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wavelet
A wavelet is a waveform of effectively limited duration that has an average value of zero and nonzero
norm
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Similarly, wavelet
analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet
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Consider a sinusoidal signal with a small discontinuity — one so tiny as to
be barely visible
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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
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There is clearly a relationship between scale and frequency
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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
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High scale a ⇒ Stretched wavelet ⇒ Slowly changing, coarse features ⇒ Low frequency ω
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It is what gives the signal its
identity
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In wavelet analysis, we often speak of approximations and details
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The details are the low-scale, high-frequency
components
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The original signal, S, passes through two complementary filters and emerges as two signals
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Then the resulting signals will each have 1000 samples, for a
total of 2000
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This is the notion of downsampling
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The process on the right, which includes downsampling, produces DWT coefficients
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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
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Looking at a signal's wavelet decomposition tree can yield valuable information
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In wavelet analysis, a signal is split into an approximation and a detail
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For
an n-level decomposition, there are n+1 possible ways to decompose or encode the signal
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This yields more
than different ways to encode the signal
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The wavelet decomposition tree is a part of this complete binary tree
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This is an example of a
representation that is not possible with ordinary wavelet analysis
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This means we look at each node of
the decomposition tree and quantify the information to be gained by performing each split

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