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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
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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
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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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