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# Tutorial 7

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ELE3410 Random Processes and DSP THE CHINESE UNIVERSITY OF HONG KONG Department of Electronic Engineering ELE3410 Random Processes and DSP Tutorial 6 Power Spectral Density 2D Gaussian and n-D Gaussian 1. Power Spectral Density What is Power Spectral Density? Parseval’s Theorem (for aperiodic signal): ∞ ∞2 2y (t) dt = |Y ( f ) | df ∫ ∫− ∞ − ∞For ergodic random process, its average power: T1 2lim x (t)dt∫T → ∞ 2T −T∞1 2= lim | X ( f ) | df∫T → ∞ 2T − ∞∞= G( f ) df∫− ∞1 2 G( f ) ≡ lim | Χ( f ) |Then . T → ∞ 2TNote: (1)G(f) ≥ 0 ∀ f (2)G(f) = G(-f) G(f) power spectral density [watt/Hz] Tutorial 6 [1/5] ELE3410 Random Processes and DSP Power spectral density function(PSD) shows the strength of the variations (energy) as a function of frequency. In other words, it shows at which frequencies variations are strong and at which frequencies variations are weak. The unit of it is energy per frequency and you can obtain energy within a specific frequency range by integrating PSD within that frequency range. How to compute PSD? Computation of PSD is done by computing autocorrelation function and then transforming it using DFT. G(f) & R(τ) constitute a Fourier Transform pair: ∞ ∞1− j2 πf τ j2 πf τG( f ) = R( τ) ⋅e d τ & R( τ) = G( f ) ⋅e df ∫ ∫2 π− ∞ − ∞ Question 1 (Assignment 6) Find the power spectral density G( ω) corresponding to the autocorrelation function 1 for| τ | < T⎧ 0 .⎨R( τ) ...
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ELE3410 RandomProcesses and DSP THE CHINESE UNIVERSITY OF HONG KONG Department of Electronic Engineering ELE3410 Random Processes and DSP Tutorial 6 Power Spectral Density 2D Gaussian and nD Gaussian 1.Power Spectral Density What is Power Spectral Density? Parseval’s Theorem (for aperiodic signal): ∞ ∞ 2 2 y(t)dt=|Y(f) |df∫ ∫ −∞ −∞ For ergodic random process, its average power: T 12 limx (t)dt T→∞2T T 12 =lim|X(f) |df T→∞2T −∞ =G(f)df −∞ 12 ≡ Χ ThenG(f)lim| (f) |. 2T T→∞ Note: (1)G(f)0f (2)G(f)= G(-f) G(f) power spectral density [watt/Hz] Tutorial 6[1/5]
ELE3410 RandomProcesses and DSP Power spectral density function(PSD) shows the strength of the variations (energy) as a function of frequency. In other words, it shows at which frequencies variations are strong and at which frequencies variations are weak. The unit of it is energy per frequency and you can obtain energy within a specific frequency range by integrating PSD within that frequency range. How to compute PSD? Computation of PSD is done by computing autocorrelation function and then transforming it using DFT. G(f) &R(τa Fourier Transform pair:) constitute ∞ ∞ j2πfτ1j2πfτ G(f)=R(τ)e dτ &R(τ)=G(f)e df ∫ ∫2π −∞ −∞ Question 1 (Assignment 6) Find the power spectral densityG(ω) corresponding to the autocorrelation function <T 1 for| |0 . R(τ) = 0 elsewhere
Is there any real random process that may has the aboveR(τwhy or why not.)? Explain Tutorial 6[2/5]
ELE3410 RandomProcesses and DSP 2. 2D Gaussian and nD Gaussian Definitions: 2 21 (xµ) p(x;µ,σ)=exp() ¾1D Gaussian:2 2 2σ 2πσ ¾2D Gaussian: Ifx,yareindependent: 2 2 ⎡ ⎤ (xµ)2 2(1 1xµ)y x ⎜ ⎟ p(x,y;µ,µ,σ,σ)=exp⎢− + x y xy2 2⎜ ⎟ 2πσ σ2σ σ x yx y⎣ ⎝⎠⎦ Ifx,yaredependentwith correlationρ: 2 2 ⎡ ⎤ yµ ρxµyµ2 2(1 1xµx) (y() 2x)(y) ⎜ ⎟ p(x,y;µ,µ,σ,σ)=exp− ⎥⎢− + x y x y 2 22 2⎜ ⎟ 2(1ρ)σ σσ σ 2πσ σ1ρx yx yx y⎠⎦⎣ ⎝ ¾nD Gaussian: Fornrandom variablesx1,x2, …xn, they can form arandom vectorx= [x1,x2, …xn]. Its T correlation matrixR:R= E[xx], T T and itscovariance matrixC:C= E[(xη)(xηwhere) ]η= E[x]. Ifx1,x2, …xnare ALL zero mean, i.e. E[x1] = E[x2] =…= E[xn] = 0, their jointp.d.f. is given by: 1T p(x,x,...,x)=exp[xCx] 1 2n n (2π) det(C) It isnD Gaussian. Properties: ¾Ifxandyare jointly Gaussian and uncorrelated, thenxandyare independent. 2 ⎡ ⎛2⎞⎤ (yµ) 1 1(xµ)y ⎜ ⎟ x ⎢ ⎥ p(x,y)=exp− + 2 2⎜ ⎟ 2πσ σ2σ σx yx y ⎣ ⎝⎠⎦ ¾Ifxandyare jointly Gaussian, thenxandyare marginally Gaussian. p(x)=p(x,y)dy −∞ ¾x1,x2, …xnare jointly Gaussian, iff the suma1x1+a2x2+ … +anxnis a Gaussian r.v. for alla1,a2, …an. ¾A real random processx(t) is said to be Gaussian (or normal), if the r.v.s.x(t1), x(t2), …,x(tn) are jointly Gaussian for anyn,t1,t2, …,tn. Q1: 2τ τ Given a Gaussian processx(t) with meanηx= 0 an( )4e form the random dRx=e, w variablesz=x(t+1) andw=x(t1). 2 (a)Find E[zw] and E[(z+w) ]; Tutorial 6[3/5]
ELE3410 RandomProcesses and DSP (b)Findfz(z), the p.d.f. of the random variablez, andP{z1}; (c)Findfzw(z,w), the joint p.d.f. of the random variableszandw.Jointly Gaussian: Definition: Two real random processesf1(t) andf2(t) are said to be jointly Gaussian, or jointly normal, if the r.v.sf1(t1,1),f1(t1, 2), …,f1(t1,m),f2(t2,1),f2(t2, 2), …,f2(t2,n), for anym,n,t1,1,t1,2, …,t1,m,t2,1, t2,2, …,t2,n; that is the joint distribution of the random variables is Gaussian. If the random process f1(t), f2(t) f3(t) and f4(t) are jointly Gaussian with E[f1(t)] = E[f2(t)] = E[f3(t)] = E[f4(t)], then E[f1(t)f2(t)f3(t)f4(t)] =E[f1(t)f2(t)]E[f3(t)f4(t)] + E[f1(t)f3(t)]E[f2(t)f4(t)] + E[f1(t)f4(t)]E[f2(t)f3(t)]. Q2: 2 4 Ifx(t) is a Gaussian process with zero mean and varianceσ, determine E[x]. 4 22 4 E[x]=3E[x]E[x]=3σQ1: 2τ τ Given a Gaussian processx(t) with meanηx= 0 andR( )=4ewe form xrandom, the variablesz=x(t+1) andw=x(t1). 2 (a) Find E[zw] and E[(z+w) ]; 4 E[zw]=E[x(t+1)x(t1)]=E[x(t)x(t+2)]=R(2)=4e x 2 22 E[(z+w) ]=E[x(t+1) ]+E[x(t1) ]+2E[x(t+1)x(t1)] 4 =2R(0)+2R(2)=8+8e x x (b) Findfz(z), the p.d.f. m variablez, andP{z1}; 2 21 (xµ) p(x;µ,σ)=exp() 2 2 2σ 2πσ 2 2 u=E x t+ = Z[ (1)] 0;σZ=E[x(t+1) ]=4;2 1z f(z)=exp()  SoZ2 2π8 Tutorial 6[4/5]
ELE3410 RandomProcesses and DSP 1 11 11 1 (1er( )erf erf P{z1}=f)+( )= +( ) 2 22 22 2 (c) Findfzw(z,w), the joint p.d.f. of the random variableszandw.2 2 1x 2ρ(τ)x x+x 1 12 2 p(x,x;τ)=exp{} 1 222 ×ρ τ 2π×4 1ρ(τ) 42[1 ()] − − 4 µE[(zz)(ww)] 4e11 4 ρ(τ)= =.= =e 2 2 µ µ4 20 02σ σ Z W 2 42 1z 2ezw+w f(z,w)=exp() ZW  So88 8(1e) 8π1e
Tutorial 6
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