Multimedia Systems: Algorithms, Standards, and Industry Practices

26 C H A P T E R 2 • Digital Data Acquisition x(t ) Impulse response y(t ) = h(t ) * x(t ) h(t ) Fourier transform Fourier transform Inverse Fourier transform X(s ) Transfer function Y(s ) = H(s ) x X(s ) H(s ) Figure 2-8 Relationship between the impulse response function in the time domain and the transfer function in the frequency domain 3.3 Useful Signals Specific signals—for example, the Delta function, box function, step function—play an important role in many signaling-processing operations, such as sampling, filter- ing, and convolution. In this section, we describe some of these functions before delv- ing into how they are used. The Delta function is a mathematical construct introduced by the theoretical physi- cist Paul Dirac and, hence, is also known as the Dirac Delta function. Informally, it is a sharp peak bounding a unit area: ␦(x) is zero everywhere except at x ϭ 0 where it becomes infinite, and its integral is 1. A useful property of the Delta function is sifting. The sifting property is used in sampling and can be shown in the following equation: q f (t) * d(t Ϫ T) ϭ L-q f (t) # d(t Ϫ T Ϫ t)dt q ϭ f (t) # d(t Ϫ (t Ϫ T))dt L-q ϭ f (t Ϫ T) The preceding derivation shows that the signal f(t) convolved with the Delta func- tion at (t Ϫ T) yields the same value of the function at (t Ϫ T). This is very useful in developing the idea of convolution and sampling. By convolving the input signal with the Delta function and using its sifting property, we can represent an approxima- tion of any system’s output. The comb is another useful function, which is an infinite series of Dirac Delta functions with a regular spacing T. During sampling, you could use the comb and convolve it with the input analog function to get sampled digital values. Other useful functions, such as the step function, the box function, and the sinc function, are shown in Figure 2-9. 3.4 The Fourier Transform The Fourier transform is due to Joseph Fourier (1768–1830), who published his initial results in 1807. Fourier proposed to represent any periodic, continuous signal as a sum

Signals and Systems 27 Delta function Comb function Step function Box function Sinc function Figure 2-9 Useful functions in signal-processing theory: Delta function (top left), comb function (top right), step function (middle left), box function (middle right), and sinc function (bottom) of individual complex sinusoids (a Fourier series expansion). In other words every periodic, continuous signal can be expressed as a weighted combination of sinusoid (sine and cosine) waves. The weights that are used to combine the sinusoids are called the Fourier series coefficients, or spectral components. More details are presented in Section 6 at the end of this chapter, and we refer the reader to this section for a more intuitive as well as an involved analysis. The Fourier formulation represents a transfor- mation of the signal into frequency space. The spectral components of a signal define

28 C H A P T E R 2 • Digital Data Acquisition a discrete set of frequencies that are combined to form the signal. This is also called a harmonic decomposition. Fourier was originally criticized for not proving the exis- tence or the convergence of his series. The formal proof and criteria was proposed by Dirichlet, but this is beyond the scope of our coverage. What is important to know is that most signals fit these criteria. An example of two signals and their Fourier trans- forms is shown in Figure 2-11. A remarkable property of the Fourier transform is duality. Informally, it means that given a function and its Fourier transform, you can interchange the labels signal and spectrum for them. For instance, the Fourier transform of a Delta function is a constant, and the Fourier transform of a constant is a Delta. Also, the Fourier trans- form of a box is a sinc, and the Fourier transform of a sinc is a box. More generally, if a signal f (t) has a Fourier transform g(␻) ϭ F{f (t)}, then h(␻) ϭ F{g(t)} ϭ f (Ϫ␻). Therefore, for symmetric signals, h(␻) ϭ f(␻). 4 SAMPLING THEOREM AND ALIASING Here we answer the questions put forth in Section 2.1, which deal with the rate at which sampling should occur. The value of a nonstatic signal keeps changing depend- ing on its frequency content. Some sample signals are shown in Figure 2-10. Figure 2-10 Examples of simple one-dimensional functions. Different numbers of samples are required to digitize them If these signals are to be digitized and reproduced back in the analog domain, the number of samples required to ensure that both starting and ending analog signals are the same is clearly different. For instance, in the first case where the signal is not changing (and, hence, has zero frequency), one single sample will suffice for this pur- pose. Two samples are required for the second signal and many more for the third one. As we go from left to right, the frequency content in the signals increases, and, therefore, the number of samples needed during the digitization process also goes up. The relationship between signals and sampling rate was established during the late 1920s by Henry Nyquist and later formalized by Claude Shannon in 1950. This relationship states that the signal has to be sampled using a sampling frequency that is greater than twice the maximal frequency occurring in the signal. Or, it can more formally be stated as follows: 1. A bandlimited signal f(t) with max frequency ␻F is fully determined from its samples f(nT ) if 2␲/T Ͼ 2␻F 2. The continuous signal can then be reconstructed from its samples f (nT) by convolution with the filter