Sampling 取樣

Discrete-time signal x[n] is obtained by sampling a continuous-time signal x_c(t):
x[n] = x_c(nT) ,   -∞ < n < ∞

Fourier Transform:

x_c(t) <--F--> X(jΩ)
x[n] <--F--> X(e^jω)

ω = ΩT

Time Shift:
x(t-t₀) <--F--> e^-jΩt₀ X(jΩ)
x[n-n₀] <--F--> e^-jωn₀ X(e^jω)

Frequency Shift:
e^jΩ₀t x(t) <--F--> X(j(Ω-Ω₀))
e^jω₀n x[n] <--F--> X(e^j(ω-ω₀))

sampling period 取樣周期 T

sampling frequency 取樣頻率 f_s or Ω_s
With respect to different units:
Samples/second: f_s = 1/T
Radians/second: Ω_s = 2π/T = 2πf_s

Nyquist Theorem
Nyquist Frequency Ω_N
Nyquist Rate 2Ω_N

aliasing 頻疊、摺疊效應

To avoid aliasing, Ω_s >= 2Ω_N

Foldover 反摺、混疊
- when the sampling rate is too low

Reference

Discrete-Time Signal Processing (2nd edition), A. Oppenheim & R. Schafer:
Fourier transform and inverse Fourier transform - p28
Time shift & frequency shift - Table 2.2, p59
Proof for time shift in z-transform - 3.4.2, p120