Convergence of the DTFT 1.2 2004/06/22 11:41:47 GMT-5 2004/07/16 09:09:14.281 GMT-5 Richard G. Baraniuk richb@rice.edu Robert Ahlfinger ahlfing@rice.edu Richard G. Baraniuk richb@rice.edu DTFT Elec 301 Converge An overview of the convergence of the DTFT. Use notation X ω here. The inverse DTFT x n ω X ω ω n 1 2 is a finite integral, and therefore well defined. The forward DTFT X ω n x n ω n is an infinite sum and therefore can exhibit strange behaviour. We would like to say that and are inverses of each other, that is, that n ω X ω ω n 1 2 ω n X ω Unfortunately, is not always true. To study , define the partial DTFT n M M x n ω n and look at n M M ω X ω ω n 1 2 ω n X m ω Let's study whether M X m ω X ω If n x n , then x M converges to x uniformly, that is M X ω X m ω 0 for all ω Although similar, uniform convergence is slightly stronger than pointwise convergence.

Uniform Convergence

as ε 0 , we can increase M to keep X m ω within ± ε of X m ω .

UNIFORM CONVERGENCE IS NICE!!! Think of DTFTs of these impulse responses of LTI systems: DELAY h n δ n n o MOVING AVERAGE h n 1 M 1 M 2 1 M 1 n M 2 0 Otherwise RECURSIVE (1st order): h n a n u n ANY STABLE SYSTEM? Unfortunately, there exists useful system/signals such that does not hold uniformly. Key Example: Ideal Lowpass Fiter

h n ω ω c ω c 1 ω n 1 2 ω c n n for all n h n ω c n n This is a sampled sinc function (well defined in terms of H )

Notes h n is noncasual n h n One practical way to get around the problems in notes 1 and 2 would be to use a truncated version of h n h M n ω c n n M n M 0 else

We would of course hope that h M ω DTFT h M n is a good approximation to the ideal response H ω . Furthermore that the approximation gets better as M . Unfortunately, this is not the case! H m ω n M M h n ω n does not converge uniformly to H ω

H M ω cannot match the discontinuities in H ω . As M , the oscillations converge to around ± ω c without their magnitude going to zero! In this case H M converges to H in mean-square (engery) sense M ω H M ω H ω 2 1 2 0 This is much weaker than uniform convergence.

ie: DTFT has convergence in norm that is not pointwise and not uniformly. Symmetry Properties of the Discrete Time Fourier Transform Sequence x n Fourier Transform X ω x n X ω x n X ω ("even") x n X e ω (conjugate-symmetric part of X ω ) x n X o ω (conjugate-antisymmetric part of X ω ) ("odd") x e n (conjugate-symmetric part of x n X R ω x o n (conjugate-antisymmetric part of x n X I ω Any real x n X ω X ω (Fourier transform is conjugate-symmetric) Any real x n X R ω X R ω (real part is even) Any real x n X I ω X I ω (imaginary part is odd) Any real x n X ω X ω (magnitude is even) Any real x n ∡ X ω ∡ X ω (phase is odd) x e n (even part of x n ) X R ω x o n (odd part of x n ) X I ω X ω X ω Discrete Time Fourier Transform Theorems Sequence x n y n Fourier Transform X ω Y ω a x n b y n a X ω b Y ω x n n d , ( n d an integer) ω n d X ω ω o n x n X ω ω o x n X ω X ω if x n real. n x n ω X ω x n y n X ω Y ω x n y n 1 2 θ X θ Y ω θ Parseval's Theorem n x n 2 1 2 ω X ω 2 n x n y n 1 2 ω X ω Y ω Modulation ω o n x n ↔ DTFT X ω ω o

Proof y n ω o n x n

Y ω k x n ω o n ω n k x n ω ω o n X ω ω o

Discrete Time Fourier Transform Pairs Sequence Fourier Transform δ n 1 δ n n o ω n .o 1 n k 2 δ ω 2 k a n u n a 1 1 1 a ω u n 1 1 ω k δ ω 2 k n 1 a n u n a 1 1 1 a ω 2 r n ω p n 1 ω p u n a 1 1 1 2 r ω p ω r 2 2 ω ω c n n X ω 1 ω ω c 0 ω c ω x n 1 0 n M 0 otherwise ω M 1 2 ω 2 ω M 2 ω o n k 2 δ ω ω o 2 k ω o n φ k φ δ ω ω o 2 k φ δ ω ω o 2 k " δ " is the felt column is different from " δ " in the right column. How/why?