Matlab: Draw an Exponential Function with exp() with a custom Y-axis at x = 0

>> x = -2:0.2:4; %incrementing by 0.2 from -2 to 4.
>> y = exp(x);
>> plot(x,y);
>> ax = gca; %Axis
>> ax.YAxisLocation = 'origin'; %Y axis at x = 0.
>> title('Exponential Function', 'FontSize', 16); %Add title to plot with custom font size

Result:

Related Information:

Exponential Function & Natural Logarithm 指數函數與自然對數
Display Axis Lines Through Origin (MathWorks)

Sinc Function / Sampling Function 取樣函數

There are two definitions for the Sinc function:

When x ≠ 0,

for mathematics,

    sinc(x) = sin(x)/x (unnormalized sinc function)

for digital signal processing,

    sinc(x) = sin(πx)/πx (normalized sinc function)

When x = 0,

    sinc(x) = sinc(0) = 1 for both mathematics and DSP.

In DSP, sinc(x) is also called the sampling function:

    Sa(πx) = sin(πx)/πx = sinc(x)

Euler's formula 歐拉公式

Leonhard Euler (1707-1783) 瑞士數學家
Euler 歐拉/尤拉

Euler's formula helps to find the exponential function of a complex number.

Euler's formula 歐拉公式:

e^jx = cosx + isinx = cisx

Therefore,

e^iθ = cosθ + isinθ (1)
e^-iθ = cos(-θ) + isin(-θ) = cosθ - isinθ (2)

(1) + (2) gives
=> e^iθ + e^-iθ =  2cosθ
=> cosθ = ( e^iθ + e^-iθ )/2

(1) - (2) gives
=> e^jθ - e^-jθ = 2isinθ
=> sinθ = ( e^iθ - e^-iθ )/2i

cosθ = Re(e^iθ) =( e^iθ + e^-iθ )/2
sinθ = Im(e^iθ) =( e^iθ - e^-iθ )/2i

Example:

eⁱ = cos(1) + isin(1) = 0.540302306 + 0.841470985i
e^iπ = cos(π) + isin(π) = -1 => e^iπ + 1 = 0

Matlab function:
exp(i*pi) or exp(j*pi)

參考資料

cis (mathematics) (Wikipedia)
Relations between cosine, sine and exponential functions
Exponential function & natural logarithm 指數函數與自然對數
Magnitude of Exponential Function exp(iθ) = ? 如何求exp(iθ)的大小/絕對值

Summation and Multiplication symbols 求和符號、乘積符號

summation 求和符號：

Σ - uppercase Sigma

multiplication 乘積符號：

Π - uppercase Pi

相關資料

數學符號表

Exponential Function & Natural Logarithm 指數函數與自然對數

exp(x) = e^x

some coordinates:

(0, 1)
(1, e)

Differentiation 微分 :
d(e^x)/dx = e^x
d(e^ax)/dx = ae^ax


Integration 積分 :

∫ e^x dx = e^x + c
∫ e^ax dx = (1/a)e^ax + c

Maclaurin Series:

e^x = 1 + x + x²/2! + x³/3! + ... + xⁿ/n!

e^a+b = e^ae^b
(e^a)^b = e^ab
e^a+bi = e^ae^bi = e^a(cosb + isinb) = e^acis(b) (Euler's formula)

Inverse of e^x:
ln(x)

natural logarithm 自然對數
log_e(x) = ln(x)

derivative 導數：
d[ln(x)]/dx = 1/x

Exponential Function plotted by Matlab:

相關資料

Exponential Function (Wikipedia)
Matlab: Draw an Exponential Function with exp() with a custom Y-axis at x = 0

DSP terms 數位訊號處理基本名詞

digital signal processing 數位訊號處理/數位信號處理

sampling 取樣
quantization 量化
reconstruction 重建

point sampling 點取樣
impulse sampling 脈衝取樣

aliasing 頻疊、混疊
Nyquist frequency 奈奎斯特頻率

zero crossing 零交叉 - 訊號剛好為零(由負到正或由正到負的瞬間) (Wikipedia)
根據Wikipedia，計算zero crossing的數量可估計語音的fundamental frequency基頻

piecewise 分段
piecewise linear function 分段線性函數

sample and hold 取樣與保持
linear interpolation 線性內插

quantization step 量化間距/量化步階
quantization error 量化誤差

Signal-to-quantization-noise ratio (SQNR) 信號-量化雜訊比
SQNR = 6.02b + 1.76 dB 量化時每增加一個bit時，SQNR可提升6.02 dB (See Signal to quantization noise in quantized sinusoidal)

unit impulse 單位脈衝 δ[n] Dirac delta function 狄拉克δ函數
unit step 單位步階/單位階 u[n]

companding 壓擴/壓伸

time domain 時域
frequency domain 頻域
complex frequency domain 複數頻域

Fourier series 傅立葉級數
discrete Fourier transform (DFT) 離散傅立葉變換
fast Fourier transform (FFT) 快速傅立葉變換

discrete Cosine transform (DFT) 離散餘弦轉換

Z-transform Z 轉換
region of convergence (ROC) 收斂域

參考資料

國家教育研究院雙語詞彙、學術名詞暨辭書資訊網

課程
DeltaMOOCx

Matlab: Draw a Sine curve with sind()

To plot a sine wave between 0° and 360° in Matlab, simply use sind() in degrees:

>> x = 0:360;
>> y = sind(x);
>> plot(x,y)

Result:

If you want to draw sine waves in radians, use sin() instead. The website below shows how to use the sin() function and how to adjust axis ranges:

Set axis limits and aspect ratio (MathWorks)

You may also try the code below:

>> s = [0:.1:2*pi];
>> y = sin(s);
>> plot(s,y)

Related Information:
How to draw a sine wave with Simulink