$2$ Implementing Matched Filters for Noisy Data

In realistic sonar and radar systems, the received signal $r[n]$ includes a noise component in addition to the signal

component, i$.$e$.,$

$r[n]=b[n-D]+$ noise,

where $b[n-D]$ is the delayed signal and Since the matched filter is a LTI system, the output due to the noisy

input will be

$p[n]=r[n]**h_{MFC}[n]=b[n-D]**h_{MFC}[n]+$ noise $**h_{MFC}[n].$

In other words, there will be a noise component added to the output we would expect from the noise$-$free

matched filtering operation. For this part of the project, the goal is to estimate the integer delay $D($or the

equivalent delay time in seconds$)$ by matched filtering the noisy reception.

$2\mathrm{.}1$ Simulating Data

In this section, you will explore simulating a noisy reception. Open up the notebook proj$-$part$2\_$simdata $.$ ipynb.

This incomplete notebook will allow you to simulate data $30$ seconds of noisy data for two types of pulse sig$-$

nals: the $13-$second Barker sequence and a $13-$second rectangular pulse. The notebook is incomplete because it

leaves out the definition of b$13($the $13-$second Barker pulse$)$ that you defined in the previous part of the project.

The notebook includes commands for saving the simulated data to a file and reading the data back from the file.

Read through the code and figure out what it is doing and how many noisy receptions it will generate.

What is the value of $D$ for these noisy receptions?

Fill in your definition of b$13.$ Add code to the notebook that allows you to plot both the rectangular and

Barker pulses for comparison.

Run the notebook to generate noisy receptions for the Barker pulse first. Plot a few noisy receptions for

the Barker pulse. Explore how the parameter noise$\_$std changes the receptions. For what values of

noise$\_$st d are you no longer able to see the Barker pulse in the reception?

Repeat the previous part for the rectangular pulse.

Change the parameter that controls D and verify that your pulses move around as you would expect.

$2\mathrm{.}2$ Implement the Matched Filter for Your Simulated Data

Pick a value of $D$ and create a set of simulated receptions for both the Barker and rectangular pulses. Process

your simulated receptions with the matched filters designed for the two different pulses. Verify that you can

estimate $D$ from the matched filter outputs. Remember that you have to account for the delay through the

causal version of the matched filter. Based on your simulation experiments can you say which type of pulse

will produce better estimates of $D$ for noisy data?

$2\mathrm{.}3$ Implement the Matched Filter for Mystery Data Sets

The course website contains two noisy data sets, one for each type of pulse $($Barker and rectangular$)$:

proj$\_$part$2\_$data$\_$barker$13.$npz

proj$\_$part$2\_$data$\_$rect$13.$npz

Each file contains $100$ noisy receptions. The value of $D$ is unknown but it is the same for all $100$ receptions for

both types of pulses. Your task is to use your matched filter to estimate $D.$

Plot the first reception for each type of pulse. Can you see the pulse in the reception? or is the noise value

too high for the pulse to be visible? $($You may want to plot a few other receptions just to get a feel for the

data set.$)$

Run your matched filter on the first reception for each type of pulse. Plot the results. What is your

estimate for $D$ based on these results?

Now automate the process of estimating $D$ for each reception. You can do this by using numpy's argmax

function $($np$.$argmax$)$ to identify the location of the peak in the matched filter output. You should be able

to use that peak location to compute your estimate for $D($accounting for the delay through the matched

filter$).$ Loop through all $100$ receptions for each data set. Save the estimated $D$ values for each reception

in a vector so that you can make one plot showing the estimates for D for each type of pulse. Can you

tell from the plot which type of pulse does a better job of estimating $D?$ Could you have predicted the

performance from the autocorrelation function for each pulse? Why or why not?