On the aliasing error upper bound for homogeneous random fields

Signal Processing 33 (1993) 127-129 Elsevier

127

Short communication

On the aliasing error upper bound for homogeneous random fields* Tibor P o g i n y Department of Applied Mathematics, Maritime Faculty, 51000 Rijeka, Studentska 2, Croatia Received 18 May 1992 Received 29 October 1992

Abstract. The Brown aliasing error upper bound in the sampling cardinal series expansion (SCSE) of weakly stationary non-bandlimited (NBL) stochastic processes is extended to NBL homogeneous random fields (HRF). The magnitude of the derived bound is ordered under some smoothness condition upon the field.

Zusammenfassung. Die Brown'sche Abschitzung der Obergrenze von Aliasing Fehlern bei der Abtastung von Cardinal-ReihenEntwicklungen (sampling cardinal series expansion, SCSE) von schwach station~en, nicht bandbegrenzten (non-band-limited, NBL) Prozessen wird auf NBL homogene Random-Felder (HRF) erweitert. Die GrrBe dieser abgeleiteten Grenze wird unter bestimmten Glittungsbedingungen ftir das Feld eingeordnet. R r s u m r . La borne suprrieure de Brown dans l'expansion en srrie cardinale d'rchantillonnage (SCSE) d'un processus stochastique faiblement stationnaire non limit6 en frrquence (NBL) est 6tendue aux champs alratoires homogrnes (HRF) HBL. Le magnitude de la borne dErivre est ordonnre suivant une certaine condition de lissage sur tout le champ. Keywords. Sampling cardinal series expansion; spectral representation; aliasing error; homogeneous random fields; non-band-limited signals.

1. Introduction Brown has shown in his introductory paper [ 1] that a weakly stationary NBL stochastic process {X(t) It ~ ~} approximated by its SCSE Xw(t) A E~_~X(n~r/w) sinc(wt-n~r), for a given bandwidth 0 < w < + ¢¢ possesses the mean-square aliasing error W~w(t)A E I X ( t ) - X w ( t ) 1 2 bounded above in terms of the spectral densityf(A) of X(t):

x

~.,(t) ~<4

I l a l > w f(A)

dA.

(1.1)

Correspondence to: Mr. T. Pog~lny, Department of Applied Mathematics, Maritime Faculty, Studentska 2, Rijeka, Croatia. *This research was supported by Grant No. 1-01-298 from the Ministry of Science, Croatia.

Here sinc(x) Ax - l sin(x), x:~0, sinc(0) A 1. Brown also proved that the constant 4 is sharp even ( 1.1 ) holds uniformly in t. It should be pointed out that ( 1.1 ) also holds whenever the spectral distribution F(A) of X(t) is continuous at all points ( 2 k + l ) w , k~7/, i.e. W~.(t)~< 4~la I >w dE(A) [5]. Due to the Paley-Wiener theorem, there do not exist signal functions which are simultaneously band-limited and duration-limited, and this fact causes complications in engineering applications, since both limitations are natural for real physical signals [ 7 ]. The way between the theory and practice will be bridged by considering signals which are approximately band-limited, on the basis of a given level of accuracy (some applications are given in [ 2-8 ] ). The point will be to choose the

0165-1684/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved

T. Pogdny / On the aliasing error bound for random fields

128

bandwidth w large enough, such that the mean-square aliasing error ~x(t) becomes sufficiently small. For example, Turbovich was proposed in this purpose the so-called iterative sampling [ 9]. The main goal of this paper is to extend ( 1.1 ) to the class of NBL HRFs and to order the magnitude of the bound with respect to the bandwidth W =~ [w~ . . . . . w~] > 0 under some smoothness conditions upon the considered field.

3. Derivation of the bound

The main tool in the derivation is the following spectral representation of the multiple SCSE of the considered field ~:(x) proved in detail in [6].

PROPOSITION 1. c])(A)is continuous on the lattice Lat(W) ~ { ( (2nl + 1)wl ..... (2nr -1- 1)Wr) Inj ~ }

if and only if m.s. ~

~w(X) =

h rk~ 1

2. Preliminaries

(3.1)

teiakxka/ 2wk d Z ( A ) .

Now, we state and prove our main results. Let { ~ ( x ) I x ~ ~r} be a zero-mean, r-dimensional NBL HRF defined on the probability space (~, 3-, P). If K(~-) ---EsC(x+ ~) ~* (x) is the correlation function of ~:(x), we have the spectral representations

THEOREM. Let qb(A) be continuous on Lat(W). Then (i) ~ w ( X ) ~=El~(x)-~w(X)12<~4 (

J lal>W

~(X) = f ~ e i dZ(h), K(~') =

(3.2)

./m(r

ei dqb(A),

where denotes the inner product Y~= 1 ajbj, Z ( h ) is the spectral field and 4)(h) is the spectral distribution function of ~(x), d4)(A) = E IdZ(A) 12. Let us introduce the multiple SCSE ~w(x) of ~(x) to the given choice of bandwidth W = (Wl ..... wr),

~w(x) ~= j=l

~_, ,~(x") --~

d(/)(A),

sinc(wkxk --nkax),

where the constant 4 is sharp; (ii) /f ~(x) is Isl-fold differentiable, follows that g(w(X) = O ( W

-2lsl) =O

Isl >0, then it

Wk 2sk .

(3.3)

PROOF. (i) On account the isometry between the Hilbert-spaces H(sc) N (~(x) Ix~ R r) and L2( Rr; dq~) ~ { q~l far I q~l2dqb} we get by (3.1)

k=l

(2.1) say, where x" runs over the set

~(x)

ei-

= ( J Rr\B [

r (eia~k)2wk 2 I-I dqS(A) k= 1

~<4 fRr\a dqb(A).

{ (nllr/wl . . . . . nr'ff/Wr) Ins E~_}. Denote Is I = Y~= 1 st >/0, sj nonnegative integers, the mixed-exponent, i.e. for some a = (a~ . . . . . a~) it is r ~ a f t ; y ~ R. Finally let us take c~rill = II~= ×j=~ ( - w j , wA. For each real xk define (et~ak)2w~ as the 2wrperiodic extension of exp(ixkAk) from ( --Wk, W~] to tiC, k = 1, ...~ r. SignalProcessing

The proof, that the constant 4 cannot be improved, would be identical to the same kind of proof in [ 1 ] for the two-dimensional case. (ii) The existence of the Islth mixed-derivative of sO(x) is equivalent to I 021sl= ox_,o,K(O)

~ 0 21~lK(0)25r ]<0~, = [OX2~---q~ 1 --... ~ox r

Therefore we get

T. Pogtiny / On the aliasing error bound for randomfields

129

Finally, we are interested in the mean-square con~ w ( X ) ~<4

r\a

vergence of ~w(x) to ~(x) as rb ~ 2 . Let us denote

~<4W--2}sl fN~

=~mini ~j~ r( Wj).

A -21sl d ~ ( A )

P R O P O S I T I O N 2. l i m , ~ ~ w ( X ) = ~ ( x ) .

~<4W-21st ff~r A -21sl dq~(A)

P R O O F . If I s I > 0, the statement directly follows from

021sl = 4 W -21sl O - - z ~ K ( 0 )

.

[]

(3.4)

(3.3).

If

Is[=0,

it

is

not

hard

to

see

J'Rr\a d ~ ( A ) m o n o t o n i c a l l y vanishes as if--* 2.

that []

R E M A R K . C o n s i d e r the practical problem: " O r d e r the b a n d w i d t h W if the m e a n - s q u a r e aliasing error level e > 0 is k n o w n , that ~ w ( x ) ~< e 2''. By the evaluation

References

(3.4) we obtain a simple, but robust formula to derive

[ 1] J.L. Brown, Jr., "On mean-square aliasing error bound in car-

the n o m i n a l b a n d w i d t h W. So for given e > 0, we have

dinal sampling expansion of random processes", IEEE Trans. Inform. Theory, Vol. IT-24, No. 2, 1978, pp. 254-256. M.K. Habib and S. Cambanis, "Sampling approximations for non-band-limited harmonizable random signals", Inform. Sci.. Vol. 23, 1981, pp. 143-152. N.S. Kambo and F.C. Mehta, "An upper bound on the meansquare error in the sampling expansion for non-band-limited processes", Inform. Sci., Vol. 21, 1980, pp. 69-73. A. Papoulis, "Error analysis in sampling theory", Proc. IEEE, Vol. 54, No. 8, 1966, pp. 947-955. T. PoDiny, "Short story about the cardinal series of weakly stationary stochastic processes", submitted. T. Pog,'iny, "Spectral representation of the sampling cardinal series expansion for non-band-limited homogeneous random fields", submitted. D. Slepian, "On bandwidth", Proc. IEEE, Vol. 64, No. 3, 1976, pp. 292-300. D.C. Stickler, "An upper bound on aliasing error", Proc. IEEE, Vol. 54, 1967, pp. 418-419. I.T. Turbovich. "Analytical representation of non-band-limited time functions", Radiotehnika, Vol. XIV, No. 3, 1959, pp. 2227 (in Russian).

wlsl>~ 2

02is I

~K(0)

1/2

.

(3.5)

Since (3.5) is not u n i q u e l y determined with respect to

[2]

[3]

coordinate b a n d w i d t h s wk, k = 1. . . . . r in W, we need some additional informations on the relations b e t w e e n the values Wk. Supposing, for example, that the set {/3k}~,=z is k n o w n in Wl=/32W2=...=/3rWr, w e clearly get the c o o r d i n a t e - b a n d w i d t h evaluations as fol-

[4] [5] [6]

lows: Wk ~

2

r

) '/'s'

O2~'

)l/21s,,

[7] [ 8]

(3.6) where k = 1 . . . . . r,/31=1.

[]

[9]

Vol.33 No. I, July 1993