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Challenge_25

# Challenge_25 - Dr Fred J Taylor Professor Figure 1...

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EEL 3135: Dr. Fred J. Taylor, Professor Lesson Title: IIR Systems Lesson Number: 25 (Section 8-9 to 8-12) Challenge A 2 nd order IIR is defined by the difference equation: y[k]=(1/2)y[k-1]-(1/4)y[k-2]+x[k]-2x[k-1]+x[k-2] Classify the stability of the system. Response Transfer function is (direct derivation): H(z)=(1-2z -1 +z -2 ) / (1-0.5z -1 +0.25z -2 ) Analyze: » n=[1, -2, 1]; {N(z)} » d=[1, -0.5, 0.25]; {D(z)} » [h,f]=freqz(n,d); » plot(f,abs(h)); » zplane(n,d); roots(n) {zeros} ans = 1 1 » roots(d) {poles} ans = 0.2500 + 0.4330i 0.2500 - 0.4330i 1

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Unformatted text preview: Dr. Fred J. Taylor, Professor Figure 1 : Magnitude frequency response and pole-zero distribution. The zeros are at z= 1and 1 (explaining in part why the filter is highpass) and the poles are z= 0.25 +j 0.433 and z= 0.25 -j 0.433. The poles are all interior to the unit circle leading to the conclusion that the filter is stable. Direct II form of H(z)=(1-2z-1 +z-2 ) / (1-0.5z-1 +0.25z-2 ) {monic} 2 x[k] ⊗ ⊕ y[k] z-1 ⊕ ⊗ b =1 b 1 =-2 a 1 =0.5 ⊕ z-1 ⊗ ⊗ ⊗ ⊕ a 2 =-0.25 b 2 =1...
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Challenge_25 - Dr Fred J Taylor Professor Figure 1...

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