# MA304: Topics in Applied Mathematics

Unit 4: Time-Frequency Analysis   The Fourier transform (FT) introduced in this unit is the analogue of the sequence of Fourier coefficients of the Fourier series discussed in Unit 3 in that the normalized integral over the “circle” in the definition of Fourier coefficients is replaced by the integral over the real line to define the FT.  While the Fourier series is used to recover the given function it represents from the sequence of Fourier coefficients, it is non-trivial to justify the validity of the seemingly obvious formulation of the inverse Fourier transform (IFT) for the recovery of a function from its FT.  This unit will introduce the notions of localized FT (LFT) and localized IFT (LIFT).  We will also establish an identity that governs the relationship between LFT and LIFT, when the sliding frequency-window function for the LIFT is complex conjugate of the Fourier transform of the sliding time-window function in for the LFT.  Because the Fourier transform of a Gaussian function remains to be a Gaussian function, any Gaussian function can be used as a time-sliding window for simultaneous time-frequency localization.  This same identity is also applied to justify the validity of the formulation of the IFT by taking the variance of the sliding Gaussian time-window to zero.  Another important consequence of this identity is the Uncertainty Principle, which states that the Gaussian is the only window function that provides optimal simultaneous time-frequency localization with area of the time-frequency window equal to 2.  Discretization of any frequency-modulated sliding time-window of the LFT at the integer lattice yields a family of local time-frequency basis functions.  Unfortunately, the Balian-Low restriction excludes any sliding time-window function, including the Gaussian, to attain finite area of the time-frequency window, while providing stability for the family of local time-frequency basis functions, called a “frame.”  This unit ends with a discussion of a way for avoiding the Balian-Low restriction by replacing the frequency-modulation of the sliding time-window function with modulation by certain cosine functions.  More precisely, a family of stable local cosine basis functions, sometimes called Malvar “wavelets,” is introduced to achieve good time-frequency localization.  As an application, undesirable blocky artifact of highly compressed JPEG pictures, as discussed in Unit 2, can be removed by replacing the 8-point DCT with certain appropriate discretized local cosine basis function for each of the 8 by 8 image tiles.

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

- Compute the Fourier transform of some simple functions. - Compute the Fourier transform of the affine transformation of some given functions. - Compute the convolution of some simple functions with certain filters. - Compute the Gabor transform of some simple functions. - Formulate local time-frequency basis functions from a given sliding time-window function. - Formulate local cosine basis functions from a given sliding time-window function.

4.1 Fourier Transform   - Lecture: MIT: Professor Gilbert Strang’s Computational Science and Engineering: “Lecture 33: Filters, Fourier Integral Transform” and “Lecture 34: Fourier Integral Transform (Part 2)” Links: MIT: Professor Gilbert Strang’s Computational Science and Engineering: “Lecture 33: Filters, Fourier Integral Transform” (YouTube) and “Lecture 34: Fourier Integral Transform (Part 2)” (YouTube)

Instructions: Please click on the links above, and view these video lectures to learn more about the essence of the Fourier Transform and filtering.  Please note that these videos cover the topics outlined for sub-subunits 4.1.1 and 4.1.2.
Viewing these lectures and pausing to take notes should take approximately 2 hours and 30 minutes to complete.

4.1.1 Definition and Essence of the Fourier Transform   - Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms” Links:University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms” (PDF)

Instructions: Please click on the link above, and then select the link to “Chapter 8: Fourier Transforms” to download the PDF file.  Study pages 283–298 on the concept and properties of the Fourier Transform.  Note that this reading covers the topics outlined for sub-subunits 4.1.1 and 4.1.2.
Studying this reading should take approximately 1 hour to complete.

4.1.2 Properties of the Fourier Transform   Note: This topic is covered by the reading and lectures assigned below sub-subunit 4.1.1.

4.1.3 Sampling Theorem   - Lecture: MIT: Professor Gilbert Strang’s Computational Science and Engineering: “Lecture 36: Sampling Theorem” Links: MIT: Professor Gilbert Strang’s Computational Science and Engineering: “Lecture 36: Sampling Theorem” (YouTube)

Instructions: Please click on the link above, and view this lecture to learn about the application of the Fourier transform and Fourier series to deriving and understanding the essence of the Sampling Theorem.
Viewing this lecture and pausing to take notes should take approximately 1 hour to complete.

4.1.4 Applications of the Fourier Transform   - Lecture: YouTube: Stanford University: Department of Electrical Engineering’s “Lecture 1: The Fourier Transforms and Its Applications,” “Lecture 6: The Fourier Transforms and Its Applications,” and “Lecture 8: The Fourier Transforms and Its Applications” Links: YouTube: Stanford University: Department of Electrical Engineering’s “Lecture 1: The Fourier Transforms and Its Applications” (YouTube), “Lecture 6: The Fourier Transforms and Its Applications” (YouTube), and “Lecture 8: The Fourier Transforms and Its Applications” (YouTube)

`````` Instructions: Please click on the links above, and view the video
lectures to learn about applications of the Fourier Transforms.
Viewing these video lectures and pausing to take notes should take
approximately 3 hours and 30 minutes to complete.

displayed on the webpages above.
``````

4.2 Convolution Filter and Gaussian Kernel   4.2.1 Convolution Filter   - Lecture: MIT: Professor Gilbert Strang’s Computational Science and Engineering: “Lecture 32: Convolution (Part 2), Filtering” Link: MIT: Professor Gilbert Strang’s Computational Science and Engineering: “Lecture 32: Convolution (Part 2), Filtering” (YouTube)

Instructions:  Please click on the above link above, and view the video lecture on the convolution filter.
Viewing this lecture and pausing to take notes should take approximately 1 hour and 15 minutes to complete.

4.2.2 Fourier Transform of the Gaussian   - Reading: Fourier Transform of the Gaussian The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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4.2.3 Inverse Fourier Transform   - Reading: Inverse Fourier Transform The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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4.3 Localized Fourier Transform   - Reading: Localized Fourier Transform The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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4.3.1 Short-time Fourier Transform (STFT)   4.3.2 Gabor Transform   4.3.3 Inverse of Localized Fourier Transform   4.4 Uncertainty Principle   - Reading: Uncertainty Principle The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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4.4.1 Time-Frequency Localization Window Measurement   4.4.2 Gaussian as Optimal Time-Frequency Window   4.4.3 Derivation of the Uncertainty Principle   4.5 Time-Frequency Bases   - Reading: Time-Frequency Bases The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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4.5.1 Balian-Low Restriction   4.5.2 Frames   4.5.3 Localized Cosine Basis   4.5.4 Malvar Wavelets