Signal Processing
Signal Processing Signal Processing
Chapter 1: Transform theory ⎧ ⎪⎨ ⎪⎩ The Fourier transform: Continuous–time aperiodic signals x(t) ∫ ∞ X(f) = x(t)e −i2πft dt −∞ ∫ ∞ x(t) = X(f)e i2πft df −∞ ⎧ ⎪⎨ ⎪⎩ The Fourier series: Continuous–time T -periodic signals ˜x(t) Discrete spectrum f k = k/T X k = 1 ∫ ˜x(t)e −i 2π T kt dt T T ∞∑ ˜x(t) = X k e i 2π T kt k=−∞ ⎧ The Discrete-Time Fourier transform: ⎧ The Discrete Fourier Transform (DFT): Discrete–time aperiodic signals x n Discrete–time N-periodic signals ˜x n ⎪⎨ 1-periodic spectrum: ˜X(ν) = ∞∑ n=−∞ x ne −i2πνn ˜X(ν + 1) = ˜X(ν) ⎪⎨ N-periodic spectrum: ˜Xk+N = ˜X k N−1 ∑ ˜X k = ˜x ne −i 2π N kn , k = 0, . . . , N − 1 n=0 ⎪⎩ ∫ 1/2 x n = ˜X(ν)e i2πνn dν −1/2 ⎪⎩ ˜x n = 1 N N−1 ∑ k=0 ˜X k e i 2π N kn , n = 0, . . . , N − 1 Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 2(28)
Transform theory ⎧ ⎪⎨ ⎪⎩ The Fourier transform: Alternative definition ω = 2πf [ rad/s] Continuous–time aperiodic signals ∫ ∞ X(ω) = x(t)e −iωt dt −∞ x(t) = 1 ∫ ∞ X(ω)e i2πωt dω 2π −∞ ⎧ ⎪⎨ ⎪⎩ The Discrete-Time Fourier transform: Alternative definition Ω = 2πν Discrete–time aperiodic signals 2π-periodic spectrum: X(Ω + 2π) = X(Ω) X(Ω) = ∞∑ x(n)e −iΩn n=−∞ x(n) = 1 ∫ π X(Ω)e iΩn dΩ 2π −π Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 3(28)
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Transform theory<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
The Fourier transform: Alternative definition ω = 2πf [ rad/s]<br />
Continuous–time aperiodic signals<br />
∫ ∞<br />
X(ω) = x(t)e −iωt dt<br />
−∞<br />
x(t) = 1 ∫ ∞<br />
X(ω)e i2πωt dω<br />
2π −∞<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
The Discrete-Time Fourier transform: Alternative definition Ω = 2πν<br />
Discrete–time aperiodic signals<br />
2π-periodic spectrum: X(Ω + 2π) = X(Ω)<br />
X(Ω) =<br />
∞∑<br />
x(n)e −iΩn<br />
n=−∞<br />
x(n) = 1 ∫ π<br />
X(Ω)e iΩn dΩ<br />
2π −π<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 3(28)
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