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Title: wavlets
Description: A wavelet is a waveform of effectively limited duration that has an average value of zero and nonzero norm. This note gives brief description about wavlets

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wavelet
A wavelet is a waveform of effectively limited duration that has an average value of zero and nonzero
norm
...
Similarly, wavelet
analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet
...
Consider a sinusoidal signal with a small discontinuity — one so tiny as to
be barely visible
...
A plot of the Fourier coefficients (as provided by the fft command) of this signal shows nothing
particularly interesting: a flat spectrum with two peaks representing a single frequency
...


There is clearly a relationship between scale and frequency
...
The more stretched the wavelet, the longer the portion of the signal with
which it is being compared, and therefore the coarser the signal features measured by the wavelet
coefficients
...

High scale a ⇒ Stretched wavelet ⇒ Slowly changing, coarse features ⇒ Low frequency ω
...
It is what gives the signal its
identity
...

In wavelet analysis, we often speak of approximations and details
...
The details are the low-scale, high-frequency
components
...


The original signal, S, passes through two complementary filters and emerges as two signals
...
Then the resulting signals will each have 1000 samples, for a
total of 2000
...
This is the notion of downsampling
...


The process on the right, which includes downsampling, produces DWT coefficients
...


Multiple-Level Decomposition
The decomposition process can be iterated, with successive approximations being decomposed
in turn, so that one signal is broken down into many lower resolution components
...


Looking at a signal's wavelet decomposition tree can yield valuable information
...

In wavelet analysis, a signal is split into an approximation and a detail
...
For
an n-level decomposition, there are n+1 possible ways to decompose or encode the signal
...
This yields more
than different ways to encode the signal
...


The wavelet decomposition tree is a part of this complete binary tree
...
This is an example of a
representation that is not possible with ordinary wavelet analysis
...
This means we look at each node of
the decomposition tree and quantify the information to be gained by performing each split
Title: wavlets
Description: A wavelet is a waveform of effectively limited duration that has an average value of zero and nonzero norm. This note gives brief description about wavlets