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2016 | OriginalPaper | Buchkapitel

3. Transforms of Discrete-Time Signals

verfasst von : Silvia Maria Alessio

Erschienen in: Digital Signal Processing and Spectral Analysis for Scientists

Verlag: Springer International Publishing

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Abstract

In this chapter, the invertible transforms used to work on discrete-time signals are discussed. Given a complex variable z, the z-transform is defined as an infinite series in the z-plane that exists in the region(s) of the plane where the series exhibits absolute convergence to an analytic function. The corresponding infinite-length signal is required to be absolutely summable. Unit-amplitude z values identify the unit circle, on which the z-transform becomes a continuous function of frequency, called the discrete-time Fourier transform (DTFT). The DTFT representation can also be extended to sequences for which the z-transform does not exist, such as signals that are only square-summable, or periodic signals like sinusoids. If a sequence has finite length, it may be represented in the frequency domain by a finite number of values obtained by properly sampling the DTF, i.e., by the discrete Fourier transform (DFT). The properties of the DFT emerge clearly if this transform is introduced passing through the discrete Fourier series (DFS) of the signal’s periodic extension. The DFT can be efficiently computed via fast Fourier transform (FFT). Each inverse transform represents an expansion of the signal in an orthogonal basis. At the end of the chapter, an appendix provides an overview of the mathematical foundations of analog and discrete-time signal expansions.

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Vorheriges Kapitel Discrete-Time Signals and Systems

Nächstes Kapitel Sampling of Continuous-Time Signals

Why do we base the convergence condition on a sum of absolute values, $|x[n] z^{-n}|$, while the z-transform is defined as a sum of complex numbers $x[n] z^{-n}$? The reason is that the value of an infinite sum can potentially depend on the order in which the elements of the series are added. In general, an infinite series $\sum _{n=1}^{\infty } a_n$ may converge (that is, yield a finite result) even when $\sum _{n=1}^{\infty } |a_n| = \infty $, but in such a case the finite value of $\sum _{n=1}^{\infty } a_n$ will vary if the order of the terms is changed. Such a series is said to be conditionally convergent. On the other hand, if the sum of absolute values is finite, the sum will be independent of the order of summation. Such a series is said to be absolutely convergent. In the z-transform we are summing the samples of a two-sided sequence and we do not want the sum to depend on the order of terms. The requirement of absolute convergence eliminates this issue and guarantees that the definition of the z-transform is unambiguous.

The ROC may also include the circle $|z| = R_-$ or the circle $|z| = R_-$ or both, but there is no general criterion for testing these possibilities.

The extended complex plane is obtained by adding a single point $z=\infty $ to the conventional complex plane. The point $z=\infty $ has modulus larger than that of any other complex number and its phase is undefined. On the other hand, the point $z=0$ has a modulus smaller than that of any other complex number and an undefined phase. The ROC may be extended to include the point $z=\infty $ if and only if the sequence is causal.

The name of this relation originates from a 1799 theorem about series, formulated by Marc-Antoine Parseval, which was later applied to the Fourier series. Analogous relations for other transforms were then derived, like the one mentioned here for the z-transform.

Parseval’s relation written in this way is meaningful only for energy signals. It does not hold for power signals having a DTFT only in a generalized sense (see Sect. 3.3). For power signal we must reason in terms of average power and write Parseval’s relation differently; this will be done in Chap. 5.

Recall from Chap. 2 that even if here choose a range of values for $\omega $ that spans $[-\pi , +\pi )$, any interval of length $2\pi $ can be used. For example, choosing $(-\pi , +\pi ]$ instead of $[-\pi , +\pi )$ would be correct as well; we could also choose $[0, 2\pi )$, etc. Actually, whenever in our discussion we will consider a DTFT over the positive frequency half-axis only, we will write $\omega \in [0, \pi ]$.

Sequences that are absolutely summable are classified as $\ell ^1$-sequences; sequences that are square-summable are classified as $\ell ^2$-sequences. See the appendix to this chapter for these topics.

In mathematics, the historical (unnormalized) Sinc function is defined as

$$\begin{aligned}\mathrm {Sinc}(x)= {\left\{ \begin{array}{ll} \frac{\sin (x)}{x} &{} \text{ for } \quad x\ne 0, \\ 1 &{} \text{ for } \quad x= 0. \\ \end{array}\right. } \end{aligned}$$

In either case (unnormalized or normalized), the value at x = 0 is defined to be the limiting value $\mathrm {Sinc}(0)= 1$.

If energy is finite, that is, if the signal is in some way transient, then this relation creates no infinity problems, and the quantity $|X(\mathrm{{e}}^{\mathrm {j}\omega })|^2$ appears as a frequency distribution of the energy of the signal: it suggests the notion of energy spectrum that will be discussed later. If energy is infinite, the notion of energy spectrum makes no sense. If the signal is a power signal, i.e., its energy is infinite but its power is finite (this could be the case of a deterministic periodic signal; see Chap. 2), another quantity may be defined in place of the energy spectrum. This quantity is the power spectrum, that we will actually study in detail in the case of random signals, rather than in the case of deterministic signals. In fact, random signals typically are power signals. For a more extended discussion of these topics in relation to deterministic signals see Chap. 5.

The expression $n{\,\text {mod}\,}N$ indicates the remainder after division of n by N. Let us make a couple of examples of how the $\mathrm {mod}$ function works:

we want to compute $340 {\,\text {mod}\,}60$. Now, 340 lies between 300 and 360, so 300 = $60\times 5$ is the greatest multiple of 60 which is less than or equal to 340; we subtract 300 from 340 and get 40;
we want to compute $-340 {\,\text {mod}\,}60$. Now, $-340$ lies between $-360$ and $-300$, so $-360 = 60\times (-6)$ is the greatest multiple of 60 less than or equal to $-340$; we subtract $-360$ from $-340$ and get 20.

The precise definition of $D_N(\omega )$ is

$$\begin{aligned}D_N(\omega )= {\left\{ \begin{array}{ll} \frac{\sin \left[ N (\omega /2)\right] }{N \sin (\omega /2)} &{} \omega \ne 2 \pi k,\qquad \qquad k=0, \pm 1, \pm 2,\pm 3 \ldots , \\ (-1)^{k\left( N-1\right) } &{} \omega =2 \pi k,\qquad \qquad k=0, \pm 1, \pm 2,\pm 3 \ldots . \end{array}\right. } \end{aligned}$$

This gives 1 at $\omega =0$.

Lebesgue integrals are somewhat more general than the basic Riemann integral. The value of a Lebesgue integral is not affected by values of the function over any countable set of values of its argument, or, more generally, a set of measure zero. For instance, a function defined as 1 on the rationals and 0 on the irrationals would have a zero Lebesgue integral. As a result of this, properties derived using Lebesgue integrals are sometimes said to be true “almost everywhere”, meaning they may not be true over a set of measure zero.

Equivalently, a sequence space is a function space whose elements are functions from the field $\mathbb {Z}$ of integer numbers (values of n) to the field $\mathbb {R}$ of real or complex numbers (values of x[n]), exactly as a continuous-time-function space is a function space whose elements are functions from the field $\mathbb {R}$ (values of t) to $\mathbb {R}$ (values of x(t)).

In this part of the book we are dealing with deterministic signals. Starting from Chap. 9, our approach to signal representation and analysis will change: we will explicitly consider signals deriving from experimental measurements, which are better described in terms of random variables/random processes and of statistical/probabilistic arguments. The present discussion covers also the random case, except that in the case of random variables, the inner product should actually be defined through the so-called ensemble average operator,

$$\begin{aligned}\left\langle x(t), y(t)\right\rangle =\mathrm {E}\left[ x y^*\right] , \end{aligned}$$

that is the “correlation” of the two random variables x and y (for zero-mean variables it is the “covariance”). This means (see Chap. 9) that the integral should include the probability density function related to each random variable. On the other hand, in the theory of wavelet representation of signals (Chaps. 13–15), the random nature of signals is not explicitly considered (see Sects. 14.2 and 14.3). Though keeping in mind that the statistical/probabilistic complication will come into play in Chap. 9, we can continue our discussion on inner-product spaces using the standard definition of inner product.

Here we do not enter in details about how passing from one basis to another one can be seen as a rotation of axes; we leave this to the intuition of the reader.

In several cases, the signal spaces encountered in wavelet theory are actually the closure of the space spanned by the basis set, meaning that the space contains not only all signals that can be expressed by a linear combination of the basis functions, but also the signals which are the limit of infinite expansions based on the considered (infinite) set. The closure of a space is usually denoted by an over-line, as in $\overline{\mathrm {Span}_k}$, but we will neglect the over-line in our notation, for simplicity.

Actually, the first equation above should be called Plancherel’s theorem; the name “Parseval’s theorem” should be reserved to the second equation, which is a particular case of Plancherel’s theorem.

More precisely, Parseval’s theorem refers to the result that the Fourier transform is unitary: loosely, this means that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval (Parseval 1806), which was later applied to the Fourier series. It is also known as Rayleigh’s energy theorem (Rayleigh 1889), or Rayleigh’s identity, after John William Strutt, Lord Rayleigh. Plancherel’s theorem (Plancherel 1910) is a more general result in harmonic analysis. It states that the integral of a function’s squared modulus is equal to the integral of the squared modulus of its frequency spectrum. This result makes it possible to speak of Fourier transforms of quadratically-integrable functions and quadratically-summable sequences, rather than just of absolutely integrable functions and absolutely summable sequences. The unitarity of any kind of Fourier transform is often called Parseval’s theorem in science and engineering fields, based on the above-mentioned earlier (but less general) result, but the most general form of this property should be called Plancherel’s theorem.

In the three-dimensional case, and thinking of the samples of x[n], with $n=0,1,2$ as the spatial coordinates $\text {x}$, $\text {y}$ and $\text {z}$ of a point with respect to a Cartesian coordinate reference system, this would be the familiar vector representation $\text {x} \varvec{i}+ \text {y}\varvec{j}+\text {z}\varvec{k}$ that uses three standard basis unit vectors along each of the three mutually perpendicular axes. Thus we would have $\varvec{e}_1=\varvec{i}$, $\varvec{e}_2=\varvec{j}$, $\varvec{e}_3=\varvec{k}$, the unit vectors indicating the direction of the axes being $[1 \, 0 \, 0]^T$, $[0 \, 1 \, 0]^T$ and $[0 \, 0 \, 1]^T$.

“Biorthogonal” means that this type of “orthogonality” requires two sets of vectors.

In order to better understand what a basis is, we may mention that in finite dimensions, analysis and synthesis operations can be represented as matrix-vector multiplications. If the expansion functions (vectors) are a basis, the synthesis matrix has these basis vectors as columns; it is square and non-singular. If the basis is orthogonal, then the synthesis matrix is orthogonal, since its rows and columns are orthogonal to one another; its inverse is equal to its transpose, and the matrix multiplied by its transpose gives the identity matrix. If the synthesis matrix is not orthogonal, then the identity matrix is the synthesis matrix multiplied by its inverse, and the dual basis consists of the rows of the inverse. If the synthesis matrix is singular, then its columns (the basis vectors) are not independent, and therefore do not form a basis: the uniqueness of the coefficients is lost. This leads to the concept of frame.

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Titel: Transforms of Discrete-Time Signals
verfasst von: Silvia Maria Alessio
Verlag: Springer International Publishing
Buch: Digital Signal Processing and Spectral Analysis for Scientists
Print ISBN: 978-3-319-25466-1

Electronic ISBN: 978-3-319-25468-5

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-25468-5_3

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