Application of DFT

Before we learn about application of DFT, first let we check it out the DFT stands for Discrete Fourier transform, it is a finite duration frequency sequence which is obtained by sampling one period of Fourier transform. So in sampling is done at 'N' equally spaced points, over the period extending from o to  2π.

• Signal analysis
• Sound filtering
• Data compression
• Partial differentiation equation
• Multiplication of large integer
• Cross correlation
• Matched filtering 
• System identification
• Power spectrum estimation
• Coherence function measurement
• Display signal and spectrum

DFT example

Before we fine DFT first let we check it out the what is DFT.

Find the DFT of the following finite equation sequence of length L.

x(n) = A for 0≤ n ≤   L-1

       = 0 otherwise

We have

  

           N-1

X(K) = ∑    x(n) . e ^–j2πkn/N

             n=0

           N-1

X(K) = ∑    A . e ^–j2πkn/N

            n=0

           N-1

X(K) = ∑    x(n) .( e ^–j2πk/N  ) ⁿ

          n=0

            
We have standard summation formula 

            N2

X(K) = ∑    a ^K   =  a ^N1   - a ^N2+1 / 1- a 

         K=N1

here N1=0, N2 = L-1 and a =  e ^–j2πk/N

          N-1

X(K) = ∑    A [ ( e ^–j2πk/N  )⁰ - .( e ^–j2πk/N  )^L-1+1  /1- e ^{–j2πk/N  ]}

             n=0

           N-1

X(K) = ∑    A [ ( 1 - .( e ^–j2πkL/N  )  / 1- e ^–j2πk/N 

             n=0

Relationship between DFT and Z transform

We already learn about what is DFT and what is Z transform, So now here this article gives the information about the relationship between DFT and Z transform to know more details about DFT as well as Z transform.

The Z transform of sequence x(n) is,

           ∞

X(Z) = ∑  x(n) . Z ^–n

           n=-∞

We know that at Z = e ^–jω

            ∞

X(Z) = ∑  x(n) . e ^–jωn

           n = -∞

It means that X(Z) is evaluated on the unit circle. 

Now suppose X(Z) is sampled at N equally spaced point on the unit circle, then we have 

ω = 2πK / N

Now if X(Z) is evaluated at Z =  e ^–jωk/n then by putting equation we get; 

            ∞

X(Z) = ∑  x(n) . e^{-j2πKn / N}

           n = -∞

At Z = e^j2πK/N

In the equation, if x(n) is a causal sequence and has N number of the sample then we can write an equation

            N-1

X(K) = ∑   x(n) . e^{-j2πKn / N}

           n = 0

At Z = e^j2πKn/N

^{X(K) = X(Z)}  At Z = e^j2πKn/N

DFT meaning

In the digital communication system, the DFT stands for Discrete Fourier transform, it is a finite duration frequency sequence which is obtained by sampling one period of Fourier transform. Sampling is done at 'N' equally spaced points, over the period extending from o to around range of 2π.

Mathematical equations :

The DFT of discrete sequence sequence is x(n) and  is denoted by X(k). It is given by,

            N-1

X(K) = ∑    x(n) . e ^–j2πkn/N

              n=0

Here k = 0,1,2,3.......N-1

Since this summation is taken for N point, it is called as N point of discrete Fourier transform called DFT.

We can obtain a discrete sequence x(n) from its DFT. It is called an inverse discrete Fourier transform. It is given by, 

                 N-1

X(n) =1/N ∑    x(k) . e ^–j2πkn/N

                    k=0

Here n = 0,1,2,3.......N-1

This is called as N point IDFT.

Properties of DFT

Before we learn about properties of DFT first we learn about the exact meaning terms of DFT. The Fourier transform can be used for the analysis of a signal. It used for transformation from the time domain to the frequency domain. Here this article gives information about properties of  DFT to know more details or learn about DFT. 

Linearity : 

Periodic signals : A x(n) + B y(n)

Fourier series coefficients : A a_k + B b_k

Time shifting :

Periodic signals : x(n - n₀)
Fourier series coefficients: a_k e^-jk(2π/N)n      

Frequency Shifting :

Periodic signals : x(n) e^jm(2π/N)n      

Fourier series coefficients : X(k - m) 

Conjugation :

Periodic signals : x*(n)       

Fourier series coefficients : ^a*_-k

Time Reversal :

Periodic signals : x(-n)       
Fourier series coefficients : ^a_-k