Some Matlab functions for random signals

Transcription

1 Some Matlab functions for random signals This document shows some examples where some Matlab functions for random signals are used. A list of useful functions: Normrnd create Gaussian random signal Unidrnd create uniform random signal Cdfplot plot the cumulative distribution function of the data Normcdf the normal cumulative distribution function, P(X x) Norminv the inverse normal cdf Mean estimate mean value Var estimate the variance Std estimate the standard deviation Pwelch estimate the spectral density Periodogram estimate the periodogram Tfestimate (in older versions also tfe) estimate the transfer function from input and output 1. Make a Gaussian signal and plot the probability density function Use the function normrnd to create a Gaussian random signal. You can check normality with the function normplot. Create a probability density function plot and cumulative distribution function plot from the data to check distribution of the data samples. Note that the function pdfun is not a standard Matlab function, see below. Example 1 Create 2000 gaussian random values with mean = 3 and standard deviation = 2 arranged in 1 row with 2000 columns. Note that the parameter x has to be a (row or column) vector, i.e. it cannot have higher dimension than 1 when functions pdfun and cdfplot are used. For dimension 2 see case 2 below. x=normrnd(3,2,1,2000); figure(1); pdfun(x) figure(2); cdfplot(x)

2 0.25 Sample pdf 1 Empirical CDF pdf(x) F(x) x x Example 2: How to handle 2-dim data as it would be 1-dim data In this example we create data arranged in a 3x3 matrix with name x_3x3: x_3x3 = [ ; ; ] This gives us x_3x3 = We convert 2-dim to 1-dim by using (:) like this: figure(3); pdfun(x_3x3(:)) figure(4); cdfplot(x_3x3(:)) The function pdfun The above examples use the function pdfun to plot the probability density function from data. function [pdfun_out, x_out] = pdfun(in,bin) % copy this m-code to a file: pdfun.m % Use it by calling without left side like this: % pdfun(x) % with data in array x. Result is in figure graph. % pdfun... computes and plots the sample prob. density function. % % pdfun(x) plots the sample pdfun of the input vector X with 100 % equally spaced bins between the minimum and maximum % values of the input vector X. % pdfun(x,n), where N is a scalar, uses N bins. Not tested % pdfun(x,n), where N is a vector, draws a pdfun using the bins % specified in N. % [f,x] = pdfun(...) does not plot the pdfun, but returns vectors % f and x such that PLOT(x,f) is the sample pdfun. % VERSION : Preliminary - all is not tested.

3 % Define parameters nx_default = 100; axis_default = 1; % Prepare absicca vector and other parameters if ((nargin ~= 1) & (nargin ~= 2)) error(eval('eval(bell),eval(warning),help pdfun')); return; if (nargin == 1) nx = nx_default; max_x = nx_default; else nx = bin; max_y = length(in); [out,x] = hist(in,nx); nx_aug = [x,x(length(x))+(x(length(x))-x(length(x)-1))]; if ( length(out(out~=0)) <= 10 ) % Discrete distribution out = out/max_y ; flag = 'discrete'; else % Continuous distribution out = (out./ diff(nx_aug))/max_y ; flag = 'continuous'; % % Output routines if (nargout == 0) axis_default = (max(x) - min(x))/2; xmin = min(x)-axis_default; xmax = max(x) + axis_default; if strcmp(flag, 'discrete') delta = max(diff(x)); nbin = length(x); xa = [ (x(1)-delta/2), (x+delta/2) ]; oa = [ out(1), 0, out(2:nbin) ]; stairs(xa,oa),... grid on,... % if( strcmp(axis('state'),'auto') ),... % axis([xmin xmax 0 1.5*max(out)]); ;... title('sample pdf') elseif strcmp(flag, 'continuous') plot(x,out,'r.'),... grid on,... % if( strcmp(axis('state'),'auto') ),... % axis([xmin xmax 0 1.5*max(out)]); ;... title('sample pdf') elseif (nargout == 1) pdfun_out = out; else pdfun_out = out; x_out = x;

4 2. Make a discrete uniform random signal Use the function unidrnd to create a uniform random signal. We illustrate with two examples. Example 1: Throwing a dice An uniform dice with the probable outcomes of 1, 2, 3, 4, 5 and 6 can be simulated by writing X = unidrnd(6,1,1000) This gives 1000 random values arranged in an array with 1 row and 1000 columns. Example 2: Binary distributed data of +4 or -4 both with probability of 50% This data is uniformly distributed with two states i.e. it is binary distributed and can be created into a row vector with 1000 values by writing: X = 8*unidrnd(2,1,1000)-12 First we create a uniformly distributed random vector with values 1 or 2 (i.e. two states), then multiply them with 8 and last subtract 12. This results in values being either -4 or Filtering a signal This example demonstrates the use of Matlab functions to filter a sinusoid signal with a linear filter. First we create the sinusoid test signal. Sample rate in this case is set to 15 samples/second. The filter is a LTI-system with transfer function H(s) = 1 s + 3 = s 3 and with a cut off frequency of ω 0 = 3 rad/s or f 0 = 3/(2π) = 0.48 Hz (bandwidth). The filter also provides an attenuation of 1/3. fs = 15 Ts = 1/fs t = [0:Ts:15]; % Study during for example 15 seconds. x = 2*sin(2.*pi.*0.3.*t); % 0.3 Hz sinus in input (before filtering) figure(1);plot(t,x); xlabel('time (s)');ylabel('signal (a.u.)') % Simulate the linear system H(s) with the Matlab function lsim: figure(2); s = tf('s') Hs = 1/(s + 3) % The result lsim(hs,x,t) % If you want the result in an array y [y,t] = lsim(hs,x,t);

5 2 Linear Simulation Results 1 Amplitude time (sec) 4. Calculate the auto-correlation function This example shows the auto-correlation function for zero-mean Gaussian noise. r x [k] = E{X[n + k]x[n]} % Autocorrelation function can be displayed for m from -20 to 20 using this code: % Comment: % m is an integer for the maximum used delay time tau in correlation calculations % tau = m * Ts where Ts = 1/(sample frequency). That is: m = max k x = randn(1000,1); % random (normal, mean 0 variance 1), 1 column of data [r_x, lags] = xcorr(x,20,'biased'); % 'none' is default stem(lags, r_x) ylabel('r_x[k]'); xlabel('k'); % Note: % XCORR has some optional meanings ( CROSS or AUTO function) % c = xcorr(x,y,'option') % which is CROSS correlation (signal x and y) % c = xcorr(x,'option') % which is AUTO correlation (same signal: x) % The latter line can be interpreted as xcorr(x,x, ) r x [k] k

6 5. Calculate the probability of P(X x) Assume that a stochastic process is normally distributed with mean = 3 and standard deviation = 2. Calculate the probability that the process has value less than or equal to 3. This is obtained from the normal cumulative distribution since Therefore P(X x) = F(x) P(X 3) = F(3) Normalizing X into Z so that it belongs to N(0, 1) gives us from table look up that the probability is 50 % With Matlab: P (Z 3 3 ) = F(0) = P = normcdf(3,3,2) We can also estimate it from data x = normrnd(3,2,1,2000); P_exp = nnz(x<=3)/length(x) 6. Estimating spectral density The following examples demonstrate the periodogram and pwelch functions. See also Matlab help by typing help periodogram and help pwelch. Example 1 randn('state',0); %Initiates random generator to same position. Ts = 1/2000; % 0.5 ms sample intervall time => fs=2000 Hz fs=1/ts; % sampl. Freq. t = 0:Ts:5; % measure here during 5 seconds x = 0.5*cos(2*pi*50*t)+1*randn(size(t)); % Some 50 Hz freq. But mostly noise figure(1); periodogram(x,[],'onesided',[],fs); % Gives plot up to frequency fs/2.

7 0 Periodogram Power Spectral Density Estimate Power/frequency (db/hz) Frequency (khz) Example 2 Compare the above to the following estimation by the Matlab function pwelch. Comment: The Hann window is also called the Hanning window. figure(2); pwelch(x,hanning(128),[],[128],fs,'onesided'); -22 Welch Power Spectral Density Estimate -24 Power/frequency (db/hz) Frequency (khz)

8 7. Window function The Hann window is also called the Hanning window. Try the following in Matlab: hann(5,'symmetric') hanning(3) An alternative way of defining a window is to use the function window window(@hann,5, 'symmetric') From its help: WINDOW(@WNAME,N) returns an N-point window of type specified by the function in a column can be any valid window function name, - Bartlett window. - Gaussian window. - Hamming window. - Hann window. - Rectangular window alias boxcar. - Triangular window.