5 The Discrete–Time Fourier Transform

– Conjugation + time reversal properties
x[n] = Re{Ev{x[n]}} + Re{Od{x[n]}} + jIm{Ev{x[n]}} + jIm{Od{x[n]}}
X(e jω )=Re{Ev{X(e jω )}} + Re{Od{X(e jω )}} + jIm{Ev{X(e jω )}} + jIm{Od{X(e jω )}}
Example:
Consider
– Observe
where
– With
and time reversal property
– Finally
x[n] = a |n| , |a| < 1
x[n] = z[n] + z[−n] − δ[n]
X(e jω ) =
z[n] = a n u[n]
z[n] F
←→
z[−n] F
←→
=
1
1
1 − ae −jω
+
1 − ae−jω 1
1 − ae jω
1 − a 2
1 − 2acos(ω) + a 2
– x[n] real and even ⇒ X(e jω ) real and even
Lampe, Schober: Signals and Communications
1
− 1
1 − aejω 150