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).

Magnitude of Exponential Function e^jθ 如何求e^jθ的大小/絕對值

How to find the magnitude of exponential function e^iθ? Or |e^iθ| = ?

According to Euler's formula,

e^iθ = cosθ + isinθ

To find the magnitude of |e^iθ|,

|e^iθ| = SQRT(e^iθ) = SQRT(cos²θ + isin²θ) = SQRT(1) = 1

Therefore,

the magnitude of e^iθ is 1.

In other words, the magnitude of complex exponential is 1.

Related Information:

Euler's formula 歐拉公式
Why is the magnitude of e^(jwt) equal to 1?
Absolute value of complex exponential

Arithmetic/Geometric Sequence, Series, and Progression 等差/等比的數列/級數

等差數列/算數數列 arithmetic sequence/arithmetic progression 1,3,5,7...
等差級數/算數級數 arithmetic series 1+3+5+7+...
等差級數和：

n(a₁ + a_n)/2

1+3+5+7+9 = 5×(1+9)/2 = 25

等比數列/幾合數列 geometric sequence/geometric progression 1,2,4,8...
等比級數/幾合級數 geometric series 1+2+4+8+...

等比級數和：

a(1-rⁿ)/(1-r) = a(rⁿ-1)/(r-1)

1+2+4+8+16 = 1×(1-2⁵)/(1-2) = 1×(2⁵-1)/(2-1) = 1×(32-1)/(2-1) = 31

公差 common difference
公比 common ratio

參考資料

https://en.wikipedia.org/wiki/Arithmetic_progression
https://en.wikipedia.org/wiki/Geometric_progression
國家教育研究院雙語詞彙、學術名詞暨辭書資訊網

Matlab: My First Time with Simulink 第一次接觸Simulink

一直不太了解Simulink是什麼
今天第一次玩玩
過程如下：

Below are instructions of a simple Simulink example for beginners. (Just like a "Hello World" in learning a new programming language.)

1. Enter the following command in Matlab:

>> simulink

After a while, the Simulink Library Browser is opened automatically.

2. Select File -> New ->Model and an "untitled" window is opened.

3. Search for "Sine Wave" and "Scope" in Simulink Library Browser and drag them into the "untitled" window. (Don't select Sine or Sine Wave Function.)

4. Connect "Sine Wave" to "Scope".

5. Double click "Scope" to open the waveform.

6. Double click "Sine Wave" to adjust the amplitude and frequency.

7. Press the ▻ button to start simulation.

Result:

Related Information:

Matlab: Draw a Sine curve with sind()