Biosignals and Systems

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1 Biosignals and Systems Prof. izamettin AYDI naydin@yildiz.edu.tr naydin@ieee.org Advanced Measurements: Correlations and Covariances More complicated measurements can be made on a signal by comparing it to other reference signals or mathematical functions. hese comparisons are implemented through an operation nown as. Correlation sees to quantify how much one thing is lie another. When comparing two mathematical functions (or signals), the technique is to multiply one by the other, then average the results. his average is often scaled by some normalizing factor. Corr = x ( t ) y ( t ) dt Corr = = Correlation he between two signals, x(t) and y(t) over a time frame is: x( ) y( ) or in discrete form Use the equation to find the (unnormalized) between the sine wave and the square wave shown. x(t) & x(t) Example -6 It is common to modify these equations by dividing by the square root of the product of the variances of the two signals. his will mae the value equal to. when the two signals are identical and - if they are exact opposites Corr normalized = Corr σ σ πt Corr x t y t dt dt πt = ( ) ( ) = sin = cos π Corr = ( cos( π) cos( ) ) = π π ote that the between a sine and cosine will be zero. 4 Covariance Correlation between different waveforms. Covariance computes the variance that is shared between two (or more) signals. Specifically, the covariance is defied as: here is no between a sine and cosine wave. Correlation: σ xy = x x y y = ( )( ) A high is seen between a sine and a triangle. Correlation:.99 - he equation for covariance is similar to the discrete form of (except that the average values of the signals have been removed). A moderate between a sine and a composite waveform Correlation:

2 Auto and Cross he lac of mathematical between a sine and a cosine can be a problem since they are intuitively similar even if they have zero. A signal could be sinusoidal (e.g., a cosine), but if you are using a sine as a reference function the would be small or negligible. o circumvent this problem, you could still use a sine reference for comparison, but shift this reference signal in time, performing the for many different time shifts. Correlating over many different time shifts is called cross. 7 Shifting one sinusoid with respect to another probes all the possible relative positions. he maximum occurs when the two are in phase and is Cross Sine Shift: deg. Corr:. -.. Sine Shift: 9 deg Sine Shift: 4 deg. Corr Correlation vs ime Shift -. When the two -. are out of - Corr: - phase the -.. is ime Shift (sec) Cross Cross he shifting in cross can be achieved mathematically by introducing a variable time delay, or time lag, or simply lag, into one of the two waveforms in the equation. (It does not matter which function is shifted) Cross rxy( τ ) = y ( t ) x ( t + τ) dt where the variable τ is a continuous variable of time used to shift x(t) with respect to y(t). he variable τ is a variable of time, but not the time variable and is sometimes called a dummy time variable, although this is a bit misleading. his variable is also called the lag or lag variable. ote that the output of this equation is itself a waveform (i.e., function) of time, τ. 9 Lower plot shows the cross function for the sinusoid and a triangle waveform given in the upper plot. ote that they are most similar (i.e. have the highest ) when one signal is shifted.8 sec. with respect to the other ime Shift or Lag (sec) Discrete Cross Auto he discrete form of the cross equation is constructed in the usual manner, (replacing continuous variables with discrete variables and integration with summation.) rxy[ n] = y( ) x( + n) = Auto is simply cross of a waveform with itself. he auto equation in continuous and discrete forms becomes: Auto rxx( τ ) = x ( t ) x ( t + τ) dt rxx[ i] = x( ) x( + i) =

3 For the Direct umerical Method via polynomial multiplication, use the following steps: directly tabulate values of the function in the second row with the increment of time above the function as in able in the previous slide; reverse tabulate the same function in the third row; do normal polynomial multiplication (from right to left); write the products in columns following normal multiplication procedures; when there are no more products, sum the columns; Auto by Polynomial Multiplication Method t x D.. x R ote: Original function ( x); ime t ; D is Direct ranscription; R is Reverse ranscription. 4 Graph of the auto function Four different signals (left side) and their auto functions (right side) ime Plot Auto A) a sinusoid; B) a slowly varying signal; C) a rapidly varying signal; ote that the auto function is an even function D) a random signal ime(sec) -. - Lags (n) 6 MALAB Implementation Covariance and Correlation Correlation and Covariance Rxx = corrcoef(x); % Signal S = cov(x); % Signal covariance where x is a matrix that contains the various signals to be compared in columns. he output of the routine corrcoef. R = xx r, r, L r, r, r, L r, M M O M r, r, L r, he output, Rxx, of the corrcoef routine will be a n-by-n matrix where n is the number of signals (ie., columns). he diagonals of this matrix represent the of the signals with themselves, Rxx (and, hence, will be ), and the off diagonals represent the s of the various combinations. he output of the cov routine is similar except the entries contain variances 7 he output of the covaraince routine cov. σ, σ, L σ, σ, σ, L σ, S = M M O M σ, σ, L σ, 8 3

4 Example -9 Determine if a sinewave and cosinewave at the same frequency are orthogonal and if sinewaves at harmonically related frequencies are orthogonal. Include one sinusoid at a non-harmonic frequency. Solution: If two signals are orthogonal they will be uncorrelated. Generate a data matrix where the columns consist of a. Hz sine and cosine, a. Hz sine and cosine, and a 3. Hz sine and a 3. Hz cosine. he six sinusoids should all be at different amplitudes. Apply the covariance (cov) and (corrcoef) MALAB functions. All of the sinusoids except the 3. cosine are orthogonal and should show negligible and covariance. 9 % Example.9: Application of the and % covariance matrices to sinusoids that are orthogonal and % nonorthogonal % Generate the sinusoids as columns of the matrix clear all; close all; = 6; fs = 6; n = (:)/fs; % umber of points in waveform % Assumed sample frequency % ime vector: sec of data x(:,) = sin(*pi*n)'; % Generate a Hz sin x(:,) = *cos(*pi*n)'; % Generate a Hx cos x(:,3) =.*sin(4*pi*n)'; % Generate a Hz sin x(:,4) = 3*cos(4*pi*n)'; % Generate a Hx cos x(:,) =.*sin(6*pi*n)'; % Generate a 3 Hx sin x(:,6) =.7*cos(7*pi*n)'; % Generate a 3. Hz cos S = cov(x) ; % Print covariance matrix Rxx = corrcoef(x) ; % and matrix Results: he output from this program is a covariance and matrix. he covariance matrix is: MALAB Implementation Auto and Cross S = he diagonals of the covariance matrix give the variance of the six signals and these differ since the amplitudes of the signals are different. he matrix shows similar results except that the diagonals are now. since these reflect the of the signal with itself. Rxx = he cross and auto operations are both performed with the same MALAB routine, with auto being treated as a special case: [r,lags] = xcorr(x,y,maxlags, options ); Only the first input argument, x, is required. If no y variable is specified, auto is performed. he optional argument maxlags limits the shifting range. he shifted waveform is shifted between ±maxlags, or the default value which is -+ to - where is length of the input vector, x. MALAB Implementation Auto- and Crosscovariance Autocovariance or crosscovariance is obtained using the xcov function: [c,lags] = xcov(x,y,maxlags, options ) he arguments are identical to those described above for the xcorr function. Auto- and crosscovariance are the same as auto- and cross if the data have zero means. Example - Determine if there is any in the variation between the timing of successive heart beats from the heart rate data shown below. Solution Load the heart rate data taen during normal resting conditions (file Hr_pre.txt). Isolate the heart rate variable (the nd column) then tae the autocovariance function. he aotocovariance function will remove the mean HR giving only the change in HR. Plot this function in such a way as to show potential over approximately 3 successive beats 3 4 4

5 Example - Example - HR(beats/sec) Preliminary HR HR(beats/sec) Meditative HR Heat rate data (beats/min against time) used in this example. Only the baseline, pre-meditative data (left plot) is used here. % EXAMPLE. and Figure. % Use of autocovariance to determine the % of heart rate variation between heart beats % clear all; close all; figure ; %load Hr_pre % Load data load zf.mat % Load data %[c,lags]=axcor(hr_pre-mean(hr_pre)); % Autocovariance (mean subtracted) [c,lags]=xcorr(zf-mean(zf)); % Autocovariance (mean subtracted) plot(lags,c,''); hold on; % Plot autocovariance %plot ( [lags () lags (end)], [ ],'') % Plot zero line for % reference xlabel('lags () ' ) ; ylabel('autocovariance') ; grid on; %axis([ ]); % Limit plot range % to ± 3 beats 6 Example -..8 Autocovariance Lags () Autocovariance function of the heart rate from one subject under normal resting conditions. Some is observed over approximately successive heart beats. 7