EECE 301 Discussion 09 - DFT Examples

EECE 301 Discussion 09 - DFT Examples - EECE 301 Signals...

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Unformatted text preview: EECE 301 Signals & Systems Prof. Mark Fowler Discussion #9 • Illustrating the Errors in DFT Processing • DFT for Sonar Processing Example #1 Illustrating The Errors in DFT Processing Illustrating the Errors in DFT processing This example does a nice job of showing the relationships between: • the CTFT, • the DTFT of the infinite-duration signal, • the DTFT of the finite-duration collected samples, • and the DFT computed from those samples. However, it lacks any real illustration of why we do DFT processing in practice. There are many practical applications of the DFT and we’ll look at one in the next example. ADC x [0] x [1] x [2] x [ N-1] ⇒ DFT processing ⇓ X [0] X [1] X [2] ⇓ ⇒ memory array Inside “Computer” sensor Recall the processing setup: ) ( t x ] [ n x (Note: no anti-aliasing filter shown… but we should have one!) ) ( ω X CTFT (theory) ) ( Ω ∞ X DTFT ∞ (theory) Zero- padding to length N zp We study the theory of these Practical computed DFT …to understand what this shows us DTFT N (theory) ) ( Ω N X memory array X [ N zp-3] X [ N zp-2] X [ N zp-1] Let’s imagine we have the following CT Signal: ) ( ) ( > = − b for t u e t x bt ) ( t x … 1 If we sample x ( t ) at the rate of F s samples/second – That is, sample every T = 1/ F s sec – we get the DT Signal coming out of the ADC is: ) ( | ) ( ] [ nT x t x n x nT t = = = For this example we get: [ ] ( ) ] [ ] [ ] [ ) ( ] [ n u a n u e n u e t u e n x n n bT bTn nT t bt Δ − − = − = = = = Note: | a | < 1 From our FT Table we find the FT of x ( t ) is: b f j f X b j X + = ⇒ + = π ω ω 2 1 ) ( 1 ) ( CTFT Result…(Theory) A t Now… analyze what we will get from the DFT processing for this signal… ow imagine that in theory...
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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 Discussion 09 - DFT Examples - EECE 301 Signals...

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