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Unformatted text preview: Summary 13
Hamid Jafarkhani Digital Signal Processing Summary 13 – p. 1/ Ideal Filters and Causality
Ideal ﬁlters are not causal. Causality implies H (ω ) cannot be zero except at a ﬁnite set of points in frequency. H (ω ) cannot be constant over a range of frequencies. A transition in H (ω ) cannot be inﬁnitely sharp. Summary 13 – p. 2/ Filter Design by Impulse Invariance
Impulse Invariance: A discretetime system is deﬁned by sampling the impulse response of a continuoustime system h[n] = ha (nT ) 1 H (ω ) = T
∞ Ha
k =−∞ ω 2πk j + T T , ω = ΩT Impulse Invariance mapping is z = esT Summary 13 – p. 3/ Impulse Invariance Properties
j Ω axis in splane is mapped to the unit circle in zplane Lefthalf plane in splane is mapped to the inside of unit circle in zplan The mapping is not onetoone Summary 13 – p. 4/ Bilinear Mapping
2 s= T 1 − z −1 1 + z −1 2 = T z−1 z+1
−1 ω 2 Ω = tan , T 2 ω = 2 tan ΩT 2 j Ω axis in splane is mapped to the unit circle in zplane Lefthalf plane in splane is mapped to the inside of unit circle in zplan The mapping is onetoone Summary 13 – p. 5/ ...
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 Fall '10
 Prof.HamidJafarkhani
 Digital Signal Processing, Frequency, Signal Processing, Impulse Invariance

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