Wavelet Transformation

$$ f(t)=\sum_{k}a_{k} \phi(t) dt $$

$$\text{Here function }\phi(t) \text{is analysis function. Depending up on the choice of }\phi(t), \text{transform is Fourier Transform if } $$ $$\phi(t) \text{ is } e^\frac{2\pi k t}{T}, \text{ is Wavelet Transform if }\phi(t) \text{ is wavelet function and Cosine Transform if } \phi(t) \text{ is cosine function.}$$

Wavelet is the newly emerging transformation theory, where you transform real signals into frequency and time domain at the same time so that the time range of occurrence for some band signals could be found. In this transformation, translated and scaled wavelet function is convoluted to the signal to be analysed.

Wavelet transform of impulse function at t=500 with haar wavelet as basis function.

Wavelet technology could be best available compression technologies with complete reconstruction of original signal with no information loss. Being always lossy compression with DCT compression technology, wavelet transform can be used for lossless compression. In any compression technologies, larger the number of transformation coefficients near to zero higher the compression ration is. Lager the number of coefficient made zero which are near to zero, higher the information is lost and the compression is lossy.

Multi-resolution Analysis

Continuous WT is done with multi-resolution analysis with change in scaling factor s and translational factor tow (t). But in case of discrete WT multi-resolutional analysis is done with different filter bank where filters composed of high pass and low pass filters. Low pass filter gives approximation where as high pass gives details of the signal to be analysed. Keep in mind the calculation time to be optimum, out put of each low pass filter in analysis filter bank is again passed to low pass and high pass filter. To keep the data bit same, down sampling in analysis filter bank is done. Again reverse process will be done in synthesis filter bank in case of reconstruction of the signal analysed.