Discrete-Time Fourier Transform Properties Quote of the Day The profound study of nature is the most fertile source of mathematical discoveries. Joseph Fourier Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, © Prentice Hall Inc.

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 2 Absolute and Square Summability Absolute summability is sufficient condition for DTFT Some sequences may not be absolute summable but only square summable To represent square summable sequences with DTFT –We can relax the uniform convergence condition –Convergence is in mean-squared sense –Error does not converge to zero for every value of  –The mean-squared value of the error over all  does

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 3 Example: Ideal Lowpass Filter The periodic DTFT of the ideal lowpass filter is The inverse can be written as Not causal Not absolute summable but it has a DTFT? The DTFT converges in the mean-squared sense Role of Gibbs phenomenon

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 4 Example: Generalized DTFT DTFT of Not absolute summable Not even square summable But we define its DTFT as a pulse train Let’s place into inverse DTFT equation

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 5 Symmetric Sequence and Functions Conjugate-symmetric Conjugate- antisymmetric Sequence Function

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 6 Symmetry Properties of DTFT Sequence x[n]Discrete-Time Fourier Transform X(e j ) x * [n]X * (e -j ) x * [-n]X * (e j ) Re{x[n]}X e (e j ) (conjugate-symmetric part) jIm{x[n]}X o (e j ) (conjugate-antisymmetric part) x e [n]X R (e j )= Re{X(e j )} x o [n]jX I (e j )= jIm{X(e j )} Any real x[n]X(e j )=X * (e -j ) (conjugate symmetric) Any real x[n]X R (e j )=X R (e -j ) (real part is even) Any real x[n]X I (e j )=-X I (e -j ) (imaginary part is odd) Any real x[n]|X(e j )|=|X(e -j )| (magnitude is even) Any real x[n] X(e j )=-X(e -j ) (phase is odd) x e [n]X R (e j ) x o [n]jX I (e j )

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 7 Example: Symmetry Properties DTFT of the real sequence x[n]=a n u[n] Some properties are

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 8 Fourier Transform Theorems SequenceDTFT x[n] y[n] X(e j ) Y(e j ) ax[n]+by[n]aX(e j )+bY(e j ) x[n-n d ] x[-n]X(e -j ) nx[n] x[n]y[n] X(e j )Y(e j ) x[n]y[n]

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 9 Fourier Transform Pairs SequenceDTFT [n-n o ] 1 a n u[n] |a|<1 u[n] cos( o n+)