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Chap 23

# Chap 23 - FrequencyResponse ofDiscreteSystems...

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Frequency Response of Discrete Systems Lindner: Chapter 23

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Frequency Response Theorem (Z‐Transform Version) Y ( z ) X ( z ) = H ( z ) Transfer Func,on BIBO stable Input Signal x ( n ) = cos Ω i n ( ) u s ( n ) y ( n ) = y tr ( n ) + y ss ( n ) Output Signal Steady State Response y ss ( n ) = H ( Ω i ) cos Ω i n + H ( Ω i ) ( ) , n 0 H ( z ) z = e j Ω i = H ( Ω i ) lim n →∞ y tr ( n ) = 0 Transient Response
Frequency Response Theorem (Fourier Version) Y ( Ω ) X ( Ω ) = H ( Ω ) Transfer Func,on Input Signal x ( n ) = cos Ω i n ( ) Output Signal = Steady State Response y ( n ) = H ( Ω i ) cos Ω i n + H ( Ω i ) ( ) H ( Ω ) Ω = Ω i = H ( Ω i )

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Frequency Response Theorem AssumpHons: 1. System is Linear 2. System is Time Invariant 3. System is BIBO stable 4. Input signal is a single sinusoid A sinusoid input signal results in a sinusoidal output signal at the same frequency The amplitude and phase of the output signal relaHve to the input signal can be computed from the transfer funcHon
Frequency Response Theorem Example 23.1.8 Y ( z ) X ( z ) = H ( z ) = 1 2 r ζ z 1 + z 2 1 2 r ζ z 1 + r 2 z 2 z = e j Ω

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