ELT COMMUNICATION THEORY

Transcription

1 ELT COMMUNICATION THEORY Matlab Exercise #2 Randm variables and randm prcesses 1 RANDOM VARIABLES 1.1 ROLLING A FAIR 6 FACED DICE (DISCRETE VALIABLE) Generate randm samples fr rlling a fair 6 faced dice fr N times (i.e. there s an equal prbability fr each value f the dice). First, rll the dice fr N=10 times (help randi) N_faces = 6; %Number f faces in the dice N_trials = 10; %Number f trials (hw many times the dice is rlled) trials = randi(n_faces,1,n_trials); % Getting randm integers Plt the histgram f the utcme values (help histgram, help hist, help bar, help histcunts) histgram functin was intrduced in MATLAB R2014b x_histgram_centers = 1:6; %x-axis fr the bin center crdinates % Befre versin R2014b hist_cunt = hist(trials,x_histgram_centers); bar(x_histgram_centers,hist_cunt) xlabel('rlled number') ylabel('ccurrence number') grid n % Histgrams can be pltted directly as "hist(trials,x_histgram_centers)", % but the pltting parameters are mre difficult t handle. % Versin R2014b r after % x-axis fr the bin edge crdinates (last edge is fr the right edge) histgram_edges = 0.5:1:6.5; % Belw the ~ sign means that we discard that specific utput argument [hist_cunt, ~]= histcunts(trials,histgram_edges); bar(x_histgram_centers,hist_cunt) xlabel('rlled number') ylabel('ccurrence number')

2 grid n % Set a grid in the pltted title('dice rlling histgram') % Histgrams can pltted directly as "h = histgram(trials,'binlimits',[0.5 % 6.5],'BinMethd','integers')", where h is the histgram bject. Nrmalize the histgram values with N_trials (number f trials) s that we get the experimental prbability density functin (pdf) fr the dice. Hence, after the nrmalizatin the sum f the histgram bin area ( integral ) is equal t ne, as required with pdfs (actually when talking abut discrete pdf, we ften refer t pmf (prbability mass functin)). pdf_experimental = hist_cunt/n_trials; Plt the nrmalized histgram, and define the true (analytic) pdf fr the fair dice and plt it n tp f the experimental pdf (help hld) bar(x_histgram_centers,pdf_experimental) xlabel('rlled number') ylabel('pdf') grid n title('dice rlling nrmalized histgram') ne_face_prbability = 1/N_faces; % prbability f ne face f the dice hld n % avids remving the previus plt % pltting true pdf (a line between tw pints) plt([ ],[ne_face_prbability ne_face_prbability],'r') hld ff % Using legend we can name different data curves in the plt (in rder f % appearance) legend('experimental pdf','true pdf') Are the true pdf and the experimental pdf perfectly equal? Repeat the prcess fr a few f times and cmpare the utcmes: Is the experimental pdf varying between simulatins? Increase the number f trials t N= and repeat the prcess fr a few times Is the experimental pdf varying between simulatins nw? 1.2 NORMAL/GAUSSIAN DISTRIBUTED RANDOM VARIABLE Generate 1000 samples fr a nrmally distributed randm variable X~(μ,σ 2 ), where the mean and variance are given as μ=3, and σ 2 =4 (help randn) N_samples = 1000; mu = 3; sigma = sqrt(4); % the variance is 4 (i.e. sigma^2=4) Calculate the fllwing statistics f the bserved samples Mean (r average, r expected value) (help mean) Standard deviatin (help std) Variance (help var) Plt the experimental pdf (i.e. the nrmalized histgram s that the sum f histgram bin area is equal t ne) Define the histgram bin center x crdinates as bin_centers = 7:bin_width:13, where the bin_width = 0.5 defines the width f ne bin bin_width = 0.5; %bin width (in the x-axis)

3 bin_centers = -7:bin_width:13; %x-axis fr the bin center crdinates % x-axis fr the bin edge crdinates (last edge is fr the right edge): % (Three dts cntinues the cmmand in the fllwing line) bin_edges = (bin_centers(1)- bin_width/2):bin_width:(bin_centers(end)+bin_width/2); % ~ means that we discard that utput argument [hist_cunt, ~]= histcunts(x,bin_edges); pdf_experimental = hist_cunt/sum(hist_cunt*bin_width); bar(bin_centers,pdf_experimental,1) % and s n with the titles, xlabels, Plt the true (analytic) pdf f X~(μ,σ 2 ) n tp f the experimental pdf pdf_true = 1/(sqrt(2*pi)*sigma)*exp(-(mu-bin_edges).^2/(2*sigma^2)); hld n plt(bin_edges,pdf_true,'r','linewidth',3) % defines a specific line width % and s n with the titles, xlabels, (remember the legend als) Repeat the experiment with different number f samples: N=100 and N= See the difference in the fitting between the experimental pdf and the analytic pdf as the number f samples changes Based n the histgram with N= samples, define the prbability P(X>5.25) Integrate (i.e. sum) histgram bins, whse x axis center values are larger than 5.25 Nte that actually 5.25 is exactly n edge between tw histgram bins b = 5.25; indices_with_bin_center_larger_than_b = bin_centers > b; cnsidered_bin_values = pdf_experimental(indices_with_bin_center_larger_than_b); %area f the cnsidered bins prbability_x_larger_than_b = sum(cnsidered_bin_values*bin_width) Check yur answer with the Q functin P(X>b)= Q((b-μ)/σ) (help qfunc) analytic_prbability = qfunc((b-mu)/sigma) 2 RANDOM PROCESSES 2.1 WHITE NOISE VS. COLORED NOISE Generate a zer mean white Gaussian nise signal with variance σ 2 =3 N = 10000; % Number f generated samples nise_var = 3; % Desired nise variance nise = sqrt(nise_var)*randn(1,n); % Nise signal generatin Plt the nise signal plt(nise) xlabel('sample index') ylabel('nise amplitude') title('white nise') xlim([0 100]) %define the x-axis limits

4 Plt the histgram f the nise signal t see that it is Gaussian distributed histgram(nise,40) xlabel('nise amplitude') ylabel('histgram cunt') title('white nise histgram') Implement a lw pass FIR filter (help firpm, recap frm the previus exercise) Use filter rder f 60 and a transitin band frm 0.1*Fs/2 t 0.2*Fs/2 N_filter = 60; %even number h = firpm(n_filter,[ ],[ ]); Plt the impulse respnse and amplitude (frequency) respnse f the filter Yu can define the amplitude respnse as a functin f nrmalized frequency (i.e. Fs/2 Fs/2 1 1) N_freq = length(nise); freq_vec_filter = -1:2/N_freq:(1-2/N_freq); %frequency vectr values nrmalized between -1 and 1,plt(freq_vec_filter,10*lg10(fftshift(abs(fft(h,N_freq))))) xlabel('nrmalized frequency (F_s/2=1)') ylabel('amplitude') title('amplitude respnse f the filter') Filter the nise signal using the abve filter (help filter) % Filter the nise signal: filtered_nise = filter(h,1,nise); Plt the white nise and filtered nise signals in the same and cmpare Remember the filter delay in rder t align the signals prperly in time filtered_nise = filtered_nise(n_filter/2+1:end); %remve the delay plt(nise(1:n_samples_plt)) hld n plt(filtered_nise(1:n_samples_plt),'r') legend('white nise','clred nise') xlabel('sample index') ylabel('nise amplitude') title('white nise and filtered (clred) nise') Plt the histgram f the filtered nise signal Hw it is distributed? (recap: What is the distributin f the utput signal f an LTI system, if the input signal is Gaussian distributed?) Cmpute and plt the autcrrelatin functin f the riginal white nise signal (help xcrr) [crr_fun, lags] = xcrr(nise); % we nrmalize the max-value t 1 and use stem-functin in rder t emphasize % the impulse-like nature f the utcme,stem(lags,crr_fun/max(crr_fun)) xlabel('\tau') % \tau gives the Greek tau-letter (\ wrks generally fr the Greek alphabet) ylabel('r(\tau)') title('autcrrelatin f white nise') xlim([-30 30]) What type f autcrrelatin functin shuld the white nise have?

5 Cmpute and plt the autcrrelatin functin f the filtered (i.e., clred) nise signal Use the abve statements, but with plt instead f stem : [crr_fun, lags] = xcrr(nise);,plt(lags,crr_fun/max(crr_fun)) %...and s n Cmpare this with the previusly pltted impulse respnse f the filter In the end, plt the pwer spectra (i.e., amplitude spectrum squared) f the tw nise signals in the same (use the decibel scale 20*lg10(abs(.))) nise_abs_spec = 20*lg10(abs(fft(nise(1:length(filtered_nise))))); filtered_nise_abs_spec = 20*lg10(abs(fft(filtered_nise))); %Define the frequency vectr values (nrmalized between -1 and 1): freq_vec = -1:2/length(nise_abs_spec):1-2/length(nise_abs_spec); plt(freq_vec,fftshift(nise_abs_spec)) hld n plt(freq_vec,fftshift(filtered_nise_abs_spec),'r') hld ff xlabel('nrmalized frequency (F_s/2=1)') ylabel('pwer [db]') title('nise spectra') legend('white nise','filtered (clured) nise') Ntice that bth signals are still Gaussian distributed (see the previus histgrams), but the pwer spectra are different. Try t remember what is the cnnectin between the crrelatin functin and the pwer spectral density functin? Make sure yu understand the difference between the fllwing cncepts: Gaussian nise and white nise I.e., here bth nise signals are Gaussian, but nly the ther ne is white 2.2 RANDOM WALK MODEL (EXAMPLE FROM THE CLASSROOM EXERCISES) Let s cnsider a randm sequence X[n], the s called randm walk mdel, as n X[ n] i W[ i] 1 where W[i] are binary i.i.d. (independent and identically distributed) randm variables with prbabilities P[W[i] = s] = p and P[W[i] = s] = 1 p Generate a randm prcess f 2000 samples and 5000 realizatins (=ensemble size) Use first p=0.5 and s=1, but write the cde s that yu can cnveniently test the prcess with ther values t N_samples = 2000; %Number f samples fr each realizatin N_ensemble = 5000; %Number f signal realizatins (i.e., the size f ensemble) %Step prbability and step size: p = 0.5; % P(Wi=s) = p, P(Wi=-s) = 1-p s = 1; %step length n = 1:N_samples; % vectr f samples indices % Generating matrix f randmly generated steps:

6 W = rand(n_ensemble,n_samples); % (i.e. unifrmly distributed randm values between 0 and 1) indices_with_psitive_s = W<p; % find ut steps ging "up" W(indices_with_psitive_s) = s; % Define steps fr ging "up" W(~indices_with_psitive_s) = -s; % Define steps fr ging "dwn" % The verall "randm walk" is achieved by taking the cumulative sum ver the % steps: X = cumsum(w,2); % (Ntice that nw each rw describes ne randm walk realizatin, s the sum % is taken ver the 2nd dimensin) Plt five example realizatins f the randm walk prcess (help fr) Use subplts inside ne (help subplt) fr ind = 1:5 subplt(5,1,ind) plt(n,x(ind,:)) xlabel('n') ylabel('x(n)') grid n title(['realizatin #' num2str(ind)]) %num2str cnverts a numerical value int a character value end % Here is a handy way t get a full screen % (therwise the might be t unclear): set(gcf,'units','nrmalized','uterpsitin',[ ]) Calculate the ensemble mean and the ensemble variance f the prcess Plt the calculated ensemble mean E[X[n]] and variance E[(X[n]- E[X[n]]) 2 ] in the same with the theretical values given as Theretical mean: E[X[n]] = ns(2p-1) Theretical variance: E[(X[n]- E[X[n]]) 2 ] = np(2s) 2 (1-p) Ntice that the ensemble mean and variance are nw functins f the sequence index n ( =time ), s there will be a specific mean and variance fr each sequence index mean_thery = n*s*(2*p-1); % Theretical mean var_thery = n*(2*s)^2*p*(1-p); % Theretical variance mean_bserved = mean(x); % Empirical mean (i.e., what we bserve) var_bserved = var(x); % Empirical variance (i.e., what we bserve) plt(n,mean_bserved,'b','linewidth',3) hld n plt(n,mean_thery,'r:','linewidth',2) hld ff legend('bserved mean','theretical mean') ylim([-2 2]) % set the axis limits in y-directin nly xlabel('n') ylabel('mean') title('mean ver the sample index') % % and the same fr the variance

7 Re run the prcess with different values f p, s, and try different number f realizatins Due t memry issues, d nt try t many realizatins at the same time (stay belw realizatins and 2000 samples) What if the number f realizatins is very lw? Is the prcess statinary (in wide sense)? Why?/Why nt?