Signal

Sampling: sensor→A/D

Digital signal VS analog signal

analog: both X Y are contiguous. digital: both are discrete.

Basic Operations on Digital Signal

\(x[n]=\{\cdots,-0.2,2.2,1.1,0.2,-3.7,\cdots\}\)

addition, multiplication, time shifting(delay), reversing(反折), stretching(拉伸)

difference(差分): \(x'[n]=x[n+1]-x[n]\)

accumulation(累加)

Convolution(卷积，非常重要): \(F(t)=x(n)*h(n)=\sum_kx[n-k]h[k]\)

Circular convolution: \(y(n)=x(n)\circledast h(n)=\sum_k(x[n-k]h[k])\times R_k(n)\)

length of results of convolution between x[n] and h[k]: n+k-1.

pulse signal: \(\delta(t)\)

step signal: \(u(n)\), \(\delta(n)=u(n)-u(n-1)\)

sine and cosine signal.

aka Linear Time Invariant System

\(y[n]=\alpha x_1[n]+\beta x_2[n]\)

accumulator is a typical LTI.

median filter is not LTI.

aka fourier transform

FT type	object
Fourier Transform	non-periodic, contiguous
Fourier Series	periodic, contiguous
Discrete Time Fourier Transform	non-periodic, discrete
Discrete Fourier Series	periodic, discrete

DTFT: \(X(e^{j\omega})=\sum_{n=0}^{N-1}x(n)e^{-j\omega n}\)

DFS: \(X(n)=\sum_{n=0}^{N-1}x(n)e^{-j\frac{2\pi}{N}kn}\)

Inverse DFS: \(x(n)=\frac{1}{N}\sum_{n=0}^{N-1}X(n)e^{j\frac{2\pi}{N}kn}\)