ch 11 summary - Summary Discrete-Time Fourier Transform...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Summary – Discrete-Time Fourier Transform (DTFT) (Chapter 11) 1. The DTFT of a DT signal x [ n ] is defined as X F ( ) = x n [ ] n = −∞ e j 2 π F n ( F is a continuous variable, called DT frequency .) 2. Key fact : Since e j 2 F n is a periodic function of F with period 1 for all n , it follows that X ( F ) must be periodic with period 1. We usually take the fundamental period to be 1 2 < F 1 2 . 3. The inverse DTFT is given by x n [ ] = X F ( ) e j 2 F n 1 2 dF . ( Examples: (i) If x [ n ] = a n u n [ ] where a is a real constant, a < 1 , then X F ( ) = a n e j 2 F n n = 0 = ae j 2 F { } n = 0 n . But recall that for any number (real or complex) b with b < 1 , b n = 1 1 b n = 0 . So, X F ( ) = 1 1 j 2 F . (ii) Let F 0 < 1 2 and define H F ( ) = 1, F F 0 ; H F ( ) = 0, F 0 < F 1 2 . (Note that this defines H F ( ) for all F , since H F ( ) must be periodic with period 1. Also note that H F ( ) is the frequency response of a DT low-pass filter (LPF) having cutoff frequency F 0 .) Then h n [ ] = H F ( ) e j 2 F n dF = 1 2 1 2 e j 2 F n dF F 0 F 0 = 2 F 0 sinc 2 F 0 n ( ) .) 4. We’ve now seen three different frequency domain representations:
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/27/2011 for the course ECE 313 taught by Professor Muschinski during the Fall '08 term at UMass (Amherst).

Page1 / 3

ch 11 summary - Summary Discrete-Time Fourier Transform...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online