On the Minimum Distance Problem for
n
Absrracr -When pulses are sent through an ideal band-limiting channel at the Nyquist rate they do not interfere with one another, and so data can be transmitted faithfully. In such a case, when the input to the channel is a binary sequence, the output is the same sequence, and two distinct such inputs give rise to outputs separated by a distance of at least two. However, when the rate of transmission is increased beyond the Nyquist, the channel produces interference among successive pulses, the output is a distorted version of the input, and distinct output sequences may now lie closer together. Motivated by recent work of Hajela, we here reconsider the problem of determining the minimum distance between output sequences of an ideal band-limiting channel, which are generated by uncoded binary sequences transmitted at a rate exceeding the Nyquist. This distance is the main parameter determining the error rate for recovery of the data in white Gaussian noise. For signaling rates up to about 25 percent faster than the Nyquist rate, we show that the minimum distance does not drop below the value which it would have in the ideal case, when there is no intersymbol interference. Mathematically, the problem is to decide if the best L2 Fourier approximation to the constant 1 on the interval ( - UT, us), 0 < U I 1, using the functions exp(inx), n > 0, with coefficients restricted to be = 1 or 0, occurs when all coefficients are zero. We show this for 0.802 . . . I U I 1, and this is best possible.
I. DEDICATION
E ARE privileged to have worked closely with S . 0.
W
Rice, if only for a part of his immensely fruitful career. We knew him as a master of classical analysis, special functions, and asymptotics, one who was not satis- fied until a number was given or a curve drawn, a col- league who would give the crucial insight to any problem brought to his attention. His office was always open, and we all took advantage of his selfless and active interest in our work. Certainly, we echo the comment made by Pollak on the occasion of Steve’s retirement from Bell Laborato- ries, “We are losing the youngest man in the department.”Our contribution to this issue concerns intersymbol in- terference in data transmission. Although this subject is not associated with the main body of Steve’s work, he did publish one paper [9] devoted explicitly to this topic.
Further, our problem had its origin [4] in work begun under his direction while he was Department Head at Bell Laboratories. We are sad that we cannot share this latest development personally with him, that we can only dedi- cate it to his memory.
Manuscript received July 25, 1987; revised November 28, 1987.
The authors are with AT&T Bell Laboratories, Murray Hill, NJ 07974.
IEEE Log Number 8824871.
11. INTRODUCTION
In binary data communication we want to send a stream of digits, each of which may have the value 1 or -1. To implement this, the stream is passed along the physical transmission medium in the form of an analog wave W ( t )
= Caks(t - a k ) , with ak = & 1 the kth digit Of the stream, and a a parameter determining the repetition rate of the pulses. In the presence of noise, a measure of the immunity of the system to errors is the quantity dLn, the smallest of the squared Euclidean distances,
between waveforms
v.(
t ) andT.(
t ) corresponding to distinct data streams { u p ) } and { u v ) } . In the ideal situa- tion when the { s ( t - ak)} are mutually orthogonal,d&, = 4 I m s 2 ( t ) dt = 4 E
achieved when exactly one data bit changes sign. Here and henceforth, the energy in the pulse s ( t ) is denoted by E
=
/Ems2(t) dt.If the basic received pulses s ( t - ak) are not mutually orthogonal, attempts to determine the component ak of s( t - ak) in the arriving wave are impeded by the presence of the other components a,s(t - aj). This difficulty, which may be considered a form of noise, is called intersymbol interference; remedies for its deleterious effects are a ma- jor concern for many data-communications media, the voiceband telephone channel- being the primary example.
A helpful circumstance here is that, because ak is re- stricted to a finite set of values, the possible distances among received signals form a discrete set of numbers.
Thus if the mutual interference among pulses is not too large, these distances are changed only slightly from the values they had without interference, and the pair that produces the minimum separation can be expected to stay unchanged. As this pair differs in only one bit, dLn remains at 4 E , and an optimum detector is capable of removing the intersymbol interference without penalty.
An idealized example of the process of data com- munication just described consists of a perfectly band- limiting or “brick-wall” channel of bandwidth [ - s, s ] on which one signals by using the mutually orthogonal pulses sin a( t - k ) / s ( t - k ) . These allow transmission of one
- m
0018-9448/88/1100-1420$01.00 01988 IEEE
MAZO AND LANDAU: ON THE MINIMUM DISTANCE PROBLEM
value per second-the Nyquist rate-without intersymbol interference. Indeed, one can recover the digit ak simply by evaluating W ( t ) at t = k. In [4], Mazo introduced the problem in which Nyquist pulses are used for the transmis- sion of uncoded binary data on such an ideally band-limit- ing channel but at a rate exceeding the distortion-free Nyquist rate. This means that the pulses used for signaling are s ( t - ak) = (sina(t - a k ) ) / n ( t - ak) with a < 1, and the excess over the Nyquist rate is 100 (a-' - 1) percent.
Analysis of the resulting pulse interference is not formally covered by Fomey [l], who assumed that the signaling pulses, when sampled at the integers (the Nyquist rate), had only a finite number of nonzero values, nor by Wyner [3], who treated the case when the pulse is integrable. For these two cases, d,, is rigorously known to be the impor- tant parameter for both upper and lower bounds on per- formance. In Mazo's problem we have only square-sum- mability of the signaling pulse, and the suitability of dmin in an upper bound is an open question. However, dmin still rigorously serves for lower bounds on performance [2].
Besides justifying the use of dmin as the performance criterion, there remains the problem of actually finding the minimum distance. In this paper we consider this question for uncoded binary signals, transmitted at a faster-than- Nyquist rate over a certain range of values of a, with the goal of seeing when dii, = 4E, the value which it has with no intersymbol interference. This consists of finding [4]
The p k in (1) are proportional to the difference between the binary data u k and a ; for two transmitted sequences and hence, when properly scaled, take the values 1,O.
The reader wishing a more explicit discussion of (1) should consult [4, derivation of eq. (15)]. The main question is whether, for U in an interval close to one, the infimum is achieved when all coefficients are zero, for then dii, = 4E.
We will show this to be true for 0.802.. 2 a 11, the lower bound being that value a, for which
This also shows that, as a decreases below U,, the best approximation in (1) is initially not a single exponential but has seven terms, a phenomenon suggested by computa- tions in [4].
In the earlier study [4], certain general results were established, including the fact that the minimum distance is not zero. Also, some upper bounds were obtained, from which one could conclude that dii, could not remain at 4E if one signaled at a rate about 25 percent above Nyquist. Although it was suggested that d 2 , = 4E up to t h s rate, the existence of even a small interval of rates where this was true could not be established at that time.
Further, it is known [4] that for a < 1 the infimum of (1) is
zero if the coefficients p k are allowed to take unrestricted real values. The coefficients would have to become un- bounded, however. Here our problem centers on the effect of restricting the coefficients to have values k 1,O.
Although considerations of complexity suggest that faster-than-Nyquist signaling is probably not to be taken seriously as a practical communication strategy [5], deter- mining the value of d,, has remained an intriguing question, and techniques derived for answering it can be expected to apply to similar problems for realistic trans- mission over channels that have severe distortion at their band edges. Recently, Hajela [6] has taken up t h s problem and has established that for some interval of rates faster than Nyquist (i.e., for 1 - 6 < a < 1 with some S ) , dkin =
4E. Here we propose a different argument which may be simpler and which shows that d i i , = 4E for 0.802 . . . I U
I 1, which is best possible. Hajela has extended his meth- ods to reach the same conclusion.
111. OVERVIEW OF THE ARGUMENT Let p ( x ) denote the trigonometric polynomial 1
+
C k = l p k e i k x K with p k =
+
1 or 0. Then, as in (l), we define1 an 2na - a n
= -\
I p ( x ) 1 2 d x .We want to minimize J,( p ) over all choices of p ( x ) ; that is, we seek
I( a ) = min J a ( P ) K ; ( p k = k l . O , O < k 2 K )
and ask for the range of values of a for which I( a ) 2 1.
We can obtain J a ( p ) explicitly in terms of the coeffi- cients { p k } by expanding I p ( x ) I 2 in (2) and integrating.
The primary difficulties we then encounter are that the samples at integral values of t of the function
sin ant
1 a n
-1
- (3)2na - 0 7 U T t
which appear in this expression decay slowly for U 2 1 (they are not absolutely summable) and that we know nothing about the number or pattern of coefficients p k to be considered. In combination, these obstacles make it hard to estimate Ja( p ) directly. We therefore first consider a smaller quantity, obtained by introducing a suitable weight into the integrand of (2). Now in the corresponding expansion our initial choice for this weight replaces the poorly behaved samples {(sin ank)/unk } by samples which decay more rapidly. This will be sufficient to show that Ja( p ) 2 1, unless the coefficients { p k } are grouped into a limited number of blocks, withm each of which the p k strictly alternate in sign. We then repeat the procedure, using a lower bound and weight tailored to exploit this information, to limit further the polynomials that can produce J a ( p ) 51. Applied a third time, this method fi-
nally leaves only a few special possibilities, which we examine individually. What is interesting about this pro- gram is that basically the same idea can be refined succes- sively all the way to a sharp bound. It is also noteworthy that, as a decreases form one, the sequences initially nearest together differ in only one digit, but the pair for which the normalized d$,/4E first drops below one differs in seven digits and hence started out much farther apart.
IV. REDUCING COMBINATORIAL COMPLEXITY Suppose G ( x ) is a function with IG(x)l11, which van- ishes for 1x1 2 UT. Then, with p ( x ) the complex conjugate of P ( X ) ,
and also
On expanding p ( x ) and G ( x ) p ( x ) on [ - a , ~ ] in the complete orthonormal system { e i n x / f i } , we obtain Fourier coefficients
where
g ( t ) =
-luT
I G ( x ) e - " " d x .2au - U T
Thus on setting
f ( t ) = C p k g ( l - k), (8) we find that the coefficients in the Fourier expansion of G ( x ) p ( x ) are { 2 a u f ( n ) / G } . Hence by Parseval's theo- rem
( X P ( X
P
( X I dx = 2 a a x p n f ( n1
(9) / _ " ! c ( x ) p ( x ) 1 2 d x = 2?ra21(f(n)l2. (10) Alternatively, we could obtain (9) by expressing the inte- gral as a quadratic form pTGp, with p the vector with components Pk and G the Toeplitz matrix having g ( k ) as its k t h diagonal. In view of (9) and(lo),
inequalities (4) and ( 5 ) yield the lower boundsJ ~ ( P ) 2 E p n f ( n ) I (11)
JJ
P I 2 a W ( n ) 1 2 . (12)Equality in (11) and (12) is obtained if G ( x ) =1, 1x1 I
UT , whereupon
sin a a t
g ( t ) =
7 =
- sinc at,but the broad tails of sinc(x) make it hard to estimate the right sides. We need a more manageable choice for G ( x ) , and so we begin with
G ( x ) = exp(ix/2)cos(x/2a) (14) for which
2 cos a a ( t - 1/2)
g ( r ) = ~ l-4a2(t-1/2)2' (15) g ( t ) being symmetric about 1/2. The general scheme of our argument will be to apply (11) or (12) with a succes- sion of functions g ( t ) , each of which gives enough infor- mation to make the next one useful.
We illustrate the utility of (15) when a = 1, in which case we have g ( 0 ) = g ( 1 ) = 1 / 2 , and all other g ( k ) vanish.
Consequently, using (8), we find
1 1
f(n) = % P n - l + Z P n .
Suppose now that an isolated alternating sequence exists among the p , which extends from i =
K,
to i = K,, inclu- sive, thatis, p K , - , = p K 2 + , = O a n d IpK,I=l, p n - , = - p n , K , e n I K,. Then f 2 ( n ) = 0 for K , < n I K , , f2(K1) =f 2 ( K 2 +1) =1/4, and C 2 + ' f 2 ( n ) =1/2. Using (12) we could then conclude J1( p ) 2 1/2; by the same reasoning, if there were N isolated alternating sequences among the p i we could conclude J , ( p ) 2. N/2. However, even more can be said. If we have two alternating sequences with no zeros between them, that is, if p K , = p K , +
,
where K , is the index of the last term of one alternating sequence and (K,+
1) is the index of the first term of the subsequent alternating sequence, then f2( K , - 1) = 1, and we obtain a larger value for X f 2 ( n ) than if the sequences had been isolated. Let us now decompose the sequence of coeffi- cients into sections, each of which consists of a maximal run of successive alternations of+
1 and - 1, and refer to these sections as alternating sequences, or blocks. Thus if the sequence { p i } is composed of N such blocks (blocks of length 1 are counted), we conclude that J1( p ) 2 N/2.For arbitrary a, direct use of (8) shows that, if P k = 1 and p k - , = 0, then If(k)l> m ( a ) where
m
(4
=d o )
-I
- l d 3 )I
. . * - l g ( - 1 ) l - l g ( - 2 ) l * - 'm
= g ( o ) - 2
c
I g W l . (16)k = 2
In writing (16), the symmetry of g ( t ) about t =1/2 has been used, so g(0) = g ( l ) , g ( - k ) = g ( k
+
l), k 2 I ; of course, we require m ( a ) > 0 for an effective lower bound.The same bound applies to f(k) if lpkl =1 and P k + l = 0.
Also, when there is no gap, that is, if P k = p k + l # 0, we
have
m
This shows that each alternating sequence of coefficients in f ( t ) contributes (from its first and last terms) at least a(2m2(a)) to the right side of (12).
Thus our first conclusion is that, for a given U, if Jo( p ) I 1, then the number of alternating sequences that are allowed among the coefficients of p is at most
where int [ y ] is the integer part of y . Table I displays g(O), m ( a ) , and Nl(a) for several values of a; we define N2(a) subsequently. The preceding argument has reduced the combinatorial complexity of the patterns of the p k in p ( x ) that need to be examined to a limited number of blocks, within each of which the coefficients alternate in sign.
TABLE I
SOME VALUES OF N , ( o ) AND N , ( o )
o R(0) m ( o ) Ni(o) N ? ( o )
1.00 0.5000 0.5000 2 -
0.96 0.5099 0.3862 3
0.94 0.5147 0.3483 4
0.92 0.5195 0.3146 5 5
0.91 0.5218 0.2986 6 5
0.90 0.5242 0.2831 6 5
0.88 0.5288 0.2537 8 6
0.87 0.5311 0.2394 10 6
0.86 0.5333 0.2250 11 6
0.85 0.5356 0.2112 13 6
0.84 0.5378 0.1975 15 6
0.83 0.5400 0.1831 17 6
0.82 0.5422 0.1709 20 7
0.81 0.5443 0.1561 25 7
0.802 0.5460 0.1440 30 8
-
-
V. REDUCING THE SIZE
We now show that the presence of alternating sequences among the coefficients { p k } of p ( x ) can be used to reduce the number of coefficients that need be considered, but for a somewhat different problem. An alternating sequence starting at position s and having overall length 1,
f z++ ( - z ) + ( - 2 ) 2 + * * + ( - z)'-1], can be summed to give
f zs
[
1 - (- z ) ' ] l + zIn (18), and henceforth, z = exp(ix). Using (18) we may restrict the general trigonometric polynomial appearing in ( 2 ) to
so that
where 4 ( x ) is a polynomial in z having at most 2N1(0) nonvanishing coefficients, { qk } ; the vector formed from the qk will be denoted by q. Typically, we may think of these coefficients as f l , but f 2 will arise whenever we have two sequences without a gap (and the number of coefficients decreases appropriately). It will be important for the sequel that we also know something about the sign pattern of the q k . Namely, from (18), each alternating sequence contributes a pair of coefficients to q in only one of the patterns ( l , l ) , ( l , O , - l), (1,0,0, l), etc.
Since 11
+
zI2 I 4, (19) can be bounded below bywhich is very much like (2), except seemingly a factor of four worse. Of course, two vital pieces of information can be applied to q ( x ) : the bounded number of coefficients and the sign pattern of certain pairs of them. Armed with this information, we reconsider our problem using, for a second time, the ideas surrounding (4) and Parseval's theorem (ll), but with a new choice for G(x). This time we introduce into (20) the G(x) corresponding to the centered function
Here we have g(1) > 0, g(2) < 0, and g(1) > lg(2)I for U in the range of interest (0.8 I U 11). Using (21) we bound from below the value of the leftmost (or rightmost) nonva- nishing term q k f ( k ) where, in analogy to (8),
Because of the sign pattern of the samples g(O), g(l), g(2), namely, (+ ,
+
, -), and the sign pattern of the first pair of coefficients in 4, namely, (1,l) or (l,O, -1) or (l,O,O,l), we see thatq a f ( 0 ) 2 ( l - k = 3
f I
1-4a2k2 cosa=k1]=s1(u)
(22)where, for notational convenience, we assume that the leftmost q k f ( k ) is at k = 0. Equation (22) also provides a lower bound for the rightmost term. Similarly, if qk # 0 and there are nonvanishmg qk, both for k ' < k and k'> k, a lower bound is
(23) If there are P pairs of coefficients, then (20), (ll), (22), and (23) yield
In writing (22) and (23) we have ignored the possibility that qk can equal two; for each q k = 2 that occurs, the contribution estimated for qkf(k) must be reduced by I g ( f ) l for some f = 3,4,. e. The total reduction thus pro-
duced in (22) or (23) cannot exceed 2CZ3g(l). On the other hand, the “diagonal terms” of q k f ( k ) + q k + l f ( k
+
1)that appear in (24) were counted as ql
+
q i + l = 2, whereas now they contribute (qk+
= 4. Thus the overall effect of every qk = 2 is to increase the right side of (24).Denote by N2( a) the largest value of P in (24) for which the right side is less than one. This is another bound on the maximum number of alternating sequences consistent with J,( p ) I 1; that is, if either Nl( U ) or N2( a) is exceeded, we surely have I( a) > 1. Several computed values of N2( a) are included in Table I, and we conclude that the maximum number of pairs that need be considered in q ( x ) is eight.
As a approaches 0.8, the values of Nl(a) would have been too large for our later methods to be effective; hence the need to reconsider the problem and obtain the smaller At this point, in the estimate (11) we pass to the choice g ( t ) = sinc(oat) of (13), for the information we now have about the number of coefficients in q(x) and their sign patterns will make it possible to limit the interference among the g ( t - k). To this end we set a = 1 - c, where- upon
N , ( a ) *
We assume that the reader is familiar with the function sinc(x) (see [8]). In particular we use ( s i n c ( x ) l ~ l , and the lobe structure of alternating signs, the main lobe corre- sponding to x in the interval [ - 1,1]. For n 2 1 we describe the values of sinc(x) when x E ( n , n
+
1) as the n th lobe, the same term also being applied to the symmetric lobe on the negative axis. Denoting the maximum of (sinc(x)l in the nth lobe by s * ( n ) , we have s*(l) = 0.21723 ..
,s*(2) = 0.12836
-
, and, for n 2 3, s * ( n ) z 2/(7r(2n+
1))to sufficient accuracy.
By (8) and (11) we are to find a lower bound for the value of q,f(n), f ( n ) itself being an inner product of the coefficient vector q = { qk} and a vector of shifted samples { g ( n - k)}. Since g(0) =1, the term q i always occurs in q k f ( k ) and contributes the value 1 (or 4 if qk = 2). To provide a lower bound for q k f ( k ) consider again (18), which shows how one alternating sequence contributes a pair of coefficients to q. The pulse samples (25) have, within the main lobe, the sign structure ( , - ,
+
, - ,+
,+ , +,
-,+
, -, ) and alternating sign structure in any other lobe. In general a sign reversal occurs between lobes.On the other hand, that portion of q determined by a pair of coefficients as in (18) must be one of the types (except for an overall equal sign): (1, l), (l,O, - l), (1,0,0, l), etc., and in forming an inner product of one of these with a strictly alternating sign structure, such as one might en- counter in a sidelobe of (25) when calculating the value of f(k), the two surviving terms cannot be of the same sign.
Bounding the contribution that the lobes of sinc(x) make can now be simplified by examining the following problem. For x E [ a , b ] , consider a positive, monotonically decreasing function h ( x ) . This is to correspond to half of a lobe (other than the main lobe) of Isinc(x)l, for example,
x E [1.5, 21. Consider evaluating h ( x ) at M pairs of points (x,, 5 J 9 where
x , < 5 , s x , + 1
and possibly also at another individual point 2
EM.
As earlier, we sum the values so obtained but require that the terms of a pair contribute with opposite sign. Thus we form the sumM
s= c
e , ( h ( x , ) - h ( 5 , ) ) + d h ( 1 7 )I = 1
where e, =
and positivity of h ( - ) ,
1 and d = & l , O . Clearly, by monotonicity
M
S I
c
( h ( x , ) - h ( 5 , ) ) + Idlh(17)1 = 1
M - 1
= %)-
c
( h ( 5 , ) - h b , + 1 ) ) - ( G 4 - IdlWT)) I h(x,) I h ( a ) .r = l
T o find an upper bound for a similar sum of such pairs over a full lobe of (sinc(x)l, plus possible individual points at each end (each corresponding to a point of a pair that straddles two lobes), divide the lobe into two halves with respect to the maximum without separating a pair. The latter requirement does not cause difficulty since the posi- tive contributing member of the pair that may be strad- dling the maximum can first be moved to the maximum, thus increasing the sum. Such a sum, then, over the nth lobe of Isinc(x)J cannot be larger than 2s*(n), or 4s*(n) when both positive and negative n th lobes are considered.
By a very similar argument, one can show that the main lobe (recall we are to take the “1” term with a positive sign) must contribute at least 1 -(c/a)sinc(c) to the inner product with q.
Assuming that P pairs of the type induced by (18) form the vector q, write
where I is an integer, I 2 0, and R = 0 or 1. It follows from the arguments just presented that the value of qkf( k ) must be at least C , where
P - l = 2 I + R
- - ( s * ( l ) + s * ( 2 ) + 4€ *e* + s * ( I ) + -R.s*(I+l)). 1
U 2
(26) Finally, from (ll), (26) is to be multiplied by 2 P to bound the sum C q k f ( k ) . Of course, certain qk may equal two, but the total weighting is always 2P. Thus, for P pairs,
1
J J p ) 2 -(2PC). 4 (27)
The bound (27) can be improved by noting that, for the first and last nonvanishing q k f ( k ) , all terms of f ( k ) are on one side of the peak of the main lobe. Thus these two
q , f ( k ) each contribute at least Consequently, 2€ p - 1
U (28) 1 dx 2 U 1 (A21
C ' = l - - s * ( m ) - f s * ( P )
-/"
Iq(l+z)l2dxm = l
and for P pairs (27) is replaced by the stronger bound
2 s -.n 1
J, ( p ) 2 - 4 (2( P - 1)
c +
2C'). (29) We note that (29) is monotonically decreasing in E for fixed P, since C and C' are. Thus it suffices to set c = 0.2 (or U = 0 . Q The lower bound (29) for p ) is displayed in Table I1 for several values of P. Using Tables I and 11, one verifies that .fa( p ) > 1 for 0.802 I U I 1 except possi- bly if P = 1, 2, or 3.TABLE I1
LOWER BOUNDS FOR Jna( p ) VERSUS P
2 0.758
3 0.951
4 1.109
5 1.207
6 1.290
7 1.330
8 1.361
9 1.358
10 1.349
or
For U = 0.8, the use of Table 11, which serves as a lower bound for the integrals of (20) and (A3), yields
o
[ &
/lq12]2 2 7.354Thus for the cases when the denominator of (A3) does not exceed 7.354 we are done. In fact, with unit coefficients in q , the denominator can take only values that are even integers from 2 to 14.
There is only one case to consider when the denominator has value 14, namely, q = (l,l,l,l), while the value 12 requires consideration of either q = (1,1, gap, k(1,l)) or (l,l,l,gap, f l ) , where "gap" means one or more zeros. The first two of these three cases arise from polynomials q ( z ) that are of the form (1
+
z)(l f z k ) . Integrating (19) in closed form, we findO n 2sinosk
Z(o) = rninL/ Ilkzk12dx=2k- > 1 (A4) if U 2 0.8.
Now let s,=sinc(kc). Then when q=(l,l,l,gap,$l), we have (for
1 on
If some of the terms in q ( x ) coalesce to produce coeffi- cients qk = 2, the bound (29) can be improved, since the component qz in q , f ( k ) now contributing four, whereas (25) counted only two for the corresponding terms sepa- rately. Thus when q ( x ) has T such coefficients,
k#O2sU - o n ask
3),
1
1
4 €
J o ( p ) 2 - ( 2 ( P - 1 ) C + 2 C ' ) + T / 2 . (30) = / - o n
+ 1 - 2 -Is,, ~ 1 - s,, + s,, + 1 I. (A5) We have thus established our claim save for the three
values of P indicated. Since treating these remaining cases seems to require special and more numerical arguments, we relegate the details to the Appendix.
By expanding sin((n l)sr), we have 2n2 cos se
ACKNOWLEDGMENT
We are indebted to D. J. Hajela for stimulating discus- sions and for rekindling our interest in this problem.
APPENDIX THE CASES P = 1,2, AND 3
We first dispose of the case P = 2. If the middle coefficient of q has magnitude two, we use (30), which augments the lower bound of Table I1 by 0.5. For four-unit coefficients we have, by Schwarz's inequality,
2n2 2 5 1
-1 +,<-s,+--,
5 lM[
E ]
n - 1 - 4 4since n 2 3 and E is sufficiently small (E < 0.2, say). Using this in (A5), we find from (A3) that for c < 0.2:
I ( U ) 2 (13.4)/12.
This concludes the necessary discussion when the denominator of (A3) has a value of 12.
When the denominator has a value of ten, the cases to be considered are q = ( l , l , g a p , f l , g a p , f l ) and (+l,gap,l,l, gap, f 1). A third possibility, ( l , l , 1, - l), can be ignored because, from (18), two adjacent coefficients of a pair cannot have oppo- site sign, while a fourth, (l,l, -1, -l), is already included in (30). Assuming the first pattern and using arguments similar to those just given, we obtain the lower bounds to qk f( k) for qr # 0 listed in Table 111. These larger bounds result from q starting with two ones of the same sign. Also, in deriving these bounds, we used s1 > s2 > s3 > s*(l) if E I 0.2. The sum of the four rows in Table I11 is a lower bound for (1/2s)/1qI2, and a plot shows
TABLE I11
LOWER BOUNDS WHEN q = (l,l,gap, & l,gap, 1) k Lower Bound to qkf( k )
€
1 2
1 + ,[SI -sj - s*(l)]
1 + :[SI - s2 -s*(l)]
a
€ 2 €
a o
1 - - - - s * ( l ) 3
4 1 - f [ 2 S * ( 1 ) + s*(2)]
a
that
1 an
- 2aa
j
- a n 1q( x) dx > 10.05 if 0 I c I 0.2.ner provides the lower bounds
Treating the case q = (+l,gap,l,l,gap, k l ) in a similar man-
for the first and last nonvanishing qkkf(k), and 1
+
:[SI - sg - s*(l)]U
for the middle two. Again, the numerator of (A3) is bounded below by a value exceeding 10.05 for 0 I c I 0.2.
Finally, when the denominator has a value of eight, if the pattern (1,l) occurs in q it is easy to see that the numerator in (A3) exceeds nine. The only remaining case is four isolated coefficients. If c = 0.181, the use of (29) and (20) in (A3) gives a value 8.07 for the numerator, and we need be concerned only with c E [0.181,0.2]. In our bounds (26) and (28) we may replace 2s*(1) by
= 0.367 where x and x mar[-+ x ; r +2nc sin x are in the first lobe. Further, x sin( x x +2nc
+
2sc)I
sin 2nc max - = 0.798
€ 2nc
Therefore, for P = 2, the estimate
t U
1--[2s*(l)+s*(2)]
is replaced by
E U
1- -[0.367+0.12836]
and
E sinm
1- - - +2s*(1)]
a
[
nc1- :[0.798+0.367] U
These estimates show that the numerator in (A3) is greater than 8.03, and conclude the investigation for P = 2.
We quickly outline P = 3 . We can again eliminate the cases when a coefficient of q equals two. From Table I1 we see that the numerator of (A3) exceeds 11.57 when c = O . 2 and hence only denominators with values 212 are of concem. Each pair must
contribute coefficients as (l,gap, k l ) , since the pattern (1,l) can be excluded from q as before. For the same reason, should any of these three pairs be adjacent, the denominator takes the value eight or ten. This leaves the case of six isolated coefficients, which is handled similarly to that of P = 2 with four isolated coefficients.
Finally, consider one alternating sequence of length L, so that p(z)
=E:;;(
- l)kzk. Summing the sequence, we finddx 1 Io"sinz [ L ( n
+
x)/2]- _ dx -
an 0 cos2 x/2 du . - _
-
We now exploit the fact that, for large L, the periodic numerator behaves like its mean value of 1/2. To express this, let M be the largest integer for which L
+
M is even and such thatM n / L I as; ('46)
(A7) by definition, ( M
+
2)n/L > an, so thatM n / L > UT - 2n/L.
By (A6) we find
1 n + ~ n / ~ l - 2 s i n L u - _ du
-
an
J,
2sin2u/2We now write
and since sir? u/2 is a decreasing function of U in .n I U I n
+
M n / L , the terms being summed are opposite in sign and increas- ing in magnitude. The choice of M makes the last term negative, hence the sum is likewise negative. Consequently,
1 Mn
J,(p)>-ttan-
an 2L
>-tan 1
UT
the last inequality by (A7). Now the right side increases in a, as a differentiation shows, and at 0=0.8, L > 50, it exceeds 1. Re- stricting consideration to L I 50, we again invoke (A7) to obtain
L Mn
Jo(p)r(M+2)s tan - 2L
with M the largest integer I aL for when M
+
L is even. For given L, the right side is monotone in M/L, hence also in a;applied with a = 0.8, this bound exceeds one when 271 L I 50 except for 31 I L I 34 and L = 41. In the remaining cases we use the exact expression
sin sna L - 1
J o ( p ) = L - 2 (L-s)-,
SITU s -1
obtained from (2) by expanding l p 1 2 and integrating term by term. Examined for the relevant values of L near
U = 0.8, this shows that J,( p ) first reaches one for L = 8.
[3] A. D. Wyner, “Upper bound on error probability for detection with unbounded intersymbol interference,” Bell Syst. Tech. J . , vol. 54, pp. 1341-1351, 1975.
[4] J. E. Mazo, “Faster-than-Nyquist signaling,” Bell Syst. Tech. J . , vol.
This completes the argument.
REFERENCES
54, pp. 1451-1462, Oct. 19751
G. J. Foschini, “Contrasting performance of faster binary signaling with QAM,” AT&T Bell Labs. Tech. J . , vol. 63, pp. 1419-1445, Oct.
1984.
D. Hajela, “Some new results on faster than Nyquist signaling,” in Proc. 1987 Conf. Information Sciences and Systems, John Hopkins Univ.. Mar. 1987.
[5]
[6]
[7] -, “On computing the minimum distance for faster than Nyquist signaling,” in Proc. 10th Symp. Information Theoty and its Applica- tions, vol. 11, Fujisawa City, Japan, Nov. 1987.
P. M. Woodward, Probability and Information Theoty, with Applica- tions to Radar, 2nd ed.
S. 0. Rice, “Distribution of Z a , / n , a , randomly equal to i 1,”
Bell Syst. Tech. J . , vol. 52, pp. 1097-1103, Sept. 1973.
[l] G. D. Fomey, Jr., “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 363-378, May 1972.
-, “Lower bounds on error probability in the presence of large intersymbol interference,” IEEE Trans. Commun., vol. COM-20, no.
1, pp. 76-71, Feb. 1972.
[8]
[9]
[2] New York: Pergamon, 1964, p. 29.
