Proc. EUSIPCO-96, VIII European Signal Processing Conference, Trieste, Italy, Sep. 1996
ON THE APPROXIMATION OF NONBANDLIMITED SIGNALS BY NONUNIFORM SAMPLING SERIES Paulo Jorge S. G. Ferreira
Departamento de Electronica e Telecomunicaco~es / INESC Universidade de Aveiro 3810 Aveiro Portugal Tel: +351-34-370502; fax: +351-34-370545 E-mail:
[email protected] ABSTRACT
sampling series. These results establish that certain classes of functions can be arbitrarily well approximated by sampling expansions, provided that the sampling period =w is suciently small. An excellent and interesting overview of these results can be found in [5]. The review papers mentioned above, as well as [8{15], are among the works that somehow address this problem (considering uniform sampling only). In this work we study nonuniform sampling expansions for not necessarily band-limited functions. The method used is an extension of the method introduced in [16], as presented in [17]. We study the approximation properties of such sampling series, having in mind the e ect of jitter, and the behavior of the L2 and L1 approximation errors, for speci c kernels. Possible connections with number theoretic concepts such as uniform distribution and discrepancy are also discussed.
The WKS sampling theorem is a central result in signal processing, but it applies to band-limited signals only. This leaves out time-limited signals, or any discontinuous signal. We consider the problem of approximating such signals, or other signals not necessarily bandlimited, using sampling series. We do not assume that the sampling instants are regularly distributed, in order to account for errors due to jitter. To the best of our knowledge, the problem of obtaining nonuniform sampling approximations for signals not necessarily bandlimited, despite its practical interest, has not been addressed in the literature. In this work we introduce a method that leads to sampling approximations with the required properties. It is shown that the sampling sums considered are capable of approximating a wide class of signals, with arbitrarily small L2 and L1 errors. Possible connections with number theory are outlined.
2 THE SAMPLING APPROXIMATIONS Let f 2 L2, and consider the convolution
1 INTRODUCTION
A sampling series is a series of the form
f (t) =
+1
X
k=?1
fw (t) =
f (tk ) k (t):
+1
sin[w(t ? k w )] f k f (t) = w w(t ? k w ) ; k=?1
We will also consider
(1)
L
?1
f ( )K (t ? ) d ;
t
cos[(w + a)t] K (t) = cos(wt) ?at 2 a 2 sin[(w + 2 )t] sin(at=2) ; = at2
the convergence being absolute and uniform. An elementary introduction to sampling theory may be found in [1]. The survey papers [2{5] and the books [6,7] give an account of the eld, its history, and contain detailed bibliographies. There are many important results concerning the approximation of not necessarily band-limited signals by 1 A function 2 2 (R) is band-limited to if its Fourier transform vanishes outside [? ]. f
+1
where the kernel K is a function of at least one parameter w, such that fw ! f as w ! 1. There are many kernels K that share this property, the most common example being the sinc kernel (2) K (t) = sin wt :
The points tk are called the sampling points, and the function is called the kernel or the interpolating function. The WKS theorem asserts that any function f band-limited to w satis es1 X
Z
(3)
and the sinc-squared kernel at=2) 2 = 1 ? cos(at) : K (t) = 2a sin(at= (4) 2 at2 In these cases fw , which approaches f in the L2 norm, also converges in the point-wise sense to f (provided
w
w; w
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Ferreira: On the approximation of nonbandlimited signals by nonuniform sampling series
3 APPROXIMATION ERRORS, JITTER
that, for example, f has bounded variation). This makes it possible to approximate f 2 L2 by fw , to any prescribed tolerance, by taking w suciently large. Our method consists in approximating fw by a nonuniform sampling series, whose coecients are the samples of f itself. Theorem 1: Let f : R ! R be a continuous function of bounded variation vanishing outside [0; 1]. Denote by fti g any N reals such that
i?1 0, there exist w and N such that the sampling series sN di ers from f by less than in the L1 norm. This statement is true for any of the kernels mentioned. Proof: Clearly,
(5)
jf (t) ? sN (t)j jf (t) ? fw (t)j + jfw (t) ? sN (t)j jf (t) ? fw (t)j + ON(w) :
for 1 i N , and let
fw (t) =
Z
1
0
f ( )K (t ? ) d :
Take any two positive numbers and whose sum does not exceed . Now take w so large that jf (t)?fw (t)j , and N so large that jf (t) ? sN (t)j . Then
(6)
Consider the sampling approximation
jf (t) ? sN (t)j + :
N X sN (t) = N1 f (tk )K (t ? tk ): k=1
We now consider the approximation error in the L2 norm in [0; 1]. Recall that the norm is de ned by kf k2 = R1 2 0 jf (t)j dt. For simplicity we consider only the sinc kernel. Theorem 3: Given > 0, there exist w and N such that the sampling approximation sN (sinc kernel) di ers from f by less than in the L2 norm in [0; 1]. Proof: In this case,
Then,
jfw (t) ? sN (t)j Vf kK k1 +NVk (t)kf k1 ;
(7)
where kk1 denotes the L1 norm, and Vk (t) denotes the variation of the function k(x) = K (x ? t) for x 2 [0; 1].
kf ? sN k kf ? fw k + kfw ? sN k:
It turns out that, for the kernels mentioned above, Vk (t) is bounded by a quantity that depends on w but not on t. This means that, given > 0, it is possible to pick N such that jfw (t)?sN (t)j < . Under these circumstances the sampling series yields a uniform approximation of fw , to within . It is a simple matter to deal with functions f whose support is contained in a compact interval [a; b], other than [0; 1]. It is slightly less straightforward, but nevertheless possible, to account for discontinuous signals f of bounded variation. For the sinc kernel it can be shown that
We evaluate each term separately. On one hand,
kf ? fw k =
k
w fw (t) ? 2N
N X k=1
f (tk )
sin[w(t ? tk )] 2
(t ? tk )
+1
?1 +1 ?1 1
Z
2
w
1=2
jf (t) ? fw (t)j2 dt
1=2
jf^(!) ? f^w (!)j2 d! jf^(!)j2 d!
1=2
1=2
;
a quantity that tends to zero as w ! 1. On the other hand, since
for 0 t 1. If t is outside [0; 1] the approximation is O(w)=N . If the kernel is of bounded variation, as the kernel (4), for example, the approximation becomes asymptotically better. In fact, for any t,
jf (t) ? fw (t)j2 dt
Z
N X 1 fw (t) ? N f (tk ) sin[w(t(?t ?t t)k )] = O(wNlog w) ; (8) k=1
0
Z
0. We have some remarks to add to these results. The rst remark concerns the e ect of jitter. It is felt only in the term jfw (t) ? sN (t)j; for which the upper bounds given in the previous section are valid. For the sinc kernel, this term is O(w log w)=N inside [0; 1], whereas for kernels of bounded variation it is O(w)=N . The e ect of the jitter is not experimentally apparent unless N is comparatively low (as N ! 1 the sum sN approaches fw uniformly). The second remark is concerned with the behavior of the approximation error as function of w, for a xed N . It follows from the previous results that the approximation error, measured using the L1 or the L2 norms, achieves a minimum value as a function of w. Indeed, the total error is the sum of two components, one of which decreases when w increases, whereas the other is an increasing function of w. This behavior is clearly seen in gures 2-1. The last remark has to do with the extension of the results given so far to signals g of bounded variation that do not necessarily vanish outside [0; 1]. Can such signals be approximated by sampling sums of the type considered? The answer is yes. It is sucient to take R; h(t) = g(0t) jjttjj > R;
k
k=1
it follows that
h(t) 2NR
N X k=1
h( k ) sin[W(t (?t ? )k )] ; k
where k = 2Rtk ? R, and W = 2wR . This concludes the argument, and shows how the sampling sums can be used to approximate L2 functions of bounded variation.
4 CONNECTIONS WITH NUMBER THEORY
The error bounds presented above are independent of the sampling set tk . Is it possible to re ne them and judge the impact that the distribution of the tk has upon the error? What are the most general distributions under which a similar theorem holds? Is the term 1=N best possible? The concept of uniform distribution modulo one (UD), introduced by Hermann Weyl in 1914 to re ne Kronecker's theorem, appears to be related to the sampling problem considered, and, indeed, it leads to partial answers to the questions that we have asked. The sequence ti 2 [0; 1] is UD i , for any A [0; 1], lim N (A; k) = m(A); k!1
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Ferreira: On the approximation of nonbandlimited signals by nonuniform sampling series
or any fti g1iN such that i?1
