tailieunhanh - Wavelet

Transforms wavelet | TRANSFORMS WAVELETS Transform Analysis Signal processing using a transform analysis for calculations is a technique used to simplify or accelerate problem solution. For example instead of dividing two large numbers we might convert them to logarithms subtract them then look-up the anti-log to obtain the result. While this may seem a three-step process as opposed to a one-step division consider that long-hand division of a four digit number by a three digit number carried out to four places requires three divisions 3-4 multiplication s and three subtractions. Computers process additions or subtractions much faster than multiplications or divisions so transforms are sought which provide the desired signal processing using these steps. Fourier Transform Other types of transforms include the Fourier transform which is used to decompose or separate a waveform into a sum of sinusoids of different frequencies. It transforms our view of a signal from time based to frequency based. Figure 1 depicts how a square wave is formed by summing certain particular sine waves. The waveform must be continuous periodic and almost everywhere differentiable. The Fourier transform of a sequence of rectangular pulses is a series of sinusoids. The envelope of the amplitude of the coefficients of this series is a waveform with a Sin X X shape. For the special case of a single pulse the Fourier series has an infinite series of sinusoids that are present for the duration of the pulse. Fundamental Third Harmonic Fifth Harmonic Digital Sampling of Waveforms In order to process a signal digitally we need to sample the signal frequently enough to create a complete picture of the signal. The discrete Fourier transform DFT may be used in this regard. Samples are taken at uniform time intervals as shown in Figure 2 and processed. Sum - Approximation of Square Wave If the digital information is multiplied by the Fourier coefficients a digital filter is created as shown Figure 3. If the sum of the .