# Wavelet Theory: Solving ∫t^pψ(t)dt=0

• omer21
In summary, the conversation discusses the derivation of the equation \sum_k (-1)^kk^pc_k=0 from the integral ∫t^pψ(t)dt=0. The conversation also mentions the approximation method and how it follows from the definitions in equations 5.12-13 on page 79. The conversation then continues to discuss the derivation of equations 5.14 and 5.15 and the steps involved in obtaining them. The conversation ends with a discussion of how to find the integral and the use of integration by parts.
omer21
I don't understand how to get the equation $\sum_k (-1)^kk^pc_k=0$ from $∫t^pψ(t)dt=0$ from here on page 80. Can somebody explain it?

It's an approximation method - follows from the definitions in eqs. 5.12-13 on p79.
Can you see how 5.14 and 5.15 are obtained?

i can see 5.14 but i can not obtain 5.15 and 5.16.
Actually i get $\sum_k (-1)^kc_k.0=0$ instead of $\sum_k (-1)^k k^p c_k=0$ from the integral $∫t^p [\sum_k (-1)^kc_k \phi(2t+k-N+1)]dt=0$.

Then you should go back over the derivations and make sure you understand what they are saying ... starting from the latest that you appear to be able to follow:

5.14 was the case for p=0.
i.e. ##\phi(t)## is capable of expressing only the zeroth monomial.
therefore, the zeroth-order moment of the wavelet function must be zero.
Which is to say...

$$\int_{-\infty}^\infty \psi(t)dt=0$$ ... which, to my mind, means that ##\psi## must be odd... which places limits on the expansion coefficients as the book says.

5.15 was the case for p=1 ... so ##\phi(t)## is capable of expressing up to the first monomial.
therefore, the zeroth and first-order moment of the wavelet function must be zero.
the zeroth order moment is above. The first order is given by...

$$\int t\psi(t)dt =0$$ ... so what steps did you follow from here?

I substituted

$\psi(t)=[∑_k(−1)^kc_k\phi(2t+k−N+1)]$

into

$∫t\psi(t)dt$.

Then i got

$∫t[∑_k(−1)^kc_k\phi(2t+k−N+1)]dt=∑_k(−1)^kc_k∫t\phi(2t+k−N+1)]dt=0$.

Actually i am not sure how to find the above integral but i did some change of variables and used integration by parts however i could not reach $∑_k(−1)^kkc_k$.

## 1. What is wavelet theory?

Wavelet theory is a mathematical framework that allows for the analysis and processing of signals and data at different scales. It involves breaking down a signal into smaller parts, called wavelets, which can be used to represent and analyze the original signal. This theory has various applications in fields such as signal processing, data compression, and image processing.

## 2. What is the significance of solving ∫t^pψ(t)dt=0 in wavelet theory?

This integral equation is known as the wavelet transform, and solving it allows for the identification of the wavelet function ψ(t) that best represents a given signal. This is an essential step in wavelet analysis, as it allows for the extraction of useful information from the original signal and can also aid in signal denoising and compression.

## 3. What is the role of the parameter p in the integral equation ∫t^pψ(t)dt=0?

The parameter p determines the type of wavelet used in the analysis. Different values of p result in different families of wavelets, each with its own set of properties and applications. For example, p=0 corresponds to the Haar wavelet, while p=1 corresponds to the Daubechies wavelet.

## 4. How is wavelet theory different from Fourier analysis?

While both wavelet theory and Fourier analysis involve decomposing a signal into smaller parts, they differ in how they do so. Fourier analysis uses sinusoidal functions to represent a signal, while wavelet theory uses wavelets which are localized in both time and frequency. This allows for more accurate representation and analysis of signals with localized features.

## 5. What are some real-world applications of wavelet theory?

Wavelet theory has various applications in fields such as signal and image processing, data compression, and pattern recognition. It is used in medical imaging to enhance and analyze images, in audio and video compression to reduce file size while maintaining quality, and in financial analysis to detect patterns in market data. It also has applications in geophysics, astronomy, and weather forecasting.

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