Chapter 2Overview of different types of transformationFourier TransformThrough Fourier Transform we will get information about the frequency spectrum, what frequencies exist in the information. Through Fourier transform, we can get perfect knowledge of the existing frequency, but we cannot get any information about the location of the frequencies in the time domain. Thus, the Fourier transform has a good response in the frequency domain and a poor response in the time domain for the input information [7]. Now, the Fourier transform decomposes a signal into complex exponential functions of different frequencies. How it does this is defined by the following two equations: _(-∞)^∞▒〖X (f)∙e^2jπft dt〗 (2.2) In the above equation, t represents time, f represents frequency, and x denotes the available signal. Note that x denotes the signal in the time domain and X denotes the signal in the frequency domain. This convention makes it possible to distinguish the two representations of the signal. Equation (2.1) is called the Fourier transform of x(t) and equation (2.2) is called the inverse Fourier transform of X(f), which is x(t). Now first here we discuss about the two types of signal system which are given below.Stationary Signal SystemNonstationary Signal SystemHere we discuss how the two types of signals will react in frequency domain and time domain by Fourier transformation of the input signal as below. x (t)= cos⁡〖(2π*10t)〗+ cos⁡〖(2π*25t)〗+cos⁡〖(2π*50t)〗+cos⁡〖(2π*100t)〗We first show the response of the stationary signal system in time and frequency...... middle of paper ......signal convolution ratio with the filter impulse response. The discrete-time convolution operation is defined as follows: x[n]*h[n]=∑_(k= -∞)^∞▒〖x[k]∙h[nk] 〗 (2.6)There exists also another method to decompose the signal into high pass and low pass signal by the lifting based scheme defined in chapter 3. Through Discrete Wavelet Transform (DWT) we can separate the high and low frequency parts of a one-dimensional signal through the use of filters [7, 8]. For a level of transformation, we need to follow the steps described below: The input signal passes through high and low component filters. Then it will be downsampled by a factor of two. Multiple levels (scales) are created by repeating the filtering and decimation process on the low-pass and high-pass outputs in the same manner.Fig. 2.9 Only one level of transformation