Negative and Reciprocal Inequalities 負數及倒數的不等式運算規則

要理解負數及倒數的不等式運算規則
其實很簡單：

Negative Number 負數：

if a < b, then -a > -b

e.g. 3 < 5 ==> -3 > -5

Reciprocal/Multiplicative Inverse 倒數：

if a < b, then 1/a > 1/b

e.g. 3 < 5 ==> 1/3 > 1/5

Reference 參考資料：

Properties of Inequalities

Voltage Divider/Potential Divider 分壓器

voltage divider/potential divider 分壓器

How to get the voltage output of a voltage divider?

Answer:

V_in ---- Z₁ ----- V_out ----- Z₂ ---- GND

i = V_in  / (Z₁ + Z₂) = V_out / Z₂

=> V_out = [Z₂ / (Z₁ + Z₂)]V_in

where Z can be resistor/capacitor/inductor.

Reference:

Voltage Divider (Wikipedia)

Inverting Amplifier 反相放大器

inverting amplifier 反相放大器

V_in ---- R_in ----- V_mid ----- R_f ---- V_out

V_mid ----- -ve of op-amp

GND ----- +ve of op-amp


What's V_out?

Answer:

V_mid  = 0 for an ideal op-amp.

i = V_in / R_in = - V_out / R_f

=> V_out = - (R_f / R_in)V_in


What's the voltage gain?

Answer:

Gain = V_out / V_in = - (R_f / R_in)

convolution and DSP

convolution 摺積/捲積/疊積/旋積
convolution theorem 捲積定理

input: x[n]
system: h[n]
output: y[n]


convolution in time domain = multiplication in frequency domain

Time Domain:

y(n) = h(n) * x(n) = Σ h(k)x(n-k) where k ∈ Z (Z: integers 整數 ... -3, -2, -1, 0, 1, 2, 3, ...)

Frequency Domain:

Y(k) = H(k)X(k)

相關資料

Convolution (Wikipedia) / 摺積 (維基百科)
Video: Introduction to the convolution (Khan Academy 可汗學院)
Meaning of R, Q, Z, and N characters in math 數學中R, Q, Z, N的意義

Meaning of R, Q, Z, and N characters in math 數學中R, Q, Z, N的意義

R - real numbers 實數 -∞ ~ ∞, no imaginary part i 不含虛部 i

Q - rational numbers 有理數 may be represented as a ratio of p/q, where q ≠ 0.

Z - integers 整數 ... -3, -2, -1, 0, 1, 2, 3, ...

N - natural numbers 自然數 1, 2, 3, 4, ...

R 包含了 Q
Q 包含了 Z
Z 包含了 N

Figure of R, Q, Z, and N relationship

Example:

point sampling

x[n] = x(nT), n ∈ Z
n 屬於整數

參考資料

https://en.wikipedia.org/wiki/Rational_number#/media/File:Number-systems.svg

Fourier Analysis Terms 傅立葉分析相關名詞

Joseph Fourier (1768-1830) - French mathematician and physicist

Fourier analysis 傅立葉分析 function ---decompose---> oscillatory components
Fourier synthesis 傅立葉合成 oscillatory components ---combine--> function

Trigonometric series 三角級數 (Wikipedia)

Fourier series 傅立葉級數 => 可表示週期函數
Fourier transform 傅立葉轉換
discrete-time Fourier transform (DTFT) 離散時間傅立葉轉換
discrete Fourier transform (DFT) 離散傅立葉轉換

Fourier series:
periodic function 週期函數
f(x) = f(x + t) where t is the period

fundamental period 基本周期 t
fundamental frequency 基本頻率/基頻 f = 1/t
fundamental angular frequency 基本頻率/基頻 ω₀ = 2πf = 2π/t

fast Fourier transform (FFT) 快速傅立葉轉換

Periodic function => Fourier series
Non-periodic function => Fourier transform
Discrete function => discrete Fourier transform

periodic signal representations:
1.  sin + cos functions
2. amplitude + phase
3. complex Fourier series

Harmonic 諧波
Harmonic series 諧波列/泛音列
Overtone 泛音

analytic signal 解析訊號

Reference

Fourier analysis (Wikipedia)
Advanced Engineering Mathematics (10th edition), p474
Engineering Mathematics Exposed page 503

Matlab: Custom Function 自定函數

To create a custom function, the function name has to be identical to the *.m file name. (函數名稱必須與m檔案的名稱一致。)

Create a new script and edit it as:

function outputName = MyFunctionName(a,b)

% This function muliplies a by b and returns the result as outputName

outputName = a*b;

Save the script as MyFunctionName.m.

If a custom function is not properly defined or the path is incorrect, Matlab may return an error like this:

Undefined function or variable 'MyFunctionName'.

Hot to set the path for M files:

Home -> Environment -> Set Path

Example of Matlab 2015 b (Mac):

Call the M file from the Matlab terminal:

>> MyFunctionName(2,3)

ans =

     6

==============
Example to return multiple vectors with unequal dimensions:

function [a, b] = MyFunc(in1, in2)
a = [in1;1;2];
b = [in2 in2*2 in2*3 in2*4];

Save the code above as MyFunc.m.

Execute the function as below:

>> [a, b] = MyFunc(1,2)

a =

     1
     1
     2


b =

     2     4     6     8

Related Information 參考資料

Common Errors When Calling Functions (MathWorks)