on the determination of the worst sampling error in a communication system for...

Math. Meth. in the Appl. Sci. 8 (1986) 256-268 A M S subject classification: 41 A 30, 41 A 45. 41 A 65

0 1986 B. G. Teubner Stuttgart

90 D 13, 90 D 45, 60 G 35, 94 A 05

On the Determination of the Worst Sampling Error in a Communication System for Pulse-Amplitude- Modulated Signals by Solving an Approximation Problem

W. Krabs, Darmstadt

Communicated by E. Meister

A communication system for pulse-amplitude-modulated signals is considered whose impulse response is bandlimited. For a certain class of sampling errors the worst case is investigated where the sampling error of the system is such that the least mean square transmission error achieved by a suitable choice of the impulse response becomes maximal. This problem is dualized in the sense of two person game theory and thereby transferred to an equivalent approximation problem for whose solution a necessary and sufficient condition is given and applied to special cases.

0 Introduction

By Shannon’s sampling theorem [3] (see also [4]) a bandlimited signal f E L2(R) can be exactly interpolated by its valuesf(k - T ) , k E N, if the bandwidth is given by 2 d T . This means that the values f ( k - T ) can be exactly sampled by a suitably designed communication system. We consider the simpler situation where the discrete signals sk, k E N, to be processed by the system are identically distributed random variables which are uncorrelated, normalized and have zero as common mean value. In this case it is easy to see that exact sampling is possible, if the impulse response of the system is given by (1.6). The starting point of our considerations is the assumption that the sampling is done with an error rk at every instant k T and that the 7k’S are identically distributed random variables with respect to the probability space of all normalized Bore1 measures defined on an error interval [-?,TI with 0 < r Q T/2. We further assume that the signals sk and the sampling errors rk are independent so that the mean square error between the input signals s k and the sampled signals y(k - T ) can be defined independently of k. This mean square transmission error (2.1) depends on the probability measure p, belonging to the sampling errors r,, k E N, and the impulse response of the system. For our purposes it is more convenient to express the mean square transmission error b y p and the Fourier trans-

On the Determination of the Worst Sampling Error in a Communication System 257

form of the impulse response which is called the transfer function of the system. In view of Shannon’s sampling theorem it is reasonable to design the system only by bandlimited impulse responses where we assume the bandwidth to be a fured multiple of 2x/T. This means that the corresponding transfer functions are all in

If a sampling error, represented by its probability measure p , is given, then the designer of the system could try to find a transfer function such that the corresponding mean square transmission error becomes as small as possible. We are now interested in the worst case where the sampling error is such that the corresponding least transmission error becomes maximal. This problem has been studied in [ 5 ] and has been completely solved for bandwidths B E [0,4x/TJ. At first one has to study the minimization of the transmission error for a fixed sampling error, i.e., for a fixed probability measure. On using results from [l] a necessary and sufficient condition for the best transfer function can be given in the form (2.11) and the least transmission error can be expressed by (2.12). The worst case problem is the max-min-part of a typical two-person-game. Therefore we also consider its dual min-max-part which is shown to be equivalent to a suitable approximation problem (see (3.14)). The main results then are the following: 1) A sufficient condition (see Theorem 3.1) is derived for a solution of the worst case max-min-problem by using the approximation problem (3.14). 2) It is shown that no duality gap can occur (see (3.18)). 3) It is shown (see Theorem 3.2) that every solution of the approximation problem (3.14) leads to a solution of the worst case max-min-problem. In Section 4 the sufficient condition of Theorem 3.1 is used in order to determine solutions of the worst case max-min-problem for the special cases N = 1, 7 E [O,T/2] and N E N, 7 = T/2. It turns out that the worst case here leads to discrete Bore1 measures concentrated on f 7, i.e., of the form

1 2

p = -(aT + s-,)

6 denoting the Dirac measure. The question is open whether a similar statement can be made for all N E N and all 7 E [0, T/2]. Also, in general, it has not yet been proved that the worst case max-min-problem has a solution. A natural way would be to prove the solvability of the approximation problem (3.14) and to use Theorem 3.2.

1 The System without Sampling Errors

The system for Pulse-Amplitude-Modulated signals, in short: PAM- system, consists of a transmitter, a channel, a receiver and a sampler. It picks up discrete signals

S, = s ( k * T ) , k € Z ,

258 W.Krabs

at a fixed time distance T > 0 which are assumed to be identically distributed random variables on some probability space (A, %,p) and satisfy (s(k T, a))keZ E I, for almost all a E A . These signals are modulated into a continuous signal

m

(1.1) y ( t ) = T 1 sjh(t - iT), t E R , IP - m

which is sampled in form of discrete signals y(k - T ) , k E Z, again. The function h = h( t ) which, mathematically, does the modulation, is called the impulse response of the system and is assumed to be at the disposal of the designer. It is to be chosen in L2(R) n C(R) such that its Fourier transform has a compact support. In general, the output signals y(k - T ) are different from the input signals sk , k E 2, and the question arises whether (1.2) y(k - T) = sk fora l lkEZ can be achieved by a suitable choice of h. An affirmative answer to this question can be given by use of Shannon’s Sampling Theorem. Under special assumptions on the input signals s,, k E Z, a simpler approach is possible. Let the signals sk be uncorrelated, normalized and have zero as common mean value, i.e., (1.3) E(s, - sk) = 0 for all i , k E Z with i =k k and

~ ( s t ) = I , €(sk) = o for all k~ Z where E denotes the mean value. Under the above assumption the sampled signals y(k - T ) are random variables on the same probability space (A,%,p) as the input signals s,. Hence, for every k E 2, the mean square error E((s,, - y(kT))’) is well defined and can be expressed, independently of k, in the form

(1.4) E((sk - y(kT))’) = (1 - Tl1(0) )~ + T 2 h ( - i T ) ’ . . i- - m

i + O

Therefore (1 -2) is satisfied which is equivalent to E((sk - y (k T))*) = 0 for all k E Z ,

if h E L2(R) n C(R) is chosen as above and such that

(1.5) h(0) = - and h ( i T ) = 0 f o r a l l i E Z , i + 0 .

The latter condition is called the Nyquist sampling condition. It is, for instance, satisfied by

1

T

1 h(r) = - T

sin (+) * t

(+ ’

t e R ,


which is in L2(R) n C(R). The quantity 2 x (1.7) o0 = - T

is called the Nyquist frequency bandwidth. The transfer function of the PAM- system with (1.6) as impulse response is then given, as the Fourier transform of h, by

1 f o r - - c o < - 0 0 "0

2 2

2 The System with Sampling Errors: The Worst Case

We assume that instead of the exact output signals y(k - T) signals of the form y(k - T + Tk) are sampled where the Tk's, representing the sampling errors, are identically distributed random variables on the probability space (Y = [ - s,7],23,p) with T E [0, T/2] where p is a Borel measure (defined on the sigma algebra 23 generated by the open subsets of Y) with p(.T) = 1. We assume that the random variables s k and rk are independent and that (1.3) holds. Then the mean value of the mean square error, E((Sk - y(k T + rk))'), is independent of k and is given in the form

= i (1 - Th(t))2dp(t) + T2 f i h ( t - iT)2dp(t). - r i = - m -r

i+ 0

In particular for p = 6 = Dirac measure (which means that no sampling error occurs) we obtain

E(6,h) = (1 - T l 1 ( 0 ) ) ~ + T f / I ( - ~ T ) ~ i - --o

i * O

= E((sk - y (k T))2 ) for all k E 2 (by (1.4)) . Therefore we also call E ( p , h ) the mean square error of the PAM-system, when sampling errors occur. Let Pr be the set of all Borel measures p (defined on the sigma algebra 23 generated by the open subsets of 7 = [ - 7, r ] ) with p ( n = 1) and let, for some given N E N, #N denote the subspace of L2(R) consisting of all h E L2(R) n C(R) whose Fourier transforms H = H ( o ) , o E R, have their support in the interval

r 1

260 W.Krabs

For everyp E p7 we define

(2.3) with E(pJ h ) given by (2.1). We are interested in the worst case where the infimum Ep,N of the mean square error E(p, h ) with respect to h attains its maximum with respect top . Thus, given T E [0, T/2] and integer N E N , we are looking for a pair (8, hp) E P7 x YN such that

(2.4)

For the solution of this problem it is more convenient to express the mean square error E ( p , h ) in terms of the Stieltjes-Fourier transform of p and the Fourier transform of h, i.e., in terms of

(2.5) FJo) = j e-j"'dp(t)

and

Ep,N = inf{E(p, h ) I h E YN}

E(8, hp) = max min E(pJ h) . p€Yr he"

- 7

OD

H ( o ) = J h(t)e-iu'dt , - O D

respectively, where j = m. For every h E YN and p E pT we then obtain

(2.6) E(p, h) = I?(P,W where

(2.7) E(p,H) = 1 - - j 1 H ( o + ioO)~,(o + ioo) 00 no N "-I i = O

+ F p ( o + iwo)H(w + iwo) - H ( o + ioo)

and

The worstcase-problem now is equivalent to finding a pair (fl, Hp) E p7 x L2(QN) such that

(2.9) ,??(flJHp) = max min Z?(p,H)

with i (p , H) defiied by (2.7) and S2, by (2.2). One can easily show by standard methods that, for a given p E p7, a function Hp E L2(sZN) satisfies

pf)r H E L ~ ( S - J ~ )


N- 1

i = O (2.11) c F,((k - i,wo)H,(o + ioo) = Fp(o + kwo)

for almost all w E @,and k = 0,. . . , N - 1 . If this is the case, then

(2.12) B(p,H,) = 1 - - 1 N-1

00 QO k = O N

C H,(o + k ~ o ) F , ( o + k o o ) d o .

3 Dualization of the Worst Case

By means of (2.5) the conditions (2.11) can be rewritten in the form N-1 T 2 5 e-j(k-~"O'Hp(o + Iwo)dp(t) = j e-j("Ck"d'dp(t) /Po - - T -T

or

C Hp(w + Iwo)&'"of - e-jo')e-jkwo'dp(t) = 0 - 7 r - l 1 1 0

for almost all o E @, and k = 0,. . . , N - 1 which is equivalent to

fo ra lmos ta l lwESZ~andk= 0, ..., N - 1 .

This implies

(3.2) I?(p,H,) = 1 - - j c Hp(o + k o o ) i &(u+k"d'dp(t)do 1 N - 1

00 QO k = O - 7 N

N - 1 - --

262 W.Krabs

A necessary and sufficient condition for vp & V to satisfy (3.5) is given by T

(3.6) j j (f(t,o) - u,(t,w))o(t,o)dwdp(t) = 0 fo ra l lue V

and is equivalent to (3.1), if we put - -T a:

N - 1

k = O (3.7) u,(t,w) = c Hp(o + koo)eikWO', t € Y , O E Q R .

As a result we have the following statement: If (2.10) is satisfied for a pair (p,H,) E Pr x L2(sZN), then u p , defined by (3.7), satisfies (3.5) and

Instead of finding a pair (fi,HH,) E Pr x L2(sZN) such that (2.9) holds we there: fore consider the equivalent problem of finding a pair (8, up) E Pr x Vsuch that

(3.9)

where

(3.10) L(P, 0 ) = j l lf( t ,*) - ~ ( t , * ) I l & ~ ~ d p ( t ) .

L(@, u6) = maxrninL(p, u ) p e r uev

- r

On the Detcrmkauon of the Worst Sampling Error in a Communication System 263

Then (B , Hb) E Yr x L’(Q,) solves the problem (2.9), if and only if (P; ub) E 9’r x V with up given by (3.7) solves the problem (3.9). As a problem which is dual to the problems (3.9) in the sense of two person games we consider the problem of finding a pair (po, 8) E Pr x Vwhich satisfies

(3.11) L(po , 8) = minmaxL(p, 0) .

By an easy argument (which is standard in game theory) it follows that

(3.12) sup inf L ( p , G ) Q inf sup L(p, 0). P€Y’, V E v UF vpss,

Therefore, by solving problem (3.1 1) one obtains, in general, an upper bound for the least mean square error of the PAM-system in the worst case, i.e., in the sense of problem (2.9). Obviously, for every u E V, we have

seY -9,

(3.13) m = ~ ( p , v ) = mUIIf(t , .) - ~(t , . ) I I tqq,) PE .9, t E j r

Therefore the search for a pair ( p e , 8) E q7 x V which satisfies (3.11) is equivalent to finding some 0 E V such that

(3.14) 11 f - G ( l < ( 1 f - ullforall V E V. As a consequence of (3.12) and (3.13) we conclude that

(3.15) sup inf L(p, v ) Q inf maxlif(t,.) - o(t,.)IItz(n,) payr uev uev I E f

= infIIf - v 1 I 2 . U E V

This gives rise to a sufficient condition for a solution (a, Hb) E 9T X L2(l?,) of (2.9), namely

Theorem 3.1 Let ( p , Hb) E Yr x L2(SaN) be given such that (3.1) holds f o r p = B and

(3.16) i i i f ( t , . ) - ua(t , . ) i i t2(nR,d~w = i i f - vfiii2. - - T

with vb by (3.7) forp = #. Then (a, Hb) solves (2.9).

P r o o f . Since (3.1) is equivalent to (3.6) we conclude that

= infL(B, v ) Q infllf - v ( I 2 . V E v E V

This implies, by virtue of (3.15), that (B, ua) E p7 x Vsolves (3.9) and therefore (J ,Hp) solves (2.9). In addition ua E Vsolves (3.14). We can prove more. The next step is to show that no “duality gap” can occur, i.e., equality holds in (3.12).

264 W. Krabs

ForthispurposeweputX= V X R,Y=C(Y)anddefineF:X+ Y,p:X-,R by

F(U,Y)(t) = Y - Ilf( t ,*) - v(t,*)lltqnR) 9 t E y,

P ( V , Y ) = Yfor(o,y)eX.

Y, = { Y € Y l Y ( t ) 2 0 If we define the ordering cone of Y by

V t E - q ,

then F is a concave mapping from X into Y. Obviously 9 defines a linear functional on X and the problem of finding some Ei E V which satisfies (3.14) is equivalent to minimizing p = ~ ( u , 7) on X subject to

F(U, E Y, H F(U, Y ) (ti 2 o v t E r. The interior of Y , is given by

If we choose, for any u E V, y > 0 such that f, = ( Y E Yly(t) > 0

Y > I l f - 412 = maxllf(t*.> - u(t,*)llLz(Q;) ,

V t E - q .

2

[ E l

then F(u, y ) E f. Therefore, by virtue of a duality theorem in abstract convex programming, (see [2]), it follows that (3.17) max inf @(u,y ,y* ) = B = infllf- vl12

where Y'EY: ( 0 . Y ) E X D€ v

Y: = {Y*E Y * I y * ( y ) 2 0

@ (0 , Y I Y*) = cp(v, Y) - Y*(F(h Y ) )

VYE Y , } and

= Y ( 1 - Y * W > + Y * ( l l f ( t , - ) - wll:qn$)) withe = 1. Let the maximum on the left-hand-side of (3.17) be attained by 9' E Y$ . Then

B < ~ ( 1 - 9*(e ) ) + 9 * ( L ( . , u ) ) f o r a l l ( v , y ) ~ X

which is only possible if 9*(e) = 1, i.e.

PW) = i N ) W ) Y -T

for some8 E 9:. This implies

(3.18) maxinfL(p,u) =/3 = inf l l f - u1I2.

Let D E Vbe a solution of the approximation problem (3.14). Then it follows that

pepr Lev VE v

mas inf L ( p , u ) = I l f - p e r DE v


which implies

m=L(p, f0) = I l f - f 0 1 I 2 9

P E Yr

hence

L(#, 6) = maxL(p, 0 ) = l l f - fill2 Pf -*r

for some f l E 9* and

L(B, 6) = inf L ( f l , u ) W V

which is equivalent to r j (f (t,o) - G(t,o))Po'dfl(t) = 0

- r (3.19)

for almost all o E Sz; and k = 0,. . . , N - 1 (see ( 3 3 , ( 3 4 , (3.1)). As a result we obtain the Theorem 3.2 If 0 E V is a solution of the approximation problem (3.14), then there exists some f l E 9-T with (3.19) and

(3.20) L(@, f0) = j I l f (t, .) - G(t,-)II&n;1dfi(t) = IIf - f i l l 2

Iff0 is of the form - - T

N- 1

k=O G(t,.) = C a(* + koo)&"O',

then (fl,Z?') solves theproblem (2.9). The latter statement is a consequence of Theorem 3.1.

4 Special Cases

We first consider the case N = 1 and r E [O, TI21 arbitrary. Then we put

Ei(t,o) = A(o) = COSTO , (tentatively)

(4.1) and obtain

o E @ , tE Y,

I l f (t,.) - G(t,.)lltz(~p) = lab) - e-jw'12do *l

3 0 0 1 sinrwo 4 2 - --+--- 2 c o s r o c o s t o d o

2 2 r - w d 2

'1 sinroo <- - Oo - - f o r a ~ t ~ [ - r , r ] . 2 2 7

Equality occurs only for t = - r and t = + r.

266 W.Krabs

1 2

If we put f i = - (a(. - r ) + 6(. + T)), then fi E Pr and (3.16) is satisfied. Further-

more, we have 1 1 j ( f i ( t ,o ) -f(tJw))dfi(t) = -(COSTO - P W r ) + -(COSTW - dWT)

- r 2 2 = COSTO - cosro = o for all o E = SZ,

which shows that (3.1) is also satisfied. By Theorem 3.1 we therefore conclude 1

that the pair (A (4.1), f i ) with f i = - (a(. - r ) + a(. + r) satisfies (2.9). 2

Second we consider the case r = - and N E N arbitrary.

Now we define (tentatively) fi by (3.7) with H, = I? and

T 2

T cos--0 form€

(4.2) A(o) = [ -9 UO]. '2 0

Then we obtain

Ilf (t> -1 - f i (L *)llt2(QoN)

.II N- 1 = 1 tf(o + koo)dkwor - e-jWr d o

QO ',I k = 0

"0 2 5 II?(o) - e-jwrI2dw, if Nis odd, 0 0 2

--

0 5 II?(o) + A(o + o ~ ) & ~ o ~ - dWr / 'do, ifNis even. =i - WO

In both cases one calculates

which implies

--- 3Wo 4 1 foral,tE[-- T T 1. 2 'T < I l f ( t> -1 - w> .)IIL2(QN) Q -

2 7 t 2

If we put@ = (6 (- -+) + 6 (. +;)) , then f i E PTI2 and (3.16) is satis- 2

fied. Let N be odd. Then, for every o E f2& and every k = 0,. . . , N - 1, we obtain


H

T - (O(f,w) - f ( f , - O ) ) & k W o ' d f i ( t )

2

- j i w o l - j w z jkwOT T 2 - e 2)e

j T w 0 r . N - 1 T T - e = L@(" + -I-"o) N - 1

2

-~-pq- .N-1 T j w y jkwo+ T - e ')e N - 1

2 . 2

N-l (I ) ( N i l k ) -00 - T - COS(O + kwo) - T = (-1) cos --w cos -- 2 2

nce (3.1) is also satisfied. By a similar calculation one can also verify that fo an even integer N the condition (3.1) is satisfied. .

1 By Theorem 3.1 we therefore conclude that the pair (A (4.2), @) with fi = -

2 (6 (- -5) + 6 (- +f> satisfies (2.9).

These results have also been proved in [5 ] by different methods and for - symme- - tric Bore1 measures. In [5 ] a full discussion of the case N = 2 for all 7 E 0, - L :I is given. Here it turns out that for sufficiently small 5 the worst sampling error has the probability measure

268 W.Krabs

p = 160

withII E 0, - 1 : depending on 7.

5 References

[l] HBnslcr , E.: Entwurf optimaler Impulse filr Puls-Amplituden-Modulationssysteme mit

121 Krabs , W.: Optimization and Approximation. New York-London-Sidney: John Wiley & Sons

[3] S h a n n o n , C. E.: Communications in the Presence of Noise. Proc. IRE (1949) 10-21 [4] de L a Vallee Pouss in , C. J.: Sur la Convergence de Formules d’Interpolation entre

[S] Vogel, P.: Bestimmung der ungilnstigsten Wahrschcinlichkeitsvcrteilung des Jitters bei PAM-

statistisch schwankendem Abtastzeitpunkt. Arch. Elektr. Obertr. 31 (1977) 349 - 354

Inc. 1978

Ordonneb Equidistantes. Bull. Acad. Roy. de Belgique (1908) 319-410

Systemen. Diplomarbeit Darmstadt 1981

Prof. Dr. W. Krabs (Fachbereich Mathematik) TH Darmstadt SchloRgartenstr. 7 6100 Darmstadt (Received April 17. 1985)

on the determination of the worst sampling error in a communication system for...

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