Question

1. Find the length of the lowpass FIR filter corresponding to the following specifications: wp- 0.3m ωs-0.4m, δp-0.01, and δ,-0.005. Use Kaiser's formula 4. Consider the design of a windowed FIR lowpass filter corresponding to the specifications given in problem #1. Determine its length if Hann, Hamming, and Blackman windows are used. Hint: refer to Equation 10.36 and Table 10.2 of the textbook. 5. With reference to the specifications in problem #1, consider the design of an FIR lowpass filter using adjustable windows. Determine the filter orders corresponding to a) Dolph- Chebyshev window, and b) Kaiser window. Figure 10.8: Relations among the fre windowed filter. quency responses of an ideal lowpass filter, a typical window, and the Table 10.2: Properties of some fixed window functions Type of Relative Sidelobe Level Ast 13.3 dB 26.5 dB 31.5 dB 42.7 dB 58.1 dB Main Lobe Width ΔML Minimum StopbandTransition Window Rectangular 4π/(201 + 1) Bartlett Hann Hamming 8T (2M 1) Blackman 12/(2M 1) Attenuation 20.9 dB See text 43.9 dB 54.5 dB 75.3 dB Bandwidth Ao 0.92T/M See text 3.11jr/M 3.32T/M 8/(2M 1) his table has been adapted from [Sar93], with the values shown in the table for oc 0.47 and M - Example 10.5 illustrates the effect of each of the above windows on the frequency response of an F lowpass filter designed using the windowed Fourier series approach ffect of Windows on the Frequency Response of an Ideal Lowpass Filter he impulse response hLP[n] of the ideal lowpass filter is given in Eq. (10.14). To create a finite- duration zero-phase FIR filter of length N , we form ht [n] = hLP[지-w[n], where M = (N 1)/2 Figure 10.9 shows the impulse response samples ht[n] for N 51 and ae-π/2. The gain response h various fixed It should be noted from this figure that the increase in h an increase in the transition in the sidelohe amnlitude results in an increase in the stopband attenuation. of each of the filters obtained by windowing the above impulse response samples wit window functions is sketched in Figure 10.10. the main lobe width of the window functions is clearly asso ciated wit 20 20 80 0.2 0.4 0.6 -l0OL 0.8 0.2 0.8 (i)/其 Blackman window 6D 0.2 0.6 0.8 Figure 10.7: Gain responses of the fixed window functions. tenuation and the expression for the transition bandwidth are not precisely known. The Bartlett window tinds applications in spectral estimation es not depend on the filter length, or the cutoff frequency de, te transition bandwidth is approximately given by the case of the window functions of Eqs. (10.30) and (10.33) to (10.34), the value of the ripple & and is essentially constant. In addition, (10.36) where c is a constant for most practical purposes and is determined from Table 10.2 after the specific window has been selected [Sar93]

Answer 1

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