5 The Discrete–Time Fourier Transform

Example:
1. Discrete-time LTI system with impulse response
and input signal
Output y[n]?
Lampe, Schober: Signals and Communications
h[n] = α n u[n] , |α| < 1
x[n] = β n u[n] , |β| < 1
– Discrete-time Fourier transforms of x[n] and h[n]
X(e jω ) =
1
1 − βe −jω
and
H(e jω 1
) =
1 − αe−jω – Discrete-time Fourier transform of y[n]
Y (e jω ) = H(e jω )X(e jω ) =
– Inverse discrete-time Fourier transform
(a) α = β
Write
Y (e jω ) =
A
+
1 − αe−jω and consider generalized function
Y (v) =
1
(1 − αv)(1 − βv)
1
(1 − αe −jω )(1 − βe −jω )
B
1 − βe −jω
= A
1 − αv
+ B
1 − βv
166