Discrete Wavelet analysis with an example

MRA of a signal is equivalently decomposing signal into average information and detail informations of the signal to be analysed. Discrete signal is decomposed into approximations and details passing through analysis bank of filters.

A random lececcum signal is analysed and its analysing signals are obtained as shown in the fig. below

fig.1

In the above plot, first plot is the original signal to be analysed which is analysed in three stage decomposition. plots 2,3 and 4 are details of the signal and plot 5th is approximation of the signal. The analysis is done with matlab routines available for one dimensional analysis for one dimensional signal. 

If informations in plots 2,3,4 and 5 are passed through reconstruction filters that is synthesis bank of filters then original signal plot in 1st can be obtained.

fig. 2

Decomposition stages are as shown above in fig.2. A3 is approximation and D1, D2, D3 are details of the signal decomposed and corresponding plots are as shown in Fig 1. Each signal after decomposition is decimated by 2 to keep bit length same applying Nyquist criteria.

Similarly 5 stage decomposition with 5 details are as follow in fig 3:

fig. 3

Multi Resolutional Analysis (MRA)

In STFT, time resolution and frequency resolution is same for particular analysis as windowing with is constant for a particular analysis, but in case of wavelet transform, scaling function changes the frequency resolution and window width correspondingly.

Continuous Wavelet Transform is give as:
$$ S(a, b)= \int f(t) \psi{(\frac{t-a}b}) dt $$

$$ \text{ changing value of b in } \psi{(\frac{t-a}b}) \text{ gives the different frequency band which means signal analysis in instant} $$ $$ \text{ of a with }\frac{1}{ b} \text{ frequency band, that is multiresolutional analysis. } $$

Similarly for digital signal, multi-resolutional analysis is performed by passing signals through analysis filters, where filters composed of high pass filters, band pass filters and low pass filters.

Out of them, output of high pass filters are to be considered as details of the signals and out put of low pass filter is to be considered as approximation. Above structural diagram briefly describe the MRA of discrete signals. d0, d1, d2.... represent details and A0 represents approximation.