com303.md

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$$
\def\sumi{\sum_{n = -\infty}^\infty}
\def\sinc{,\text{sinc}}
\def\sgn{,\text{sign}}
\def\arg{,\text{arg}}
\def\min{,\text{min}}
\def\Z{\mathbb{Z}}
\def\C{\mathbb{C}}
\def\N{\mathbb{N}}
\def\R{\mathbb{R}}
\def\H#1{\mathcal H{#1}}
\def\abs#1{,\lvert #1\rvert}
\def\norm#1{,\lVert #1\rVert}
$$

COM303

GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.

Spring 2016: Signal Processing

[TOC]

Introduction

• signal : description of the evolution of a physical phenomenon
  • processing
    • analysis : understanding the information carried by the signal
    • synthesis : creating a signal to contain the given information
  • communication
    • reception : analysis of an incoming signal
    • transmission : synthesis of an outgoing signal
• digital model
  • discrete time : $\overline{x}=\frac{1}{N}\sum _{n=0}^{N-1}x[n]$
  • discrete amplitude : important for storage, processing, transmission
  • storage : $0,1$
  • data transmission : $x_1(t)=G\text{sgn}[x(t)+G\sigma (t)]$ as quantized signals
• analog model
  • continuous time : $\overline{x}=\frac{1}{b-a}{\int }_a^{b}f(t)dt$
  • storage : paper, vinyl, VHS, silver plates
  • data transmission : $x_N(t)=x(t)+NG\sigma (t)$ with $N$ repeaters ($sign$ after each repetion to avoid noise multiplication)
• sampling theorem : $x(t)=\text{sumi}x[n]\text{sinc}(\frac{t-n{T}_s}{T_s})$
• discrete-time signal : a sequence of complex numbers $x:\text{Z}\to \text{C}$ noted as $x[n]$
  • analysis : periodic measurement
  • synthesis : stream of generated samples
  • delta signal : $x[n]=\delta [n]$
  • unit step : $x[n]=u[n]=1_{n\ge 0}$
  • exponential decay : $x[n]=\text{abs}{a}^{n}u[n]$ with $\text{abs}a<1$
• signal classes
  • finite-length : $\overrightarrow{x}=[x_0\dots x_{N-1}{]}^T$
  • infinite-length : $x[n]$
  • periodic : $\tilde{x}[n]=\tilde{x}[n+kN]$, replicated finite-lenght
  • finite-support : finite-length completed with $0$'s
• elementary operators
  • scaling : $y[n]=\alpha x[n]$
  • sum : $y[n]=x[n]+z[n]$
  • product : $y[n]=x[n]\cdot z[n]$
  • shift by $k$ (delay) : $y[n]=x[n-k]$, best with periodic extension (circular)
• energy : $E_x=\text{sumi}\text{abs}{x[n]}^2$
  • periodic : $E_x=\mathrm{\infty }$
• power : $P_x=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{2N+1}\sum _{n=-N}^N\text{abs}{x[n]}^2$
  • periodic : $P_x=\frac{1}{N}\sum _{n=0}^{N-1}\text{abs}{\tilde{x}[n]}^2$
• DSP as building block
  • adder : $x[n]+y[n]$
  • multiplier : $\alpha x[n]$
  • delay : $z^{-N}$ giving $x[n-N]$
  • moving average : $y[n]=\frac{x[n]+x[n-1]}{2}$
    • zero initial conditions : set a start time $n_0=0$, assume input and output are zero for all time before $n_0$
    • recursion : $y[n]=x[x]+\alpha y[n-1]$
• Karplus-Strong algorithm : $y[n]=\alpha y[n-3]+\overline{x}[n]$ with $\overline{x}[n]=\delta [n]+2\delta [n-1]+3\delta [n-2]$
  • build a recursion loop with a delay of $M$ ($M$-tap) : controls pitch
  • choose a signal $\overline{x}[n]$ that is nonzero only for $0\le n<M$ : controls color (timbre)
  • choose a decay factor : $\alpha$ controls envelope (decay)
  • input $\overline{x}[n]$ to the system
  • play the output
  • frequency : $f=\frac{1}{MT}Hz$ with time $T$ associated to $M$ sample interval

Vector spaces

• vector spaces : usefull in approximation and compression, easy sampling, interpolation, Fourier Transform, fundamental in communication system design
  • euclidian : ${\text{R}}^2,{\text{R}}^3$
  • linear algebra : ${\text{R}}^N,{\text{C}}^N$
  • space of square-summable infinite sequences : $l_2(\text{Z})$
  • space of quare-integrable functions over an interval : $L_2([a,b])$
    • inner product : $<x,y>={\int }_b^{a}x(t)y^{\ast }(t)dt$
  • once you know the properties are satisfied, you can use all the tools for the space
  • ingredients
    • set of vectors : $V$
    • set of scalars : $\text{C}$
  • properties : $x,y,z$ are vectors
    • $x+y=y+x$
    • $(x+y)+z=x+(y+z)$
    • $\alpha (x+y)=\alpha y+\alpha x$
    • $(\alpha +\beta )x=\alpha x+\beta x$
    • $\alpha (\beta x)=(\alpha \beta )x$
    • $\mathrm{\exists }0\in V\mid x+0=0+x=x$
    • $\mathrm{\forall }x\in V\mathrm{\exists }(-x)\mid x+(-x)=0$
  • inner (dot) product : $<\cdot ,\cdot >:V\times V\to \text{C}$
    • $<x+y,z>=<x,z>+<y,z>$
    • $<x,y>=<y,x{>}^{\ast }$
    • $<\alpha x,y>={\alpha }^{\ast }<x,y>$
    • $<x,\alpha y>=\alpha <x,y>$
    • $<x,x>=\text{norm}{x}^2\ge 0$
    • $<x,x>=0⟺x=0$
    • $<x,y>=\text{norm}x\text{norm}y\cos \alpha$
    • $<x,y>=0$ with $x,y\ne 0$ : orthogonal (maximally different)
    • defines a norm : $\text{norm}x=\sqrt{<x,x>}$
    • norm defines a distance : $d(x,y)=\text{norm}x-y$
    • mean square error : $\text{norm}{x-y}^2$
    • for signals : $<x,y>=\sum _{n=0}^{N-1}{x}^{\ast }[n]y[n]$, requires square-summable $\sum \text{abs}{x[n]}^2<\mathrm{\infty }$
    • pythagorean theorem : $\text{norm}{x+y}^2=\text{norm}{x}^2+\text{norm}{y}^2$ for $x\perp y$
• canonical basis : set of $K$ vectors in a vector space with basis $W={w^{(k)}}_{k=0,\dots ,K-1}$
  • linear combination : $\mathrm{\forall }x;x=\sum _{k=0}^{K-1}{\alpha }_k{w}^{(k)}$ with ${\alpha }_k\in \text{C}$ unique
  • unique representation implies linear independence : $\sum _{k=0}^{K-1}{\alpha }_k{w}^{(k)}=0⟺{\alpha }_k=0;k=0,\dots ,K-1$
  • basis for functions over an interval : $v^{(v)}=\sin (\pi (2k+1)t)$ with $\sum _{k=0}^N(1/N)v^{(k)}$
  • orthogonal basis : $<w^{(k)},w^{(n)}>=0$ for $k\ne n$
  • orthonormal bassis : $<w^{(k)},w^{(n)}>=\delta [n-k]$ for $k\ne n$ (using Gram-Schmidt)
  • basis expansion : ${\alpha }_k=<w^{(k)},x>$
  • change of basis : $x=\sum _{k=0}^{K-1}{\alpha }_k{w}^{(k)}=\sum _{k=0}^{K-1}{\beta }_k{v}^{(k)}$
    • if $v^{(k)}$ orthnormal
    • ${\beta }_h=<v^{(h)},x>=<v^{(h)},\sum _{k=0}^{K-1}{\alpha }_k{w}^{(k)}>=\sum _{k=0}^{K-1}{\alpha }_k<v^{(h)},w^{(k)}>=\sum _{k=0}^{K-1}{\alpha }_k{c}_{hk}$
    • parseval's theorm : energy is conserved across change of basis $\text{norm}{x}^2=\sum _{k=0}^{K-1}\text{abs}{{\alpha }_k}^2$
    • rotation matrix : $R=\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$ with new coefficient $\beta =R\alpha$
• subspace : subset of vectors closed under addition and scalar multiplication
  • approximation : less dimensionnal projection
  • least-square approximation for subset $S$ : $\hat{x}=\sum _{k=0}^{K-1}<s^{(k)},x>s^{(k)}$ (orthogonal projection)
    • has minimum-norm error : $\arg \underset{y\in S}{min}\text{norm}x-y=\hat{x}$
    • error orthogonal to approximation : $<x-\hat{x},\hat{x}>=0$
• Gram-schmidt : original set $s^{(k)}\to u^{(k)}$ orthonormal set
  1. $p^{(k)}=s^{(k)}-\sum _{n=0}^{k-1}<u^{(n)},s^{(k)}>u^{(n)}$
  2. $u^{(k)}=p^{(k)}/\text{norm}{p}^{(k)}$ for each step $k$
• Legendre polynormials : orthonormal basis for $P_N([-1,1])$, ${\alpha }_k=<u^{(k)},x>$ with naive basis $s^{(k)}=t^k$
  • $u^{(0)}=\sqrt{1/2}$
  • $u^{(1)}=\sqrt{3/2}t$
  • $u^{(2)}=\sqrt{5/8}(3t^2-1)$
• Hilbert space : vector space $H(V,\text{C})$, inner product $<\cdot ,\cdot >:V\times V\to \text{C}$, completeness (limiting operations must yield vector space elements)
  • finite-length and periodic signals live in ${\text{C}}^N$
  • inifinte-length signals live in $l_2(\text{Z})$
  • different bases are different observation tools for signals
  • subspace projections useful in filtering and compression

Fourier analysis

• time domain : linear combination of atomic time units $x[n]=\sum _{k=0}^{N-1}x[k]{\delta }^{(k)}$
• oscillations are everywhere
  • sustainable dynamic systems exhibit oscillatory behavior
  • things that don't move in circles can't last
  • human receptors
    • cochlea : ear, air pressure, frequencies between 20Hz and 20KHz
    • rods and cones : retina, electromagnetic sinusoids, frequencies between 430THz to 790THz
  • oscillatory heartbeat : $x[n]=A\cos (\omega n+\varphi )$ with frequency $\omega$, initial phase $\varphi$, amplitude $A$
  • complex exponential : $x[n]=A{e}^{j(\omega n+\varphi )}=A[\cos (\omega n+\varphi )+j\sin (\omega n+\varphi )]$
  • periodic in $n$ : $e^{j\omega n}$ with $\omega =\frac{M}{N}2\pi$ with $M,N\in \text{N}$
  • wagonwheel effect : max speed at $\omega =\pi$
  • digital frequency : how many samples before pattern repeats, no unit
  • physical frequency : how many secondes before pattern repeats, hertz, $\frac{1}{M{T}_s}$ with $M$ samples of $T_s$ system clock
• frequency domain : as important as the time domain, hidden signal properties
• fourier : simple change of basis, change of perspective
  • analysis : express signal as combination of periodic oscillations, from time to frequency domain
  • synthesis : create signal with frequency content, from frequency to time domain
  • orthogonal basis : $w_k[n]=w_n^{(k)}=e^{j\frac{2\pi }{N}nk}$ (normalization $1/\sqrt{N}$)
  • proof = $<w^{(k)},w^{(h)}>=N$ for $h=k$ otherwise $0$
  • wrapping the phase : $[-\pi ,\pi ]$ by adding $2\pi$
• discrete fourier transform : DFT, numerical algorithm
  • analysis : $X_k=<w^{(k)},x>$ or $X[k]=\sum _{n=0}^{N-1}x[n]e^{-j\frac{2\pi }{N}nk}$
  • synthesis : $x=\frac{1}{N}\sum^{N-1}{k=0}X_kw^{(k)}$ or $x[n]=\frac{1}{N}\sum{k=0}^{N-1}X[k]e^{j\frac{2\pi}{N}nk}$
  • parseval : $\sum _{n=0}^{N-1}|x[n]{|}^2=\frac{1}{N}\sum _{k=0}^{N-1}|X[k]{|}^2$
  • finite-length time shifts : $IDFT{e}^{-j\frac{2\pi }{N}Mk}X[k]=x[(n-M)\mathrm{mod}N]$ circular
  • real signal : symmertic in magnitude $|X[k]|=|X[N-k]|$ for $k=1,\dots ,\lceil N/2\rceil$
  • DFT of length $M$ step : $X[k]=\frac{\sin (\frac{\pi }{N}Mk)}{\sin (\frac{\pi }{N}k)}{e}^{-j\frac{\pi }{N}(M-1)k}$
  • $X[0]=M$
  • $X[k]=0$ if $Mk/N$ integer
  • DFT of $L$ periods : $X_L[k]=L\overline{X}[k/L]$ if $k$ multiple of $L$ otherwise $0$ with $\overline{X}[k/L]=\sum _{n=0}^{M-1}\overline{x}[n]e^{-j\frac{2\pi }{M}n\frac{k}{L}}$
• matrix form basis change : $\begin{bmatrix}1 & 1 & 1 & \cdots & 1\\ 1 & W^1 & W^2 & \cdots & W^{N-1}\\ 1 & W^2 & W^4 & \cdots & W^{2(N-1)}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & W^{N-1} & W^{2(N-1)} & \cdots & W^{(N-1)^2}\end{bmatrix}$ with $W[n,m]=W_N^{nm}=e^{-j\frac{2\pi }{N}nm}$
  • analysis : $X=Wx$
  • synthesis : $\frac{1}{N}{W}^{H}X$
  • hermitian : $W^H$ conjugate transpose
  • dft : $W_N^m=W_N^{(m\mathrm{mod}N)}$
• short-time fourier transform : STFT
  • time obfuscates frequency : $\mathrm{\Delta }t\mathrm{\Delta }f=2\pi$
  • small singal pieces of length $L$ : $X[m;k]=\sum _{n=0}^{L-1}x[m+n]e^{-j\frac{2\pi }{L}nk}$
  • spectrogram : magnitude (dark small, white large), $10{\log }_{10}(|X[m,k]|)$
    • system clock : $F_s=1/T_s$
    • highest positive frequency : $F_s/2$ Hz
    • frequency resolution : $F_s/L$ Hz
    • width of time slices : $L{T}_s$ s
    • wideband (short window) : precise location of transistions, poor frequency resolution
    • narrowband (long window) : more frequency resolution, less precision in time
• discrete fourier series : DFS, $N$-periodic, equivalent to DFT of one period
  • analysis : $X[k]=\sum _{k=0}^{N-1}x[n]e^{-j\frac{2\pi }{N}nk}$ with $n\in \text{Z}$
  • synthesis : $x[n]=\frac{1}{N}\sum _{k=0}^{N-1}X[k]e^{j\frac{2\pi }{N}nk}$ with $n\in \text{Z}$
• discrete-time fourier transform : DTFT, infinite signals, mathematical tool
  • analysis : $F(\omega )=X(e^{j\omega })=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]e^{-j\omega n}=<e^{j\omega n},x[n]>$, represented over $[-\pi ,\pi ]$ with $x[n]\in l_2(\text{Z})$
  • synthesis : $x[n]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F(\omega )e^{j\omega n}d\omega$
  • periodicity : $F(\omega )$ is $2\pi$-periodic, can be rescales $M$ times $X(e^{j\omega M})$
  • splitting into chunks : $\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}y[n]=\sum _{p=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=0}^{M-1}y[pM+m]$
  • signal symmetric : DTFT symmetric $x[n]=x[-n]⟺X(e^{j\omega })=X(e^{-j\omega })$
  • signal real : DTFT hermitian symmetric $x[n]=x^[n]\iff X(e^{j\omega})=X^(e^{-j\omega})$ (symmetric magnitude)
  • signal symmetric real : DTFT symmetric and real
• review DFT : numerical algorithm (computable)
• review DFS
• review DTFT : mathematical tool (proofs)
• pulse train : $\tilde{\delta }(\omega )=2\pi \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\omega -2\pi k)$ (Dirac delta in space of $2\pi$-periodic functions)
• embedding finite-length signals : $N$-tap signal, spectral DFT to infinite sequence
  • periodic extension : $\tilde{x}[n]=x[n\mathrm{mod}N]$
    • $\tilde{X}(e^{j\omega })=\frac{1}{N}\sum _{k=0}^{N-1}X[k]\tilde{\delta }(\omega -\frac{2\pi }{N}k)$
  • finite-support extension : $\overline{x}[n]=x[n]$ for $0\le n\le N$ otherwise $0$
    • interval indicator signal : $\overline{r}[n]=1$ when $0\le n<N$ else 0
    • $\overline{X}(e^{j\omega })=\sum _{k=0}^{N-1}X[k]\mathrm{\Lambda }(\omega -\frac{2\pi }{N}k)$
    • $\mathrm{\Lambda }(\omega )=(1/N)\overline{R}(e^{j\omega })$ : smooth interpolation of DFT values
    • $\overline{R}(e^{j\omega })=\frac{\sin (\frac{\omega }{2}N)}{\sin (\frac{\omega }{2})}{e}^{-j\frac{\omega }{2}(N-1)}$ : DTFT of interval signal
• zero-padding : pad data vector with 0's to obtain nicer plots, no information added
• modulation : $x[n]$ baseband, $DTFTx[n]\cos ({\omega }_{c}n)=\frac{1}{2}[X(e^{j(\omega -{\omega }_c)})+X(e^{j(\omega +{\omega }_c)})]$, caution if too large
• demodulation : $DTFTx[n]\cos ({\omega }_{c}n)2\cos ({\omega }_{c}n)=X(e^{j\omega })+\frac{1}{2}[X(e^{j(\omega -2{\omega }_c)})+X(e^{j(\omega +2{\omega }_c)})]$
• fast fourrier transform : FFT
  • original complexity : $O(N^2)$ for $X=W_{n}x$
  • divide and conquer : $O(N\log (N))$

Linear systems

• LTI
  • linearity : $\text{H}{a}_1{x}_1[n]+a_2{x}_2[n]=a_1\text{H}{x}_1[n]+a_2\text{H}{x}_2[n]$
  • time invariance : $\text{H}x[n]=y[n]⟺\text{H}x[n-n_0]=y[n-n_0]$
• impulse response : $h[n]=\text{H}\delta [n]$
• convolution : $x[n]h[n]=\sum_{k=-\infty}^\infty x[k]h[n-k]=<x^[k],h[n-k]>$
  • algorithm : time-reverse $h[m]$, center it in $n$, compute inner product
• moving average : MA, $y[n]=\frac{1}{M}\sum _{k=0}^{M-1}x[n-k]$
  • impulse reponse : $h[n]=1/M$ for $0\le n<M$
  • smoothing effect : proportional to $M$, # operations and storage
• leaky integrator : LI, $y[n]=\lambda y[n-1]+(1-\lambda )x[n]$ with leak $\lambda <1$
  • impulse response : $h[n]=(1-\lambda ){\lambda }^{n}u[n]$
  • smoothing effect : depends on $\lambda$, constant operations and storage
• stability
  • finite impulse response (FIR) : finite support, finite # of samples involved in the computation of each output, always stable
  • infinite impulse response (IIR) : infinite support, infinite # of samples involed in the computation of each output (finite computation though)
  • causal : impulse response zero for $n<0$, only past samples
  • noncausal : impulse response nonzero for $n<0$, need future samples
  • bounded-input bounded-output (BIBO) : require impulse response absolutely summable
  • nice signal : bounded signal
• frequency reponse : DTFT of impulse response determines the frequency characteristic of a filter
  • eigensequences : complex exponentials, cannot change frequency of sinusoids
  • convolution theorem : multiplication in frequency space
  • amplitude : amplification when $\text{abs}H(e^{j\omega })>1$ or attentuation when $\text{abs}H(e^{j\omega })<1$
  • phase : overall delay, shape change
    • zero phase : $\mathrm{\angle }H(e^{j\omega }=0$
    • linear phase : $\mathrm{\angle }H(e^{j\omega })=d\omega$ (almost linear phase on region of interest can do also the trick)
    • nonlinear phase : for example $1/2\sin ({\omega }_{0}n)+\cos (2{\omega }_{0}n+2{\theta }_0)$

Ideal filters and approximation

• filter types (magnitude response)
  • lowpass : basebands, MA, LI
  • highpass
  • bandpass
  • allpass
• filter types (phase response)
  • linear phase
  • nonlinear phase
• ideal lowpass filter : $H(e^{j\omega })=1$ for $|\omega |\le {\omega }_c$ otherwise $0$ ($2\pi$-periodic implicit)
  • zero-phase : no delay
  • impulse response : $h[n]=\frac{\sin ({\omega }_{c}n)}{\pi n}$
  • cannot compute in finite time : infinite support, two-sided
  • approximation : decay slowly in time
  • sinc-rect pair : $rect(x)=1$ for $|x|\le 1/2$ and $sinc(x)=\frac{\sin (\pi x)}{\pi x}$ for $x\ne 0$, $1$ if $x=0$ ($0$ if $x$ nonzero integer, not summable)
• ideal highpass filter : $H(e^{j\omega })=1$ for $\pi \ge |w|\ge {\omega }_c$ ($2\pi$-periodic implicit)
  • impulse response : $H_{hp}(e^{j\omega })=1-H_{lp}(e^{j\omega })$ give $h_{hp}[n]=\delta [n]-\frac{{\omega }_c}{\pi }sinc(\frac{{\omega }_c}{\pi }n)$
• ideal bandpass filter : $H(e^{j\omega })=1$ for $|\omega \pm {\omega }_0|\le {\omega }_c$ ($2\pi$-periodic implicit)
  • impulse response : $h_{bp}[n]=2\cos ({\omega }_{0}n)\frac{{\omega }_c}{\pi }sinc(\frac{{\omega }_c}{\pi }n)$
• impulse truncation
  • FIR appromixation of length $M=2N+1$ : $\hat{h}[n]=h[n]$ for $|n|\le N$
  • MSE minimized by symmetric impulse truncation around zero : $MSE=\text{sumi}|h[n]-\hat{h}[n]{|}^2$
  • gibbs phenomenon : maximum error around 9% regardless of $N$
    • A, B integrals : $(A+B)/2\pi \approx 1.09$ with $A\approx \pi$ and $B\approx 2\pi 0.589$ (independant of $M$)
• window method : narrow mainlobe (transition sharp), small sidelobe (small error), short window (FIR efficient)
  • MSE minimized, simple
  • cannot control max error
• frequency sampling : take $M$ of desired frequency response $S[k]=H(e^{j{\omega }_k})$ where ${\omega }_k=(2\pi /M)k$, compute inverse DFT of $S[k]$, use $s[n]$ to build $\hat{h}[n]$ (finite support)
  • simple, works with arbitrary frequency response
  • cannot control max error
  • sampling in the frequency : periodization in time domain $s[n]=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}h[n+mM]$
• z-transform : power series, convergence always absolute
  • analysis : $X(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]z^{-n}$
  • synthesis :
  • region of convergence : ROC, circular symmetry
    • finit support signals : converge everywhere
    • causal signals : extends outwards from circle touching largest magnitude pole, converge if ROC include unit circle
    • independent of values of $H(z)$ : assume $Y(z)$ exists for $x[n]=\delta [n]$
    • $z_n$ : zeros of transfert function, glue sheet to ground
    • $p_n$ : poles of transfert function, push sheet up
  • constant coefficient difference equation : $\sum _{k=0}^{N-1}{a}_{k}y[n-k]=\sum _{k=0}^{M-1}{b}_{k}x[n-k]$ give $Y(z)\sum _{k=0}^{N-1}{a}_k=X(z)\sum _{k=0}^{M-1}{b}_{k}x[n-k]$ with $Y(z)=H(z)X(z)$

Filter structures

• abstract filter implementation
• resonator : narrow bandpass filter, detect presence of sinusoid of given frequency, shift passband of LI
  • $H(z)=\frac{G_0}{1-2\lambda \cos ({\omega }_0{z}^{-1})+|\lambda {|}^2{z}^{-2}}$
  • $a_1=2\lambda \cos {\omega }_0$ and $a_2=-|\lambda {|}^2$
• DC removal : DC balanced signal has zero sum, kill zero frequency
• hum removal : remove a specific nonzero frequency $H(z)=\frac{(1-e^{j{\omega }_0}{z}^{-1})(1-e^{-j{\omega }_0}{z}^{-1})}{(1-\lambda e^{j{\omega }_0}{z}^{-1})(1-\lambda e^{-j{\omega }_0}{z}^{-1})}$
• fractional delay : compute in-between values for sample, ideal, approximated with local interpolation
• Hilbert filter : allpass, $H(e^{j{\omega }_0})=-j$ for $0\le \omega <\pi$ or $j$ for $-\pi \le \omega <0$, $h[n]=\frac{2}{\pi n}$ for $n$ odd otherwise $0$
  • Hilbert demodulation : $(jy[n]\ast h[n]+y[n])e^{j{\omega }_{0}n}=x[n]$ for $y[n]=x[n]\cos ({\omega }_{0}n)$

Filter design

• requirements
  • frequency response : passband, stopband
  • phase : overall delay, linearity
  • limit on computational resources or precision
  • determine $N$, $M$, $a_k$, $b_k$ : $H(z)=\frac{b_0+b_1{z}^{-1}+\cdots +b_{M-1}{z}^{-(M-1)}}{a_0+a_1{z}^{-1}+\cdots +a_{N-1}{z}^{-(N-1)}}$, hard nonlinear problem
• limitations
  • transitions cannot be infinitely sharp : $H(z)=B(z)/A(z)$ is ${\text{C}}^{\mathrm{\infty }}$
  • magnitude response cannot be constant over an interval : $B(z)-cA(z)$ infinite number of roots, need to be $0$ for all $z$
  • higher filter order : small transition band, small error, expensive, large delay
  • equiriplle error
• IIR
  • pros : computationally efficient, strong attenuation easy, good for audio
  • cons : stability issues, difficult to design for arbitrary response, nonlinear phase
  • method : conversion fo analog design, deeply studied field
  • butterworth : lowpass
    • magnitude response : maximally flat, monotonic over $[0,\pi ]$
    • design parameters : order $N$ ($N$ poles and zeros), cutoff frequency
    • design test criterion : width of transition band, passband error
  • chebyshev : lowpass
    • magnitude response : equiriplle in passband, monotonic in stopband
    • design parameters : order $N$ ($N$ poles and zeros), passband max error, cutoff frequency
    • design test criterion : width of transition band, stopband error
  • elliptic : lowpass
    • magnitude response : equiriplle in passband and in stopband
    • design parameters : order $N$ ($N$ poles and zeros), passband max error, cutoff frequency, stopband min attenuation
    • design test criterion : width of transition band
  • magnitude response in decibels : filter attenuation expressed $A_{dB}=20{\log }_{10}(|H(e^{j\omega })|/G)$ with filter max passband $G$
• FIR
  • pros : always stable, optimal design techniques exist, can be designed with linear phase
  • cons : computationally much more expensive, may sound harsh
  • method : optimal minimax filter design, algorithm for linear phase, equiriplle error in passband and stopband (proceeds by minimizing the maximum error)
  • half length : $L$
  • even FIR : $M=2L$ taps
  • odd FIR : $M=2L+1$ taps
  • delay : $(M-1)/2$
  • FIR only have zeros : if $z_0$ zero, so is $z_0^{\ast }$ and $1/z_0$
  • type I : $h[L+n]=h[L-n]$, $h_d[n]=h[n+L]$, $h_d[n]=h_d[-n]$
    • $H(z)=z^{-L}(h_d[0]+\sum_{n=1}^Lh_dn)$
    • zero locations : $H(z^{-1})=z^{2L}H(z)$
  • type II : $h[n]=h[2L-1-n]$
    • $H(z)=z^{-C}\sum_{n=0}^{L-1}hn$ with $C=L-1/2$
    • zero locations : $H(z^{-1})=z^{2C}H(z)=z^{2L-1}H(z)$
    • always have zero : at $\omega =\pi$
  • type III
    • $H(z)=z^{-L}\sum_n h_dn$
    • always have zero : at $\omega =0,\pi$
  • type IV
• Parks-McClellan algorithm : optimal, polynomial fitting $H_d(e^{j\omega })=h_d[0]+2\sum _{n=1}^L{h}_d[n]\cos \omega n$
  • magnitude response : equiriplle in passband and stopband
  • design parameters : order $N$ (number of taps), passband edge ${\omega }_p$, stopband edge ${\omega }_s$, ratio of passband to stopband error ${\delta }_p/{\delta }_s$
  • design test criterion : width of transition band, stopband error
  • Chebyshev polynomials : $T_n(\cos \omega )=\cos n\omega$
    • $T_0(x)=1$
    • $T_1(x)=x$
    • $T_2(x)=2x^2-1$
    • $\dots$
    • $T_n(x)=2x{T}_{n-1}(x)-T_{n-2}(x)$
  • convert the space to $x=\cos \omega$
    • $l_p=[0,{\omega }_p]\to l_p^{\prime }=[\cos {\omega }_p,1]$
    • $l_s=[{\omega }_s,\pi ]\to l_s^{\prime }0[-1,\cos {\omega }_s]$
  • minimize maximum error : $E=\underset{P(x)}{min}\underset{x\in l_p^{\prime }\cup l_s^{\prime }}{max}|P(x)-D(x)|$
    • $P(x)=h_d[0]+\sum _{n=1}^{L}2h_d[n]T_n(x)$
    • $D(x)=1$ when $x\in l_p^{\prime }$ else 0
  • alternation theorem : min max works only if $P(x)-D(x)$ alternates $M+2$ times in $l_p^{\prime }\cup l_s^{\prime }$ between $\pm E$
    • works also with weighting function : $W(x)=1$ when $x\in l_p^{\prime }$ or ${\delta }_p/{\delta }_s$ else ($x\in l_s^{\prime }$)
    • weighted min max : $min\underset{x\in l_p^{\prime }\cup l_s^{\prime }}{max}\text{abs}W(x)[P(x)-D(x)]$
    • max number of alternations : $L+3$
      • polynomial of degree $L$ : $L-1$ local extrema
      • ${\omega }_p$ and ${\omega }_s$
      • sometimes : $\omega =0$ or $\omega =\pi$
    • in-band alternations : $L-1$
  • Remez algorithm : guess positions of alternations, if not satisfied find extrema or error and repeat
  • algorithm : $M$ filter coefficients, stopband and passband tolerances ${\delta }_s$ and ${\delta }_p$, if error too big, increase $M$ and retry
  • eliptic vs 51-tap minimax
• design bandpass and highpass by modulating lowpass

Real-time processing

• system clock : at most $T_s$
  • record value : $x_i[n]$
  • process value in causal filter
  • output value : $x_o[n]$
• buffering : interrupt for each samples would be too much overhead
• input
• output
• double buffering
  • delay : $d=T_s\frac{L}{2}$ seconds
  • underflow : CPU does not fill fast enough
• multiple buffering (> 2) for protection
  • delay : $d=T_{s}L$ seconds
  • underflow : more protection
• echo
• reverb : superposition of many echos with different delays and magnitudes, costly, cheap alternative is allpass $H(z)=\frac{-\alpha +z^{-N}}{1-\alpha z^{-N}}$
• non-LTI effects
  • distortion : clip signal $y[n]=trunc(ax[n])/a$
  • tremolo : sinusoidal amplitude modulation $y[n]=(1+\cos ({\omega }_{0}n)/G)x[n]$
  • flanger : sinusoidal delay $y[n]=x[n]+x[n-\lfloor d(1+\cos ({\omega }_{0}n))\rfloor ]$
  • wah : time-varying bandpass filter

Stochastic signal processing

• gaussian random variable : $f(x)=N(m,{\sigma }^2)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp -\frac{(x-m{)}^2}{2{\sigma }^2}$
  • mean : $m$
  • variance : ${\sigma }^2$
• uniform random variable : $f(x)=U(A,B)=\frac{1}{B-A}$
  • mean : $m=\frac{A+B}{2}$
  • variance : ${\sigma }^2=\frac{(B-A{)}^2}{12}$
• discrete uniform random variable : $f(x)=UA,B=\frac{1}{B-A+1}\sum _{k=A}^B\delta (x-k)$
  • mean : $m=\frac{A+B}{2}$
  • variance : ${\sigma }^2=\frac{(B-A+1{)}^2-1}{12}$
• autocorrelation : $R_{XY}=E[XY]$
• covariance : $C_{XY}=E[(X-m_X)(Y-m_Y)]$
  • uncorrelated elements : $E[XY]=E[X]E[Y]\Rightarrow C_{XY}=0$
  • independent elements : $f_{XY}(x,y)=f_X(x)f_Y(y)\Rightarrow C_{XY}=0$
• random vector
  • cdf : $F_X(\alpha )=P[X_n\le {\alpha }_n,n=0,\dots ,N-1]$ with $\alpha \in {\text{R}}^N$
  • pdf : $f_X(\alpha )=\frac{{\partial }^N}{\partial {\alpha }_0\cdots \partial {\alpha }_{N-1}}{F}_X({\alpha }_0,\dots ,{\alpha }_{N-1})$
  • mean : $E[X]=m_x=[E[X_0]\dots E[X_{N-1}]{]}^T\in {\text{R}}^N$
  • cross-correlation : $R_{XY}=E[X{Y}^T]\in {\text{R}}^{N\times N}$
  • covariance matrix : $C_{XY}=E[(X-m_X)(Y-m_Y{)}^T]\in {\text{R}}^{N\times N}$, symmetric, postive-definite, diagonal if uncorrelated or independent
• gaussian random vector : $f(x)=\frac{1}{\sqrt{(2\pi {)}^n\text{abs}\mathrm{\Lambda }}}\exp -\frac{1}{2}(x-m{)}^T{\mathrm{\Lambda }}^{-1}(x-m)$
  • mean : $m\in {\text{R}}^N$
  • covariance matrix : $\mathrm{\Lambda }\in {\text{R}}^{N\times N}$
  • uncorrelated elements : independent elements
• random process
  • full characterization : all possible sets of $k$ indices for all $k\in \text{N}$, too much
  • first-order : $f_{X[n]}(x[n])]\to$ time-varying mean $\overline{x}[n]=E[x[n]]$
  • second-order : first order + $f_{X[n]X[m]}(x[n],x[m])\to$ time-varying cross-correlation ${\sigma }_x[n,m]=E[x[n]x[m]]$
  • ... : time-varying further moment
• stationarity : time-invariant partial-order descriptions $f_{X[n_0]\cdots X[n_{k-1}]}(\cdots )=f_{X[n_0+M]\cdots X[n_{k-1+M}]}(\cdots )$
  • mean : time-invariant
  • correlation : depends only on time lag
  • higher-order moments : depend only on relative time differences
• wide-sense stationarity (WSS) : stationarity up to second-order descriptions
  • mean : $E[X[n]]=m$
  • correlation : $E[X[n]X[m]]=r[n-m]$
• independent, identically distributed process (IID) : $f_{X[n_0]\cdots X_{[}{n}_{k-1}]}(x)={\mathrm{\Pi }}_{n=0}^{k}f(x_n)$
  • mean : $E[X[n]]=m$
  • correlation : $E[X[n]X[m]]={\sigma }^2\delta [n-m]$
• power spectral density (PSD) : independent of pdf
  • intuition : $P[k]=E[\text{abs}{X_N[k]}^2/N]$
    • energy distributions in frequency : $\text{abs}{X_N[k]}^2$
    • power distributions in frequency : $\text{abs}{X_N[k]}^2/N$, aka density
    • frequency-domain representations for stochastic processes is power
    • $P[k]=1$ : power is equally distributed over all frequencies, cannot predict how fast signal moves
  • truncated DTFT : $X_N(e^{j\omega })=\sum _{n=-N}^{N}x[n]e^{-j\omega n}$
  • for signal : $P(e^{j\omega })=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{2N+1}\text{abs}{X_N(e^{j\omega })}^2$
  • for WSS random process : $P_X(e^{j\omega })=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{2N+1}E[\text{abs}{X_N(e^{j\omega })}^2]=\underset{N\to \mathrm{\infty }}{lim}\sum _{k=-2N}^{2N}\frac{2N+1-\text{abs}k}{2N+1}{r}_X[k]e^{-j\omega k}$
    • $P_X(e^{j\omega })=\underset{N\to \mathrm{\infty }}{lim}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{w}_N[k]r_X[k]e^{-j\omega k}=DTFT{r}_X[k]$
    • $\underset{N\to \mathrm{\infty }}{lim}{w}_N[k]=1$
• white noise : IDD
  • zero mean : autocorrelation = covariance
  • $P_X(e^{j\omega })={\sigma }^2$
• additive white gaussian noise (AWGN) : IID, zero mean gaussian process, WSS $\Rightarrow$ full stationary
• filter random process : output is WSS if input WSS
  • mean : $m_Y=m_{X}H(e^{j0})$
  • correlation : $r_Y[n]=h[n]\ast h[-n]\ast r_X[n]$
  • power distribution : $P_Y(e^{j\omega })=\text{abs}{H(e^{j\omega })}^2{P}_X(e^{j\omega })$
  • deterministict signal filter : works in magnitude in stochastic case but concept of phase lost

Interpolation

• bridging two world : sampling and interpolation
• continuous-time signal processing : $x(t)\in L_2(\text{R})$ complex function of real variable with finite energy
• analog LTI filters : $y(t)=(xh)(t)=<h^(t-\tau),x(\tau)>$
• fourier
  • frequence : $F=\frac{\mathrm{\Omega }}{2\pi }$ Hz with $\mathrm{\Omega }$ rad/s and period $T=\frac{1}{F}$
  • analysis : $X(j\mathrm{\Omega })=<e^{j\mathrm{\Omega }t},x(t)>={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x(t)e^{-j\mathrm{\Omega }t}dt$ not periodic
  • synthesis : $x(t)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}X(j\mathrm{\Omega })e^{j\mathrm{\Omega }t}d\mathrm{\Omega }$
• convolution theorem : $Y(j\mathrm{\Omega })=X(j\mathrm{\Omega })H(j\mathrm{\Omega })$
• bandlimitedness : $X(j\mathrm{\Omega })=0$ for $\text{abs}\mathrm{\Omega }>{\mathrm{\Omega }}_N$
  • total bandwidth : ${\mathrm{\Omega }}_B=2{\mathrm{\Omega }}_N$
  • prototypical function : $T_s=\frac{2\pi }{{\mathrm{\Omega }}_B}=\frac{\pi }{{\mathrm{\Omega }}_N}$
    • $\phi (t)=\text{sinc}(\frac{t}{T_s})$
    • $\mathrm{\Phi }(j\mathrm{\Omega })=\frac{\pi }{{\mathrm{\Omega }}_N}rect(\frac{\mathrm{\Omega }}{2{\mathrm{\Omega }}_N})$
• interpolation : fill the gaps between samples
  • requirement : decide interval $T_s$, make sure $x(n{T}_s)=x[n]$, make sure $x(t)$ is smooth (infinitely differentiable)
• polynomial interpolation : $N$ points, polynomial of degree $N-1$, $p(t)=a_0+a_{1}t+\cdots +a_{N-1}{t}^{N-1}$
  • $N$-degree polynomial bases in interval
    • naive : $1,t,\dots ,t^N$
    • Legendre basis : orthonormal, increasing degree
    • Chebyshev basis : orthonormal, increasing degree
    • Lagrange : interpolation property, same degree
  • consider symmetric interval : $l_N=[-N,\dots ,N]$
  • naive : $p(0)=x[0]$, $p(T_s)=x[1]$, $p((N-1)T_s)=x[N-1]$
  • space of degree-$2N$ polynomial over $l_N$ : $P_N$
  • $P_N$ basis : $2N+1$ Lagrange polynomials $L_n^{(N)}(t)={\mathrm{\Pi }}_{k=-N,k\ne n}^N\frac{t-k}{n-k}$ for $n=-N,\dots ,N$
  • Lagrange interpolation : $p(t)=\sum _{n=-N}^{N}x[n]L_n^{(N)}(t)$
    • polynomial of degree $2N$ through $2N+1$ points unique
    • satifies : $p(n)=x[n]$ for $-N\le n\le N$ since $L_n^{(N)}(m)=1$ only if $n=m$ else $0$
  • pros : maximally smooth
  • cons : interpolation bricks depend on $N$
• zero-order interpolation : $-N\le t\le N$, $x(t)=\sum _{n=-N}^{N}x[n]rect(t-n)$
  • $x(t)=x[\lfloor t+0.5\rfloor ]$
  • kernel : $i_0(t)=rect(t)$ (support $1$)
  • not continuous
• first-order interpolation : $x(t)=\sum _{n=-N}^{N}x[n]i_1(t-n)$
  • kernel : $i_1(t)=1-\text{abs}t$ for $\text{abs}t\le 1$ else 0 (support $2$)
  • continuous but derivative is not
• third-order interpolation : $x(t)=\sum _{n=-N}^{N}x[n]i_3(t-n)$
  • kernel : splicing two cubic polynomials (support $4$)
  • continous up to second derivative
• local interpolation : $x(t)=\sum _{n=-N}^{N}x[n]i_c(t-n)$
  • requirements : $i_c(0)=1$ and $i_c(t)=0$ for $t$ nonzero integer
  • pros : same interpolating function independently of $N$
  • cons : lack of smoothness
  • limit : local = global interpolation, $\underset{N\to \mathrm{\infty }}{lim}{L}_n^{(N)}(t)=f(t-n)$
• sinc interpolation : $x(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]\text{sinc}(\frac{t-n{T}_s}{T_s})$
  • convergence : $\text{sinc}(t-n)$ and $L_n^{(\mathrm{\infty })}(t)$ share an infinite number of zeros
  • spectral representation : $X(j\mathrm{\Omega })=(\pi /{\mathrm{\Omega }}_N)X(e^{j\pi (\mathrm{\Omega }/{\mathrm{\Omega }}_N)})$ for $\text{abs}\mathrm{\Omega }\le {\mathrm{\Omega }}_N$ else $0$
  • fast interpolation : $T_s$ small, wider spectrum
  • slow interpolation : $T_s$ large, narrower spectrum
• space of bandlimited functions : ${\mathrm{\Omega }}_N$-BL$\subset L_1(\R)$, hilbert space
• sampling theorem
  • special case : $T_s=1$ and ${\mathrm{\Omega }}_N=\pi$
    • basis : ${\phi }^{(n)}(t)=\text{sinc}(t-n)$ with $n\in \text{Z}$
    • sampling as basis expansion : $<{\phi }^{(n)}(t),x(t)>=<\text{sinc}(t-n),x(t)>=<sinc(n-t),x(t)>=(\text{sinc}\ast x)(n)\text{=}\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}rect(\frac{\mathrm{\Omega }}{2\pi })X(j\mathrm{\Omega })e^{j\mathrm{\Omega }n}d\mathrm{\Omega }=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X(j\mathrm{\Omega })e^{j\mathrm{\Omega }n}d\mathrm{\Omega }=x(n)$
    • analysis : $x[n]=<\text{sinc}(t-n),x(t)>$
    • synthesis : $x(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]\text{sinc}(t-n)$
    • sufficient representation : $x[n]=x(n)$ with $n\in \text{N}$
  • general case : $T_s=\pi /{\mathrm{\Omega }}_N$
    • basis : ${\phi }^{(n)}(t)=\text{sinc}((t-n{T}_s)/T_s)$
    • analysis : $x[n]=<\text{sinc}(\frac{t-n{T}_s}{T_s}),x(t)>=T_{s}x(n{T}_s)$
    • synthesis : $x(t)=\frac{1}{T_s}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]\text{sinc}(\frac{t-n{T}_s}{T_s})$
    • coefficient : $T_{s}x(n{T}_s)$ for any $x(t)\in\Omega_N-$BL
    • sufficient representation : $x[n]=x(n{T}_s)$ for any $x(t)\in {\mathrm{\Omega }}_N$-BL
  • corollary : ${\mathrm{\Omega }}_N$-BL$\subset\Omega$-BL for any $\mathrm{\Omega }\ge {\mathrm{\Omega }}_N$
    • sufficient representation : $x[n]=x(n{T}_s)$ for any $x(t)\in {\mathrm{\Omega }}_N$-BL and any $T_s\le \pi /{\mathrm{\Omega }}_N$
  • hertz : any signal $x(t)$ bandlimited to $F_N$Hz can be sampled with no loss of information using sampling frequency $F_s\ge 2F_N$ (i.e. sampling period $T_s\le 1/2F_N$)

Sampling and application

• sinc sampling
• sinc sampling for ${\mathrm{\Omega }}_N$-BL signals
• raw sampling
• raw sampling continuous-time complex expoential : $x(t)=e^{j{\mathrm{\Omega }}_{0}t}$
  • all angular speed allowed
  • periodic : $T=2\pi /{\mathrm{\Omega }}_0$
  • $FT{e}^{e^{j{\mathrm{\Omega }}_{0}t}}=2\pi \delta (\mathrm{\Omega }-{\mathrm{\Omega }}_0)$
  • bandlimited : to ${\mathrm{\Omega }}_0^{+}$
  • raw sample : snapshots at regular intervals of rotating point $x[n]=e^j{\mathrm{\Omega }}_0{T}_{s}n$
    • digital frequency : ${\omega }_0={\mathrm{\Omega }}_0{T}_s$
    • easy : $T_s<\pi /{\mathrm{\Omega }}_0\Rightarrow {\omega }_0<\pi$
    • tricky : $\pi /{\mathrm{\Omega }}_0<T_s<2\pi /{\mathrm{\Omega }}_0\Rightarrow \pi <{\omega }_0<2\pi$
    • trouble : $T_s>2\pi /{\mathrm{\Omega }}_0\Rightarrow {\omega }_0>2\pi$
  • aliasing
• raw sampling arbitrary signal
  • spectrum : $x[n]=x_c(n{T}_s)$, start with inverse Fourier Transform
    • split intergration interval (aliased) : $x[n]=\frac{1}{2\pi }\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{(2k-1){\mathrm{\Omega }}_N}^{(2k+1){\mathrm{\Omega }}_N}{X}_c(j\mathrm{\Omega })e^{j\mathrm{\Omega }n{T}_s}d\mathrm{\Omega }$
    • periodzation : ${\tilde{X}}_c(j\mathrm{\Omega })=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{X}_c(j(\mathrm{\Omega }-2k{\mathrm{\Omega }}_N))$ give $x[n]=\frac{1}{2\pi }{\int }_{-{\mathrm{\Omega }}_N}^{{\mathrm{\Omega }}_N}{\tilde{X}}_C(j\mathrm{\Omega })e^{j\mathrm{\Omega }{T}_{s}n}d\mathrm{\Omega }$
    • set $\omega =\mathrm{\Omega }{T}_s$ : $x[n]=IDTFT\frac{1}{T_s}{\tilde{X}}_C(j\frac{\omega }{T_s})$
    • final : $X(e^{j\omega })=\frac{\pi }{{\mathrm{\Omega }}_N}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{X}_C(j\frac{\omega {\mathrm{\Omega }}_N}{\pi }-2jk{\mathrm{\Omega }}_N)$
    • bandlimited to ${\mathrm{\Omega }}_0$ : ${\mathrm{\Omega }}_N>{\mathrm{\Omega }}_0$
    • bandlimited to ${\mathrm{\Omega }}_0$ : ${\mathrm{\Omega }}_N={\mathrm{\Omega }}_0$
    • bandlimited to ${\mathrm{\Omega }}_0$ : ${\mathrm{\Omega }}_N<{\mathrm{\Omega }}_0$
    • non-bandlimited
• sampling stategies : given a sampling period $T_s$
  • bandlimited to $\pi /T_s$ or less : raw sampling is fine (equivalent to sinc sampling up to scaling factor $T_s$)
  • non-bandlimited
    • bandlimit via a lowpass filter in continuous-time domain before sampling
      • raw sample and incur aliasing - bandlimiting : optimal with respect to least squares approximation
  • aliasing : error cannot be controlled, better bandlimit
• sinc sampling and interpolation
• least squares approximation with sinc sampling and interpolation
• processing of analog signals
  • impulse invariance : $H_C(j\mathrm{\Omega })=H(e^{j\pi \mathrm{\Omega }/{\mathrm{\Omega }}_N})$
  • duality : $H(e^{j\omega })=H_C(j\omega )$
  • fractional delay : $H(e^{j\omega })=e^{-j\omega d}$ with $d\notin \text{Z}$
    • discrete time : ideal filter (approximation with cubic local interpolation)
  • digital differentiator : $H(e^{j\omega })=j\omega$, $FT{x}_c^{\prime }(t)=j\mathrm{\Omega }{X}_C(j\mathrm{\Omega })$
    • discrete time : $h[n]=\frac{(-1{)}^n}{n}$ for $n\ne 0$ else $0$, ideal filter (approximations exist)

Multirate signal processing

• advanced sampling
  • from continous to discrete
  • discrete to continuous
  • practial interpolation :
    • time : $x(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]i(\frac{t-n{T}_s}{T_s})$
    • frequency : $X(j\mathrm{\Omega })=T_s\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]I(j{T}_s\mathrm{\Omega })e^{jn{T}_s\mathrm{\Omega }}=\frac{\pi }{{\mathrm{\Omega }}_N}I(j\pi \frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_N})X(e^{j\pi \frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_N}})$
  • sinc interpolation : $i(t)=\text{sinc}(t)$ and $I(j\mathrm{\Omega })=rect(\frac{\mathrm{\Omega }}{2\pi })$
  • zero-order hold : $i(t)=rect(t)$ and $I(j\mathrm{\Omega })=sinc(\frac{\mathrm{\Omega }}{2\pi })$ ($X(j\Omega)=\frac{\pi}{\Omega_N}\sinc(\frac{\Omega}{2\Omega_N})X(e^{j\pi\Omega/\Omega_N})$)
  • first-order
• bandpass sampling : sampling theorem is only sufficient condition, in theory ${\mathrm{\Omega }}_N>{\mathrm{\Omega }}_{max}$ what if signal is bandpass
  • bandpass : $X(j\mathrm{\Omega })=0$ for $\text{abs}\mathrm{\Omega }-{\mathrm{\Omega }}_c>{\mathrm{\Omega }}_0$
  • no aliasing : requires at least ${\mathrm{\Omega }}_N\ge {\mathrm{\Omega }}_0$
  • baseband condition : ${\mathrm{\Omega }}_N={\mathrm{\Omega }}_c/k$ for some $k\in \text{N}$
  • samples : for $T_s=\pi /{\mathrm{\Omega }}_N$, $x[n]=x(n{T}_s)$ is alias-free, baseband DT version of $x(t)$
  • AM channel
• upsampling : zoom, $X_{NU}[n]=x[k]$ for $n=kN$ else $0$ with $k\in \text{Z}$
  • zero-fill
  • spectral representation : $X_{NU}(e^{j\omega })=X(e^{j\omega N})$
    • time domain : zeros between nonzero samples not natural
    • frequency domain : extra replicas of spectrum not natural
  • zero-order interpolator : for $N$-upsampling, $i_0[n]=u[n]-u[n-N]$ and $\text{abs}{I}_0(e^{j\omega })=\text{abs}\frac{\sin (\frac{\omega }{2}N)}{\sin (\frac{\omega }{2})}$
  • first-order interplator : $i_1[n]=(i_0[n]\ast i_0[n])/N$
  • ideal digital interpolator
• downsampling : dezoom, $X_{ND}[n]=x[nN]$
  • spectral representation : $X_{ND}(e^{j\omega })=\frac{1}{N}\sum _{m=0}^{N-1}X(e^{j\frac{w-2\pi m}{N}})$
  • without aliasing
  • with aliasing
  • with aliasing filter
  • caution with highpass signal
  • rational sampling rate change : in pratice, time-varying local interpolation
  • subsample interpolation : compute $x(n+\tau )$ with $\text{abs}\tau <1/2$
    • local Lagrange approximation : $x(n+\tau )\approx x_L(n;\tau )=\sum _{k=-N}^{N}x[n-k]L_k^{(N)}(\tau )=(x\ast d_{\tau })[n]$ with $2N+1$-tap FIR $d_{\tau }[k]=L_k^{(N)}(\tau )$

Quantization

• quantitzation : map range of signal onto finite set
  • factors : storage budget (bits per sample), storage scheme (fixed, floating), input properties (range, distribution)
  • irreversiable : quantization noise
  • simplest quantizer : each sample encoded individually (scalar), quantized independently (memoryless), encoded using $R$ bits
  • scalar : $A\le x[n]\le B$, $2^R$ intervals
  • error : $e[n]=Qx[n]-x[n]=\tilde{x}[n]-x[n]$, model error as white noise
  • uniform : interval $\mathrm{\Delta }=(B-A)2^{-R}$
    • hypothesis : $f_x(\tau )=\frac{1}{B-A}$
    • MSE : ${\sigma }_e^2=\sum _{k=0}^{2^R-1}{\int }_{l_k}{f}_x(\tau )({\tilde{x}}_k-\tau {)}^{2}d\tau$
    • minimizing error : $\frac{\partial {\sigma }_e^2}{\partial {\tilde{x}}_m}=0$ for ${\tilde{x}}_m=A+m\mathrm{\Delta }+\frac{\mathrm{\Delta }}{2}$, interval midpoint
    • error energy : ${\sigma }_e^2=\frac{{\mathrm{\Delta }}^2}{12}$
    • signal energy : ${\sigma }_x^2=(B-A{)}^2/12$
    • signal to noise ratio : $SNR={\sigma }_x^2/{\sigma }_e^2=2^{2R}$ ($SN{R}_{dB}=10{\log }_{10}{2}^{2R}\approx 6R$ db)
  • unbounded : clip samples to $[A,B]$ (linear distortion) vs smoothly saturate input (simulate saturation curves of analog electronics)
  • gaussian : ${\sigma }_e^2=\frac{\sqrt{3}\pi }{2}{\sigma }^2{\mathrm{\Delta }}^2$
  • Lloyd-Max algo : design optimal quantizer for distribution
  • companders
• A/D converter : sampling discretizes time, quantization discretises amplitude
  • oversampling : reduce quantization error
• D/A converter
  • oversampling : cheaper hardware for interpolation, undo in-band distortion in analog domain, significant out-of-band distortion, minimal D/A rate
  • oversampling using ZOH

Image processing

• digital image : two-dimensional $M_1\times M_2$ finite-support signal $x[n_1,n_2]$, pixel grid
  • grayscale : scalar pixel value
  • color : multidimensional pixel value
• still works : linearity, convolution, fourier, interpolation, sampling
• breaks : fourier analysis less relevant, filter design hard, IIRs rare, linear operators only mildly useful
• news : affine transforms, finite-support signals, causality loses meaning
• rect : $rect(\frac{n_1}{2N_1},\frac{n2}{2N_2})=1$ only if $\text{abs}{n}_1<N_1$ and $\text{abs}{n}_2<N_2$ else $0$
• separable signals
  • dirac : $\delta [n_1,n_2]=\delta [n_1]\delta [n_2]$
  • rect : $rect(\frac{n_1}{2N_1},\frac{n2}{2N_2})=rect(\frac{n_1}{2N_1})rect(\frac{n_2}{2N_2})$
• nonseparable signal : $x[n_1,n_2]=1$ if $\text{abs}{n}_1+\text{abs}{n}_2<N$ else $0$
• 2D convolution : $x[n_1,n_2]\ast h[n_1,n_2]=\sum _{k_1=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{k_2=-\mathrm{\infty }}^{\mathrm{\infty }}x[k_1,k_2]h[n_1-k_1,n_2-k_2]$
  • operations : $M_1\ast M_2$ per output sample
• 2D convolution for separable signals : $x[n_1,n_2]h[n_1,n_2]=\sum_{k_1=-\infty}^\infty h_1[n_1-k_1]\sum_{k_2=-\infty}^\infty x[k_1,k_2]h_2[n_2-k_2]=h_1[n_1](h_2[n_2]*x[n_1,n_2])$
  • operations : $M_1+M_2$ per output sample
• affine transforms : mapping ${\text{R}}^2\to {\text{R}}^2$ that reshapes coordinate system $t^{\prime }=At-d\in {\text{R}}^2\ne {\text{Z}}^2$
  • translation : $A=I$, $d=\begin{bmatrix}d_1 \\ d_2\end{bmatrix}$
  • scaling : $A=\begin{bmatrix}a_1 & 0 \\ 0 & a_2\end{bmatrix}$, $d=0$
  • rotation : $A=\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$, $d=0$
  • horizontal flips : $A=\begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix}$, $d=0$
  • vertical flips : $A=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, $d=0$
  • horizontal shear : $A=\begin{bmatrix}1 & s \\ 0 & 1\end{bmatrix}$, $d=0$
  • vertical shear : $A=\begin{bmatrix}1 & 0 \\ s & 1\end{bmatrix}$, $d=0$
  • discrete-space solution
    • inverse transform : $t=A^{-1}(m+d)$
    • interpolate mid-points : $(t_1,t_2)=({\eta }_1+{\tau }_1,{\eta }_2+{\tau }_2)$ with ${\eta }_{1,2}\in \text{Z}$ and $0\le {\tau }_{1,2}<1$
• bilinear interpolation
  • first-order interpolator : $y[m_1,m_2]=(1-{\tau }_1)(1-{\tau }_2)x[{\eta }_1,{\eta }_2]+{\tau }_1(1-{\tau }_2)x[{\eta }_1+1,{\eta }_2]+(1-{\tau }_1){\tau }_{2}x[{\eta }_1,{\eta }_2+1]+{\tau }_1{\tau }_{2}x[{\eta }_1+1,{\eta }_2+1]$
• 2D DFT
  • analysis : $X[k_1,k_2]=\sum _{n_1=0}^{N_1-1}\sum _{n_2=0}^{N_2-1}x[n_1,n_2]e^{-j\frac{2\pi }{N_1}{n}_1{k}_1}{e}^{-j\frac{2\pi }{N_2}{n}_2{k}_2}$
  • synthesis : $x[n_1,n_2]=\frac{1}{N_1{N}_2}\sum _{k_1=0}^{N_1-1}\sum _{k_2=0}^{N_2-1}X[k_1,k_2]e^{-j\frac{2\pi }{N_1}{n}_1{k}_1}{e}^{-j\frac{2\pi }{N_2}{n}_2{k}_2}$
  • orthogonal basis : $N_1{N}_2$ vectors, $w_{k_1,k_2}[n_1,n_2]=e^{j\frac{2\pi }{N_1}{n}_1{k}_1}{e}^{j\frac{2\pi }{N_2}{n}_2{k}_2}$
  • basis for $k_1=3$ and $k_2=230$
  • matrix form
  • HDR images : need to compress the levels
    • remove flagrant outliers
    • nonlinear mapping : $y=x^{1/3}$ after normalization $x\le 1$
  • magnitude : does not carry much information
  • phase : carry some info
• image filtering
  • lowpass : smoothing
  • highpass : enhancement, edge detection
  • 2D IIRs : nonlinear phase, border effects, stability (fundamental theorem of algebra does not hold in multiple dimensions)
  • noncomputable CCDE
  • 2D FIR : generally zero centered, odd number of taps in both directions, stable
    • moving average
    • gaussian blur : $h[n_1,n_2]=\frac{1}{2\pi {\sigma }^2}{e}^{-\frac{n_1^2+n_2^2}{2{\sigma }^2}}$
    • sobel
    • Laplacian : $\mathrm{\Delta }f(t_1,t_2)=\frac{{\partial }^{2}f}{\partial t_1^2}+\frac{{\partial }^{2}f}{\partial t_2^2}$

Image compression

• compression : exploit physical redundancy, allocate bits for things that matter
  • lossy : some information discarded
  • key : compressing at block level, using suitable transform (change of basis), smart quantization, entropy coding
• level compressing
  • pixel level : reduce number bits per pixel, coarser quantization, limit 1bpp
  • block level : divide in blocks, code average value with 8 bits, less than 1bpp, exploit local spatial correlation, compress remote regions independently
  • transform coding
  • discrete cosine transform : $C[k_1,k_2]=\sum _{n_1=0}^{N-1}\sum _{n_2=0}^{N-1}x[n_1,n_2]\cos [\frac{\pi }{N}(n_1+1/2)k_1]\cos [\frac{\pi }{N}(n_2+1/2)k_2]$
• smart quantization
  • standard : $\hat{x}=floor(x)+0.5$
  • deadzone : $\hat{x}=round(x)$
  • variable step : fine to coarse
• JPEG standard
  • split image into 8x8 non-overlapping blocks
  • computed DCT of each block
  • quantize DCT coefficients according to psycovisually-tuned tables
  • zigzag scan to maximize ordering
  • run-length encoding : each nonzero value encoded as $[(r,s),c]$, 8 bit per pair
    • runlength : $r$ number of zeros before current value
    • size : $s$ number of bits needed to encode value
    • actual value : $c$
    • $(0,0)$ : indicates only zeros from now
  • huffman coding

Digital communication system

• success factors
  • DSP paradigm : integer easy to regenerate, good phase control, adaptive algo
  • algo : entropy coding, error correction, trellis-coded modulation, viterbi
  • hardware : miniatuziation, general-purpose platforms, power efficiency
• analog channel : affect capacity
  • bandwidth constraint : grow as $1/T_s$
  • power constraint : limit max amplitude
  • AM radio channel : from $530$kHz to $1.7$MHz, each channel $8$kHz, power limited by law (daytime, nighttime, interference, health)
• channel capacity : maximum amount of information that can be reliably delivered over a channel (bit per second)
• all-digital paradigm : keep everything digital until we hit physical channel
• transmitter design
  • white random sequence : $a[n]$
  • shaping bandwidth : shrink support of full-band signal with upsampling (does not change data rate)
  • Baud rate : available bandwidth $W=F_{max}-F_{min}$, $F_s=KW>2F_{max}$ thus ${\omega }_{max}-{\omega }_{min}=2\pi \frac{W}{F_s}=\frac{2\pi }{K}$
  • raised cosine : $1/n^2$ decay
• transmission reliability : $â[n]=a[n]+\eta [n]$
  • error : depends on power of noise vs of signal, decoding strategy, alphabet
  • signaling alphabet
    • mapper : split incoming bitstream into chunks, assign symbol $a[n]$ from finite alphabet $A$ to each
    • slicer : receive value $â[n]$, decide which symbol is closest, piece back blocks
  • two-level signaling :
    • alphabet : $\pm G$
    • transmitted power : ${\sigma }_s^2=G^2$
    • error probability : $Q(G/{\sigma }_0)=Q({\sigma }_s/{\sigma }_0)=Q(\sqrt{SNR})$
• signaling scheme
  • PAM : chunks of $M$ bits (corresponding $k[n]\in 0,\dots ,2^M-1$), $a[n]=G((-2^M+1)+2k[n])$, $a^{\prime }[n]=\arg \underset{a\in A}{min}[\text{abs}â[n]-a]$, zero-mean, distance $2G$
  • QAM : chunks of $M$ even bits, $M/2$ for PAM $a_r[n]$ and $a_i[n]$, $a[n]=G(a_r[n]+j{a}_i[n]$, $a^{\prime }[n]=\arg \underset{a\in A}{min}[\text{abs}â[n]-a]$
    • error probability : circular approximation $P_{err}\approx \exp -\frac{G^2}{{\sigma }_0^2}\approx \exp -3SNR/2^{M+1}$
    • transmitted power : ${\sigma }_s^2=G^2\frac{2}{3}(2^M-1)$
    • recipe : $M={\log }_2(1-\frac{3}{2}\frac{SNR}{\ln (p_e)})$
    • passband : $c[n]=b[n]e^{j{\omega }_{c}n}$, $s[n]=\mathrm{\Re }c[n]=b_r[n]\cos {\omega }_{c}n-b_i[n]\sin {\omega }_{c}n$
• receiver
  • issues
    • propagation delay : handshake, delay estimation
    • linear distortion : $D(j\mathrm{\Omega })$, adapative equalization
    • interference : line probing
    • clock drifts : timing recovery
  • Hilbert demodulation
  • throughput : $WM$ bps
• shannon capacity upperbound : $C=W{\log }_2(1+SNR)$
• delay compensation : $D(j\mathrm{\Omega })=e^{-j\mathrm{\Omega }d}$, $d=(L+\tau )T_s$ with $\text{abs}\tau <1/2$ and $L\in \text{N}$
  • bulk delay : $L$
  • fractional delay : $\tau$
    • transmit : $b[n]=e^{j{\omega }_{0}n}$, $s[n]=\cos (({\omega }_c+{\omega }_0)n)$
    • receive : $\hat{s}[n]=\cos (({\omega }_c+{\omega }_0)(n-L-\tau ))$
    • after demodulation : $\hat{b}[n]=e^{j{\omega }_0(n-\tau )}$ (known frequency)
    • compensating : $\hat{s}[n]=s((n-\tau )T_s)$ (after offsetting bulk delay), in theory compensate with sinc fractional delay ($h[n]=\sinc(n+\tau)$), in practive filter local Lagrange approx
• adaptive equalization : in theory $E(z)=1/D(z)$ but change over time
  • bootstrapping using training sequence
• ADSL channel : $N$ sub-channel, $M$-QAM
  • discrete multitone modulation : insteadd of lowpassing, use $2N$-tap interval indicator
  • complex output signal : $c[n]=\sum _{k=0}^{N-1}{a}_k[\lfloor n/2N\rfloor ]e^{j\frac{2\pi }{2N}nk}$
  • specs : $F_{max}=1104$KHz, $N=256$, $M=4$, voice channel 0-7, upstream data: 7-31, max throughput 14.9Mbps (downstream)