EECE 301 Note Set 25 DFT - DTFT - CTFT Relations

# EECE 301 Note Set 25 DFT - DTFT - CTFT Relations - EECE 301...

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1/22 EECE 301 Signals & Systems Prof. Mark Fowler Note Set #25 • D-T Signals: Relation between DFT, DTFT, & CTFT • Reading Assignment: Sections 4.2.4 & 4.3 of Kamen and Heck

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2/22 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).
3/22 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 ) ( t x ] [ n x “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

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4/22 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 “neat” 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
5/22 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.

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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.

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EECE 301 Note Set 25 DFT - DTFT - CTFT Relations - EECE 301...

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