1.8.4 A causal LTI system is implemented with the di↵erence equationy(n) = 0.5x(n) + 0.2x(n-1) + 0.5y(n-1)-0.1y(n-2).(a) Find the frequency response of this system.(b) Compute and plot the frequency response magnitude|Hf(!)|using the MATLAB commandfreqz.(c) Find the output produced by the inputx(n) = cos(0.2⇡n). Compare your answer with the output signalfound numerically with the MATLAB commandfilter.(d) Use the Matlab commandzplaneto make the pole-zero diagram. 1.8.4) Solution
- 1 - 0.5 0 0.5 1 - 1 - 0.5 0 0.5 1 Real Part Imaginary Part Pole - Zero Diagram 125
1.8.5 Two discrete-time LTI systems are used in series.x(n)HGy(n)The frequency responses are shown.!-⇡-23⇡-13⇡013⇡23⇡⇡Hf(!)2!-⇡-23⇡-13⇡013⇡23⇡⇡Gf(!)11(a) Accurately sketch the frequency response of the total system.(b) Find the output signaly(n) produced by the input signalx(n) = 5 + 3 cos⇣⇡2n⌘+ 2 cos✓2⇡3n◆+ 4 (-1)n.1.8.5) Solution 126
1.8.8 The mangitude and phase of the frequency response of a discrete-time LTI system are:|Hf(!)|=⇢2for|!|<0.5⇡1for 0.5⇡<|!|<⇡.\Hf(!) =⇢0.3⇡for-⇡<!<0-0.3⇡for⇡<!<0.(a) Sketch the frequency response magnitude|Hf(!)|for|!|⇡.(b) Sketch the frequency response phase\Hf(!) for|!|⇡.(c) Find the output signaly(n) produced by the input signalx(n) = 2 sin(0.2⇡n) + 3 cos(0.6⇡n+ 0.2⇡).1.8.8)
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