1.8.4 A causal LTI system is implemented with the di↵erence equationy(n) = 0.5x(n) + 0.2x(n1) + 0.5y(n1)0.1y(n2).(a) Find the frequency response of this system.(b) Compute and plot the frequency response magnitudeHf(!)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 polezero 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 discretetime 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 discretetime LTI system are:Hf(!)=⇢2for!<0.5⇡1for 0.5⇡<!<⇡.\Hf(!) =⇢0.3⇡for⇡<!<00.3⇡for⇡<!<0.(a) Sketch the frequency response magnitudeHf(!)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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 Fall '14
 Frequency, Signal Processing, LTI system theory, 2J, 2 J, 124 Pole