Chapter 2
Overview of different types of Transformation
Fourier Transform
By the Fourier transformation, we will get the information about the frequency spectrum, which frequencies exist in the information. From the Fourier transform, we can get the perfect knowledge of what frequency exists, but we can’t get any information about where the frequencies are located in the time domain. So the Fourier transformation has good frequency domain response and bad time domain response for the input information [7]. Now, Fourier transformation decomposes a signal to complex exponential functions of different frequencies. The way it does this, is defined by the following two equations:
X(f)=∫_(-∞)^∞▒〖x (t)∙e^(-2jπft) dt〗 (2.1) x(t)=∫_(-∞)^∞▒〖X (f)∙e^2jπft dt〗 (2.2)
In the above equation, t stands for time, f stands for frequency, and x denotes the signal at hand. Note that x denotes the signal in time domain and the X denotes the signal in frequency domain. This convention is used 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, here we first discuss about the two types of signal system which are given below.
Stationary signal system
Non-stationary signal system
Here we discuss about how both type of signals will react in frequency domain and time domain by Fourier transformation of the input signal like below. x (t)= cos〖(2π*10t)〗+ cos〖(2π*25t)〗+cos〖(2π*50t)〗+cos〖(2π*100t)〗
First we show the response of the stationary signal system in both time and frequ...
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...ration of convolution of the signal with the impulse response of the filter. The convolution operation in discrete time is defined as follows: x[n]*h[n]=∑_(k= -∞)^∞▒〖x[k]∙h[n-k] 〗 (2.6)
There is also one other method to decompose the signal into high pass and low pass signal by the lifting based scheme which is defined in the chapter 3. By the Discrete Wavelet Transform (DWT), we can separate the high and low-frequency portions of one dimensional signal through the use of filters [7, 8].
For one level of transform we have to follow steps derived below:
Input signal is passed through High & Low component filters
Then it will be down sampled by a factor of two
Multiple levels (scales) are made by repeating the filtering and decimation process on low-pass and high-pass outputs same.
Fig. 2.9 One level of transformation
This equation shifts from the parent function based on the equation f(x) = k+a(x-h) . In this equation, k shifts the parent function vertically, up or down, depending on the value of k. The h value shifts the parent function to the left or right. If h equals 1, it goes to the right 1 unit, if it is negative 1, it goes to the left 1 unit. If a is negative, the parent function is reflected on the x-axis. If x is negative, the parent function is reflected on the y-axis.
In other words T(n) can be expressed as sum of T(n-1) and two operations using the following recurrence relation:
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