003_BME343_Discrete_Time_Systems

003_BME343_Discrete_Time_Systems - Time-Domain Analysis of...

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Time-Domain Analysis of Discrete-Time Systems • Vision • Digitized signals • Numerical (computer) TD analysis of DT systems 3-1 (p ) modeling
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Objectives • Determine the characteristic equation, characteristic roots, and characteristic modes of a linear constant coefficient ifference equation difference equation • Calculate the impulse response for a linear constant coefficient difference equation • Calculate the zero-input and zero-state response of a linear constant coefficient difference equation Calculate and plot the convolution between two functions in discrete time se the fact that exponentials are eigenfuctions of linear Use the fact that exponentials are eigenfuctions of linear time invariant systems to simplify convolution • Be able to intuitively explain system behavior in terms of the TD analysis of DT systems 3-2 characteristic modes and time constants
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Discrete-Time Signals • Discrete-time signals are only defined at certain time (or space) points • Discrete time (DT) signals often arise from sampling of continuous-time (CT) signals • DT signals are defined at fixed intervals •S pacing between intervals = T pg •x ( n T ) = x [ n ] x(t) x [ n ] Continuous-time signal Discrete-time signal 3 4 3 4 0 1 2 0 1 2 TD analysis of DT systems 3-3 0 1 2 3 4 5 6 7 -2 -1 t 0 1 2 3 4 5 6 7 -2 -1 n
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Signal Energy and Power • Generally, need a measure of signal size – include both time and amplitude •S i gnal energy 2 () Ex t d t Discrete-time signal Continuous-time signal 2 [] n x  /2 1 T x n  N Signal power 2 lim x T T P xt dt T  2 1 lim 21 x N N x n N P • Discussion TD analysis of DT systems 3-4
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Signal Operations • Shifting T CT T () ( ) tx t T  – DT [] [ 5 ] s nx n x • Time reversal –CT – DT ( ) t TD analysis of DT systems 3-5 [ ] r bn xx
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Signal Operations • Decimation and Interpolation [] [ ] xn xM n d [ / ] 0, , 2 , 0 otherwise e xn L n L L xn  1 zeros 1 zeros 1 zeros [0],0,0,. ..,0,0, [1],0,0,. ..,0,0, [2],0,0,. ..,0,0,. .. LL L xxx        TD analysis of DT systems 3-6
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Discrete-Time Functions • Discrete-time impulse function [ n ] 1 0 [] 0 0 n n n • Discrete-time unit step nction function u [ n ] for 0 n un TD analysis of DT systems 3-7 for 0 n
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Discrete-Time Functions: Quiz 0.8 1 0.8 1 2 0.4 0.6 2 0.4 0.6 -5 0 5 0 0.2 n -5 0 5 0 0.2 n 0.8 1 0.8 1 0.2 0.4 0.6 0.2 0.4 0.6 TD analysis of DT systems 3-8 -10 -5 0 5 10 0 n -10 -5 0 5 10 0 n
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Discrete-Time Exponential n • A continuous-time exponential can be expressed in alternate form λλ (o r λ =ln ) tt ee   • Similarly, … the discrete-time exponential … λ n λ r λ =l n ) n • Nature of discrete-time exponential function jb jb a a e e beca 1 use aj b a j b e e 1 n n    TD analysis of DT systems 3-9
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Discrete-Time Exponential n Continuous-time signal Discrete-time signal aginary axis i Right half-plane (RHP) Left half-plane (LHP) nal Imaginary axis Exponentially
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This note was uploaded on 09/28/2009 for the course BME 343 taught by Professor Emelianov during the Fall '09 term at University of Texas at Austin.

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003_BME343_Discrete_Time_Systems - Time-Domain Analysis of...

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