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EE3TP4_23_DTFT_Details_v3_Lecture 30

# EE3TP4_23_DTFT_Details_v3_Lecture 30 - Discrete-Time...

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X = n =−∞ x [ n ] e j n x [ n ]= 1 π π X  e jn d Discrete-Time Fourier Transform (and Inverse DTFT) These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.

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Physical Relationship of DTFT Im pulse Gen CT LPF DAC x ( t ) x [ n ] = x ( nT ) “Hold” Sample at t = nT ADC x t x t f X ( f ) B –B A CTFT of x ( t ) Ω 2 π 4 π 2 π 4 π X  A/T π π DTFT of x [ n ] f F s 2F s –F s 2 F s X f A/T CTFT of x t F s /2 –F s /2
Motivating D-T System Analysis using DTFT D-T System DAC x ( t ) x [ n ] ADC y ( t ) y [ n ] f X ( f ) B –B CTFT of x ( t ) f X ( f ) B –B CTFT of y ( t ) X  π π DTFT of x [ n ] X  π π DTFT of y [ n ]

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Section 4.1 continued: The Details X = n =−∞ x [ n ] e j n In rad/sample radians X ω = −∞ x t e jωt dt In rad/sec radians Compare to CTFT : Define the DTFT : Very similar structure … so we should expect similar properties!
Example of Analytically Computing the DTFT n x [ n ] -3 -2 -1 1 2 3 4 5 6 x [ n ]= { 0, n 0 a n , 0 n q 0, n q q = 4 If a ∣ 1, x [ n ] decays If a ∣ 1, x [ n ] explodes If a 0, x [ n ] oscillates By definition: x t = n = 0 q a n e j n = n = 0 q ae j n Given this signal model, find the DTFT. X = 1 ae j q 1 1 ae j n = q 1 q 2 r n = r q 1 r q 2 1 1 r General Form for Geometric Sum: Periodic in frequency!

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Characteristics of DTFT 1.Periodicity of
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