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)