Summary13 - Summary 13 Hamid Jafarkhani Digital Signal...

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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 discrete-time system is defined by sampling the impulse response of a continuous-time 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 s-plane is mapped to the unit circle in z-plane Left-half plane in s-plane is mapped to the inside of unit circle in z-plan The mapping is not one-to-one 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 s-plane is mapped to the unit circle in z-plane Left-half plane in s-plane is mapped to the inside of unit circle in z-plan The mapping is one-to-one Summary 13 – p. 5/ ...
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Summary13 - Summary 13 Hamid Jafarkhani Digital Signal...

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