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Transform theory Convolution theorem for the Discrete Fourier Transform (periodic convolution in discrete frequency) Let Then Proof: { 1 ˜z n = DFT −1 ˜X k ⊗ N Ỹk ˜z n = ˜x nỹ n ˜Z k = 1 ˜X k ⊗ N Ỹk = 1 N−1 ∑ ˜X k−q Ỹ q N q=0 } = 1 N ⎧ ⎫ N−1 ∑ ⎨ N−1 1 ∑ ⎬ ˜X ⎩ k−q Ỹ q N ⎭ ei 2π N kn k=0 q=0 = {(p, q) = (k − q, q)} = 1 N−1 ∑ N−1−q ∑ N 2 = 1 N q=0 N−1 ∑ p=0 p=−q ˜X pe i 2π N pn 1 N ˜X p Ỹ qe i 2π N (p+q)n N−1 ∑ q=0 Ỹ qe i 2π N qn = ˜x nỹ n Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 22(28)
Time-shift in the Fourier transform Transform theory Proof: F{x(t − t 0 )} = e −iωt 0 X(ω) ∫ ∞ F{x(t − t 0 )} = x(t − t 0 )e −iωt dt = {u = t − t 0 , du = dt} ∫ ∞ −∞ ∫ ∞ = x(u)e −iω(u+t0) du = e −iωt 0 x(u)e −iωu du = e −iωt 0 X(ω) −∞ −∞ Frequency-shift in the Fourier transform F{x(t)e iω0t } = X(ω − ω 0 ) Proof: ∫ ∞ ∫ ∞ F{x(t)e iω0t } = x(t)e iω0t e −iωt dt = x(t)e −i(ω−ω0)t dt = X(ω − ω 0 ) −∞ −∞ Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 23(28)
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Page 3 and 4: Transform theory ⎧ ⎪⎨ ⎪⎩
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Signal Processing

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