3 Fourier Series Representation of Periodic Signals

Proofs:
1. Continuous–time LTI system with input x(t) = ejωt ∞
∞
y(t) = h(τ)x(t − τ) dτ = h(τ)e jω(t−τ) dτ
−∞
= e jωt
∞
−∞
= H(jω)e jωt
h(τ)e −jωτ dτ
−∞
with the definition of the amplitude factor
∞
H(jω) = h(τ)e −jωτ dτ
−∞
2. Discrete–time LTI system with input x[n] = ejωn y[n] =
∞
h[k]x[n − k] =
∞
k=−∞
= e jωn
∞
k=−∞
= H(e jω )e jωn
h[k]e −jωk
with the definition of the amplitude factor
H(e jω ∞
) =
Lampe, Schober: Signals and Communications
k=−∞
k=−∞
h[k]e −jωk
h[k]e jω(n−k)
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