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]={⋯ ,−0.2,2.2,1.1,0.2,−3.7,⋯ }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]x'[n]=x[n+1]-x[n]

accumulation(累加)

⚠️Convolution(卷积​，非常重要): F(t)=x(n)∗h(n)=∑kx[n−k]h[k]F(t)=x(n)*h(n)=\sum_kx[n-k]h[k]

  • commutative 交换律
  • associative 结合律
  • distributive 分配律

Circular convolution: y(n)=x(n)⊛h(n)=∑k(x[n−k]h[k])×Rk(n)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.

Signal Types

details

pulse signal: δ(t)\delta(t)

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

sine and cosine signal.

Signaling System

LTI

aka Linear Time Invariant System

y[n]=αx1[n]+βx2[n]y[n]=\alpha x_1[n]+\beta x_2[n]

accumulator is a typical LTI.

median filter is not LTI.

Fourier Transform

aka fourier transform

FT typeobject
Fourier Transformnon-periodic, contiguous
Fourier Seriesperiodic, contiguous
Discrete Time Fourier Transformnon-periodic, discrete
Discrete Fourier Seriesperiodic, discrete

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

DFS: X(n)=∑n=0N−1x(n)e−j2πNknX(n)=\sum_{n=0}^{N-1}x(n)e^{-j\frac{2\pi}{N}kn}

Inverse DFS: x(n)=1N∑n=0N−1X(n)ej2πNknx(n)=\frac{1}{N}\sum_{n=0}^{N-1}X(n)e^{j\frac{2\pi}{N}kn}