Find Filters for Wavelets Analysis Homework: p_k Coefficients & Relations

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SUMMARY

The discussion focuses on deriving four specific filters for wavelet analysis based on given p_k scaling coefficients and wavelet relations. The filters in question are low-pass decomposition, high-pass decomposition, low-pass reconstruction, and high-pass reconstruction, defined by the equations φ(x) and ψ(x). Participants express confusion regarding the design of these filters and inquire about the possibility of constructing time-series data using wavelets, similar to Fourier transforms. The consensus emphasizes the need for a solid understanding of wavelet theory to tackle these problems effectively.

PREREQUISITES
  • Understanding of wavelet theory and its applications
  • Familiarity with scaling coefficients and wavelet relations
  • Knowledge of filter design in signal processing
  • Basic proficiency in Fourier transforms and time-series analysis
NEXT STEPS
  • Study the derivation of wavelet filters from scaling coefficients
  • Learn about the construction of time-series using wavelets
  • Explore the differences between wavelet transforms and Fourier transforms
  • Investigate practical applications of wavelet analysis in signal processing
USEFUL FOR

Students and researchers in signal processing, mathematicians focusing on wavelet theory, and professionals involved in time-series analysis will benefit from this discussion.

eemath_tamu
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Homework Statement


Given the following p_k scaling coefficients and the following wavelet relations, find all four filters corresponding to these coefficients: low-pass decomposition, high-pass decomposition, low-pass reconstruction, and high-pass reconstruction.

Homework Equations


\phi (x)= p_0\phi (2x) + p_1\phi (2x-1) + p_2\phi (2x-2) + p_3\phi (2x-3)
\psi (x)=p_3\phi (2x+2) - p_2\phi (2x+1) + p_1\phi (2x) - p_0\phi (2x-1)

The Attempt at a Solution


I just need help knowing how to start this problem. I've been looking throughout my book for the equations of the filters. I don't really understand how to design the filters from the wavelet relations.
 
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Does anyone know if i can construct-compose time-series consist of wavlets? e.g. if i have as data a characteristic wave height and a wave length i can produce time-series with Fourier transform which equals of a superposition of sinus waves. i want to do the same but with wavelets. possible?
 

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