The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula: The Gaussian function has infinite range and it is impractical for implementation. However, a level of significance can be chosen (for instance 0.00001) for the distribution of the Gaussian function. Outside these limits of integration the Gaussian function is small enough to be ignored. Thus the Gabor transform can be satisfactorily approximated as This simplification makes the Gabor transform practical and realizable. The window function width can also be varied to optimize the time-frequency resolution tradeoff for a particular application by replacing the with for some chosen . (Wikipedia).
Introduction to the z-Transform
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces the definition of the z-transform, the complex plane, and the relationship between the z-transform and the discrete-time Fourier transfor
From playlist The z-Transform
The Spectrogram and the Gabor Transform
Here I introduce the spectrogram, which is a moving-window Fourier transform, giving insight into the time-frequency content of a data set. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 2 from: "Data-Driven Science and E
From playlist Data-Driven Science and Engineering
Karlheinz Gröchenig: Gabor Analysis and its Mysteries (Lecture 1)
The lecture was held within the framework of the Hausdorff Trimester Program Mathematics of Signal Processing. In Gabor analysis one studies the construction and properties of series expansions of functions with respect to a set of time-frequency shifts (phase space shifts) of a single fu
From playlist HIM Lectures: Trimester Program "Mathematics of Signal Processing"
Laplace transform: sin(at) and cos(at)
Playlist: https://www.youtube.com/watchv=5RCijylGyK4&list=PLN2B6ZNu6xmfaT2IBigUylhfpFFA1L8Mp German version: https://youtu.be/ABX0kUx2WuM Update video: https://youtu.be/Hh6mR_vwWuw Let's, once again, kill two birds with one stone! We are taking a look at the laplace transformation of a ti
From playlist Laplace transform
PUSHING A GAUSSIAN TO THE LIMIT
Integrating a gaussian is everyones favorite party trick. But it can be used to describe something else. Link to gaussian integral: https://www.youtube.com/watch?v=mcar5MDMd_A Link to my Skype Tutoring site: dotsontutoring.simplybook.me or email dotsontutoring@gmail.com if you have ques
From playlist Math/Derivation Videos
Karlheinz Gröchenig: Gabor Analysis and its Mysteries (Lecture 4)
The lecture was held within the framework of the Hausdorff Trimester Program Mathematics of Signal Processing. In Gabor analysis one studies the construction and properties of series expansions of functions with respect to a set of time-frequency shifts (phase space shifts) of a single f
From playlist HIM Lectures: Trimester Program "Mathematics of Signal Processing"
Nicki Holighaus: Time-frequency frames and applications to audio analysis - Part 1
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Analysis and its Applications
Karlheinz Gröchenig: Gabor Analysis and its Mysteries (Lecture 2)
Due to technical problems the blackboard is not visible. The lecture was held within the framework of the Hausdorff Trimester Program Mathematics of Signal Processing. In Gabor analysis one studies the construction and properties of series expansions of functions with respect to a set of
From playlist HIM Lectures: Trimester Program "Mathematics of Signal Processing"
Gaussian Integral 10 Fourier Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I show how the Gaussian integral appears in the Fourier transform: Namely if you t
From playlist Gaussian Integral
Speech Analysis and a Tour of the Math Behind it #SoME2
What you are looking at right now is a spectragram of my voice as I am speaking into a microphone. When I first saw this I was quite perplexed by all the strange shapes and features embedded within the image. It made me wonder just how speech analysis is done? I was so intrigued by this t
From playlist Summer of Math Exposition 2 videos
Jean-Claude Risset: Sound, music and wavelets in Marseille: A reminder of early sonic [...]
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist 30 years of wavelets
Karlheinz Gröchenig: Gabor Analysis and its Mysteries (Lecture 3)
Due to technical problems the blackboard is not visible. The lecture was held within the framework of the Hausdorff Trimester Program Mathematics of Signal Processing. In Gabor analysis one studies the construction and properties of series expansions of functions with respect to a set of
From playlist HIM Lectures: Trimester Program "Mathematics of Signal Processing"
Laplace transform: The Unit Step Function ℒ{1} and ℒ{0}
Playlist: https://www.youtube.com/watch?v=5RCijylGyK4&list=PLN2B6ZNu6xmfaT2IBigUylhfpFFA1L8Mp German version: https://youtu.be/EygP8LO8YjI Let's dive into a new series' of videos! Welcome to the world of relaxation processes and laplace transforms =) Help me create more and better conten
From playlist Laplace transform
This video describes how to compute the Spectrogram in Python. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 2 from: "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and
From playlist Data-Driven Science and Engineering
Hans Feichtinger: Wavelet theory, coorbit spaces and ramifications
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist 30 years of wavelets