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EE3TP4_25_DFT-DTFT-CTFT_Relations_v3_Lecture 32

# EE3TP4_25_DFT-DTFT-CTFT_Relations_v3_Lecture 32 -...

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X = n =−∞ x [ n ] e j n x [ n ]= 1 π π X  e jn d Discrete-Time Fourier Transform (DTFT) X [ k ]= n = 0 N 1 x [ n ] e j2π kn / N k = 0, 1, 2, ... , N 1 x [ n ]= 1 N k = 0 N 1 X [ k ] e j2π kn / N n = 0, 1, 2, ... , N 1 Discrete Fourier Transform (DFT) These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.

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We can use the DFT to implement numerical FT processing This enables us to numerically analyze a signal to find out what frequencies it contains!!! A CT signal “comes in” through a sensor & electronics (e.g., a microphone & amp) The ADC creates samples (taken at an appropriate F s ) ADC DFT Processing (via FFT) X [0] X [1] X [2] X [ N -1] Inside “Computer” memory array x t x [ n ] “H/W” or “S/W on processor” x [0] x [1] x [2] x [N-1] memory array FFT algorithm computes N DFT values DFT values in memory array (they can be plotted or used to do something “neat”) N samples are “dumped” into a memory array
If we are doing this DFT processing to see what the original CT signal x ( t ) “looks” like in the frequency domain… … we want the DFT values to be “representative” of the CTFT of x ( t ) Likewise … If we are doing this DFT processing to do some processing to extract some information from x ( t ) or to modify it in some way… … we want the DFT values to be “representative” of the CTFT of x ( t ) So … we need to understand what the DFT values tell us about the CTFT of x ( t )… We need to understand the relations between … CTFT, DTFT, and DFT

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We’ll mathematically explore the link between DTFT & DFT in two cases: …0 0 x [0] x [1] x [2] ... X [ N – 1] 0 0 N “non-zero” terms (of course, we could have some of the interior values = 0) 1. For x [ n ] of finite duration : For this case … we’ll assume that the signal is zero outside the range that we have captured. So … we have all of the meaningful signal data. 2. For x [ n ] of infinite duration … or at least of duration longer than what we can get into our “DFT Processor” inside our “computer”. So … we don’t have all the meaningful signal data. What effect does that have? How much data do we need for a given goal?
DFT and DTFT: Finite Duration Case If x [ n ] = 0 for n < 0 and n N then the DTFT is: X = n =−∞ x [ n ] e j n = n = 0 N 1 x [ n ] e j n we can leave out terms that are zero Now … if we take these N samples and compute the DFT (using the FFT, perhaps) we get: X [ k ]= n = 0 N 1 x [ n ] e j2π kn / N k = 0, 1, 2, ... , N 1 Comparing these we see that for the finite-duration signal case: X [ k ]= X k N X  1 k π 2 π π /2 - π /2 - π 0 2 3 4 5 6 7 X [ k ] DTFT & DFT : DFT points lie exactly on the finite-duration signal’s DTFT!

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Summary of DFT and DTFT for a finite duration x [ n ] x [ n ] DFT DTFT X  X [ k ]= X k N Points of DFT are “samples” of DTFT of x [ n ] Zero-Padding Trick” After we collect our N
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EE3TP4_25_DFT-DTFT-CTFT_Relations_v3_Lecture 32 -...

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