FOURIER SERIES AND FOURIER TRANSFORM , Definition , Complex Exponential , Inverse

By   November 14, 2019

In the field of communication engineering, we need to analyze a given signal. To do so, we have to express the signal in its frequency domain. The translation of a signal from time domain to frequency domain is obtained by using the tools such as Fourier series and Fourier transform.
1.18.1. Fourier Series: Definition and Basic Concept
Sine waves and cosine waves are the basic building functions for any periodic signal. This means that any periodic signal basically consists of sine waves having different amplitudes, of different frequencies and having different relative phase shifts.
Fourier series represents a periodic waveform in the form of sum of infinite number of sine and cosine terms. It is a representation of the signal in a time domain series form.
Fourier series is a tool used to analyze any periodic signal. After the analysis, we obtain the following information about the signal :
(i) What all frequency components are present in the signal ?
(ii) Their amplitudes and
(iii) The relative phase difference between these frequency components.
All the frequency components are nothing else but sine waves at those frequencies.
1.18.2. Exponential Fourier Series (i.e. Complex Exponential Fourier Series)
          Substituting the sine and cosine functions in terms of exponential function in the expression for the quadrature Fourier series, we can obtain another type of Fourier series called the exponential Fourier series.
A periodic signal x(t) is expressed in the exponential Fourier series form as under :
Amplitude and phase spectrums
The amplitude spectrum of the signal x(t) is denoted by,
The phase spectrum of x(t) is given by,
The amplitude spectrum is a symmertic or even function. This means that  Cn = C-n. But the phase spectrum is an asymmetric or odd function. This means that arg (Cn) = -arg (C-n)
1.18.3 Fourier Transform: Mathematical Representation
A Fourier transform is the limiting case of Fourier series. It is used for the analysis of non-periodic signals. The Fourier transform of a signal x(t) is defined as follows:
Fourier transform
or                                                   Equation
These equations are known as analysis equations.
1.18.4 Inverse Fourier Transform : Mathematical Representation
The signal x(t) can be obtained back from Fourier transform X(t) by using the inverse Fourier transform. The inverse Fourier transform (IFT) is defined as under:
Inverse Fourier transform
Or                                                  Equation
Amplitude and phase spectrum
(i)      The amplitude and phase spectrums are continuous rather than being discrete in nature. Out of them, the amplitude spectrum of a real valued function x(t) exhibits an even symmetry.
Therefore, we have
X(t) = X(-f)
(ii)     Also, the phase spectrum has an odd symmetry. This means that
1.18.5 Various Properties of Fourier Transform
Table 1.4    Various properties of Fourier Transform

Sr. No. Name of Property Mathematical expression
1. linearity or superposition Equation
2. Time scaling Equation
3. Daulity or symmetry Equation
4. Time shifting Equation
5. Area under x(t) Equation
6. Area under X(f) Equation
7. Frequency shifting Equation
8. Differentiation in time domain  
9. Integration in time domain  
10. Conjugate functions  
11. Multiplication in time domain  
12. Convolution in time domain  

1.18.6 Fourier Transforms of Few Standard Signals
The Fourier transform of few standard signals is given in table 1.5.
Table 1.5    Fourier transform of few standard signals

Sr. No. Name of signal Mathematical representation Fourier transfrom
1.       Rectangular pulse of amplitude A and duration T    
2.       Sinc pulse    
3.       Decaying exponential signal for t > 0    
4.       Rising exponential pulse for t < 0    
5.       Double exponential pulse    
6.       Unit impulse    
7.       DC signal    
8.       Cosine signal    
9.       Sine Signal    
10.              Signum function    
11.              Unit step