Tài liệu Bài báo cáo- thiết kế loc số

Chapter 8 IIR Filter Design 1. Problem P 8.1 Analog Butterworth lowpass filter design: Ω M ATLAB Script: 111 p = 30 rad/s, R p=1 dB, Ωs = 40 rad/s, As = 30 dB. 112 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 The system function is given by Ha (s) = 2:8199 1022 1122 s + 57:338s + 984:84 s + 50:777s + 984:84 11 2 + 41:997s + 984:842 + 31:382s + 984:84ss 11 s2 + 19:395s + 984:84s2 + 6:5606s + 984:84 1 s2 + 61:393s + 984:84 1 s + 31:382 The filter design plots are given in Figure 8.1. 2. Problem P 8.2 Analog Elliptic lowpass filter design: Ω p = 10 rad/s, R M ATLAB Script: p=1 dB, Ωs = 15 rad/s, As = 40 dB. A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 113 Analog Butterworth Lowpass Filter Design Plots in P 8.1 Magnitude Response 0 1 0.89 Magnitude 30 Decibels Log−Magnitude Response 0 0 30 40 Analog frequency in rad/sec 100 100 0 30 40 Analog frequency in rad/sec 100 Phase Response 0 0.1 Impulse Response Phase in pi units 0.05 −3.47 −4.89 ha(t) 0 0 30 40 Analog frequency in rad/sec 100 −0.05 0 0.5 1 t (sec) 1.5 2 Figure 8.1: Analog Butterworth Lowpass Filter Plots in Problem P 8.1 114 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 The system function is given by Ha (s) = 46978s4 + 220:07s2 + 2298:5 s5 + 9:23s4 + 184:71s3 + 1129:2s2 + 7881:3s + 22985 : The filter design plots are given in Figure 8.2. Analog Elliptic Lowpass Filter Design Plots in P 8.2 Magnitude Response 1 0.89 log−Magnitude in dB Magnitude 0 Log−Magnitude Response 40 0 0 1015 Analog frequency in rad/sec 30 100 0 1015 Analog frequency in rad/sec 30 Phase Response 0 3 2 ha(t) −1.26 −1.64 1 0 −1 −2 Impulse Response Phase in pi units 0 1015 Analog frequency in rad/sec 30 0 2 4 t (sec) 6 8 10 Figure 8.2: Analog Elliptic Lowpass Filter Design Plots in P 8.2 3. Problem P 8.3 A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 115 The filter passband must include the 100 Hz component while the stopband must include the 130 Hz component. To obtain a minimum-order filter, the transition band must be as large as possible. This means that the passaband cutoff must be at 100 Hz while the stopband cutoff must be at 130 Hz. Hence the analog Chebyshev-I lowpass filter specifications are: Ω p = 2π (100) rad/s, R p = 2 dB, Ωs = 2π (130) rad/s, As = 50 dB. M ATLAB Script: 116 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 The system function is given by Ha (s) = 7:7954 1022 1 1 s2 + 142:45s + 51926 s2 + 75:794s + 301830 The magnitude response plots are given in Figure 8.3. 1 s2 + 116:12s + 168860 11 2 + 26:323s + 388620ss + 75:794 Analog Chebyshev−I Lowpass Filter Design Plots in P 8.3 Magnitude Response 1 0.79 Magnitude 0 0 100130 Analog frequency in Hz 200 Log−Magnitude Response 0 Decibels 50 100 0 100130 Analog frequency in Hz 200 Figure 8.3: Analog Chebyshev-I Lowpass Filter Plots in Problem P 8.3 4. Problem P 8.4 Analog Chebyshev-II lowpass filter design: Ω p = 2π (250) rad/s, R p = 0:5 dB, Ωs = 2π (300) rad/s, As = 45 dB. M ATLAB Script: A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 117 118 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 The filter design plots are given in Figure 8.4. Digital Butterworth Filter Design Plots in P 8.7 Log−Magnitude Response 0 Decibel 50 0 0.4 Frequency in Hz 0.6 1 Impulse Response 0.1 ha(t) 0 −0.1 0 10 20 30 40 5060 time in seconds 70 80 90 100 Figure 8.4: Analog Chebyshev-II Lowpass Filter Plots in Problem P 8.4 5. Problem P 8.5 M ATLAB function afd. 6. Problem P 8.6 A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 119 Digital Chebyshev-1 Lowpass Filter Design using Impulse Invariance. M ATLAB script: (a) Part (a): T = 1. M ATLAB script: The filter design plots are shown in Figure 8.5. (b) Part (b): T = 1=8000. M ATLAB script: 120 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 Filter Design Plots in P 8.6a Log−Magnitude Response 0 Decibel 40 60 0 1500 2000 Frequency in Hz 4000 Impulse Response 0.3 0.2 0.1 0 −0.1 −0.2 h(n) 0 10 20 30 40 50 n 60 70 80 90 100 Figure 8.5: Impulse Invariance Design Method with T =1 in Problem P 8.6a The filter design plots are shown in Figure 8.6. (c) Comparison: The designed system function as well as the impulse response in part 6b are similar to those in part 6a except for an overall gain due to Fs = 1=T = 8000. This problem can be avoided if in the impulse invariance design method we set h (n) = T ha (nT ) 7. Problem P 8.7 Digital Butterworth Lowpass Filter Design using Impulse Invariance. MATLAB script: A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 121 Filter Design Plots in P 8.6b Log−Magnitude Response 0 Decibel 40 60 0 15002000 Frequency in Hz Impulse Response 4000 3000 2000 1000 0 −1000 −2000 h(n) 0 10 20 30 40 50 n 60 70 80 90 100 Figure 8.6: Impulse Invariance Design Method with T = 1=8000 in Problem P 8.6b The filter design plots are shown in Figure 8.7. Comparison: From Figure 8.7 we observe that the impulse response h (n) of the digital filter is a sampled version of the impulse response ha (t ) of the analog proptotype filter as expected. 122 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 Digital Butterworth Filter Design Plots in P 8.7 Log−Magnitude Response 0 Decibel 50 0 0.4 Frequency in Hz 0.6 1 Impulse Response 0.1 ha(t) 0 −0.1 0 10 20 30 40 5060 time in seconds 70 80 90 100 Figure 8.7: Impulse Invariance Design Method with T =2 in Problem P 8.7 8. M ATLAB function dl A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 123 M ATLAB verification using Problem P8.7: 124 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 The filter design plots are given in Figure 8.8. Digital Butterworth Filter Design Plots in P 8.8 Log−Magnitude Response 0 Decibel 50 0 0.40.6 Frequency in pi units 1 Magnitude Response 0.5 |H| 0.0 0 0.40.6 Frequency in pi units 1 Figure 8.8: Digital filter design plots in Problem P8.8. 9. Problem P 8.11 Digital Butterworth Lowpass Filter Design using Bilinear transformation. M ATLAB script: (a) Part(a): T = 2. M ATLAB script: A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 125 The filter design plots are shown in Figure 8.9. Comparison: If we compare filter orders from two methods then bilinear transformation gives the lower order than the impulse invariance method. This implies that the bilinear transformation design method is a better one in all aspects. If we compare the impulse responses then we observe from Figure 8.9 that the digital impulse response is not a sampled version of the analog impulse response as was the case in Figure 8.7. (b) Part (b): Use of the function. butt M ATLAB script: 126 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 Digital Butterworth Filter Design Plots in P 8.11a Log−Magnitude Response 0 Decibel 50 0 0.4 Frequency in Hz 0.6 1 Impulse Response 0.2 ha(t) 0 −0.2 0 10 20 30 40 5060 time in seconds 70 80 90 100 Figure 8.9: Bilinear Transformation Design Method with T =2 in Problem P 8.11a The filter design plots are shown in Figure 8.10. Comparison: If we compare the plots of filter responses in part 9a with those in part 9b, then we observe that the butt function satisfies r stopband e specifications exactly at  s . Otherwise the both designs are essentially similar. 10. Problem P 8.13 Digital Chebyshev-1 Lowpass Filter Design using Bilinear transformation. M ATLAB script: A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 127 Digital Butterworth Filter Design Plots in P 8.11b Log−Magnitude Response 0 Decibel 50 0 0.4 Frequency in Hz 0.6 1 Impulse Response 0.2 h(n) 0 −0.2 0 5 10 15 20 25 n 30 35 40 45 50 Figure 8.10: Butterworth filter design using the function butt in Problem P 8.11b = 1. (a) Part(a): T M ATLAB script: (b) Part(b): T = 1=8000. The filter design plots are shown in Figure 8.11. M ATLAB script: 128 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 Filter Design Plots in P 8.13a Log−Magnitude Response 0 Decibel 40 60 0 1500 2000 Frequency in Hz 4000 Impulse Response 0.3 0.2 0.1 0 −0.1 −0.2 h(n) 0 10 20 30 40 50 n 60 70 80 90 100 Figure 8.11: Bilinear Transformation Design Method with T =1 in Problem P 8.13a The filter design plots are shown in Figure 8.12. (c) Comparison: If we compare the designed system function as well as the plots of system responses in part 10a and in part 10a, then we observe that these are exactly same. If we compare the impulse invariance design in Problem 6 with this one then we note that the order of the impulse invariance designed filter is one higher. This implies that the bilinear transformation design method is a better one in all aspects. 11. Digital lowpass filter design using elliptic prototype. M ATLAB script using the function: bi A PRIL 98 S OLUTIONS M ANUAL FOR DSP USING M ATLAB 129 Filter Design Plots in P 8.13b Log−Magnitude Response 0 Decibel 40 60 0 1500 2000 Frequency in Hz 4000 Impulse Response 0.3 0.2 0.1 0 −0.1 −0.2 h(n) 0 10 20 30 40 50 n 60 70 80 90 100 Figure 8.12: Bilinear Transformation Design Method with T = 1=8000 in Problem P 8.13b 130 S OLUTIONS M ANUAL FOR DSP USING M ATLAB A PRIL 98 The filter design plots are shown in Figure 8.13. Digital Elliptic Filter Design Plots in P 8.14a Log−Magnitude Response 0 Decibel 60 80 0 0.4 Frequency in Hz 0.6 1 Impulse Response 0.2 ha(t) 0 −0.2 0 10 20 30 40 5060 time in seconds 70 80 90 100 Figure 8.13: Digital elliptic lowpass filter design using the bilinear function in Problem P8.14a. M ATLAB script using the function: elli