AMATH 582/482, Winter 2022 Homework 1
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Homework 1
AMATH 582/482, Winter 2022
Directions, Reminders and Policies
Read these instructions carefully:
∙ You are required to upload a PDF report to Canvas along with a zip of your code.
∙ The report should be a maximum of 6 pages long with references included. Minimum font size 10pts
and margins of at least 1inch on A4 or standard letter size paper.
∙ Do not include your code in the report. Simply create a zip file of your main scripts and functions,
without figures or data sets included, and upload the zip file to Canvas.
∙ Your report should be formatted as follows:
– Title/author/abstract: Title, author/address lines, and short (100 words or less) abstract. This is not meant to be a separate title page.
– Sec. 1. Introduction and Overview
– Sec. 2. Theoretical Background
– Sec. 3. Algorithm Implementation and Development
– Sec. 4. Computational Results
– Sec. 5. Summary and Conclusions
– Acknowledgments ( no more than four or five lines, also see the point below on collaborations)
– References
∙ I suggest you use LATEX(Overleaf is a great option) to prepare your reports. A template is provided on
Canvas under the Syllabus tab. You are also welcome to use Microsoft Word or any other software that properly typesets mathematical equations.
∙ I encourage collaborations, however, everything that is handed in (both your report and your code)
should be your work. You are welcome to discuss your assignments with your peers and seek their advice but these should be clearly stated in the acknowledgments section of your reports. This also
includes any significant help or suggestions from the TAs or any other faculty in the university. You don’t need to give all the details of the help you received, just a sentence or two.
∙ Your homework will be graded based on how completely you solved it as well as neatness and little things like: did you label your graphs and include figure captions. The homework is worth 20 points. 10 points will be given for the overall layout, correctness and neatness of the report, and 10 additional points will be for specific technical things that the TAs will look for in the report itself.
∙ Late submissions will not be accepted on Canvas, send them to [email protected] directly. Late
reports are subject to a 2 points/day penalty up to five days. They are no longer accepted afterwards.
For example, if your report is three days late and you managed to get 16/20, your final grade will be
16 − 6 = 10. Basically, you will lose 2% of your overall course grade for each day the report is late. So be careful.
Problem Description: Finding Submarines
Your goal in this homework is to locate a submarine in the Puget Sound using noisy acoustic data. We do not know much about this submarine as it is a new technology that emits an unknown acoustic frequency that you need to detect. Unfortunately the submarine is moving so its location and path need to be determined. Broad spectrum recording of acoustics data obtained over 24 hours in half-hour increments is available to you. You can download the data from using the Google drive links on Canvas; either of the data files subdata.npy for Python users, subdata.mat for MATLAB users or subdata.csv in text format if previous two formats are insufficient. These files contain a matrix with 49 columns of data corresponding to the
measurements of acoustic pressure taken over 24 hours. The measurements themselves are 3D and taken on a uniform grid of size 64 × 64 × 64. The provided notebook will visualize this data for you. If the 3D plots are slow and you cannot see the dynamic behavior of the data consider downloading the GIF file subdata.gif on Canvas. This should help you better understand the information in the data set.
Some comments and hints
Here are some useful comments and facts to guide you along the way.
1. First, observe that you are provided three dimensional dynamic data, that is acoustic pressure measurements in 3D and as a function of time. This makes visualization difficult as the data set is truly four dimensional. If you would like to see temporal variations then I suggest looking at 2D slices of the data as a function of time. This is if the GIF is not enough.
2. On the same note, we only saw 1D and 2D Fourier transforms in class but nothing changes in the 3D setting except that you need to use the fftn function. fftshift remains valid and is still needed.
3. Here is a useful fact about noise and Fourier transform that will help you in this assignment and in many applications.
It is known that adding mean zero white noise to a signal is equivalent to adding mean zero white noise to its Fourier series coefficients.
This fact enables one to devise a very simple and effective noise filtering technique in situations where multiple measurements are available that are subject to the same noise. Such is the case in many imaging or acoustics applications like our submarine problem. Since the noise is random and mean
zero it should average to zero over many samples. Hence, averaging the measurements in the Fourier domain should already reduce the noise.
The more measurements you have the better the effect of this averaging. But in the case of the submarine data you only have few measurements so you still need to do some kind of filtering.
Tasks
Below is a list of tasks to complete in this homework and discuss in your report.
1. Through averaging of the Fourier transform and visual inspection, determine the frequency signature (center frequency) generated by the submarine.
2. Design and implement a Filter to extract this frequency signature in order to denoise the data and determine the path of the submarine. You can also plot the 3D path of the submarine if you like.
3. Determine and plot the 
,
coordinates of the submarine during the 24 hour period. This information can be used to deploy a sub-tracking aircraft to keep an eye on your submarine in the future.
2022-01-26