Contents:-

 

  Introduction to Fourier analysis methods

 

        ♦ Continuous-time Fourier series (CTFS)

 

        ♦ Discrete-time Fourier series (DTFS)

 

        ♦ Continuous-time Fourier transform (CTFT)

 

        ♦ Discrete-time Fourier transform (DTFT)

 

        ♦ Salient properties of Fourier analysis method

 

    Relationship among Fourier analysis methods

 

        ♦ Relationship between CTFS and CTFT

 

        ♦ Relationship between DTFS and DTFT

 

        ♦ Relationship between CTFT and DTFT

 

 

                   

 

                                     Introduction to Fourier analysis method

In the following we first describe different Fourier methods available for analysing continuous-time and discrete-time signals. It would be followed by listing of salient properties of these Fourier analysis methods.

 

Continuous-time Fourier series (CTFS):

 

For a continuous-time signal x(t), the Fourier series representation of a signal over a representation time is defined as

                                                                   

where X[k] is the harmonic function, k is the harmonic number and f _F = 1/ T_F. The harmonic function is computed as

                                                                    

The signal and its harmonic function form a pair which is indicated by the notation 

                                                                

The left hand side of the above relation represents the signal in time domain while the right hand side represents the transformation of the signal to a "harmonic-number" domain.

 

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Discrete-time Fourier series (DTFS):

 

For a discrete-time signal x(n), the Fourier series representation over a representation time    is defined as

                                                            

 where X[k] is the harmonic function and the harmonic function can be found as 

 

                                                           

The discrete-time signal and its harmonic function form a pair which is indicated by the notation below

                                                                    

where 'N_F' is the representation time and the notation means summation over any range of consecutive 'k' exactly N_F in length.

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Continuous-time Fourier transform (CTFT):

 

The continuous-time Fourier transform(CTFT) is defined as

 

                                                                 

 

                                                                    

 

The forward and inverse transforms  are almost same, only the sign of the exponent and the variable of integration change.

The continuous-time signal x(t) and its Fourier transform X(f) pairs are indicated by the notation below.

                                                                    

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Discrete-time Fourier transform (DTFT):

 

The discrete-time Fourier transform is defined by

                                                                   

   and                                                          

The signal and its harmonic function form a pair which is indicated by the notation below

                                                                    

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Salient properties of Fourier analysis methods:

 

Some of the salient properties exhibited by the continuous-time as well as discrete-time Fourier transform / Fourier series are given below.

               

        Linearity property: 

For CTFS	For DTFS
For  CTFT	For DTFT

  

          Time shifting property: 

For CTFS	For DTFS
For CTFT	For DTFT

   

         Frequency shifting property:

For CTFS	For DTFS
For CTFT	For DTFT

  

 

Multiplication-convolution property:

 

            Multiplication:

For CTFS	For DTFS
For CTFT	For DTFT

  

            Convolution:

For CTFS	For DTFS
For CTFT	For DTFT

 

            Parseval's theorem:

For CTFS	For DTFS
For CTFT	For DTFT

 

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    Relationship among Fourier analysis mehods: 

 

There are many similarities among different Fourier analysis methods and their relationship are discuss below. These relationships provide the basis of processing of the analog signal in digital domain.

Relationship between CTFS and CTFT:

 

For a periodic signal x(t) with fundamental period T₀ = 1/ f₀ can be represented for all time using CTFS representation  

                                                                              

Now using the frequency shifting property and the CTFT pair we can find the CTFT of x(t) as

                                                  

The CTFT of a periodic function is a continuous frequency function, which consists of a sum of impulses spaced apart by the fundamental frequency of the signal, whose strengths are the same as the CTFS harmonic function at the same harmonic number multiple of the fundamental frequency. The CTFS is just a special case of the CTFT.

                                                   

                                                                   

                                              

                              Fig.1 CTFS harmonic function and CTFT for a normalized gaussian function

 

  The information equivalence of a harmonic function X[k] and CTFT function X(f) is shown in Fig.1. Note, here X[K] is a function of discrete variable, harmonic number k whereas X(f) is a function of continuous variable frequency f. These two functions are equivalent in the sense that X(f) is non zero only at the integer multiples of 'k' of the fundamental frequency f₀ and X[k] is only defined at integer values of k. 

 

We can observe that the values of X[k] at the integer values of 'k' are the same as the strengths of the impulses in X(f) that occur at 'kf₀'. For a periodic function x(t) 

                                                        ........................(1)

The relationship between CTFS and CTFT can also be estblished for aperiodic signal and is achieved by finding relationship between the CTFT of an aperiodic signal and the CTFS function of  periodic extention of that signal. Let us consider x(t) be an aperiodic function of time and let x_p(t) is a periodic extension of x(t) with fundamental period T_p defined by

                                     

       

                         

        Fig.2. A signal and its CTFT and the periodic repetition of the signal and its CTFS harmonic function 

                             

The CTFT of x(t) is. Using the multiplication-convolution duality of the CTFT, the CTFT of x_p(t) is

                                                ........................................(2)

                                                          ,

where   now using equation (1),

                                                ........................(3)

Now combining equations (2) & (3), we get

                                  

We can say that if an aperiodic function is periodically extended to form a periodic function x_p(t) with fundamental period 'T_p' the values of the CTFS harmonic function X_p[K] of x_p(t) are samples of the CTFT X(f) of x(t) taken at the frequencies Kf_p and then multiplied by the fundamental frequency of the CTFS 'f_p'. This forms an equivalance between sampling in the frequency domain and periodic repetition in the time domain. Figure 2 shows a signal and its CTFT and periodic repetition of the signal and its CTFS harmonic function.

 

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Relationship between DTFS and DTFT:

 

The DTFS of a periodic function x[n] with fundamental period N₀ is defined as

                                   

Using the frequency shifting property and the DTFT transform pair , we can find the DTFT of x[n] as

                                   

Then                                 

The DTFS is simply a special case of the DTFT for periodic functions. If a function x[n] is periodic, its DTFT consists only of impulses occurring at K/N₀ with strengths X[K].

                                                   

                                                                     

                                                

                           Fig.3 Harmonic function and DTFT of

 

For a periodic function x[n] with fundamental period N₀

                                                           ...............................(4)

Another important point is the relationship between the DTFT of an aperiodic signal and the DTFS harmonic function of a periodic extension of that signal. Let x[n] be an aperiodic function and its DTFT is X(F). Let x_p[n] be a periodic extension of x[n] with fundamental period N_p such that

                                               

Using the multiplication-convolution duality of the DTFT,

                                    ...................(5)

Now using equation (4) we get

                                                     ........................................(6)

Combining equation(5) and (6) we have

                                                          

From the above equation, it says that if an aperiodic signal x[n] is periodically repeated with fundamental period N_p to form a periodic signal x_p[n], the values of its DTFS harmonic function X_p[k] can be found from X(F), which is the DTFT of x[n], evaluated at the discrete frequencies k / N_p. This forms an equivalence between sampling in the frequency domain and periodic repetition in the time domain. Figure 4 shows a signal and its DTFT and periodic repetition of the signal and its DTFS harmonic function.

 

                            

                

                

             Fig.4 A signal and its DTFT and the periodic repetition of the signal and its DTFS harmonic function.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Top

 

Relationship between CTFT and DTFT:

 

The CTFT is the Fourier transform of a continuous-time function and DTFT is the Fourier transform of a discrete-time function. By multiplying a continuous time function x(t) by a periodic train of unit impulses spaced T_s seconds apart, we can create the continuous-time impulse function

                                             

If we form a function x[n] whose values are the values of the original continuous time function x(t) at integer multiples of T_s and these are also the strengths of the impulses in the continuous time impulse function , we get the relationship x[n] = x(nT_s). The two functions x[n] and are completely defined by the same set of numbers and contain the same information. By finding the CTFT of the above equation we have,

                                                 

 Or                                             

where f_s = 1/T_s, if we make the change of variable f→f_sF, we get

                                                  

The above equation is exactly the definition of the DTFT of x[n], which is X_F(F). If x[n] = x(nT_s) and 

                                                      ,

Then                                                   

 or                                      

  Also                                                    ...................................(7)

We have a correspondence between a function x[n] of a discrete independent variable and an impulse function of a continuous independent variable, here x[n] in case of discrete time 'n' and in case of continuous time 't'. There is also an information equivalence between the DTFT of the function x[n] and the CTFT of the function . Figure 5 given below shows this information.

 

                     

                      

                      

                            Fig.5 DTFT of a sinc function x[n] and CTFT of a impulse function 
                                              

There is also some equivalence between the CTFT of the original function x(t) and the DTFT of the function x[n] through equation (7). Given X_f(f), we can find X_F(F). However the reverse of this statement is not always true. Given  X_F(F), we cannot always be sure of being able to find X_f(f).                                                                                                                                                                                                                                                                                Top