An application of a general sampling theorem

3 downloads 0 Views 154KB Size Report
23] and fundamental in information theory and communication, particularly since the advent of modern digital ... Theorem to the analysis of a speech compression system. ... leads to an apparent paradox related to the uncertainty principle of quantum mechanics. Despite this, the ..... [25] H. Taub and D. L. Schilling. Principles ...
An application of a general sampling theorem M. G. Beaty

M. M. Dodson

Dedicated to Paul Butzer to mark his seventieth birthday. Abstract: An extension of the sampling theorem to multi-band signals is discussed and an application to the compression of speech outlined. 1991 Mathematics Subject Classification: 94A05, 42A99 Keywords: Multi-band sampling, speech compression

1

Introduction

A signal carries information from a sender to a receiver. The signal might consist of a sequence of discrete symbols, such as letters from an alphabet or digits, or might be a continuous reading such as a voltage, temperature or pressure. It is a remarkable fact that under certain circumstances, continuous or analogue signals and discrete signals are equivalent. This is the content of the Sampling Theorem, associated with C. E. Shannon [22, 23] and fundamental in information theory and communication, particularly since the advent of modern digital computers. This and its numerous applications [17] have generated a great deal of interest in the mathematics surrounding the Sampling Theorem (excellent accounts are given in the survey articles [5, 15]). Naturally there are particularly close connections with interpolation theory and Fourier analysis (see for example [16, 18, 19]) but in addition there are unexpected links with other branches of mathematics [6, 7, 20] (Paul Butzer has played an important part in establishing these links, see [16, §9.2]). This paper will be concerned with the relationship between analogue and discrete signals which underlies much of modern communication and will discuss applications of an extension of the Sampling Theorem to the analysis of a speech compression system. An analogue signal can be described mathematically as a continuous function f : R → R which is square-integrable, corresponding to the signal having finite energy [24]. The Fourier transform F (f ) of f (in the Fourier-Plancherel sense) will also be written f ∧ : R → C and is given by Z ∧ f (x) = F (f )(x) = f (t)e−2πitx dt. R

This places the study of analogue signals firmly into the realm of Fourier analysis and the Hilbert space L2 (R). For physical reasons, the frequencies of analogue signals are taken to be bounded above, for example in the speech of the adult male the frequencies are taken to

be below 4kHz. Signals with bounded spectra are called band-limited and have the representation Z w f (t) = f ∧ (x)e2πitx dx, −w

where w is the maximum frequency (strictly speaking the supremum of the frequencies, a technicality which we will usually disregard). Signals of this type must be the restriction to the real axis of entire functions; this implies that the signals cannot be zero on any interval. It might seem natural to suppose signals are also of bounded duration (in time) but this leads to an apparent paradox related to the uncertainty principle of quantum mechanics. Despite this, the conventional band-limited approach has proved highly effective in signal processing (for a very interesting discussion on this problem see [16, Chapter 17]). An alternative is to seek a different class of functions to model signals arising in engineering applications. As well as studying the rˆole of the Sampling Theorem in mathematics, Butzer and his school at Aachen have worked with a class of not necessarily band-limited functions which leads in the direction of approximation theory (see for example [4, 7, 8, 9]).

2

The Sampling Theorem

The Sampling Theorem tells us that a band-limited analogue signal f with maximum frequency w is determined by its values (samples) f (k/2w), k ∈ Z, taken every 1/2w seconds. The intuitive justification is that an analogue signal with maximum frequency w cannot vary substantially in a time interval less than one half a cycle, i.e., in 1/2w seconds. This had been recognised since the 1920’s when radio transmission was becoming widely used but the Sampling Theorem provides a formula which expresses a signal in terms of the discrete values or samples taken every 1/2w seconds:     1 X k k sin 2πw(t − k/2w) X k f (t) = f sinc 2πw(t − = f ), (1) 2w k∈Z 2w π(t − k/2w) 2w 2w k∈Z where sinc x = (sin x)/x for x 6= 0 and 1 at x = 0. This formula corresponds to a sampling rate of 2w (often called the Nyquist rate), twice the maximum frequency of the signal. Shannon’s proof was based on tessellating the frequency domain with copies of the spectrum (spectral repetition) and identifying the coefficients in the resulting Fourier series for the periodic function with the sample values. In much of electronics, sampling and processing are implemented instantaneously (in ‘real time’) or at least with a minimal delay. This excludes the direct use of the Sampling Theorem in which the value of the signal at each instant is given from (1) by an infinite (slowly converging) sum. Nevertheless the choice of suitable sampling rates, the avoidance of aliasing [16, Chapter 11] and the effectiveness of processing techniques rely on the Sampling Theorem. Indeed, (1) can be interpreted as the result of passing a train of delta functions weighted by the sample values through an ideal filter (one which passes those frequencies below w without change and blocks higher frequencies). It is not possible to generate weighted delta functions in an engineering application but this idea is closely related to the method commonly used to reconstruct an analogue signal from digital samples (see §3.3 below).

By contrast, in X-ray crystallography the 3-dimensional version of the Sampling Theorem can be used directly. To simplify the explanation, we shall confine ourselves to one-dimensional crystals. The electron density d(x) describing a single molecule of a onedimensional crystal vanishes outside the interval [−a/2, a/2] where a is the length of the cell. The regular repetition of the crystals causes selective interference of the radiation scattered from an X-ray beam. When collected on a photographic film, the image consists of an array of enhanced points at the reciprocal lattice points Z/a. The Sampling Theorem allows the complete reconstruction of the Fourier transform F of d from the observed values of F (k/a), k ∈ Z. Taking the inverse Fourier transform, the original density function d can be reconstructed and the molecular structure determined. The argument extends to three dimensions but unfortunately only the modulus of the points can be observed. Getting round this very serious limitation is the major problem in determining molecular structure using X-ray crystallography but the Sampling Theorem can be used to reduce the number of observations and calculations required [21]. The use of the Sampling Theorem in electronic engineering will be illustrated by a speech compression system which exploits the spectral structure of the voice through a generalisation of the Sampling Theorem to signals with more than one frequency band.

3

Multi-band signals

Band-limited signals fall into different types. Signals which have a spectrum consisting of a single band about the origin, with no gaps, are called low-pass. This is the type of signal most often associated with the description band-limited. For purposes of transmission, many signals such as radio and television are confined to a band centred about a high frequency carrier wave. This type of signal is called band-pass. More generally, signals which have spectra made up of several distinct bands are called multi-band. These can arise naturally in applications or through harmonics associated with a fundamental [26]. The Sampling Theorem has been extended to band-pass signals [25] and is used to avoid aliasing in radio [26] and interferometry [14]. A further extension is to a larger class of multi-band signals [12]. 3.1

A disjoint translates condition

The property that translates by integer multiples of the sampling rate of the Fourier transform of a signal do not overlap is critical to sampling theory and allows the extension of the Sampling Theorem to certain multi-band signals. Let A be a measurable subset of R and let s be a positive number. The disjoint translates condition   ( ∅, k ∈ Z \ {0} k A∩ A+ (2) = s A, k = 0 is central to our analysis. Using the equivalent and more convenient difference set condition 1 D(A) ∩ Z = {0}, s

where D(A) = {a − a′ : a, a′ ∈ A}, it is shown in [12] that given A, there exists a number s > 0 satisfying |A| ≤ 1/s ≤ 2 sup{|a| : a ∈ A}. Sets satisfying (2) are clearly more general than intervals and in addition are not necessarily bounded. An algorithm to find the largest number s which satisfies (2) is given in [13]. 3.2

A multi-band Sampling Theorem

The class of continuous finite energy functions whose Fourier transform vanishes outside A will be denoted by B(A), i.e., B(A) = {ϕ ∈ C(R) ∩ L2 (R) : ϕ∧ (x) = 0,

x∈ / A},

where C(R) is the class of continuous functions. We consider multi-band signals ϕ ∈ B(A) when A satisfies (2). Note that since |A| ≤ 1/s, the Fourier transform of a function in B(A) is integrable as well as square integrable. The difference between the maximum frequency and the measure of the frequency bands for a multi-band signal can be exploited in applications to reduce the sampling rate required for reconstruction. Note further that when ϕ is real (as will be the case in the system discussed below), A is symmetric, i.e., A = −A. The characteristic function of a set S will be denoted χS and the inverse Fourier transform F −1 (g) of a function g will also be written g ∨. Theorem 1. Let ϕ ∈ B(A) and suppose A ⊂ R satisfies the disjoint translates condition (2) for s ∈ R. Then Z X 2 |ϕ(sk)|2 kϕk = |ϕ(t)|2 dt = s R

k∈Z

and for each t ∈ R, ϕ(t) = s

X

ϕ(sk)χ∨A (t − sk),

(3)

k∈Z

where the convergence is absolute and uniform. A proof of this result and further references are given in [12] which includes further references. 3.3

Reconstructing the processed signal

As noted above in §2, the sampling formula cannot be implemented directly in electronic engineering to reconstruct an analogue signal such as speech. Instead, a two stage process called ‘sample and hold’ is adopted. This approximates the recipe indicated by the sampling formula (1) of passing a train of delta functions weighted by the samples through an ideal filter. The first stage is to construct a ‘box-car’ signal, in which the value of each sample in turn is held for the duration of the sampling interval. (In the analysis below, the boxcar signal will be represented mathematically by a step-function.) The second stage of the

reconstruction process is to process the box-car signal by a filter which passes only those frequencies contained in the original signal. The result is that the reconstructed signal is actually a smoothed step-function approximation. The method is ingenious in its simplicity and uses the Sampling Theorem as a guideline for choosing appropriate sampling rates. Although it is not used explicitly, the formula (1) provides a theoretical justification for the sample and hold technique. However the method does introduce errors and these are now analysed for multi-band signals and the corresponding formula (3). In the speech compression system described below, a multi-band approximation ϕ to a (low-pass) speech signal f is obtained by removing frequencies of low amplitude. This process can be modelled by the following composition of transformations: f → f ∧ → Tδ → (Tδ )∨ = ϕ where the threshold function Tδ is given by ( 0, Tδ (x) = f ∧ (x)

if |f ∧(x)| < δ otherwise,

i.e., ϕ = F −1 ◦ Tδ ◦ F (f ). Let supp Tδ = A and choose s so that A satisfies (2). Then by Theorem 1, X (Tδ )∨ (t) = ϕ(t) = s ϕ(sk)χ∨A (t − sk). k∈Z

In the speech compression scheme, the samples ϕ(sk), k ∈ Z are transmitted together with information about A. The first stage of the sample-and-hold construction is to generate from the samples of the thresholded signal ϕ a step-function σϕ given by X X ϕ(sk)χ[sk,s(k+1))(t) (4) ϕ(sk)χ[0,s) (t − sk) = σϕ (t) = k∈Z

k∈Z

and which coincides with ϕ at the left hand edge sk of each interval [sk, s(k + 1)), k ∈ Z. Lemma 1. Let ϕ ∈ B(A) and suppose that A satisfies (2) for s. Then X |ϕ(sk)|2 = kϕk2 . kσϕ k2 = s k∈Z

Proof. For Z

2

|σϕ (t)| dt =

R

Z XX

=

X

|ϕ(sk)|

k∈Z

But by Theorem 1, s

ϕ(sk)ϕ(sk ′ )χ[0,s) (t − sk)χ[0,s) (t − sk ′ ) dt

R k∈Z k ′ ∈Z

P

k∈Z |ϕ(sk)|

2

2

Z

s(k+1)

dt = s sk

X

|ϕ(sk)|2.

k∈Z

= kϕk2 and the lemma follows.

Thus the energy of the step-function σϕ is the same as that of the signal ϕ if the sampling rate s is chosen appropriately. The second part of the sample and hold technique is to remove frequencies outside A by a suitable filter in order to obtain a signal with the same frequency support as ϕ. In practice A is bounded, so that this operation removes high frequencies and has the effect of smoothing σϕ . The resulting function ρϕ can be represented by the following composition: ρϕ = F −1 χA F (σϕ ) = (χA σϕ∧ )∨ .

(5)

Now by (4), σϕ∧ has the representation X ϕ(sk)(χ[sk,s(k+1)) )∧ (x), σϕ∧ (x) ∼

(6)

k∈Z

where (χ[sk,s(k+1)) )∧ (x) = s e−πisx sinc πsxe−2πiskx . Hence the Fourier transform ρ∧ϕ of ρϕ has the representation X ϕ(sk)e−2πiskx , ρ∧ϕ (x) ∼ χA (x)(σϕ )∧ (x) = g(x)χA (x)s k∈Z

where g(x) = e−πisx sinc πsx. But by Theorem 1, ϕ∧ is represented by X ϕ(sk)e−2πiskx χA (x). ϕ∧ (x) ∼ s k∈Z

Hence for almost all x, ρ∧ϕ (x) = e−πisx sinc πsx ϕ∧ (x) = g(x) ϕ∧ (x), i.e., ρ∧ϕ = g ϕ∧ almost everywhere. It follows that Z X ρϕ (t) = s e−πisx ϕ(sk) sinc πsx e2πi(t−sk)x dx. A

3.4

(7)

(8)

k∈Z

Error analysis

The signals are now taken to be real, so that the set A is symmetric (and satisfies (2)). Note that for each ϕ ∈ B(A), the approximating function ρϕ is also in B(A). Let ε(t) = ϕ(t) − ρϕ (t). Then ε ∈ B(A) and by (7) for almost all x, ε∧ (x) = (1 − e−πisx sinc πsx)ϕ∧ (x). Theorem 2. The error in reconstruction introduced by the sample and hold method satisfies Z  2 kεk = 1 − 2 sinc πsx cos πsx + sinc2 πsx |ϕ∧ (x)|2 dx. (9) A

Proof. The functions ϕ, ρϕ are in L2 . By Plancherel’s theorem and (7), kεk2 = kϕ∧ − ρ∧ϕ k2 = kϕ∧ − gϕ∧k2 = k(1 − g)ϕ∧k2 . Hence 2

kεk =

Z

|1 − e−πisx sinc πsx|2 |ϕ∧ (x)|2 dx

A

and the theorem follows.

Corollary 1. Let A be bounded. Then for s sufficiently small,

where C = π 2

kεk = kϕ − ρϕ k = Cs2 + O(s4) R 4 ∧ 1/2 x |ϕ (x)|2 dx . A

Proof. Since A is bounded, A satisfies (2) for s ≤ 1/α, where α = sup{|a| : a ∈ A}. Also for s sufficiently small, 1 − 2 sinc πsx cos πsx + sinc2 πsx = π 2 s2 x2 + O(sx)4 and the corollary follows on substituting in (9). It is thus enough to bound the fourth moment of |ϕ∧ |2 to get an error term which is quadratic in the sampling interval s. Thus reducing the sampling interval reduces the error in the reconstructed signal ρϕ . The error in the case of a low-pass signal with maximum frequency w can be shown to be at most Ks2 w 2 , where K is an absolute positive constant. Sampling at the Nyquist rate (where s = 1/2w) does not in general eliminate the error and consequently over-sampling is often employed. In [2] similar estimates have been obtained in which the sample and hold construction has been symmetrised by taking the value σϕ (s(k + 1/2)) at the mid-point of the horizontal segment of the step function to be ϕ(sk). This simplifies the calculations and reduces the errors a little, for instance in Corollary 2.2 of [2], the constant C is smaller (by a factor of 1/6) than in Corollary 1 above. This is of no significance in practice. The attenuation of the signal ϕ is more severe at higher frequencies and if s is not very small, the errors can be quite large. Corollary 2. Suppose the infimum v of the positive frequencies of ϕ is positive and that s > 1/πv. Then   1 kϕk. kεk ≥ 1 − πsv Proof. By the definition of sinc and for x > 1/πs,  2 1 1 − 2 sinc πsx cos πsx + sinc πsx ≥ 1 − 2| sinc πsx| + sinc πsx ≥ 1 − πsx 2

2

and so from (9), 2 2  Z  1 1 ∧ 2 kεk ≥ 1− |ϕ (x)| dx ≥ 1 − kϕk2 . πsx πsv A 2

This has implications for using reduced sampling rates to achieve data compression. For example, consider a band-pass signal of the bandwidth w where w is small compared to v. Then the maximum frequency v + w is comparable to v and the error will be of the order of the norm of the signal ϕ for sampling rates of 2w. If the complement of A includes an interval about the origin then any function in B(A) automatically has infinitely many real zeros [10, Theorem 1]. Accordingly we content ourselves with the following pointwise upper bound for ε. Theorem 3. For each t ∈ R, 2

|ε(t)| ≤ kϕk

2

Z

A

 1 − 2 sinc πsx cos πsx + sinc2 πsx dx.

Proof. Since ε is integrable, the difference ε = ϕ − ρϕ is given by Z Z 2πitx ∧ ε(t) = e ε (x) dx = e2πitx (1 − g(x))ϕ∧(x) dx, ∧

A

R

where g(x) = e sinc πsx. Hence by Cauchy’s inequality, |ε(t)| ≤ kϕ∧ k khk, where h(t) = e2πitx (1 − g(x)), and the theorem follows. −πisx

3.5

Running averages

The function ρϕ obtained from the function ϕ can be realised as a ‘running average’ of ϕ. Theorem 4. For each t ∈ R, 1 ρϕ (t) = s

Z

t

ϕ(v) dv.

t−s

Proof. By Parseval’s theorem Z Z Z 1 t 1 1 ϕ(v)χ[t−s,s) dv = ϕ∧ (x)χ∧[t−s,s) (x) dx ϕ(v) dv = s t−s s s R Z R = ϕ∧ (x)e2πitx−πisx sinc πsx dx. R

But by (8), ρϕ (t) = s

Z

=s

Z

e−πisx

A

=

ϕ(sk) sinc πsx e2πi(t−sk)x dx

k∈Z

e2πi(t−s/2)x sinc πsx

A

Z

X

X

ϕ(sk)e−2πiskx

k∈Z

!

dx

e−2πi(t−s/2)x sinc πsx ϕ∧ (x) dx,

R

as claimed. A similar result is given in [2, §2] but for the symmetrised step-function approximation and with the range of integration [t − s/2, t + s/2].

4

A speech compression system

Typically the human voice has three significant frequency bands, each one associated with some physical characteristic of the mouth and throat. If frequencies with amplitude below a suitable threshold are discarded, then one is left with three well separated frequency bands. Under favourable circumstances they will satisfy the disjoint translates condition (2) for a small sampling rate. When this happens, the thresholded signal can be reconstructed from samples taken at the low rate and hence data compression is possible. In practice, signal reconstruction will be via the sample and hold technique analysed in §3, and the errors this introduces are illustrated below. 4.1

Preparation of compressed data

This speech compression system consists of two parts; the first prepares compressed data for transmission via a communications channel, the second receives the data and reconstructs the speech. It is considered to be a real time system, i.e., incoming speech is processed almost as it is received and is not stored in advance. From a mathematical viewpoint, this is not very satisfactory as Fourier analytic methods require complete knowledge of the input signal; this is not possible in practice. Instead, the incoming speech is initially sampled at the Nyquist rate of 8kHz, and the stream of samples divided into groups (or frames, in engineering terminology) of 128 for subsequent processing. It is convenient that the number of samples in each frame is a power of two as the discrete Fourier Transform is a essential tool in the system. A frame of 128 samples corresponds to a time segment of about 16 milliseconds. This use of the word frame has no connection with frames in Hilbert space theory. The purpose of this system is to compress speech by reducing the number of samples necessary for reconstruction. To achieve this aim, the system seeks to find a lower sampling rate than the Nyquist rate for each frame, and then sub-sample accordingly (for instance if it is found that a sampling rate of one-quarter the Nyquist rate suffices, then only every fourth sample need be transmitted). The first step is to take the (discrete) Fourier Transform of the frame in the expectation that the resultant spectrum will consist of distinct peaks, as mentioned above. If this is the case, then a threshold is applied to isolate the areas of high amplitude; these lie on a multi-band set to which the theory and algorithms discussed earlier can be applied. The treatment of each frame separately means that the natural setting to analyse the errors is that of real trigonometric polynomials, rather than B(A), the band-limited subset of L2 . With 128 samples per frame, the input speech signal can be considered to be a low-pass trigonometric polynomial, containing frequencies in the range {1, 2, . . . , 64}. The trigonometric polynomial 64 X f (t) = cn cos 2πnt n=1

can thus represent the speech signal for the duration of the frame. Following [12], the set containing the significant high amplitude frequencies is denoted J, i.e., those j ∈ J with

|cj | greater than the chosen threshold. Then f has a multi-band approximation ϕ given by X ϕ(t) = cj cos 2πjt. (10) j∈J

Various algorithms can be employed to select a suitable threshold. For example, the threshold could be set by a simple proportion of the largest amplitude within the spectrum, or set at a level which retains a certain proportion of the energy of the frame. Next, to achieve compression, sampling rates q < 128 are sought that satisfy the difference set condition (J ∪ −J) ∩ qZ = {0}.

(11)

The algorithm given in [13] can be used to find the smallest number q which satisfies this equation. A practical problem now arises: if the reduced sampling rate is itself a power of two, then sub-sampling is particularly convenient technically. It seems natural to increase the calculated smallest possible rate to the next higher power of two, but as discussed in detail in [1], this is not certain to give a rate which also satisfies the difference set condition (11). One possible approach is to only test powers of two in the difference set condition, finding the smallest available. Then the set J can be augmented to a set J˜ with cardinality q, as outlined in [12]. In this way as few as possible coefficients are discarded, which helps to minimise the error in signal reconstruction. 4.2

Transmission of data

A reduced set of data for the frame can now be transmitted. The data required is the set of samples {f (m/q) : m = 0, . . . , q − 1} along with band information describing the set J. This data is sufficient to recover ϕ, the multi-band approximation to f . The necessity of transmitting a description of the set J used in each frame is an extra overhead, but in practice this will not be significant. 4.3

Signal reconstruction

At the receiver the frame is reconstructed by the sample and hold method. Each sample value is used in turn to generate one step in a step-function signal. The step-function is then passed through a ‘comb’ filter which passes those frequencies in the set J, and blocks other frequencies. The system is illustrated by a typical frame. This contained two significant bands, one ‘base-band’ and one at around 3kHz. Figure 1 shows a short time segment (1/10 of a frame, corresponding to 1.6 milliseconds) of the original speech signal f , the multi-band approximation ϕ and the final reconstructed signal ρϕ . After a threshold was applied, a suitable reduced sampling rate was found to be 32. The frequencies discarded contributed about 26% of the original signal energy, and 30% of the L1 norm of the spectrum. Therefore, the frame was not particularly favourable, as to get a good compression rate of four has meant that a significant part of the content of the signal has been discarded.

400

200

f 0

ρ -200

ϕ

-400 0.8

0.85

0.9

Figure 1: Part of an example frame

4.4

Error analysis

This system exhibits many of the errors discussed in sampling theory. The initial sampling at the Nyquist rate will in practice introduce a quantisation error, which is usually a relatively minor problem. The reconstructed signal suffers aliasing: the Fourier transform of ϕ has support on a subset J of the support of f ∧ . Aliasing can be a severe problem in applications. In this example, the underlying justification of the method is that only those frequencies with a relatively low amplitude are omitted from the reconstruction. Unfortunately, the pointwise aliasing error is bounded by the L1 sum of those amplitudes, and this estimate is sharp even in the case of multi-band signals (see [3] for details). In Figure 1, there are indeed points at which f and ϕ differ by large values. However, the dominant error in the reconstruction ρϕ is that due to the sample and hold implementation of digital to analogue conversion. In this setting of real trigonometric polynomials, the analysis of sample and hold in §3 leads to the equation X ρϕ (t) = cj sinc(πj/q) cos 2πj(t − 1/2q), (12) j∈J

the finite dimensional analogue of (8). By comparing (12) with (10), it can be seen that essentially each coefficient cj is attenuated by a factor sinc(πj/q). This is very serious with the reduced sampling rate q, as the frequencies j in the upper bands of ϕ∧ will be greater than q. Figure 1 shows how much better the multi-band function ϕ approximates f than the final reconstruction ρϕ .

With a sampling rate of twice the maximum frequency contained in the signal, this attenuation is not very serious. Nevertheless the reconstruction benefits by increasing the sampling rate, i.e., by reducing the sampling interval s, as shown in Corollary 1. This reduction of attenuation is one of the advantages of ‘over-sampling’, as in compact disc players. The alternative formulation of this error, in terms of a moving average, is also helpful. In this example, the window length of the moving average becomes 1/32, as compared to 1/128 which would be the case with reconstructing f with the original samples taken at the Nyquist rate. The window length of 1/32 is about one-third of the portion of the frame displayed in Figure 1, and it is not surprising that ρϕ does little more than display the trend of f . Though not shown, a reconstruction using sample and hold at the Nyquist rate follows f very closely with small errors at turning points. 4.5

Conclusion

The outlined speech compression system seeks to exploit physical attributes of the voice to achieve compression. The attenuation which causes the large reconstruction error might be mitigated by boosting the higher frequencies, though this is not certain to be effective. In real communications systems there might be other factors which take precedence over the aim of maximum compression. For example, noise in the communication channel might require that the sampling rate is increased to maintain signal quality. Indeed, instead of simply a compression scheme, the system could monitor and adapt to the prevailing conditions, thus maintaining efficiency with good quality signal reconstruction. This in fact was the purpose of a communication system which relies on meteor trails to reflect the signal over the horizon [11]. The trails are transient so that it is important that the signal is encoded efficiently. In this system, multi-band sampling theory was implemented as outlined above to reduce sampling rates and in conjunction with other encoding techniques gave significant data compression. It is evident however that the multi-band theory has to be applied with caution when dealing with real time signal processing where the formula (3) in Theorem 1 cannot be used directly in the reconstruction. In particular, the advantages of a reduced sampling rate are offset by increased errors in the sample and hold reconstruction. Nevertheless a substantial part of these errors is due to high frequency attenuation and phase shift which do not greatly affect comprehension of speech and indeed the system [11] using the techniques discussed above proved successful. When (3) is used, the multi-band theory offers scope for further reductions in the sampling rate and data compression.

5

Acknowledgement

We are very grateful to Paul Butzer for his continued interest and encouragement. His friendship and support over the years have meant a great deal to us.

References [1] M. G. Beaty and M. M. Dodson. The distribution of sampling rates for signals with equally spaced, equally wide spectral bands. SIAM J. Appl. Math., 53 (1993), 893–906. [2] M. G. Beaty, M. M. Dodson, and J. R. Higgins. Approximating Paley-Wiener functions by smoothed step functions. J. Approx. Th., 78 (1994) 433–445. [3] M. G. Beaty and J. R. Higgins. Aliasing and Poisson summation in the sampling theory of Paley-Wiener spaces. J. Fourier Analysis and Applic., 1 (1994), 67–85. [4] P. Butzer and A. Gessinger. The approximate sampling theorem, Poisson’s sum formula, a decomposition theorem for Parseval’s equation and their interconnections. Workshop on Sampling Theory and Applications, Jurmala, Latvia, 1995. [5] P. L. Butzer. A survey of the Whittaker-Shannon sampling theorem and some of its extensions. J. Math. Res. Exposition, 3 (1983), 185–212. [6] P. L. Butzer, M. Hauss, and R. L. Stens. The sampling theorem and its unique role in various branches of mathematics. In Mathematical Sciences, Past and Present, 300 years of Mathematische Gesellschaft in Hamburg. Mitteilungen Math. Ges. Hamburg, Hamburg, 1990. [7] P. L. Butzer, W. Splettst¨oßer, and R. L. Stens. The sampling theorem and linear prediction in signal analysis. Jber. d. Dt. Math.-Verein., 3 (1988), 1–70. [8] P. L. Butzer and R. L. Stens. Prediction of non-bandlimited signals from past samples in terms of splines of low degree. Math. Nachr., 132 (1987), 115–130. [9] P. L. Butzer and R. L. Stens. Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Review, 34 (1992), 40–53. [10] J. Clunie, Q. I. Rahman, and W. Walker. On entire functions of exponential type bounded on the real axis. Preprint, Universit´e de Montr´eal, 1997. [11] M. Darnell, M. M. Dodson, B. Honary, and W. He. Adaptive-rate sampling applied to the storage and transmission of multiple band-pass signals. In Proc. Sixth Int. Conf. on System Engineering N. W. Bellamy (ed.), pp. 109–119, Coventry Polytechnic ICSE, 1988. [12] M. M. Dodson and A. M. Silva. Fourier analysis and the sampling theorem. Proc. Royal Irish Acad., 85A (1985), 81–108. [13] M. M. Dodson and A. M. Silva. An algorithm for optimal regular sampling. Signal Process., 17 (1989), 169–174. [14] R. Dodson. The Mauritius radio telescope and a study of selected supernova remnants associated with pulsars. PhD thesis, Durham University, Durham, UK, 1997. [15] J. R. Higgins. Five short stories about the cardinal series. Bull. Amer. Math. Soc., 12 (1985), 45–89. [16] J. R. Higgins. Sampling theory in Fourier and signal analysis: Foundations. Clarendon Press, Oxford, 1996. [17] A. J. Jerri. The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE, 65 (1977), 1565–1596. [18] R. J. Marks II. Introduction to Shannon sampling and interpolation theory. Springer-Verlag, New York, 1991. [19] R. J. Marks II (ed.). Advanced topics in Shannon sampling and interpolation theory. SpringerVerlag, New York, 1993.

[20] Q. I. Rahman and G. Schmeisser. The summation formulae of Poisson, Plana, Euler-Maclaurin and their relationship. J. Math. Sci. (Part I), 28 (1994), 151–171. [21] D. Sayre. Some implications of a theorem due to Shannon. Acta Cryst., 5 (1952), 834. [22] C. E. Shannon. A mathematical theory of communication. Bell Sys. Tech. J., 27 (1948), 379–423. [23] C. E. Shannon. Communication in the presence of noise. Proc. IRE, 37 (1949), 10–21. [24] D. Slepian. On bandwidth. Proc. IEEE, 64 (1976), 292–300. [25] H. Taub and D. L. Schilling. Principles of communication systems. McGraw Hill, New York, second edition, 1986. [26] R. G. Vaughan, N. L. Scott, and D. R. White. The theory of bandpass sampling. IEEE. Trans. Signal Processing, 39 (1991), 1973–1983.

M. G. Beaty School of Mathematics and Statistics University of Newcastle Newcastle-upon-Tyne NE1 7RU United Kingdom M. M. Dodson Department of Mathematics University of York York Y01 5DD United Kingdom Eingegangen am 23. Januar 1998