$3\mathrm{.}3$ Automating the estimates

Once we have a process for doing the estimation we would like to automate it$.$ For this part generate signals for tin$[0,5\mathrm{.}0].$

$($a$)$ Now we would like to automate the process of finding the delays. Use the max function to find the

peak in the autocorrelation and convert the corresponding lag into an estimate of the relative time

shift. Write a MATLAB function lab$1$est $($A$,$ B$,$ y$1,$ y$2)$ that outputs both the estimated angle and

L given the model parameters and the received signals.

In this function, replace $(\backslash$tau $\_(1)-\backslash$tau $\_(2))($c$\_(*))/($A$)$ with $.$

$($b$)$ Using evenly spaced values of L from $20$ meters to L$=120$ meters in increments of $5$ meter, write

a loop that generates signals from a speaker at location L using lab$1$sim$()$ and then estimates L

using lab$1$est $().$ Plot the relative error between the true Lhat$($L$)$ :

$\backslash$epsi $\_($rel $)=1-(($hat$($L$)))/($L$)$

Plot $|\backslash$epsi $\_($rel $)|$ as a function of L$.$ Use axis to make the vertical axis go up to $0\mathrm{.}5.$ You should see the

relative error is no more than $5\%.$ Here is my code so far but it doesn't work and is very scrambled. Parameters A $=0\mathrm{.}025$; $\%$ Vertical distance between microphones $($in meters$)$ B $=200$; $\%$ Horizontal distance from microphones to sound source $($in meters$)$ fs $=1$e$6$; $\%$ Sampling frequency $($Hz$)\%\%$ Signal function s$($t$)$ sig $=$ @$($t$)100*$ cos$(2*$ pi $*5000*$ t$).*($heaviside$($t$)-$ heaviside$($t $-1$e$-3))$; $\%\%$ Initialize storage for L estimates L$\_$values $=20$:$5$:$120$; $\%$ True distances from $20$m to $120$m in increments of $5$m estimated$\_$L$\_$values $=$ zeros$($size$($L$\_$values$))$; $\%$ Preallocate for estimated L $\%\%$ Loop through each true L value to estimate L$\_$hat for i $=1$:length$($L$\_$values$)$ L$\_$true $=$ L$\_$values$($i$)$; $\%$ Current true L value $\%$ Generate the signals using lab$1$sim $[$y$1$sig, y$2$sig$]=$ lab$1$sim$($A$,$ B$,$ L$\_$true, sig$)$; $\%$ Generate time vector for simulation t $=0$:$1/$fs:$5$e$-3$; $\%5$ ms of data $\%$ Compute y$1($t$)$ and y$2($t$)$ y$1\_$t $=$ y$1$sig$($t$)$; y$2\_$t $=$ y$2$sig$($t$)$; $\%$ Ensure y$1\_$t and y$2\_$t are column vectors for xcorr compatibility y$1\_$t $=$ y$1\_$t$($:$)$; y$2\_$t $=$ y$2\_$t$($:$)$; $\%$ Estimate angle and L using Lab$1$est function $[$~$,$ L$\_$est$]=$ Lab$1$est$($A$,$ B$,$ y$1\_$t$,$ y$2\_$t$,$ fs$)$; $\%$ Capture the estimated L $\%$ Store the estimated L estimated$\_$L$\_$values$($i$)=$ L$\_$est; end

figure; plot$($L$\_$values, estimated$\_$L$\_$values, $\text{'}-$o$\text{'},$ 'LineWidth', $2)$; $\%$ Plot estimated L hold on; plot$([$min$($L$\_$values$)$ max$($L$\_$values$)],[$min$($L$\_$values$)$ max$($L$\_$values$)],\text{'}$r$--\text{'},$ 'LineWidth', $1\mathrm{.}5)$; $\%$ Diagonal line for perfect estimation xlabel$(\text{'}$True L $($meters$)\text{'})$; ylabel$(\text{'}$Estimated $\backslash$hat$\{$L$\}($meters$)\text{'})$; title$(\text{'}$True L vs$.$ Estimated $\backslash$hat$\{$L$\}\text{'})$; grid on; legend$(\text{'}$Estimated $\backslash$hat$\{$L$\}\text{'},$ 'Perfect Estimation'$)$; hold off; $\%\%$ Calculate and display relative error relative$\_$error $=$ abs$(($estimated$\_$L$\_$values $-$ L$\_$values$)./$ L$\_$values$)$; disp$(\text{'}$Relative Error for Each True L Value:$\text{'})$; disp$($relative$\_$error$)$; $\%\%$ Function to generate signals with delays function $[$y$1$sig, y$2$sig$]=$ lab$1$sim$($A$,$ B$,$ L$,$ sig$)\%$ Creates tau$1$ and tau$2$ using the A$,$ B$,$ and L in meters t$1=($sqrt$(($B$)^2+($L $-$ A$)^2)/343)$; t$2=($sqrt$(($B$)^2+($L $-(2*$ A$))^2)/343)$; $\%$ Creates y$1$sig and y$2$sig by using the input signal sig and delays it y$1$sig $=$ @$($t$)$ sig$($t $-$ t$1)$; y$2$sig $=$ @$($t$)$ sig$($t $-$ t$2)$; end $\%\%$ Function to estimate theta and L using cross$-$correlation function $[$theta$\_$hat, L$\_$hat$]=$ Lab$1$est$($A$,$ B$,$ y$1\_$t$,$ y$2\_$t$,$ fs$)\%$ Calculate the cross$-$correlation between y$1\_$t and y$2\_$t $[$c$,$ lags$]=$ xcorr$($y$1\_$t$,$ y$2\_$t$)$; $\%$ Normalize the cross$-$correlation by the energy of the two signals cNorm $=$ c $/($sqrt$($sum$($y$1\_$t$.^2)*$ sum$($y$2\_$t$.^2)))$; $\%$ scales cross$-$correlation between $-1$ and $1\%$ Convert lags to time by dividing by fs lags$\_$fs $=$ lags $/$ fs; $\%$ Plot the normalized cross$-$correlation c versus time lags figure; plot$($lags$\_$fs$,$ cNorm$)$; xlabel$(\text{'}$Time Lag $($s$)\text{'})$; ylabel$(\text{'}$Normalized Cross$-$Correlation'$)$; title$(\text{'}$Normalized Cross$-$Correlation between y$1($t$)$ and y$2($t$)\text{'})$; grid on; $\%$ Find the peak in the cross$-$correlation to estimate the relative time shift $[$~$,$ peak$\_$index$]=$ max$($cNorm$)$; $\%$ Find the index of the maximum value in cNorm relative$\_$time$\_$shift $=$ lags$\_$fs$($peak$\_$index$)$; $\%$ Convert the peak index to time shift $\%$ Calculate the estimated angle theta$\_$hat using the relative time shift cs $=343$; $\%$ Speed of sound in air in m$/$s ratio $=$ min$(($cs $*$ relative$\_$time$\_$shift$)/$ A$,1)$; theta$\_$hat $=$ asin$($ratio$)$; $\%$ Estimate the distance L L$\_$hat $=$ A $*$ cos$($theta$\_$hat$)+$ B; fprintf$(\text{'}$Estimated theta$\_$hat: $\%\mathrm{.}8$f radians

$\text{'},$ theta$\_$hat$)$;

fprintf$(\text{'}$Estimated L$\_$hat: $\%\mathrm{.}8$f meters

$\text{'},$ L$\_$hat$)$;

end