Applied & Computational Harmonic Analysis Homework Page (Spring 2023)

Homework #1 Due Monday April 24, 2023 PDF file
Please take a look at my comments on HW1.

Homework #2 Due Monday, May 15, 2023 PDF file
My comments to HW2 is here. Also, please take a look at my Julia codes and run them!

prob2.jl

prob3.jl

prob4.jl

Homework #3 Due Monday, June 5, 2023 PDF file
Please read my comments to HW3, which contain useful and important information.

Final Project Report Due 3pm Friday, June 9, 2023

Please use a word processing system to write your report (LaTeX strongly recommended).
You should not write more than 10 pages.
Submit your pdf file of your report to me via email.
The best possible scenario is to write a report on a project you yourself come up with. Examples are:
- Description of your Ph.D. research projects where you plan to use the tools you learned (or will learn) in this course.
- A survey of some particular area of applications of the ideas learned (or to be learned) in this course, e.g., image compression, denoising, interpolation, fast algorithms, numerical analysis, statistics, etc.

If you have difficulty specifying your project, I would suggest that you do one of the following possible projects. Regardless of which project you choose, I would recommend to use some of the following Julia packages for your project:
1. Wavelets.jl;
2. WaveletsExt.jl;
3. SignalAnalysis.jl (for the Windowed/Short-Time Fourier Transform);
4. ContinuousWavelets.jl;
5. Gabriel Peyré's The Numerical Tours in Julia.
- Project A: Discriminant Signal Analysis
  - Record your voice (some words) via microphone.
  - Get the digital waveforms of your voice.
  - If you are male, then ask a female student (not necessarily in this class) to get her voice recording of the same words as you recorded. If you are female, get voices of a male student.
  - Apply the Windowed Fourier Transform to these voice recordings to compute the spectrograms.
  - Interpret the results. Can you identify male voices and female voices by looking at the spectrograms?
  - Repeat the same experiments using the Continuous Wavelet Transform and the scalograms. Which are easier to interpret the voices, spectrograms or scalograms?
  - Repeat the same experiments using the Scattering Transform. You need to read the following paper for the scattering transform:
    J. Andén, V. Lostanlen, and S. Mallat: "Joint time-frequency scattering for audio classification," 2015 IEEE 25th International Workshop on Machine Learning for Signal Processing (MLSP). The scattering transform software can be downloaded from here.
  - Describe your further thoughts for recognizing male/female voices.
  - Instead of male/female voice signals, you can use any two classes of signals that show different time-frequency characteristics. Examples include: comparison of historical records of some stock prices of two companies or market indices of two countries, or seismograms (seismic signals) caused by earthquakes vs those by nuclear explosions, to name a few.
- Project B: Generalized Morse Wavelet Transform Experiments
  - Get your favorite time series data such as music signals, stock market price curves, etc.
  - Read the following papers very carefully and learn what the generalized Morse wavelet transforms are:
    S. C. Olheda and A. T. Walden: "Generalized Morse wavelets," IEEE Trans. Signal Process., vol.50, no.11, pp.2661-2670, 2002.
    J. M. Lilly and S. C. Olheda: "Higher-order properties of analytic wavelets," IEEE Trans. Signal Process., vol.57, no.1, pp.146-160, 2009.
    J. M. Lilly and S. C. Olheda: "On the analytic wavelet transform," IEEE Trans. Inform. Theory, vol.56, no.8, pp.4135-4156, 2010.
    J. M. Lilly and S. C. Olheda: "Generalized Morse wavelets as a superfamily of analytic wavelets," IEEE Trans. Signal Process., vol.60, no.11, pp.6036-6041, 2012.
  - Apply the generalized Morse wavelet transform with various different parameters (β, γ) to your favorite data you prepared above, plot the time-frequency plane figure of your data.
  - Interpret these time-frequency plane figures, analyze your data on these time-frequency planes, and discuss what you have found out about your data, and discuss which parameter pair (β, γ) gives you the best result in terms of data interpretation.
- Project C: Coifman-Meyer Gaussian Bases Experiments
  - Get your favorite time series data such as music signals, stock market price curves, etc.
  - Read the following paper very carefully and learn what the Gaussian bases of Coifman and Meyer are:
    R. R. Coifman and Y. Meyer: "Gaussian Bases," Applied and Computational Harmonic Analysis, vol.2, no.3, pp.299-302, 1995.
  - Implement the Gaussian Bases using Julia.
  - Expand your favorite data you prepared above into the Gaussian Basis with a particular parameter ζ, plot the magnitude of the expansion coefficients in the nonincreasing order. Then do the same for different values of ζ and compare the results. In particular, examine the relationship between the speed of the decay of the magnitude of the coefficients and the parameter ζ.
  - Repeat the above experiments for a different signal and check your previous explanation and conclusion on the relationship between the coefficient decay speed and ζ still hold or not.