Elsevier

Signal Processing

Volume 81, Issue 3, March 2001, Pages 581-592
Signal Processing

The comb signal and its Fourier transform

https://doi.org/10.1016/S0165-1684(00)00233-4Get rights and content

Abstract

In this paper, we study the aperiodic comb signal from the point of view of the Fourier transform. The comb is very important in the theory of ideal sampling. The knowledge of its properties is crucial for the establishment of suitable interpolation schemes. Here, we present sufficient conditions so that the Fourier transform of an aperiodic comb is an aperiodic comb. We use this result to propose: (1) an alternative approach to the definition of an almost periodic signal and its anharmonic Fourier series; (2) a generalisation of the Shannon–Whittakker sampling/reconstruction for the irregular sampling case. Application of this theory to pulse duration modulation and pulse position modulation is also presented.

Introduction

The comb signal is one of the most important entities in Signal Processing, because of its connections with Fourier Series (FS) and ideal sampling [8]. The usual comb is a periodic repetition of the Dirac's delta (generalised) function [10], [12]. As is well known, its Fourier transform (FT) is also a periodic comb [1]. In this paper we will study the FT of the general aperiodic comb signal and formulate conditions to guarantee that its FT is an aperiodic comb, too. To see the importance of this subject, let us consider the following practical situation. One of the objectives in electrocardiogram (ECG) processing is the study of the variability of the cardiac frequency. This is usually done from the so-called RR intervals that are the time intervals between peaks of consecutive cardiac beats. These values constitute a time series. We can model the excitation of the heart as a pulse frequency modulation signal. With this, the RR interval signal can be considered as being proportional to the modulating signal. Therefore, we have a signal that is sampled at a non-uniform spacing: the beat peak positions. Let dn (n=1,2,…,L) be a sequence of RR intervals. Taking 0 as the time origin reference, we define a set of sampling instants:

tn=tn−1+dnt0=0,n=1,2,…,Land a signal, v(t)

v(tn)=dnthat is proportional to the modulating signal. In the available commercial systems, the signal v(t) is treated as if it was obtained by uniform sampling. To analyse the error we are making, we took a signal dn obtained from an ECG signal and constructed the sequence of instants, tn, through (1.1). We sampled a sinusoid at those instants and at a uniform spacing nT, where the sampling interval, T, is the mean value of dn. For these two signals, we computed their FT by using the FFT. The results are shown in Fig. 1 (top and middle pictures). As it is clear, the use of FFT to compute the FT of the non-uniformly sampled signal is incorrect. To avoid the problem, we assumed that the signal v(t) was ideally sampled by a non-periodic combpt=−∞+∞δt−tnobtaining the signalvst=−∞+∞vtnδt−tnthat has the following FT:Vsω=−∞+∞vtnejωtn.We used this expression to obtain the spectrum shown at the bottom picture in Fig. 1 that shows a good agreement with the picture in the middle. This fact means that the available approaches to studying the variability of cardiac frequency are intrinsically wrong. These considerations served as motivation for the study we present in this paper.

The problem of non-uniform sampling has received increasing attention due to practical applications in real life. The theory of frames [6] has being the most important tool for dealing with the problem. Here we adopt a more general point of view. We intend to state conditions generalising some of the current results on ideal sampling and reconstruction.

In Section 2, we present the main result of this paper: under stated conditions, the FT of an aperiodic comb is an aperiodic comb [11]. We precise such (sufficient) conditions. The proof is in Appendix A. That result has interesting implications in some well-known fields, as: almost periodic functions and non-uniform sampling. These subjects are treated in Section 3. In Section 4 we present two applications to communication theory: the pulse duration modulation (PDM) and pulse position modulation (PPM) [2]. Most of the mathematical base of the theory is in distribution theory. We present in Appendix B a brief overview of the Axiomatic Theory of Distributions [7], [13].

In the following, we represent the sets of integer and real numbers by Z and R, respectively. The Dirac's symbol will always be represented by δ(t).

Section snippets

The FT of a comb

Consider a set of instants tn (n=−∞,…,0,…,+∞) assumed to form a, as fast as n, increasing sequence such that t±∞=±∞ (for Theorems 2.1 and 2.2, we only need to assume that the sequence increases faster than n).

Definition 2.1

A comb is a distribution, c(t), defined byct=−∞+∞δt−tn.

Theorem 2.1

The series −∞+∞δt−tn is convergent.

To prove it, let us introduce the function s(t) given byst=0+∞rt−tn−∞−1r−t−tn,where r(t) is the ramp functionrt=tt⩾00t<0It is not hard to see that s(t) is a continuous function. In fact, for

Consequences

Let xb(t) a signal with FT Xb(ω) and tn an ALS. Let us assume that Xb(ω) is a bounded function. We define an almost periodic function, x(t), as the generalised function resulting from convoluting xb(t) with c(t) given by (2.1). Attending to the properties of the δ function, we can writext=n=−∞xbt−tn.By the use of Eq. (2.19) we conclude that x(t) is represented by the anharmonic FS [3], [9]:xt=−∞+∞xnejωntwith

Xn=Cn.Xbn)However, using (2.1) and the definition of FT, we obtainxn=ctejωnt−∞+∞xb

Applications to pulse modulation

In the following, we are going to study two important cases that fall inside the theory we presented in previous sections. We will present the expressions for the modulated signals in pulse duration modulation (PDM) and in pulse position modulation (PPM) [13].

We will consider the PDM signal with trailing-edge modulation of the pulse duration. The modulating signal dependence is on the location, tk, k=−∞, …, +∞, of the pulse edge. According to the generalisation of Nyquist criterion stated in

Conclusions

In this paper, we studied the aperiodic comb signal and its FT. We showed how we can guarantee that the FT of an aperiodic comb is an aperiodic comb. We used this result to propose an alternative approach to the definition of an almost periodic signal and computed its anharmonic Fourier series. The original first goal of this theory was the generalisation of the Shannon–Whittakker sampling/reconstruction for the irregular sampling case was also presented. Based on the previous results we

References (13)

  • R. Bracewell

    The Fourier Transform and Its Applications

    (1965)
  • A.B. Carlson

    Communication Systems

    (1983)
  • C. Cordoneanu

    Almost Periodic Functions

    (1968)
  • A. M. Davis, Almost periodic extension of band-limited functions and its application to nonuniform sampling, IEEE...
  • A.M. Davis, K∞ Generalised functions, Proceedings of the IEEE International Conference on Circuits and Systems,...
  • R.J. Duffin et al.

    A class of nonharmonic Fourier series

    Trans. Amer. Math. Soc.

    (1952)
There are more references available in the full text version of this article.

Cited by (12)

View all citing articles on Scopus
1

Also with INESC.

View full text