NEW ZEALAND JOURNAL OF MATHEMATICS Volume 23 (1994), 147 155

A SA M PL IN G FO R M U LA IN SIG N AL PR O C E SSIN G A N D TH E PR IM E N U M B E R TH EO R E M

J.L. S c h i f f a n d W .J . W a l k e r

(Received February 1994)

Abstract. An algorithm known as the Arithmetic Fourier Transform, which is used in the computation of Fourier coefficients, is shown to be related to the Prime Number Theorem. The algorithm turns out to be applicable to step functions and so can be applied to a phase filter related to the Buys-Ballot filter.

1. IntroductionIn two previous works [5], [6 ] the authors demonstrated a sampling formula for

the computation of the Taylor coefficients of an analytic function by means of evaluating the function at a countable set of points on the boundary of a circle. More specifically, if f ( z ) is analytic in the unit disk \z\ < 1, f ' ( z ) G Lipi on \z\ = 1, and f ( z ) = cn~n, (co = 0), then the Taylor coefficients cn are given by theformula

where // is the Mobius function from number theory (cf. Section 2). The represen­tation (1) can also be utilized to compute the coefficients of the inverse z-transform[7]. Moreover, the truncation of the infinite series in (1) has become known as the Arithmetic Fourier Transform (AFT) [4], [8 ], [9], [10], and as it essentially in­volves only addition operations, the calculation of the AFT can be accomplished by parallel processing.

In the present work, we demonstrate the validity of the representation (1) for the Fourier coefficients of even step-functions. These functions are important in the field of signal processing and in time series analysis. Our method of proof is quite different from that of [5], [6 ], in that we invoke a deep result from number theory known as the Davenport Formula. In Section 2 we show that this formula is equivalent to the Prime Number Theorem, thus providing a new connection between the Prime Number Theorem and the sampling formula.

In Section 4 we discuss the application of the representation (1) to a new approach in the field of digital filters.

Acknow ledgem ent. We would like to thank Garry Tee who read our earlier papers and suggested a possible connection with methods developed by Vinogradov. Chris Triggs suggested that we look at the Buys - Ballot filter.

n — 1,2, 3 ,... (1)

1991 AM S Mathematics Subject Classification: Primary 42A16, Secondary 11A25.

148 J.L. SCHIFF and W.J. WALKER

2. P rim e N um ber T heoremBefore we can establish the connection between the Prime Number Theorem

(PNT) and signal processing, a discussion of some preliminaries is required.Instrumental in what follows is the Mobius function fi defined on the positive

integers by(i) /i(l) = 1;

(ii) /i(j) = 0 if there is a prime p such that p 2 \ j;(iii) /i(j) = (—1)̂ if j = P1P2 ■ ■ -Pe is the prime factorization of j , and the p's

are all distinct.Although the PNT states that

x7v(x) ~ ---- , as x —> 00 ,

m x

(that is, —► 1, as x —»■ 00), where t t ( x ) is the number of prim es less thanor equal to x , we shall make use of a well-known (cf. [2 ]) asymptotic result from number theory which is equivalent to the PNT, namely that

^ kk- 1

It is next necessary to introduce the first Bernoullian function , namely

t ^ [ t ]

0 , t = [t]{t} =

where [t] is the greatest integer part of t. This function was invoked by Davenport[1] to establish what we term the Davenport Formula (DF),

'' k 7Tk= 1

the convergence being uniform in 9.With this background, we are ready to demonstrate

T heorem 1. The D F implies the P N T .

Proof. In fact, given any £ > 0, there exists a positive integer no = no(e) suffi­ciently large such that

----sin 27:9 —fc= 1

< e,

for all 9 G [0, 27r], by the DF. Moreover, for all 9 0 sufficiently small, we can make | — ^ sin27T0| < e, and hence

A SAMPLING FORMULA IN SIGNAL PROCESSING 149

Now, by choosing 6 > 0 possibly smaller, we can achieve

{M} = kO — - , k = 1, 2 , . . . , no (i.e. [kO] = 0),

and <n0

As a consequence, for 9 suitably small,

no m2e >

>

E mk= i

no / 7 \ -t no

k= 1

Ek — \

^(k)

k=i

2 ^ kk= 1

giving

y - M*0^ fck=l

< 2e + ^|/x(fc)|6»fc=i

< 2s -f~ 9ng

< 3s.

Therefore Y1T= l = which is equivalent to the PNT.

In a technically difficult proof, Davenport [1] derives the DF invoking the PNT as well as some of the theory of Vinogradov. It is this “equivalence” between the PNT and the DF which engenders the connection between the former and the field of signal processing. In a sense however, the DF is a much deeper result than the PNT.

3. Step-FunctionsThe Davenport Formula will now be used to prove the following theorem.

T heorem 2. Let f (0 ) be an even step-function defined on [—7r, 7r] and extended to be periodic of period ‘lix. Suppose that f (0 ) also satisfies:

(i) A t a discontinuity 9

m = \ (/(»+ ) + / (» - ) ) ,

(ii) / is normalized so that f {9 )d 9 = 0. If fo r any positive integer N

150 J.L. SCHIFF and W.J. WALKER

then the Fourier cosine coefficients of f are given by

y , rk m Tl ^ L.k=1

(Note. Defining F n on an equally spaced array on [0,27r), is consistent with established notation and also obviates the necessity of distinguishing between even and odd values of N . In the proof it will be convenient to use periodicity and consider an equally spaced array on (—7r, 7r]).

P roof. We first consider the special case of the even step-function

f ! - ; ■ I» l< 6 A(0) = < 5 - ; . » = ± b

[ b < \0\ < jt,

which satisfies the hypotheses (i) and (ii). The function fi, arises from the nor­malization of

' 1, \6\ < b

±, 9 = ±b

0, b < \0\ < 7r,9b{0) =

by the subtraction of the constant

1 f n ba0 = — / gb(0)d,0 =

2?r J - j, tt

For n > 1, fb and gj, have the same Fourier cosine coefficients,

1 ra n — — / cos n0gb(6)d6

* J -tt2 . .= — sin nb.

mr

Hence to prove the special case we must show that

p , 2 .> , — Fkn = — sin wo,' kn mrk=l

-1 r (2 7T777.\wherem= 0 ^ '

We claim Fn = —2 {^ r} . There are two cases to consider.

Case 1. If b = for some positive integer m, then b itself is a sample point and

A SAMPLING FORMULA IN SIGNAL PROCESSING 151

On the other hand, using (i) and (ii) of the hypothesis, it is easy to show that because 6 is a sample point,

m—0 y 7

Case 2 . -If b is not a sample point then the number of positive multiples of ^ which are less than b is [ ^ ] • Including the origin itself there are 2 [ |^ ] +1 sample points on {—b.b) at which fb takes the value (l — £). Hence there are N — 2 [^~] — 1 sample points at which it takes the value (-^ ), and therefore

Fn = 2bN

= 2

2n

bN

+ 1 1 + I N - 2bN2n

- 1 - -

2irbN

-------+ 17T

= - 2bN2n

This completes the proof of the two cases. Then by the DF, it follows that

A := ln *—' kk=l/ —2 \ / 1 . o bn

= I — I I ----sm 2tt —\ n J \ 7r 2tt

= — sin bn, mv

proving the special case.

In general every even step-function on [— is a finite linear combination of the constant function and functions of the form gb{0). The normalization of the constant function is zero and the normalisation of g b { 0) is f b ( 0) . Therefore it suffices to consider a finite linear combination of step-functions f b { 0) , and Theorem 2 follows from the special case just considered.

A comment about the notation is appropriate. Depending on whether N is even or odd, the point ir may or may not be a sample point in Fjsj. For this reason it is convenient to operate on [0 , 27r] in both the theorem itself and the applications which follow. It is not easy to extend the theorem to a full Fourier series defined on [0,27r]. The sine part of the series corresponds to an odd function defined on [—7r, 7r]. If / is odd then F n = 0 for each N , and more sample points are required.

4. A pplicationsIn applications such as image processing a function / is defined on an interval

[0, L] and represented by its Fourier cosine series. Suppose that / is sampled at L equally spaced points with sample values (zj)i<j<£* Conventionally, a spectral analysis is carried out by taking the Discrete Cosine Transform of this data. Here we

152 J.L. SCHIFF and W.J. WALKER

shall adopt a different approach and represent the discrete data by a step function extended to [—L, L] as an even function. The Fourier cosine coefficients of the step function may then be calculated by using the AFT and Theorem 2. The advantages of this approach are those inherent in various versions of the AFT; namely, the AFT requires only additions and can be accomplished by parallel processing. The N equally-spaced values which arise in S n can be read directly from the step function, whereas generally, the implementation of the AFT requires interpolation between sample values.

A possible disadvantage is that convergence of the series representation for an may be slow. However it should be possible to accelerate this by an averaging procedure.

All these comments aside, our primary objective is to use the AFT to investigate a particularly interesting digital filter.

Let the discrete data (zj)i<j<L, be represented by the step-function

where Z (t) is defined at the discontinuities so that condition (i) of Theorem 2 is satisfied. Further Z ( t ) is extended to be even, and then normalised by the subtraction of the constant

Then, by setting Xj = Zj — c, 1 < j < L , we see that the general situation reduces to the consideration of normalised data {x j ) i< j< L , represented by a step-function

which satisfies the hypothesis of Theorem 2 and which is extended to be even and periodic of period 2L. By a change of variable in Theorem 2,

A spectral analysis may then be implemented by finding an for selected values of n, where the period associated with an is ^ .

Suppose now that there is a dominant periodicity component of period T in the discrete data. For convenience we let L — T R . The periodicity is reflected in two ways. Firstly, by the term a2R cos 4^ which together with higher order terms a2Rp cos captures the “cosine” component. Secondly, there is a “sine”

Z (t ) = Z j , j - l < t < j , 1 < j < L ,

X ( t ) = x j , j - K t < j , l < j < L ,

OC

X (t ) = 2_^ a n c o s —j — : 0 < t < 2Ln = 1

wherek=l N - l

and

A SAMPLING FORMULA IN SIGNAL PROCESSING 153

component which is largely captured by the coefficients a,2R - \ and CL2R+ 1 , which occur with similar magnitudes but with opposite signs. To explain this second feature write

(2R — l)7r£ ( 2 R + l ) 7 v t n t 2 R n tcos------ ---------- cos------- ------- = 2 sm — sin ------L L L L

n . ixt . 2nt— 2 sm — sin ----.

L TObserve that since sin j > 0 on [0, L], integrating against the two cosine terms on [0, L\ approximates integrating against sin

The filter which we are about to define has the property that the coefficients r p are reduced to zero while computer experiments show that the coefficients

a 2R - i and CL2R+1 remain virtually unaltered. That is, the filter is a “phase” filter which filters out the “cosine” component of period T but which leaves the “sine” component of period T unchanged.

We shall first define the filter for the discrete data {x j ) i< j< L , extended to [0,2L],by

1 ^Vk = op Xk+ iT ’ 1 - k - T -

j =o

Then is defined to be the periodic extension of y k of period T on [0, 2L\ and the spectral analysis is applied to the residual data

Uj = x j - y j , 1 < j < 2 L.

L

Recall that (^ j)i< j< l has been normalised so that ^ Xj = 0. A straightforwardj=i

L

calculation shows that = 0 . Associated with ( y j ) we have a step function3 = 1

Y T (t) which is even and periodic of period T. Further Y T (t) must satisfy condition(i) of Theorem 2 and is uniquely defined in terms of X ( t ) by

1 2R—1

y7'W = 2R E X ( t + kT).k= 0

Then U{t) = X ( t ) - Y T ( t ) , 0 < t < 2L, is the step function representing the residual data. By Theorem 2,

. v— 717ft U ( t ) — 2_J a n c o s —j — ̂ 0 < t < 2 L ,

n= 1

where an = ^ j ^ - S kn,k= l N - l

and

r‘* (T ) -£ yT(T)m =0 ' ' m = 0 ' '

>Nm=0N - l r x N - l

154 J.L. SCHIFF and W.J. WALKER

The action of the filter is described by the following.

Theorem 3. In the Fourier cosine representation of the residual function U (t ) ,0-2RP = 0 fo r every positive integer P .

Proof. The theorem will follow from the AFT algorithm if we show that for each positive integer M, S 2r m = 0 - We write

2 R M —1 / o r \ 2 R M — 1 / r j i \ M — l / r p \

£ rT S ) - £ ’' Im = 0 ' 7 m = 0 ' 7 7 n = 0 ' '

y ~ 1 2-̂ ~ 1 / m T \ Z R M - l /

■ E E ' g * " - E * t771 = 0 f c=0 7 7 1 = 0 ' /

2HM-1 , 2Lm v

= E *7 7 1 = 0

\2 R A 1 J

It is now immediate that S 2r m — 0 and the proof is complete.

R em arks. Theorem 3 may also be proved by trigonometric formulas. The “phase” filter defined above was motivated by the Buys-Ballot filter ([3] pp.196-200) which is used to suppress a strong seasonal component of period T. This filter is defined for the discrete data ( x j ) i < j < L , by

R3 = 0

Then y is defined to be the periodic extension of y k of period T on [0, L\ and the spectral analysis is applied to the residual data

Uj = Xj - y j , 1 < j < L.

In this case the spectral analysis is applied to the full Fourier series

. ^ ( 2nnt 2n7rt\U(t) = 2_^ I an cos —-----h bn sin ——— 1 , 0 < t < L.

71 = 1 ' '

By using the step function construction and the AFT algorithm it can be shown that a u p = 0 for every positive integer P . The coefficients bpp cannot be easily analysed using the AFT but may be shown to be zero by trigonometric formulas. In conclusion we emphasise that the spectral analysis associated with the “phase” filter is implemented by taking the Fourier cosine series on [0, L], and the filter itself is defined in terms of the extended data on [0 , 2L\.

A SAMPLING FORMULA IN SIGNAL PROCESSING 155

References

1. H. Davenport, On some infinite series involving arithmetical functions, I and II, Quart. J. Math. 8 (1937), 8-13 and 313-320.

2. H.G Diamond, Elementary methods in the study of the distribution of prim e numbers, Bull. Amer. Math. Soc. 7 (1982), 553-589.

3. L.H. Koopmans, The Spectral Analysis of Time Series, Academic Press, 1974.4. I.S. Reed, D.W. Tufts, X. Yu, T.K. Truong, M.-T. Shih and X. Yin, Fourier

Analysis and Signal Processing by use of the Mobius inversion formula , IEEE Trans, on ASSP 38 (1990), 458 470.

5. J.L. Schiff and W.J. Walker, A sampling theorem for analytic functions , Proc. Amer. Math. Soc. 99 (1987), 737-740.

6 . J.L. Schiff and W.J. Walker, A sampling theorem and W in tn er’s results on Fourier coefficients, J. Math. Anal. Appl. 133 (1988), 466-471.

7. J.L. Schiff, T.J. Surendonk, and W.J. Walker, An algorithm for computing the inverse z- transform , Trans. IEEE 40 (9) (1992), 2194-2198.

8. J.L. Schiff and W.J. Walker, The Arithm etic Fourier Transform , in G.F.B. Riemann - A Mathematical Legacy, Hadronic Press (to appear).

9. D.W. Tufts and G. Sadasiv, The Arithm etic Fourier Transform , IEEE ASSP Magazine 5 (1988), 13-17.

10. D.W. Tufts, Z. Fan and Z. Cao, Image Processing and the Arithmetic Fourier Transform, SPIE/IST Conference (January 1989).

J.L. Schiff and W.J. Walker Department of Mathematics The University of Auckland Auckland NEW ZEALAND