Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

18-8 Passive, Active, and Digital FiltersH(e jv ) ¼ 12pð p pH 0 d (e ju )W r (e j(v u) )du (18:37)where W r (e ju ) is the spectrum of the rectangular window.Since the two functions in the integral are periodic, a circular convolution results and the limits ofintegration are taken over one period. Thus the frequency response H(e jv ) will be a ‘‘smeared’’ version ofthe desired response H 0 d (e jv ) and the discontinuities in the desired frequency response become transitionbands of H(e jv ). To understand this, it is instructive to examine the frequency response for the causalrectangular window, that is,W r (e jvT ) ¼ XN 1e jvnTn¼0¼ e jv(N1)T=2 sin (vNT=2)sin (vT=2)(18:38)The spectrum W r (e jvT ) for N ¼ 31 is shown in Figure 18.3. The spectrum W r (e jvT ) has two features thatare worth noting, the mainlobe width and the sidelobe amplitude. The mainlobe width is defined asthe distance between the two points closest to v ¼ 0, where W r (e jvT ) is zero. For a rectangular window,the mainlobe width is equal to 4p=N. The maximum sidelobe amplitude for W r (e jvT ) is equal toapproximately 13 dB relative to the maximum value at v ¼ 0.Figure 18.4 shows the log–magnitude response of applying a 31-point rectangular window to approximatean ideal low-pass filter with a cutoff frequency equal to p=4. It can be seen that the sharp transitionin the ideal response at v ¼ v c has been converted into a gradual transition. Also, in the passband a seriesof overshoots and undershoots occur, and in the stopband, where the desired response is zero, the FIRfilter has a nonzero response. These are the results of the convolution between W r (e jvT ) and H d (e jvT ).The mainlobe of W r (e jvT ) causes the smearing of the desired response and the sidelobes of W r (e jvT )appear as overshoots and undershoots to the desired response. It is interesting to note that there will0–10Amplitude spectrum (dB)–20–30–40–50–60–70–800 0.5 1 1.5 2 2.5 3 3.5Frequency (rad)FIGURE 18.3Fourier transform of the rectangular window.