EECE 301 Note Set 9 DT Convolution

EECE 301 Note Set 9 DT Convolution - EECE 301 Signals &...

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1/23 EECE 301 Signals & Systems Prof. Mark Fowler Note Set #9 • Computing D-T Convolution • Reading Assignment: Section 2.2 of Kamen and Heck
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2/23 Ch. 1 Intro C-T Signal Model Functions on Real Line D-T Signal Model Functions on Integers System Properties LTI Causal Etc Ch. 2 Diff Eqs C-T System Model Differential Equations D-T Signal Model Difference Equations Zero-State Response Zero-Input Response Characteristic Eq. Ch. 2 Convolution C-T System Model Convolution Integral D-T System Model Convolution Sum Ch. 3: CT Fourier Signal Models Fourier Series Periodic Signals Fourier Transform (CTFT) Non-Periodic Signals New System Model New Signal Models Ch. 5: CT Fourier System Models Frequency Response Based on Fourier Transform New System Model Ch. 4: DT Fourier Signal Models DTFT (for “Hand” Analysis) DFT & FFT (for Computer Analysis) New Signal Model Powerful Analysis Tool Ch. 6 & 8: Laplace Models for CT Signals & Systems Transfer Function New System Model Ch. 7: Z Trans. Models for DT Signals & Systems Transfer Function New System Model Ch. 5: DT Fourier System Models Freq. Response for DT Based on DTFT New System Model Course Flow Diagram The arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).
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3/23 2.2 “Computing” D-T convolution -We know about the impulse response h [ n ] -We found out that h [ n ] interacts with x [ n ] through convolution to give the zero-state response: −∞ = = i i n h i x n y ] [ ] [ ] [ How do we “work” this? This is needed for understanding how: (1) To analyze systems (2) To implement systems Don’t forget…The design process includes analysis Two cases, depending on form of x [ n ]: 1. x [ n ] is known analytically 2. x [ n ] is known numerically or graphically Analytical Convolution (used for “by-hand” analysis): Pretty straightforward conceptually: - put functions into convolution summation - exploit math properties to evaluate/simplify
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4/23 Example: ] [ ] [ n u a n x n = ] [ ] [ n u b n h n = Recall this form from 1 st -order difference equation example ] [ n u b n ] [ n u a n ? ] [ = n y −∞ = = i i n h i x n y ] [ ] [ ] [ −∞ = = i i n i i n u b i u a ] [ ] [ ) ( a function of n i gets “summed away” < = 0 , 0 0 , 1 ] [ i i i u = = 0 ) ( ] [ i i n i i n u b a Now use: > = n i n i i n u , 0 , 1 ] [ = = = = n i i n n i i n i b a b b a 0 0 ) ( Now use: You should be able to go here directly
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5/23
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This note was uploaded on 02/29/2012 for the course EECE 301 taught by Professor Fowler during the Fall '08 term at Binghamton University.

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EECE 301 Note Set 9 DT Convolution - EECE 301 Signals &amp;...

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