Real-Time Digital Signal Processing - Chapter 4: Frequency Analysis

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Real-Time Digital Signal Processing - Chapter 4: Frequency Analysis

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Real-Time Digital Signal Processing Sen M Kuo, Bob H Lee Copyright # 2001 John Wiley & Sons Ltd ISBNs: 0-470-84137-0 (Hardback); 0-470-84534-1 (Electronic) Frequency Analysis Frequency analysis of any given signal involves the transformation of a time-domain signal into its frequency components The need for describing a signal in the frequency domain exists because signal processing is generally accomplished using systems that are described in terms of frequency response Converting the time-domain signals and systems into the frequency domain is extremely helpful in understanding the characteristics of both signals and systems In Section 4.1, the Fourier series and Fourier transform will be introduced The Fourier series is an effective technique for handling periodic functions It provides a method for expressing a periodic function as the linear combination of sinusoidal functions The Fourier transform is needed to develop the concept of frequency-domain signal processing Section 4.2 introduces the z-transform, its important properties, and its inverse transform Section 4.3 shows the analysis and implementation of digital systems using the z-transform Basic concepts of discrete Fourier transforms will be introduced in Section 4.4, but detailed treatments will be presented in Chapter The application of frequency analysis techniques using MATLAB to design notch filters and analyze room acoustics will be presented in Section 4.5 Finally, real-time experiments using the TMS320C55x will be presented in Section 4.6 4.1 Fourier Series and Transform In this section, we will introduce the representation of analog periodic signals using Fourier series We will then expand the analysis to the Fourier transform representation of broad classes of finite energy signals 4.1.1 Fourier Series Any periodic signal, x(t), can be represented as the sum of an infinite number of harmonically related sinusoids and complex exponentials The basic mathematical representation of periodic signal x(t) with period T0 (in seconds) is the Fourier series defined as 128 FREQUENCY ANALYSIS x…t† ˆ X ck e jkO0 t , …4:1:1† kˆ where ck is the Fourier series coefficient, and V0 ˆ 2p=T0 is the fundamental frequency (in radians per second) The Fourier series describes a periodic signal in terms of infinite sinusoids The sinusoidal component of frequency kV0 is known as the kth harmonic The kth Fourier coefficient, ck , is expressed as … ck ˆ x…t†e jkV0 t dt: …4:1:2† T0 T0 This integral can be evaluated over any interval of length T0 For an odd function, it is easier to integrate from to T0 For an even function, integration from T0 =2 to T0 =2 is commonly used The term with k ˆ is referred to as the DC component because „ c0 ˆ T10 T0 x…t†dt equals the average value of x(t) over one period Example 4.1: The waveform of a rectangular pulse train shown in Figure 4.1 is a periodic signal with period T0 , and can be expressed as  x…t† ˆ kT0 t=2  t  kT0 ‡ t=2 otherwise, A, 0, …4:1:3† where k ˆ 0,  1,  2, , and t < T0 Since x(t) is an even signal, it is convenient to select the integration from T0 =2 to T0 =2 From (4.1.2), we have ck ˆ T0   kV0 t t # sin A e jkV0 t At jkV0 t : Ae dt ˆ ˆ T0 kV0 t T0 T0 jkV0 t 2 " … T0 …4:1:4† This equation shows that ck has a maximum value At=T0 at V0 ˆ 0, decays to as V0 ! 1, and equals at frequencies that are multiples of p Because the periodic signal x(t) is an even function, the Fourier coefficients ck are real values For the rectangular pulse train with a fixed period T0 , the effect of decreasing t is to spread the signal power over the frequency range On the other hand, when t is fixed but the period T0 increases, the spacing between adjacent spectral lines decreases x(t) A −T0 −T0 Figure 4.1 − T0 t t 2 t T0 Rectangular pulse train 129 FOURIER SERIES AND TRANSFORM A periodic signal has infinite energy and finite power, which is defined by Parseval's theorem as Px ˆ T0 … T0 X zˆ0:75 and c2 ˆ z 0:75 zˆ 0:5 ˆ 0:8: Thus we have X …z† ˆ 0:8z z 0:75 0:8z : z ‡ 0:5 The overall inverse z-transform x(n) is the sum of the two inverse z-transforms From entry of Table 4.3, we obtain x…n† ˆ 0:8‰…0:75†n … 0:5†n Š , n  0: The MATLAB function residuez finds the residues, poles and direct terms of the partial-fraction expansion of B…z†=A…z† given in (4.2.9) Assuming that the numerator and denominator polynomials are in ascending powers of z , the function [c, p, g ]= residuez(b, a); finds the partial-fraction expansion coefficients, cl , and the poles, pl , in the returned vectors c and p, respectively The vector g contains the direct (or polynomial) terms of the rational function in z if L  M The vectors b and a represent the coefficients of polynomials B(z) and A(z), respectively If X(z) contains one or more multiple-order poles, the partial-fraction expansion must Pm gj include extra terms of the form jˆ1 …z pl †j for an mth order pole at z ˆ pl The coefficients gj may be obtained with 139 THE Z-TRANSFORM gj ˆ …m dm j†! dzm  …z j pl †m X …z† z j  zˆpl : …4:2:14† Example 4.8: Consider the function X …z† ˆ z2 ‡ z 1†2 …z : We first express X(z) as X …z† ˆ g1 …z 1† ‡ g2 1†2 …z : From (4.2.14), we have " d …z g1 ˆ dz g2 ˆ …z # 1†2 X …z† d ˆ …z ‡ 1† ˆ 1, z dz zˆ1 zˆ1 1† X …z† ˆ …z ‡ 1†jzˆ1 ˆ 2: z zˆ1 Thus X …z† ˆ z …z 1† ‡ 2z 1†2 …z : From Table 4.3, we obtain h z i x…n† ˆ ZT ‡ ZT z 1 " # 2z 1†2 …z ˆ ‡ 2n, n  0: The residue method is based on Cauchy's integral theorem expressed as 2pj ‡ c z k m  dz ˆ if k ˆ m if k ˆ m …4:2:15† Thus the inversion integral in (4.2.8) can be easily evaluated using Cauchy's residue theorem expressed as ‡ X …z†zn dz x…n† ˆ 2pj c X ˆ residues of X …z†zn at poles of X …z†zn within C: …4:2:16† ... of signal, the two-sided z-transform defined in (4.2.1) becomes a one-sided z-transform expressed as X …z† ˆ X x…n†z n : ? ?4:2 :3† nˆ0 Clearly if x…n† is causal, the one-sided and two-sided z-transforms... transform is the z-transform The z-transform yields a frequency- domain description of discrete-time signals and systems, and provides a powerful tool in the design and implementation of digital filters... two time-domain signals to the multiplication of their corresponding z-transforms Some of the commonly used signals and their z-transforms are summarized in Table 4.3 136 FREQUENCY ANALYSIS

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