Signals in Natural Domain
Chapter 6 :  Discrete fourier series and Discrete fourier transform
 

Linear convolution using the Discrete Fourier Transform

Output of a linear time invariant-system is obtained by linear convolution of input signal with the impulse response of the system. If we multiply DFT coefficients, and then take inverse transform we will get circular convolution. From the examples it is clear that result of circular convolution is different from the result of linear convolution of two sequences. But if we modify the two sequence appropriately we can get the result of circular convolution to be same as linear convolution. Our interest in doing linear convolution results form the fact that fast algorithms for computing DFT and IDFT are available. These algorithms will be discussed in a later chapter. Here we show how we can make result of circular convolution same as that of linear convolution.

If we have sequence 1of length L and a sequence 2of length M , the sequence 3obtained by linear convolution has length ( L + M - 1).

This can be seen from the definition

                                                                                                                  (6.31)

as x[k] = 0 for. For 2 hence. Similarly for 4, so. Hence 6  is possibly nonzero only for.
Now consider a sequence 8, DTFT is given by

writing

We get

If we take
we see that
Comparing this with the DFT equation (6.), we see that
can be seen as DFT coefficients of a sequence
                                                                                           (6.32)

obviously if has length less then or equal to N , then

However, if the length of 1 is greater than 2 may not be equal to 3 for all values of l.
The sequence 4 in equation (6.31) has the discrete Fourier transform

The N-point DFT of sequence is

                                                                                
                                                                                

where 1 and 2 are N-point DFTs of 3 and 4 respectively. The sequence resulting as the inverse DFT of 5 is then by equation (6.32).

From the circular convolution property of the DFT we have

Thus, the circular convolution of two-finite length sequences can be viewed as linear convolution, followed time aliasing, defined by equation (6.32). If N is greater than or equal to ( L + M - l ), then there will be no time aliasing as the linear convolution produces a sequence of length ( L + M - l ). Thus we can use circular convolution for linear convolution by padding sufficient number of zeros at the end of a finite length sequence. We can use DFT algorithm for calculating the circular convolution.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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