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Wavelets: a mathematical microscope

Artem Kirsanov・26 minutes read

Signal processing involves analyzing noisy signals in the real world using mathematical operations like Fourier transform and wavelet transform to extract structure and frequency components for better understanding and analysis. Wavelet transform provides a compromise between time and frequency resolution, offering a unique analysis tool for signals by adjusting the mother wavelet through time and frequency for a two-dimensional analysis space.

Insights

  • Signal processing is crucial for objectively characterizing and quantifying the structure within noisy real-world signals, with Fourier transform offering insights into frequency components but sacrificing time information for frequency details.
  • Wavelet transform, utilizing short-lived oscillations localized in time and satisfying constraints like zero mean and finite energy, provides a unique analysis tool that offers a compromise between time and frequency resolution, allowing for a more optimal analysis of signals in a two-dimensional space.

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Recent questions

  • What is signal processing?

    The field dedicated to analyzing noisy signals.

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Summary

00:00

Analyzing Signals with Wavelet Transform

  • The real world is filled with noisy and irregular signals that possess a certain structure.
  • Signal processing is a field dedicated to analyzing these signals, like electrical activity recordings from the mouse brain.
  • Scientists need precise mathematical operations to objectively characterize and quantify the structure within signals.
  • The concept of time-frequency duality involves representing data through alternative pairs like x1, x2 and y1, y2.
  • Fourier transform decomposes functions into sines and cosines, providing insights into frequency components.
  • Fourier transform sacrifices time information for frequency details, limiting its application in time-sensitive scenarios.
  • Wavelets offer a compromise between time and frequency resolution, using short-lived oscillations localized in time.
  • Wavelets are a family of functions satisfying constraints like zero mean and finite energy, providing a unique analysis tool.
  • Wavelet transform involves modifying the mother wavelet through time and frequency adjustments to analyze signals in a two-dimensional space.
  • The contribution of a wavelet to a signal is determined by how well it matches the signal's frequencies, assessed through mathematical operations like integration.

18:28

Understanding Dot Product and Wavelet Transform

  • Dot product is the product of vector lengths and cosine of the angle between them, indicating similarity.
  • Dot product is large and positive when vectors are similar and close, with a small angle.
  • When vectors are orthogonal, dot product is zero, showing independence.
  • If vectors are anti-correlated and opposite, dot product is negative.
  • Dot product can be calculated by multiplying corresponding coordinates of vectors and summing them.
  • Convolution is the operation of sliding a wavelet across a signal, calculating dot products.
  • Wavelet transform involves varying scaling and translation parameters to analyze signal frequencies.
  • Complex numbers, involving real and imaginary components, are used to describe rotations and wavelet functions.
  • Wavelet transform provides a compromise between time and frequency resolution, offering optimal analysis of signals.
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