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

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) 4 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 character-istics 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 7. 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 xt ckejkO0t, 4:1:1 k where ck is the Fourier series coefficient,and V0 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 xtejkV0tdt: 4:1:2 0 T0 This integral can be evaluated over any interval of length T0. For an odd function,it is easier to integrate from 0 to T0. For an even function,integration from T0=2 to T0=2 is commonly used. The term with k 0 is referred to as the DC component because c0 T0 T0 xtdt 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 A, kT0 t=2 t kT0 t=2 0,otherwise, 4:1:3 where k 0, 1, 2, ...,and t < T0. Since x.t) is an even signal,it is con-venient to select the integration from T0=2 to T0=2. From .4.1.2),we have T0 jkV0t t sinkV0t c Ae 0 dt : 4:1:4 0 0 0 0 2 0 20 This equation shows that ck has a maximum value At=T0 at V0 0,decays to 0 as V0 ,and equals 0 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 2 2 −T0 − 2 0 2 T0 t Figure 4.1 Rectangular pulse train FOURIER SERIES AND TRANSFORM 129 A periodic signal has infinite energy and finite power,which is defined by Parseval`s theorem as Px 1 xt2dt ck2: 4:1:5 0 T0 k Since ck2 represents the power of the kth harmonic component of the signal,the total power of the periodic signal is simply the sum of the powers of all harmonics. The complex-valued Fourier coefficients, ck,can be expressed as ck ckejfk: 4:1:6 A plot of ck versus the frequency index k is called the amplitude .magnitude) spectrum, and a plot of fk versus k is called the phase spectrum. If the periodic signal x.t) is real valued,it is easy to show that c0 is real valued and that ck and ck are complex conjugates. That is, ck ck, ck ck and fk fk: 4:1:7 Therefore the amplitude spectrum is an even function of frequency V,and the phase spectrum is an odd function of V for a real-valued periodic signal. If we plot ck2 as a function of the discrete frequencies kV0,we can show that the power of the periodic signal is distributed among the various frequency components. This plotis called the power density spectrum of the periodicsignal x.t). Since the power in a periodic signal exists only at discrete values of frequencies kV0,the signal has a line spectrum. The spacing between two consecutive spectral lines is equal to the funda-mental frequency V0. Example 4.2: Consider the output of an ideal oscillator as the perfect sinewave expressed as xt sin2pf0t, f0 2p : We can then calculate the Fourier series coefficients using Euler`s formula .Appendix A.3) as sin2pf0t 1 ej2pf0t ej2pf0t ckejk2pf0t: k We have 1=2j, k 1 ck 1=2j, k 1 4:1:8 0,otherwise. 130 FREQUENCY ANALYSIS This equation indicates that there is no power in any of the harmonic k 1. Therefore Fourier series analysis is a useful tool for determining the quality .purity) of a sinusoidal signal. 4.1.2 Fourier Transform We have shown that a periodic signal has a line spectrum and that the spacing between two consecutive spectral lines is equal to the fundamental frequency V0 2p=T0. The number of frequency components increases as T0 is increased,whereas the envelope of the magnitude of the spectral components remains the same. If we increase the period without limit .i.e., T0 ),the line spacing tends toward 0. The discrete frequency components converge into a continuum of frequency components whose magnitudes have the same shape as the envelope of the discrete spectra. In other words,when the period T0 approaches infinity,the pulse train shown in Figure 4.1 reduces to a single pulse,which is no longer periodic. Thus the signal becomes non-periodic and its spectrum becomes continuous. In real applications,most signals such as speech signals are not periodic. Consider the signal that is not periodic .V0 0 or T0 ),the number of exponential components in .4.1.1) tends toward infinity and the summation becomes integration over the entire continuous range .,. Thus .4.1.1) can be rewritten as xt XVejVtdV: 4:1:9 This integral is called the inverse Fourier transform. Similarly,.4.1.2) can be rewritten as XV xtejVtdt, 4:1:10 which is called the Fourier transform .FT) of x.t). Note that the time functions are represented using lowercase letters,and the corresponding frequency functions are denoted by using capital letters. A sufficient condition for a function x.t) that possesses a Fourier transform is xtdt < : 4:1:11 That is, x.t) is absolutely integrable. Example4.3:CalculatetheFouriertransformofthefunctionxt eatut,where a > 0 and u.t) is the unit step function. From .4.1.10),we have FOURIER SERIES AND TRANSFORM 131 XV eatutejVtdt eajVtdt 0 a jV: The Fourier transform XV is also called the spectrum of the analog signal x.t). The spectrum XV is a complex-valued function of frequency V,and can be expressed as XV XVejfV, 4:1:12 where XV is the magnitude spectrum of x.t),and fV is the phase spectrum of x.t). In the frequency domain, XV2 reveals the distribution of energy with respect to the frequency and is called the energy density spectrum of the signal. When x.t) is any finite energy signal,its energy is Ex xt2dt 1 XV2dV: 4:1:13 This is called Parseval`s theorem for finite energy signals,which expresses the principle of conservation of energy in time and frequency domains. For a function x.t) defined over a finite interval T0,i.e., xt 0 for t > T0=2,the Fourierseriescoefficientsck canbeexpressedintermsofXVusing.4.1.2)and.4.1.10)as ck 1 XkV0: 4:1:14 0 For a given finite interval function,its Fourier transform at a set of equally spaced points on the V-axis is specified exactly by the Fourier series coefficients. The distance between adjacent points on the V-axis is 2p=T0 radians. If x.t) is a real-valued signal,we can show from .4.1.9) and .4.1.10) that FTxt XV and XV XV: 4:1:15 It follows that XV XV and fV fV: 4:1:16 Therefore the amplitude spectrum XV is an even function of V,and the phase spectrum is an odd function. Ifthetimesignalx.t)isadeltafunctiondt,itsFouriertransformcanbecalculatedas XV dtejVtdt 1: 4:1:17 ... - tailieumienphi.vn