Why transform is Wavelet ?

In Fourier Transform, signal is decomposed as the constituent of sine and cosine functions losing time information about the signal signal that is FT can not give the time of occurrence for the particular frequency sine or cosine signal.

Where as in Wavelet Transform, in Continuous Wavelet Transform analysis function is correlated to small segment of signal to be analysed translating wavelet function to calculated wavelet coefficient and again wavelet is dilated and then again correlated to segments of signal to be analysed translating the dilated wavelet along the time axis. This seems that small segment is formed which is wavelet of the signal to be analysed.

In Discrete Wavelet Transform, signal is passed through bank of analysis filters which separates the signal of particular frequency band signal that is signal of particular band of frequency is separated which is small wavelet of the signal to be analysed. Thus this kind of analysis is sub band coding or multi resolutional analysis.

De-Noising a Signal with WT

Wavelet Transform is also one of the most useful tool to remove noise from the communication signal. Significant de-noising can be obtained with higher level of approximation coefficients in wavelets.

Fig. 1

In the above fig first plot is original signal with noise when sample area is 3000 and the signal is decomposed with wavelet transform and then reconstructed. As we go downwards smoother signal is reconstructed. Second Last plot is reconstructed plot with approximation lavel-5.

In addition to that, wavelet transform can be used for compression techniques which can give good compression with lossy or lossless type compression. 

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.