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

Partial Fraction 部分分式

Partial fraction decomposition may be required for dealing with z-transform.

partial fraction decomposition 部分分式分解
partial fraction expansion 部分分式展開

Solution:

Let X/( 1 )( 2 ) = A/( 1 ) + B/( 2 )

Multiply ( 1 ) and ( 2 ) at both sides and get

X = A ( 2 ) + B ( 1 )

Let ( 1 ) = 0 to get A:
A = X / ( 2 ) | _{( 1 ) = 0}

Let ( 2 ) = 0 to get B:
B = X / ( 1 ) | _{( 2 ) = 0}

Base and Exponent 底數與指數

For xⁿ

中文讀法：x的n次方
英文讀法：x to the power of n

where

x:
base 底數

n:
index/exponent 指數
power 次方

Linear, Time-Invariant, Causality, Stability 線性、非時變、因果性、穩定性

linear system 線性系統
time-invariant system 非時變系統
causality 因果性
stability 穩定性

For stability,
bounded-input bounded-output (BIBO) 有界輸入有界輸出

unit circle 單位圓
region of convergence 收斂域

When the ROC of z-transform X(z) includes the unit circle, x[n] is:

stable 穩定
absolutely summable 絕對可加

Eigendecomposition, Eigenvalue, Eigenvector, Eigenfunction 特徵分解、特徵值、特徵向量、特徵函數/固有值、固有向量、固有函數

Eigendecomposition 特徵分解
Spectral decomposition 譜分解

scalar 純量
vector 向量

For n×n matrix A,

if Ax = λx

where
λ is a scalar.

x is a non-zero vector with dimension N.

Then

Av = λv
Eigenvalue 特徵值/固有值 λ
Eigenvector 特徵向量/固有向量 v

Af = λf
Eigenfunction 特徵函數/固有函數 f

For an LTI system, an input signal x[n] which is an eigenfunction of the system such as x[n] = e^jωn appears at the output of the system with the eigenvalue H(e^jω).