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Signal Processing

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Transform theory<br />

Convolution theorem for the Discrete-Time Fourier transform (periodic<br />

convolution in continuous frequency)<br />

Let<br />

Then<br />

Proof:<br />

z n = x ny n<br />

˜Z(ν) = ˜X(ν) ⊗ Ỹ (ν) = ∫ 1/2<br />

−1/2<br />

˜X(ν − α)Ỹ (α) dα<br />

{ } ∫ { 1/2 ∫ } 1/2<br />

z n = F −1 ˜X(ν) ⊗ Ỹ (ν) =<br />

˜X(ν − α)Ỹ (α) dα e i2πνn dν<br />

−1/2 −1/2<br />

= {(β, α) = (ν − α, α), dβ dα = dν dα}<br />

∫ 1/2 ∫ 1/2−α<br />

=<br />

˜X(β)Ỹ (α)ei2π(β+α)n dβ dα<br />

α=−1/2 β=−1/2−α<br />

∫ 1/2<br />

∫ 1/2<br />

= ˜X(β)e i2πβn dβ Ỹ (α)e i2παn dα = x ny n<br />

−1/2<br />

−1/2<br />

Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 20(28)

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