5 The Discrete–Time Fourier Transform

Example:
1. Signal
Lampe, Schober: Signals and Communications
x[n] = a n u[n], |a| < 1
– Fourier transform
X(e jω ∞
) = x[n]e −jωn =
=
n=−∞
1
1 − ae −jω
∞
n=−∞
– Discrete-time Fourier transform pair
a n u[n] F
←→
– Magnitude of spectrum
|X(e jω
)| =
1
1
− ae−jω
=
a n u[n]e −jωn =
1
1 − ae −jω
1
√ 1 + a 2 − 2acosω
– a ≈ 1: x[n] ≈ u[n]
⇒ very slowly varying
⇒ spectrum concentrated around 2πk, k ∈ Z
|X(e jω )|
1/(1 − a)
1/(1 + a)
∞
n=0
−2π −π
π 2π ω
136
(ae −jω ) n