low-pass 低通
high-pass 高通
band-pass 帶通
band-stop 帶止/帶阻
passband 通帶/通頻帶
stopband 止帶/阻帶/抑制帶
transition band 過度帶/過渡頻帶
cutoff frequency 截止頻率
ripple 漣波
Matlab Filter Tools:
filterDesigner, sptool
2017/10/25
2017/10/23
Matlab: Play an audio file with a directory
To play a sound file e.g. a wav file in a folder, type the following commands:
>> [y,Fs]= audioread('MyFolder/song.wav'); %Use slash for both Windows and Mac. Backslash works with Windows only.
>> sound(y,Fs);
>> [y,Fs]= audioread('MyFolder/song.wav'); %Use slash for both Windows and Mac. Backslash works with Windows only.
>> sound(y,Fs);
2017/10/21
Matlab: Save and load variables/matrices in MAT file
To store and then read variables/matrices with Matlab, use the save and load commands with a *.mat file.
Example:
>> var1 = 1;
>> var2 = 2;
>> save var.mat var1 var2
>> clear
>> load var.mat
Related Information:
Matlab: Clear Commands (Study EECC)
Example:
>> var1 = 1;
>> var2 = 2;
>> save var.mat var1 var2
>> clear
>> load var.mat
Related Information:
Matlab: Clear Commands (Study EECC)
2017/10/12
Fundamental Frequency, Harmonic and Overtone 基頻、諧波、泛音
Fundamental Frequency 基本頻率/基頻
Fundamental Tone 基音/根音
Harmonic 諧波/諧音
Harmonic series 諧波列/泛音列
Overtone 泛音
Standing Wave 駐波
Periodic Tone 週期音
Pure Tone 純音
Complex Tone 複合音/複音
Higher Harmonics 高次諧波 (f×2, f×3...)
Partial 分音 - 出現於complex tone中，不一定是F0的integer multiple的sine wave
Example of a harmonic series:
f = fundamental frequency / fundamental tone / 1st harmonic 第一諧音/波
f×2 = 1st overtone 第一泛音 / 2nd harmonic 第二諧音/波
f×3 = 2nd overtone 第二泛音 / 3rd harmonic 第三諧音/波
參考資料
Harmonic (Wikipedia)
駐波(standing waves)(小小整理網站)
Harmonics (諧波)、Overtones (泛音)、Partials (分音)
Fourier Analysis Terms 傅立葉分析相關名詞
Fundamental Tone 基音/根音
Harmonic 諧波/諧音
Harmonic series 諧波列/泛音列
Overtone 泛音
Standing Wave 駐波
Periodic Tone 週期音
Pure Tone 純音
Complex Tone 複合音/複音
Higher Harmonics 高次諧波 (f×2, f×3...)
Partial 分音 - 出現於complex tone中，不一定是F0的integer multiple的sine wave
Example of a harmonic series:
f = fundamental frequency / fundamental tone / 1st harmonic 第一諧音/波
f×2 = 1st overtone 第一泛音 / 2nd harmonic 第二諧音/波
f×3 = 2nd overtone 第二泛音 / 3rd harmonic 第三諧音/波
參考資料
Harmonic (Wikipedia)
駐波(standing waves)(小小整理網站)
Harmonics (諧波)、Overtones (泛音)、Partials (分音)
Fourier Analysis Terms 傅立葉分析相關名詞
2017/10/04
Matlab: Unit Impulse δ[n] 單位脈衝函數
The unit impulse, or the Dirac delta function, has zero values for all input values except a value of one at zero.
>> t = -10:10;
>> s = (t==0); %s is one only when t equals 0
>> stem(t,s)
Result:
For δ[n-2], the Matlab code is
>> t = -10:10;
>> s2 = (t==2); %s is one when t equals 2
>> stem(t,s2)
Result:
Reference:
DSP terms 數位訊號處理基本名詞 (StudyEECC)
Dirac Delta Function (Wikipedia)
2017/09/30
Matlab: Plot multiple curves with legend
Example 1: Draw sin(x), sin(x/2) and sin(x)+sin(x/2)
>> x = 0:360;
>> y1 = sind(x);
>> y2 = sind(x/2);
>> y3 = y1+y2;
>> plot(x, y1, x, y2, x, y3);
>> legend('sin(x)','sin(x/2)','sin(x)+sin(x/2)')
Result:
Example 2: Draw log(x), log2(x), log10(x)
>> x = 0:0.01:5;
>> y1 = log(x);
>> y2 = log2(x);
>> y3 = log10(x);
>> plot(x, y1, x, y2, x, y3)
>> legend('log(x)','log2(x)','log10(x)')
Result:
Example 3: 3e-0.25*tsin(5t)
>> t = 0:0.01:30;
>> x1 = 3*exp(-0.25*t).*sin(5*t);
>> x2 = 0.2*cos(2*t);
>> x3 = 3*exp(-0.25*t).*sin(5*t)+0.2*cos(2*t);
>> plot(t,x1,t,x2,t,x3);
>> legend('x1','x2','x3');
Result:
Related Information:
Draw an Exponential Function with exp() with a custom Y-axis at x = 0
Exponential Function & Natural Logarithm 指數函數與自然對數
>> x = 0:360;
>> y1 = sind(x);
>> y2 = sind(x/2);
>> y3 = y1+y2;
>> plot(x, y1, x, y2, x, y3);
>> legend('sin(x)','sin(x/2)','sin(x)+sin(x/2)')
Result:
Example 2: Draw log(x), log2(x), log10(x)
>> x = 0:0.01:5;
>> y1 = log(x);
>> y2 = log2(x);
>> y3 = log10(x);
>> plot(x, y1, x, y2, x, y3)
>> legend('log(x)','log2(x)','log10(x)')
Result:
Example 3: 3e-0.25*tsin(5t)
>> t = 0:0.01:30;
>> x1 = 3*exp(-0.25*t).*sin(5*t);
>> x2 = 0.2*cos(2*t);
>> x3 = 3*exp(-0.25*t).*sin(5*t)+0.2*cos(2*t);
>> plot(t,x1,t,x2,t,x3);
>> legend('x1','x2','x3');
Result:
Related Information:
Draw an Exponential Function with exp() with a custom Y-axis at x = 0
Exponential Function & Natural Logarithm 指數函數與自然對數
2017/09/28
Matlab: Hello GUI with guide
To draw a window or the graphical user interface (GUI) with Matlab, type:
>> guide
Select Blank GUI (Default):
A window for GUI design is displayed:
Draw something such as a "Hello World!" static text and some buttons.
Save the design figure as your preferred name, e.g. gui_layout.fig
A corresponding m file such as gui_layout.m is automatically created.
In the Command Window, type the name of the m-file e.g. type gui_layout and the window designed previously is displayed.
>> guide
Select Blank GUI (Default):
A window for GUI design is displayed:
Draw something such as a "Hello World!" static text and some buttons.
Save the design figure as your preferred name, e.g. gui_layout.fig
A corresponding m file such as gui_layout.m is automatically created.
In the Command Window, type the name of the m-file e.g. type gui_layout and the window designed previously is displayed.
2017/09/02
OSI Model 開放式系統互聯通訊參考模型
OSI - Open System Interconnection
Open System Interconnection reference model 開放式系統互聯通訊參考模型 (Wikipedia/維基百科)
Layer 7 - Application 應用層
Layer 6 - Presentation 表達層/展示層
Layer 5 - Session 會議層/交談層
Layer 4 - Transport 傳輸層
Example: TCP (Transmission Control Protocol). Connection-oriented links for conveying segments of data.
Example: UDP (User Datagram Protocol). Connectionless transmission model. For Multimedia traffic.
Layer 3 - Network 網路層 Example: IP (Internet Protocol). Data transported as packets.
Layer 2 - Data Link 資料鏈結層
Logical Link Control (LLC) sublayer
Media Access Control (MAC) sublayer - e.g. CSMA/CD (802.3), CSMA/CA (802.11)
Layer 1 - Physical 實體層
References:
Communication Systems (4th Edition) by Simon Haykin, Wiley 2001, p11-12.
What is the difference between TCP and IP protocols?
Transmission Control Protocol(Wikipedia)
User Datagram Protocol(Wikipedia)
OSI七層簡介
Open System Interconnection reference model 開放式系統互聯通訊參考模型 (Wikipedia/維基百科)
Layer 7 - Application 應用層
Layer 6 - Presentation 表達層/展示層
Layer 5 - Session 會議層/交談層
Layer 4 - Transport 傳輸層
Example: TCP (Transmission Control Protocol). Connection-oriented links for conveying segments of data.
Example: UDP (User Datagram Protocol). Connectionless transmission model. For Multimedia traffic.
Layer 3 - Network 網路層 Example: IP (Internet Protocol). Data transported as packets.
Layer 2 - Data Link 資料鏈結層
Logical Link Control (LLC) sublayer
Media Access Control (MAC) sublayer - e.g. CSMA/CD (802.3), CSMA/CA (802.11)
Layer 1 - Physical 實體層
References:
Communication Systems (4th Edition) by Simon Haykin, Wiley 2001, p11-12.
What is the difference between TCP and IP protocols?
Transmission Control Protocol(Wikipedia)
User Datagram Protocol(Wikipedia)
OSI七層簡介
2017/08/24
2017/08/23
Tensor 張量
Array of numbers that may have
- scalar 純量
- vector 向量
- matrix 矩陣
- more dimensions.
Refernces
Lecture slides for Chapter 2 of Deep Learning, Ian Goodfellow
張量 (Tensor) - 金門大學陳鍾誠的網站
- scalar 純量
- vector 向量
- matrix 矩陣
- more dimensions.
Refernces
Lecture slides for Chapter 2 of Deep Learning, Ian Goodfellow
張量 (Tensor) - 金門大學陳鍾誠的網站
2017/08/19
Data Center Locations - Google, Akamai, CloudFlare, Amazon CloudFront
Below data are recorded on August 19, 2017
Google - https://www.google.com/about/datacenters/inside/locations/index.html
Taiwan 台灣 - Chunghua County 彰化縣
Akamai - https://www.akamai.com/us/en/locations.jsp
Taiwan 台灣 - Taipei 台北市
CloudFlare - https://www.cloudflare.com/network/
Taiwan 台灣 - Taipei 台北(亞洲第28個/全球第77個data center)
Amazon CloudFront Edge InfraStructure - https://aws.amazon.com/cloudfront/details/
Taiwan 台灣 - Taipei 台北
Google - https://www.google.com/about/datacenters/inside/locations/index.html
- USA x 7
- Asia x 2 (Taiwan, Singapore)
- Europe x 4
Taiwan 台灣 - Chunghua County 彰化縣
Akamai - https://www.akamai.com/us/en/locations.jsp
Taiwan 台灣 - Taipei 台北市
CloudFlare - https://www.cloudflare.com/network/
- 116 data centers
Taiwan 台灣 - Taipei 台北(亞洲第28個/全球第77個data center)
Amazon CloudFront Edge InfraStructure - https://aws.amazon.com/cloudfront/details/
- 22 edge locations / 11 regional edge caches
Taiwan 台灣 - Taipei 台北
2017/07/31
Rectifiers 整流器
rectifiers 整流器
half-wave rectifiers 半波整流器
full-wave rectifiers 全波整流器
diode rectifier 二極體整流器
half-wave rectifiers 半波整流器
full-wave rectifiers 全波整流器
diode rectifier 二極體整流器
2017/07/24
Matlab: Play Demo Sounds
It is very easy to play some demo sounds with Matlab commands using the sound function.
To play a female speaker saying 'Matlab' type:
>> load mtlb
>> sound(mtlb,Fs)
Note that the workspace contains mtlb and Fs, so sound(mtlb,Fs) is called.
You may clear the workspace with the the clear command:
>> clear
To play a snippet of Handel's Hallelujah Chorus, type:
>> load handel
>> sound(y,Fs)
There are also other sounds you may load:
load chirp
load gong
load laughter
load train
References:
Formant Estimation with LPC Coefficients
Record and Play Audio
Sound + FFT
To play a female speaker saying 'Matlab' type:
>> load mtlb
>> sound(mtlb,Fs)
Note that the workspace contains mtlb and Fs, so sound(mtlb,Fs) is called.
You may clear the workspace with the the clear command:
>> clear
To play a snippet of Handel's Hallelujah Chorus, type:
>> load handel
>> sound(y,Fs)
There are also other sounds you may load:
load chirp
load gong
load laughter
load train
References:
Formant Estimation with LPC Coefficients
Record and Play Audio
Sound + FFT
2017/06/13
Vectors 向量
dot product / scalar product / inner product / projection product 點積/內積/純量積 a·b
A·B = |A| |B| cos(θ)
AB = |A| cos(θ)
For vectors a = (a1, a2, ..., an) and b = (b1, b2, ..., bn) ,
a·b = a1b1 + a2b2 + ... + anbn
For example, vectors a = (1, 2, 3) and b = (0, 4, 5),
a·b = 1×0 + 2×4 + 3×5 = 23
Matlab code:
>> a = [1 2 3];
>> b = [0 4 5];
>> dot(a,b)
ans =
23
cross product/vector product 點積/向量積 a×b
參考資料
內積與外積
A·B = |A| |B| cos(θ)
AB = |A| cos(θ)
For vectors a = (a1, a2, ..., an) and b = (b1, b2, ..., bn) ,
a·b = a1b1 + a2b2 + ... + anbn
For example, vectors a = (1, 2, 3) and b = (0, 4, 5),
a·b = 1×0 + 2×4 + 3×5 = 23
Matlab code:
>> a = [1 2 3];
>> b = [0 4 5];
>> dot(a,b)
ans =
23
cross product/vector product 點積/向量積 a×b
參考資料
內積與外積
2017/05/13
Cent and Semitone 音分和半音
cent 音分
semitone/half step/half tone 半音
whole tone 全音
accidental 變音記號
1 semitone = 100 cents
octave 八度音
一個八度音 = 12個半音 = 5個全音 + 2個半音
Twelve-tone equal temperament (十二平均律)
- divide an octave into 12 equal parts
Number of Semitones between Two Frequencies (兩個頻率間相差的半音數):
n = abs(log2(f1/f2)/log2(2(1/12))) = abs(12log2(f1/f2))
Reference
How to find the number of semitones between two frequencies?
semitone/half step/half tone 半音
whole tone 全音
accidental 變音記號
1 semitone = 100 cents
octave 八度音
一個八度音 = 12個半音 = 5個全音 + 2個半音
Twelve-tone equal temperament (十二平均律)
- divide an octave into 12 equal parts
Number of Semitones between Two Frequencies (兩個頻率間相差的半音數):
n = abs(log2(f1/f2)/log2(2(1/12))) = abs(12log2(f1/f2))
Reference
How to find the number of semitones between two frequencies?
2017/04/06
Matlab: Draw a grayscale
>> a = [0 40 80 120 160 200 220 255];
>> b = [a;a;a;a;a;a;a;a];
>> colormap('gray');
>> imagesc(b)
Result:
>> b = [a;a;a;a;a;a;a;a];
>> colormap('gray');
>> imagesc(b)
Result:
2017/04/03
Modulation terms 調變相關名詞
modulation 調變
modulator 調變器
demodulator 解調器/解調變器
amplitude modulation (AM) 振幅調變/調幅
angle modulation 角度調變
- frequency modulation (FM) 頻率調變/調頻
- phase modulation (PM) 相位調變/調相
modulation index 調變指數
Reference
S. Haykin, Communication Systems, 4th Edition, Wiley, pp106
國家教育研究院雙語詞彙、學術名詞暨辭書資訊網
modulator 調變器
demodulator 解調器/解調變器
amplitude modulation (AM) 振幅調變/調幅
angle modulation 角度調變
- frequency modulation (FM) 頻率調變/調頻
- phase modulation (PM) 相位調變/調相
modulation index 調變指數
Reference
S. Haykin, Communication Systems, 4th Edition, Wiley, pp106
國家教育研究院雙語詞彙、學術名詞暨辭書資訊網
2017/03/18
2017/03/13
Matlab: Length and Size of a Vector/Matrix
Given that
>> a = [1 2 3;4 5 6]
>> b = 1:5
Length
>> length(a)
ans =
3
>> length(b)
ans =
5
Size
>> size(a)
ans =
2 3
>> size(b)
ans =
1 5
>> a = [1 2 3;4 5 6]
>> b = 1:5
>> length(a)
ans =
3
>> length(b)
ans =
5
Size
>> size(a)
ans =
2 3
>> size(b)
ans =
1 5
Matlab: How to use Quantiz with partition and codebook
The Quantiz function requires the Communications System Toolbox.
This quantization function requires at least an input signal and a partition vector.
Partition
The example below shows an input signal between 1 and 10. The partition vector equals [2 5 7]. When the signal is quantized, values become:
y = 0 if x <= 2
y = 1 if x <= 5
y = 2 if x <= 7
y = 3 if x > 7
>> x = 1:10
x =
1 2 3 4 5 6 7 8 9 10
>> partition = [2 5 7]
partition =
2 5 7
>> y = quantiz(x, partition)
y =
0 0 1 1 1 2 2 3 3 3
Codebook
Using the same input signal x and partition above, add a codebook as:
>> codebook = [-4 0 2 4]
Insert the codebook as the third parameter in the Quantiz function. The index and quantized value quants are output:
>> [index,quants] = quantiz(x,partition,codebook)
index =
0 0 1 1 1 2 2 3 3 3
quants =
-4 -4 0 0 0 2 2 4 4 4
Reference
Quantization (MathWorks)
This quantization function requires at least an input signal and a partition vector.
Partition
The example below shows an input signal between 1 and 10. The partition vector equals [2 5 7]. When the signal is quantized, values become:
y = 0 if x <= 2
y = 1 if x <= 5
y = 2 if x <= 7
y = 3 if x > 7
>> x = 1:10
x =
1 2 3 4 5 6 7 8 9 10
>> partition = [2 5 7]
partition =
2 5 7
>> y = quantiz(x, partition)
y =
0 0 1 1 1 2 2 3 3 3
Codebook
Using the same input signal x and partition above, add a codebook as:
>> codebook = [-4 0 2 4]
Insert the codebook as the third parameter in the Quantiz function. The index and quantized value quants are output:
>> [index,quants] = quantiz(x,partition,codebook)
index =
0 0 1 1 1 2 2 3 3 3
quants =
-4 -4 0 0 0 2 2 4 4 4
Reference
Quantization (MathWorks)
2017/03/12
PCM, DPCM, ADPCM
2017/02/13
Sampling 取樣
Discrete-time signal x[n] is obtained by sampling a continuous-time signal xc(t):
x[n] = xc(nT) , -∞ < n < ∞
Fourier Transform:
xc(t) <--F--> X(jΩ)
x[n] <--F--> X(ejω)
ω = ΩT
Time Shift:
x(t-t0) <--F--> e-jΩt0 X(jΩ)
x[n-n0] <--F--> e-jωn0 X(ejω)
Frequency Shift:
ejΩ0t x(t) <--F--> X(j(Ω-Ω0))
ejω0n x[n] <--F--> X(ej(ω-ω0))
sampling period 取樣周期 T
sampling frequency 取樣頻率 fs or Ωs
With respect to different units:
Samples/second: fs = 1/T
Radians/second: Ωs = 2π/T = 2πfs
Nyquist Theorem
Nyquist Frequency ΩN
Nyquist Rate 2ΩN
aliasing 頻疊、摺疊效應
To avoid aliasing, Ωs >= 2ΩN
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
x[n] = xc(nT) , -∞ < n < ∞
Fourier Transform:
xc(t) <--F--> X(jΩ)
x[n] <--F--> X(ejω)
ω = ΩT
Time Shift:
x(t-t0) <--F--> e-jΩt0 X(jΩ)
x[n-n0] <--F--> e-jωn0 X(ejω)
Frequency Shift:
ejΩ0t x(t) <--F--> X(j(Ω-Ω0))
ejω0n x[n] <--F--> X(ej(ω-ω0))
sampling period 取樣周期 T
sampling frequency 取樣頻率 fs or Ωs
With respect to different units:
Samples/second: fs = 1/T
Radians/second: Ωs = 2π/T = 2πfs
Nyquist Theorem
Nyquist Frequency ΩN
Nyquist Rate 2ΩN
aliasing 頻疊、摺疊效應
To avoid aliasing, Ωs >= 2ΩN
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
2017/02/12
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
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
2017/02/09
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
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
2017/02/08
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
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
2017/02/07
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] = ejωn appears at the output of the system with the eigenvalue H(ejω).
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] = ejωn appears at the output of the system with the eigenvalue H(ejω).
2017/01/30
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 eiθ? Or |eiθ| = ?
According to Euler's formula,
eiθ = cosθ + isinθ
To find the magnitude of |eiθ|,
|eiθ| = SQRT(eiθ) = SQRT(cos2θ + isin2θ) = SQRT(1) = 1
Therefore,
the magnitude of eiθ 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
According to Euler's formula,
eiθ = cosθ + isinθ
To find the magnitude of |eiθ|,
|eiθ| = SQRT(eiθ) = SQRT(cos2θ + isin2θ) = SQRT(1) = 1
Therefore,
the magnitude of eiθ 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
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