later by Foschini [3] and reproduced in a number of subsequentworks (e.g., [4]-[6]). This argument was simplified in various ways by Mazo [7], I. INTRODUCTION Acampora [8], and Viterbi and Omura [9], [lo]. The HE FORNEY upper bound on the error probability Forney bound (8) is an infinite series whose convergence of maximum likelihood sequence detection is the and computation are nontrivial. Foschini [3] showed local cornerstone of the analysis of optimum demodulators for convergence (i.e., for sufficiently low noise levels) of the channels with intersymbol interference and additive Gauss- bound for any finite-length intersymbol interference probian noise. In this paper we derive a tighter upper bound by lem,’ and Forney [l] showed how to apply Viterbi’s symbolic flowgraph technique to compute the bounding series. applying the method of error-sequencedecomposition. The new upper bound presented in Section II admits a Due to the dynamical component introduced by the simple proof and shows that a substantial number of terms presence of intersymbol interference into the data demodcan be excluded from the Forney bound.2 The main ulation problem, optimum signal detection cannot be contribution of this result is to show the most general and achieved on the basis of the independent observation of sharpest method yet to obtain bounds on the error probthe time interval of each transmitted symbol. Rather it is ability of minimum distance sequencedetectors. In Section necessary to treat the problem as one of sequencedetecIII a new bound is shown to converge for a sufficiently tion, whereby observation of the whole received waveform high signal-to-noise ratio even if the intersymbol interis required to produce a sufficient statistic. Since in this ference length is not restricted to be finite. The validity of case the transmitted symbols are not independent this bound is proved under restrictions that relax those of a posteriori, there is not a unique optimality criterion even Wyner [ll] and that are satisfied by all cases of interest in though all sequencesare assumed to be equiprobable. In practice. Note that even though suboptimum implementapractice the main optimality criterion is maximum likelitions are unavoidable if the interference has infinite length, hood sequence detection; i.e., the detector selects the seit is important to have bounds on the minimum achievable quence of symbols corresponding to the minimum energy noise realization. The preeminenceof maximum likelihood uncoded error probability that do not hinge on the strong sequencedetection is due to two main reasons: first, unless assumption that the receivedsignal has finite support. the background noise is dominant, it achieves near-optiII. NEW UPPER BOUND mum error probability, and second, it can be implemented via the Viterbi algorithm in time-complexity per binary The starting point of the proof of the new upper bound decision, which is independent of the number of trans- will be a conceptually straightforward derivation of the mitted symbols and exponential in the number of interfer- Forney bound which will lead naturally to the new result. ing symbols at any given time. Forney obtained upper [l] Suppose that the receiver observes an antipodally modand lower bounds [2] on the error probability which are ulated sequence of equiprobable and independent bits imbedded in additive white Gaussiannoise whose two-sided Abstruct-Maximum likelihood sequence detection of digital signaling subject to intersymbol interference and additive white Gaussian noise is considered. A new approach to the error probability analysis results in an upper bound which is tighter than the Forney bound. In addition, in the case of infinite-length intersymbol interference a locally convergent bound is shown to hold under fairly general assumptions on the signal autocorrelation.

T

Manuscript received June 12, 1985; revised October 4, 1985. This work was supported in part by the National Science Foundation under Grant ECS-8504752. The author is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA. IEEE Log Number 8610113.

r Global convergence (i.e., for any signal-to-noise ratio) occurs special cases (e.g., when only two symbols interfere at a time). *For example, if the interference length is equal to two, the bound is composed of all the finite sequences drawn from ( while the new bound only allows sequences of alternate + 1 and

001%9448,‘87/01000-0062$01.00 01987 IEEE

only in Forney 1, + 11, - 1.

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sequencesB, c E, will henceforth be referred to as simple. To justify that the resulting series is still an upper bound to the error probability, i.e.,

power spectral level is equal to a*; i.e., rt =

5 b(i)st-,T + II,, j= -M

t E I, b(i) E { -1, +l}.

P,(k) s

(1) The objective of the maximum likelihood sequencedetector is to select the most likely sequenceb = {b(i) E { -1, +l}, i = -M;.., M} given rt, t E I. Since the noise is Gaussian and all sequencesare a priori equiprobable, this is equivalentto the mean-;quarecriterion whereby the detector outputs the sequenceb that maximizes3 Q(b) = 2 < S(b), r > -jlS(b)ll’

(2) where S,(b) = XE -M b(i)stdlr, t E I and s E L,(O, co). The performance of any detector that maximizes (2) is obviously independent of the implementation of the decision algorithm; whether this is the Viterbi algorithm, brute force, or any other approach is immaterial in the sequel. W e are interested not in the probability that the detector outputs an erroneoussequencebut in the probability that there is an error in the output sequencekth component; i.e., P,(k) = P[b’(k) # g(k)] if b’ is the transmitted sequence.Note that for every M , P,(k) need not be independent of k; however, as it4 -+ cc) it convergesto the sought-after bit error rate since it is a bounded monotone increasing sequence. Let us proceed to our deriv_ationof the Forney upper bound on P,(k). If b’(k) # b(k), then there exists an error sequencee, i.e., a vector whose 244 + 1 components are drawn from { - LO, l} such that e(k) # 0, O(b’- 2~) = max,,fi(b) and such that if c(i) # 0, then b’(i) = c(i), i.e., b’ - 2~ is a sequenceof + 1. For every sequencein E, = {e E {-1,0,1}2M+1, c(k) # 0}, the probability of the latter event is equal to 2~“(‘), where w(e) is the number of nonzero components of e (recall that b’ is drawn equiprobably from the ensemble{ - 1, + 1}2M+1). Unfortunately, unlesswe deal with the one-shotcaseM = 0, the probability that b’- 2e is the most likely sequence does not a d m it an explicit expression.However, this event is upper-bounded by {Q(b’ - 2r) 2 Q(b’)}, whose probability is easily seento equal

P [(n> s(d/lls(dll) < -IIS(e)lll = Q(ll~(~h’~)~ (3) Hence we have obtained an upper bound on F ’,(k), n a m e ly, f’,(k) 5 c 2-w”‘Q(ll~(411/4.

(4

= lrfis,dtandlIftI*= (f>f>.

c

2-w’c’ew(wJ)~

(5)

CSB,

we will show that if e E E, is not simple and Q(br - 2~) = maxb Q(b), then we can find a simple sequencel 1 E B, such that Q(b’ - 2~‘) 2 Q(b’). To that end, note that we can always write l as the sum of simple sequences.The desired sequencee1 E B, is the unique simple subsequence of e whose kth componentis not equal to zero. If b’- 2r is the sequenceselectedby the maximum likelihood sequence detector, then necessarilywe have Q(b’ - 24 2 Q(b’ - 2(c - 6’)).

(6) Moreover, it is easyto show for any arbitrary pair en,cb of error sequencesthat Q(b’-

2(ca + d’)) - Q(b’ - 2~~) + Q(b’) -Si(b’ - 24

= -8 < S(C), (S(cb)).

(7) Since S(r’) and S( Q - r’) are orthogonal (recall that e1 is a subsequenceof e flanked by at least K - 1 zerosat each side), when we particularize (7) to the case ea = el, eh = e - cl, its right side is equal to zero; together with (6), it implies that Q(b’- 2~‘) 2 Q(b’) as we wanted to show. In Forney’s work [l] the upper bound on the error probability of the maximum likelihood sequencedetector is not given as in (5); rather, it is expressedin the form P, < c Q(d/2u) dED

c

We-”

Es&j

in the case of binary m o d u lation. This correspondsto a rearrangementof the terms in the summation of (5) when M = co. In (8), E, is the set of error eventswith Euclidean weight equal to d*, and D is the set of square roots of Euclidean weights attained by the error events. An error event is a finite string of elementsdrawn from { - LO, l} such that the first and the last elementsare nonzero and such that there are no more than K - 2 consecutivezeros between any pair of nonzeroelements.Clearly, if M = 00, there exists a one-to-onecorrespondencebetweenthe error events and a partition of the simple error sequences according to the equivalencerelation of being a shifted version of one another. The Euclidean weight of the error event 5 is equal to d2([) = 4llS(e)(1*, where e is any simple sequencebelonging to the equivalenceclass associated to 6. Since w(t) is equal to the number of elementsin the intersection of B, and the equivalenceclass associated to E, it follows that the right sidesof (5) and (8) coincide. Note further that to be able to apply the symbolic flowgraph technique to the computation of (8), the function Q(x) is substituted in [l] by its upper bound l/2 exp ( - x2/2), resulting in the bound P, I i

c

exp( -d2/8u2)

c

We-“.

(9

64

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Let us return to the derivation of (5) and see how to tighten the upper bound by the elimination of additional sequences. The main step of the foregoing proof is the identification of a set of sequences, B,, such that if E E E, - B,, and b’ - 2r is most likely a posteriori, then e1 E B, exists such that Q(b’ - 2~‘) 2 O(b’). Now we will find a proper subset Fk c B, that still satisfies this property. Notice that we only used the fact that e was not simple in order to decomposeit into C’E B, and (E - E’) such that S(r’) and S(e - el) are mutually orthogonal. However, had (S(E~), S(c - e’)) 2 0, the proof would have held verbatim since the nonpositivity of the right side of (7) is enough, along with (6), to conclude &?(b’) < Q(bt - 2~‘) (Fig. 1). Therefore, we can exclude from the bound all the sequencese that already have a subsequencee1 in the series satisfying (S( cl), S(E - cl)) 2 0. How can we characterize the sequencesthat remain in the bound? Let us say that an error sequencel E E, is decomposable(cf. [12]) into nonzero error sequencese1 and E* if z = e1 + e*, E(i) = 0 implies that c’(i) = c2(i) = 0 (i.e., there are no cancellations in the addition e1 + r*) and (S(e’), S(e*)) 2 0. Then we claim that the set of sequences in the bound is equal to Fk, the subset of indecomposable sequencesin E,. Obviously, if a sequenceis indecomposable, then it does not have any subsequenceE’ such that (S( e’), S(e - e’)) 2 0, and hence it belongs to the bound. The converse,i.e., any decomposablesequence E E E, is decomposable into E’+ (E - e’) where E’E Fk, is not obvious becausedecomposability is not a transitive property; if e is decomposable into ea + cb and en is decomposable into cc + ed it does not follow that l is decomposable into ec + (cd + eb). However, it can be shown [12] by induction on w(e) that for any e E E, - Fk we can indeed find such a subsequencel ’ E Fk. Therefore, all decomposable sequencescan be excluded from the bound, and we have shown the following result.

.

s(b-2 0 (respectively, Js,s,_r -C 0), the simple sequencesare those that do not contain any zero amid nonzero components, while I;k consists exclusively of the simple sequenceswhose nonzero components have alternate signs (respectively, the same sign). The symbolic flowgraph technique [l] to compute the bounding series(9) is not suitable for computing (10) since this series is no longer over the set of simple sequences.A branch-andbound combinatorial approach, which is of independent interest, is proposed in [13] to compute efficiently the bounding series up to any prespecified degreeof accuracy. In the limit as M + cc, the right side of (10) is an upper bound to the bit-error rate of the maximum likelihood sequence detector. In the infinite-horizon case, an infinite number of indecomposable sequencesexist, and the convergence of the seriesin (10) needs to be elucidated. Local convergence of the Forney bound, and a fortiori of (lo), was proved by Foschini [3]. However, recall that the Forney bound holds only for finite-length intersymbol interference problems, while the new bound holds for any noise level regardless of whether the interference length is finite (notice that K does not play any role in the derivation of Proposition 1) Nevertheless,if the signal does not ‘have finite support, the seriesin (10) may fail to converge. Fortunately, it is shown in the next section that it is possible to obtain a locally convergent bound that holds under mild restrictions on the signal autocorrelation.

...

III. \

I

S(b) = TRANSMITTED SIGNAL n = PROJECTION OF NOISE REALIZATION

Fig. 1. If c is decomposable into l ’ and C” and b - 2~ is most likely sequence then both b - 2~’ and b - 2~” are more likely than b.

INFINITE-LENGTH INTERSYMBOL INTERFERENCEBOUND

The reason (10) may diverge when the received signal does not have finite support is that we are including a large number of almost-decomposable sequencesin the series; i.e., indecomposable sequencesthat can be put as the sum of two components that form an angle slightly greater than 90 O. For example, supposethe signal autocor-

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relation function is positive and consider the sum of assumeCondition 2, we have 2-""'QC IIS(e>ll/ u 1 over the error sequenceswith only two Kc’) &r2) nonzero components + i and + 1 respectively,separated ll~(~‘)ll* = (l/m ) C C ~Yi>f2(m> by an arbitrary number of zeros. These sequencesare i=/(G) m=f(c*) indecomposableand the above sum divergesfor any noise . “G(ej”) ,Ml-m) dw level becauseboth w(r) and llS( z) 1)are upper-boundedfor J0 all such sequences. Our approach to circumvent this problem is to trade a slight increasein the effective noise level for the elimina= (l/r) (“G(ei”) ‘$) c’(i) ejwi 2 dw JO tion of a large number of almost decomposablesequences 1 i=f(r’) in such a way that the resulting bound convergeslocally. Proposition2: Let Bf c E, denote the subsetof N-simple sequences,i.e., those containing no more than N - 1 zeros a m id nonzero components.Denote the sampled autocorrelation function by r,, = /?‘, sIslenTdt. Supposethat either = ,r:ofTIG(ejw) c lc’(i)j* (14) i=f(r’) 1) 5 nlr,l -c 00, or where the energy spectral density is denoted by G(ej”) = il=l Cr= _ (x rn e-jwn, and the last equation follows from 00 03 Parseval’sformula. Now since C r e-jwn > 0. 2) c Ir,l < a, and inf ~EP,~l

n=l

fi=

-m

”

Then for every 0 < (Y< 1 there exists a positive integer N such that

P,(k) 5 (

c 2-"(')Q( Ils(r)ll(l - +J>.

(11)

csB,N

i=f(r’)

m=l(r’)+N

it follows from (14) that

I(S(E~), ~(~2))1/11~(~1)112 2 [ ,$fv,G(@“)]pl

f

I~J,

n=N

Proof: The assumptionson the autocorrelation function are placed to ensure that the inner product between any pair of sequencesthat are sufficiently far apart can be m a d e as small as desired. More precisely,for every cx> 0 there exists an integer N such that for any pair el, e* that satisfies Z(r’) + N I f(e*), (f(r) and Z(r) denote the position of the first and last nonzero componentsof e, respectively), ICWL

G2))l

5 w4)IIw)l12.

(

2)

To seethis, consider the following l(S(El), S(E*))l =

Kc’) Kr2) C El(i) C “(j)Y,-i i=/(G) i=/(2)

l(r’) s C i=f(r’)

I

f

(15) which can be m a d e as small as desired according to Condition 2. Hence it readily follows from (12) that for any (Y> 0 there exists an N such that if e E E, - BF, then E has a subsequenceE’E Bt that satisfies

Im’L

SG - f’))l 5 ww(~‘)112.

(16)

To complete the proof we just need to m o d ify slightly the argument that led to (5). F ix 0 < (Y< 1, and select an N that satisfies the above condition. Supposethat E E E, - Br and Q( 6’ - 2~) = max,, Q(b), and select the subsequence e’E Bf that satisfies (16). If we particularize (7) to en = r’, cb = e - e’, it follows from Q(bt - 2~) 2 il(b’ - 2(e - e’)) that G(b* - 2~‘) - G(b’) 2 8l(S(e’), S(c - >}

n) 2 (1 - ~)ll~(~‘)ll*}

and (11) follows in the sameway as (3).

(19)

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sequence e there are at most Z(e) - f(e) sequencesin Br that can be obtained by shifting e. Thus we can write 1

l S(b-ZE)

2-“(‘)exp ( -]]S(E)](~/~U~)

fEBkN

I

E (n + 1)

c

n=O

l EB/

2-4’) ed-IIWl12/2~2).

f (l;)L;ki n (22) Now we lower-bound w(e) and ]]s(r)]] in the right side of (22) as follows. On one hand we have S(b).

S(b-2e”l

S(C)

n I N(w(c) - 1)

I-----

Fig. 2. If h - 2e is most likely sequence, then projection of noise realization along direction of S(C’) is greater than IlS(r’)ll - a.

It is worth mentioning that the bound in Proposition 2 can be tightened further by considering any subset G c Br that satisfies the following property: if e E E, - G, then there exists a subsequencee1 E G such that (S(E~), S(E El)) 2 -a&s(E’)l12. In Wyner [ll] it was shown that the limit as (I + 0 of the Forney bound (dominated by the minimum-distance terms in the sum) also holds in the case of infinite-length intersymbol interference. Under a slightly stronger condition than Condition 2 it is proved in [ll] that liic inf e2 In P, 5 - yjn/[S(r)l12.

(20)

Proposition 2 strengthens this result in two directions. First, it provides a bound valid for all noise levels rather than an asymptotic result (which readily follows from (11)). Second, the bound also holds under Condition 1, which does not appear to impose any limitations in practice since it is satisfied by the output of any asymptotically stable linear system driven by a finite-length L, signal. This is in contrast to Condition 2, which may fail to hold in important partial responseproblems (such as duobinary signaling through linear channels). It remains to prove that the series in (11) converges. Although the sum therein is over a set of simple sequences, convergence does not follow from Foschini’s result [3] because his method hinges on the finiteness of the intersymbol interference length. Here we will take a different approach to prove the following result which implies local convergence of the bound in Proposition 2. Proposition 3: Suppose that s E L,(O, cc) has nonzero energy. For every positive integer N there exists a, > 0 such that if 0 < u < uO,then 1

2-w(‘)exp ( -]]S(e)]]2/2u2)

< 00.

(21)

because otherwise r contains more than N - 1 zeros amid nonzeros. On the other hand, recalling that the signal s is causal and defining the energy of S,(e) in [jr, (j + l)T] by T,(c(k - n);..,

= J (j+l)T[i$p~(i).Y-iT]i jT

4d)

for j 2 k - n, we have

f

lls(e)l12 =

rj(c(k - n>,.: ., c( j))

J=k-n

2

5

Ti(c(k - n);..,E(j)).

(23)

j=k-n

Consequently, (22) is overbounded by n + 1)2-“lN

-

5

T,(c(k-

n);..,c(j))/2u

j=k-n

2 i

... c(k-n)s{-l,O,l} c c(k)s{-1,O.l)

. exp -

t

‘Tj(c(k-n);.*,c(j))/2u2

j=k-n

If we can show that there exists a scalar h > 0 such that c

**-

c(k&n)E(-l,O,l)

c c(k)s{-l,O,l)

LEBf

Proof: If two error sequencescoincide up to a shift, then both their weight and energy coincide and hence their contributions to the bound are identical. For any given

dt

. exp i

i

Tj(c(k - n);.*,c(j))/2u2

j=k-n

5 [l + 2exp ( -h2/2u2)]

‘+l,

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then it follows that (24) is overboundedby the series

IV.

CONCLUSION

The method of error sequencedecomposition,originally developed to analyze optimum m u ltiuser detectors [12], has been successfullyapplied to the bit error rate analysis which convergesfor 0 < u2 =Cut = (1/2)h2[ln(2/(21/N of maximum likelihood sequencedetection of signals iml))]-l. To prove (25), we assumewithout loss of generality bedded in white Gaussiannoise and subject to intersymbol that /o’~,?dt # 0 (since the energyof the signal is positive, interference. This method reducesthe analysis of the seif /o’s,?dt = 0, an equivalent m o d e l to (1) can be obtained quence detector (an m-ary hypothesis testing problem) to by replacing s, by an appropriately shifted version), and the analysis of a collection of binary hypothesis testing we show by induction that (25) holds for h* = /o’sFdt. For problems. n = 0 it is easy to check that the result holds with equality. Perhaps the simplest example that captures the essence Now fix n > 0 and c(k - n); . ., c(k - l), and let of this method is the following. Supposewe observe a finite-dimensional vector r = b,v, + b,v, + n where vi is (k+l)T k-1 a 2-c c(i)s,-,, ‘dt, a known vector and b, E { -1, l}, i = 1,2 and n is a kT i=k-n random vector. The probability that the m inimum distance decision rule incurs in an error in the first coefficient given ah cos/? = C(i)S,-i, St-kT dt. that (b,, b2) = (- 1, - 1) can be bounded by 21/N-1 E m [2-‘/“(1

+ 2exp ( -h2/2u2)]

m

WI=1

1 1

J I

Then we can write

P [ 6, f b,lb = -q

exp(-‘YIk(c(k - n);.*,c(k))/2u2)

c c(k)E(-l,O,l}

I P[jlr + v1 + v,jl 2 Ilr - v1 + v,ll Ir = -q

- v, + n]

+ v1 + 02112 jlr - v1 - v211Ir = -q

- v2 + n].

+P[l(r

=

c c(k)E(-l,O,l]

exp(-[a2

- v2 + n]

+ Ic(k)lh2

(28)

Obviously, if vi and v, are orthogonal, then the second + 2ahc(k)cos/?]/2u2) term in the right side of (28) is superfluous.However, as some algebra (or F ig. 1) indicates, the same is true if = exp(-a2/2u2) + exp(h2(cos2P - 1)/2u2) vTv22 0. . [exp (-[a - h COSP]~/~U~) Numerical examplescomparing the Forney bound (9) computed through the transfer function of the state di+ exp(-[a + hcosfi]2/2u2)]. (26) agram, the new upper bound (10) computed via the It is straightforward to show that since exp ( h2(cos2/3 branch-and-bound technique of 1131,and the one-shot 1)/2u*) 2 1, the right side of (26) attains its global maxi- error probability Q( w 1/2/u) appearin F igs. 3 and 4. F ig. 3 m u m as a function of the scalar a at the origin, and hence correspondsto the simplest intersymbol interferenceprobit is overboundedby [l + 2 exp ( - h2/2u *)I. Therefore,we can write the left side of (25) as c

**.

c(k-n)E{-l,O,l)

c c(k-l)E{-l,O,l}

k-l

. exp -

c

‘T,(c(k - n);..,c(j))/2u*

/=k-n

i

I

exp(-T,(c(k

c

- n);..,c(k))/2u2)

c(k)E(-l,O,l)

2 [l + 2exp(-h2/2u2)]

c

...

c(k-n)E{-l,O,l)

c c(k-l)=[-l,O,l) k-l

. exp i

c

‘T,(c(k - n);..,c(j))/2u2

j=k-n

I [I + 2exp ( -h2/2u2)]“+l

(27)

where the last inequality follows from the induction hypothesis and the fact that the left side of (25) does not depend on k for any n.

I 2

Fig. 3.

I 4

I 6 SNR (dB)

, 8

I IO

Bit error rate for duobinary signaling with rectangular pulse

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becomes noticeable, enlarging appreciably the region on which the upper and lower bounds provide a tight approximation to the uncoded bit error rate.

I-

-.? _

REFERENCES 111 G. D. Forney, “Maximum likelihood sequence estimation of digital PI

1

[31 [41 .6

[51

[61 -8 0

Fig. 4.

I

I

2

4

I

I

I

6 SNR (dE)

0

IO

Bit error rate for truncated exponential pulse (K = 5)

lem, namely, duobinary signaling with a rectangular pulse (K = 2, pi = w/2). Fig. 4 corresponds to an exponential pulse truncated to 5T and whose time constant is equal to 3T/2 (K = 5, rl = 0.512w, r, = 0.26Ow,r, = O.l26w, r4 = 0.051~). In the high SNR region (in which the Forney bound is tight), the upper bounds are dominated by the minimum Euclidean distance (or error energy llS( r) 11 2, terms of the series; in this region the difference between both upper bounds is due to the substitution of Q(x) by l/2 exp (-x2/2) in (9). As the SNR decreasesthe effect of the error sequences eliminated from the Forney bound

[71

PI [91 [lOI [ill

WI [I31

sequences in the presence of intersymbol interference,” IEEE Trans. Znform. Theory, vol. IT-18, pp. 363-378, May 1972. --, “Lower bounds on error probability in the presence of large intersymbol interference,” ZEEE Trans. Commun., vol. COM-20, pp. 76-77, Feb. 1972. G. J. Foschini, “Performance bound for maximum-likelihood reception of digital data,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 47-50, Jan. 1975. J. G. Proakis, Digital Communications. New York: McGraw Hill, 1983. G. Ungerboeck, “Adaptive maximum likelihood receiver for carrier-modulated data transmission systems,” IEEE Trans. Commun., vol. COM-22, pp. 624-636, May 1974. W. Van Etten, “Maximum likelihood receiver for multiple channel transmission systems,” IEEE Trans. Commun., vol. COM-24, pp. 276-283, Feb. 1976. J. E. Maze, “A geometric derivation of Fomey’s upper bound,” Bell Syst. Tech. J., vol. 54, pp. 1087-1094, Aug. 1975. A. S. Acampora, “Maximum likelihood decoding of binary convolutional codes on band-limited satellite channels,” IEEE Trans. Commun., vol. COM-26, pp. 766-776, June 1978. A. Viterbi and J. Omura, “Error bounds for intersymbol interference channels,” The Information Theory Approach to Communications, G. Longo, Ed. New York: Springer-Verlag, 1978. _ Principles of Digital Communication and Coding. New York: McGraw-Hill, 1979. A. D. Wyner, “Upper bound on error probability for detection with unbounded intersymbol interference,” Be/l Syst. Tech. .Z., vol. 54, pp. 1341-1351, Sept. 1975. S. Verdu, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 85-96, Jan. 1986. _ “New bound on the error probability of maximum likelihood sequence detection of signals subject to intersymbol interference,” in Proc. 1985 Conf. Information Sciences and Systems, pp. 413-418, March 1985.