E.1. Convolution - JulTob/Analysis GitHub Wiki

Convolution Sum

x[n] = βˆ‘ βˆ€k x[k] Β· ẟ[n-k]

x[n] βˆ— h[n] = βˆ‘ βˆ€k x[k] Β· h[n-k] = y[n]

Convolution Integral

x(t) = ∫ βˆ€πœ x(𝜏) Β· ẟ(t-𝜏) d𝜏

x(t) βˆ— h(t) = ∫ βˆ€πœ x(𝜏) Β· h(t-𝜏) d𝜏 = y(t)

Properties

Conmutative

$$x βˆ— h = h βˆ— x aⁿ·x βˆ— h = x βˆ— aⁿh = aⁿ (x βˆ— h)$$
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