Chapter 2. Simulation Techniques. References:
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1 Simulatio Techiques Refereces: Chapter 2 S.M.Kay, Fudametals of Statistical Sigal Processig: Estimatio Theory, Pretice Hall, 993 C.L.Nikias ad M.Shao, Sigal Processig with Alpha-Stable Distributio ad Applicatios, Joh Wiley & Sos, 995 V.K.Igle ad J.G.Proakis, Digital Sigal Processig Usig MATLAB V.4, PWS Publishig Compay, 997 E.Part-Eader, A.Sjoberg, B.Meli ad P.Isaksso, The MATLAB Hadbook, Addiso-Wesley, 996 S.K.Park ad K.W.Miller, Radom umber geerators: good oes are hard to fid, Commuicatios of the ACM, vol.3, o.0, Oct. 988
2 Simulatio Techiques Sigal Geeratio. Determiistic Sigals It is trivial to geerate determiistic sigals give the sythesis formula, e.g., for a sigle real toe, it is geerated by MATLAB code: x( Acos( ω + θ, 0,, L, N N0; % umber of samples is 0 A; % toe amplitude is w0.2; % frequecy is 0.2 p; % phase is for :N x(a*cos(w*(-+p; % ote that idex should be > 0 ed 2
3 A alterative approach is 0:N-; % defie a vector of size N x A.*cos(w.*+p; % the first time idex is also %.* is used i vector multiplicatio Both give x Colums through Colums 8 through Q.: Which approach is better? Why? 3
4 Example 2. Recall the simple mathematical model of a musical sigal: x( t a( t cm cos(2πmf0t m + φ m A further simplified form is x ( t cos(2 πf 0 t where each music ote has a distict f 0. Let's the followig piece of music: A A E E F# F# E E D D C#C# B B A A E E D D C# C# B B (repeat oce (repeat first two lies oce 4
5 The America Stadard pitch for each of these otes is: A: Hz B: Hz C#: Hz D: Hz E: Hz F#: Hz Assumig that each ote lasts for 0.5 secod ad a samplig frequecy of 8000 Hz, the MATLAB code for producig this piece of music is: asi(2*pi*440*(0: :0.5; % frequecy for A bsi(2*pi*493.88*(0: :0.5; % frequecy for B cssi(2*pi*554.37*(0: :0.5; % frequecy for C# dsi(2*pi*587.33*(0: :0.5; % frequecy for D esi(2*pi*659.26*(0: :0.5; % frequecy for E fssi(2*pi*739.99*(0: :0.5; % frequecy for F# 5
6 lie[a,a,e,e,fs,fs,e,e]; % first lie of sog lie2[d,d,cs,cs,b,b,a,a]; % secod lie of sog lie3[e,e,d,d,cs,cs,b,b]; % third lie of sog sog[lie,lie2,lie3,lie3,lie,lie2]; % composite sog soud(sog,8000; % play soud with 8kHz samplig frequecy wavwrite(sog,'sog.wav'; % save sog as a wav file Note that i order to attai better music quality (e.g., flute, violi, we should use the more geeral model: x( t a( t cm cos(2πmf0t m + φ m Q.: How may discrete-time samples i the 0.5 secod ote with 8000 Hz samplig frequecy? Q.: How to chage the samplig frequecy to 6000 Hz? 6
7 2. Radom Sigals Uiform Variable A uiform radom sequece ca be geerated by x( seed ( a seed mod( m,,2,l where seed 0, a ad m are positive itegers. The umbers geerated should be (approximately uiformly distributed betwee 0 ad ( m. A set of choice for a ad m which geerates good uiform variables is a 6807 ad m This uiform PDF ca be chaged easily by scalig ad shiftig the geeratio formula. For example, a radom umber which is uiformly betwee 0.5 ad 0.5 is give by seed x( ( a seed mod( m seed m 0.5 7
8 The power of x( is var( x x 2 p( x dx Note that x( is idepedet (white. x 2 dx To geerate a white uiform umber with variace σ 2 x : seed a seed ( mod( m seed x( σ x m MATLAB code for geeratig zero-mea uiform umbers with power 2: N5000; % umber of samples is 5000 power 2; % sigal power is 2 u (rad([,n]-0.5.*sqrt(2*power; % rad give a uiform umber % i [0,] 8
9 Evaluatio of MATLAB uiform radom umbers: m mea(u % * mea computes the time average m p mea(u.*u % compute power p y mea((u-m.*(u-m % compute variace v
10 plot(u; % plot the sigal
11 hist(u,20 % plot the histogram for u % with 20 bars Q.: Is the radom geerator acceptable? Does ergodicity hold?
12 a xcorr(u; % compute the autocorrelatio plot(a % plot the autocorrelatio
13 axis([4990, 500, -500, 2000] % chage the axis The time idex at 5000 correspods to R uu (0 white 3
14 Gaussia Variable Give a pair of idepedet uiform umbers which are uiformly distributed betwee [0,], say, ( u, u2, a pair of idepedet Gaussia umbers, which have zero-mea ad uity variace, ca be geerated from: w 2l( u cos(2πu 2 w 2 2l( u si(2πu This is kow as the Box-Mueller trasformatio. Note that the Gaussia umbers are white. MATLAB code for geeratig zero-mea Gaussia umbers with power 2: N5000; % umber of samples is 5000 power 2; % sigal power is 2 w rad([,n].*sqrt(power; % rad give Gaussia umber 2 4
15 % with mea 0 ad variace Evaluatio of MATLAB Gaussia radom umbers: m mea(w % * mea computes the time average m p mea(w.*w % compute power p y mea((w-m.*(w-m % compute variace v
16 plot(w; % plot the sigal
17 hist(w,20 % plot the histogram for w % with 20 bars
18 a xcorr(w; % compute the autocorrelatio plot(a % plot the autocorrelatio
19 axis([4990, 500, -500, 2000] % chage the axis The time idex at 5000 correspods to R ww (0 9
20 Impulsive Variable The mai feature of impulsive or impulse process is that its value ca be very large. A mathematical model for impulsive oise is called α-stable process, where 0 < α 2. α -stable process is a geeralizatio of Gaussia process ( α 2 ad Cauchy process ( α The variable is more impulsive for a smaller α A α -stable variable is geerated usig two idepedet variables: Φ which is uiform o ( 0.5π,0. 5π, ad W which is expoetially distributed with uity mea, where W is produced from W l(u where u is a uiform variable distributed o [0,] 20
21 MATLAB code for 0 < α < 2 ad α alpha.8; % alpha is set to.8 beta 0; % beta is a symmetric parameter N5000; phi (rad(,n-0.5*pi; w -log(rad(,n; k_alpha - abs(-alpha; beta_a 2*ata(beta*ta(pi*alpha/2.0/(pi*k_alpha; phi_0-0.5*pi*beta_a*k_alpha/alpha; epsilo - alpha; tau -epsilo*ta(alpha*phi_0; a ta(0.5.*phi; B ta(0.5.*epsilo.*phi./(0.5.*epsilo.*phi; b ta(0.5.*epsilo.*phi; z (cos(epsilo.*phi-ta(alpha.*phi_0.*si(epsilo.*phi./(w.*cos(phi; d (z.^(epsilo./alpha -./epsilo; i (2.*(a-b.*(+a.*b - phi.*tau.*b.*(b.*(-a.^2-2.*a.*(+epsilo.*d./((-a.^2.*(+b.^2+tau.*d; 2
22 plot(i;
23 MATLAB code for α N5000; phi (rad(,n-0.5*pi; a ta((0.5.*phi; i 2.*a./(-a.^2; plot(i
24 PDF for differet α 24
25 The impulsiveess is due to the heavier tails, i.e., PDF go to zero slowly 25
26 AR, MA ad ARMA Processes MA process is geerated from x b w( + b w( + L + bn w( ( 0 N where { w( } is a white oise sequece. Oly the trasiet sigal is eeded to remove. e.g., for a secod-order MA process Q w ( 0, < 0 x ( b0w( + b w( x ( 0 b0w(0 + bw( b0w(0 x ( b0w( + bw(0 x ( 2 b0w(2 + bw(... The trasiet sigal is x(0. We should choose { x (, x(2, L} 26
27 MATLAB code for geeratig 50 samples of MA process with b, b 2: 0 b0; b2; N50; wrad(,n+; % geerate N+ white oise samples for :N x( b0*w(++b*w(; % shift w by oe sample ed Alteratively, we ca use the covolutio fuctio i MATLAB: b0; b2; N50; wrad(,n+; % geerate N+ white oise samples b [b0 b]; % b is a vector ycov(b,w; % sigal legth is N xy(2:n+; % remove the trasiet sigals 27
28 From (.44, the PSD for MA process is Φ xx ( ω + 2e jω 2 σ 2 w + 2e jω 2 It ca be plotted usig the freqz commad i MATLAB: b0; b2; b [b0 b]; a; [H,W] freqz(b,a; % H is complex frequecy respose PSD abs(h.*h; plot(w/pi,psd; 28
29
30 To evaluate the MA process geerated by MATLAB, we use (.38: 2 N jω Φ xx ( ω lim E x( e N N 0 N N 00 E {} average of 00 idepedet simulatios MATLAB code: N00; b [ 2]; % b is a vector for m:00 % perform 00 idepedet rus wrad(,n+; % geerate N+ white oise samples ycov(b,w; % sigal legth is N xy(2:n+; % remove the trasiet sigals p(m,: abs(fft(x.*fft(x; ed psd mea(p./00; idex /50:/50:2; plot(idex,psd; axis([0,, 0 0]; 30
31 3
32 N N 0000 E {} average of 0000 idepedet simulatios
33 Trasiet sigals are also eeded to remove i AR & ARMA processes because of o-statioarity due to the poles: x( ax( + a2 x( 2 + L + am x( M + w( x( a x( + b + 0 a w( 2 x( + b w( 2 + L + a M + L + b x( N w( e.g., for a first order AR process: x ( ax( + w( 2( m + a 2 Rxx (, m a + σ 2 w a ostatioary because R xx (, + m depeds o 2( + M N for sufficietly large, say, a <<, we ca cosider it statioary. sice a is the pole, extesio to geeral AR ad ARMA processes: 2( + i p <<, for all poles { } p i 33
34 2( + Suppose a is required ad the AR parameter is a The required is calculated as ( 0.9 2( ( MATLAB code for geeratig 50 samples of the AR process: M 43; N 50; a -0.9; y( 0; for 2:M+N y( a*y(-+rad; ed xy(m+:m+n; plot(x; 34
35
36 Digital Filterig Give a iput sigal x ( ad the trasfer fuctio H (z, it is easy to geerate the correspodig output sigal, say, y ( For FIR system, we ca follow the MA process, while for IIR system, we ca follow the ARMA process. The trasiet sigals ca be removed if ecessary as i the MA, AR ad/or ARMA processes. Give H (z, the impulse respose ca be computed via iverse DTFT: h ( 2π π π H ( ω e jω d ω Frequecy spectrum for H (z impulse respose { h( } y( h( x( h( k x( k x( k h( k k k 36
37 Example 2.2 Compute the impulse respose for H d (z with the followig DTFT spectrum, ad ω 0. 2π ad ω 0. 4π. o c Η (ω d 0.5 π ω c ω o 0 ω o ω c π h d ( 2π 2π π π ω H c d ( ω e jω jω dω 0.5 e dω + ω c 2π ω o jω 0.5 e dω ω o si( ω c 2π + si( ω o 2π 37
38 si(0.2π + si(0.4π h d (, L,,0,, L 2π Note that h d (0 ca be obtaied by usig L Hospital s rule or: h d (0 Combiig the results: h d 2π 2π π H ( ω e dω π ω c 0.5dω + ω c d jω 0 2π ω o 2π 0.5dω ω o π π 0.3, ( si(0.2π + si(0.4π, 2π H ω d c ( ω dω + ω 2π o 0 otherwise 0.3 y( h ( x( x( k h ( k x( k h ( k d k d M k M d 38
39 Example 2.3 Compute the impulse respose of a time-shift fuctio which time-shifts a sigal by a o-iteger delay D. ( ( D x y exp( (, ( ( D j H X e Y D j ω ω ω ω ω sic( 2 2 ( 2 ( ( D d e d e e d e H h D j j D j j ω π ω π ω ω π π π ω π π ω ω π π ω where x x x π π si( sic( sic( ( sic( ( sic( ( ( D k k x D k k x D x y M M k k 39
40 Questios for Discussio. Observe that the followig sigal: y( 0 k 0 x( ksic( k D x( which depeds o future data { x( +, x( + 2, L, x( + 0}. This is referred to as a o-causal system. How to geerate the output of the o-causal system i practice? 2. The spectrum for the Hilbert trasform is D H ( ω j, j, 0 < ω π π ω < 0 Use a FIR filter with 5 coefficiets to perform the Hilbert trasform of a discrete-time sigal x []. Let the resultat sigal be y []. 40
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