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If we split the N point data sequence into two N/2 point data sequences f_{1}(n) and f_{2}(n) corresponding to the even numbered and odd numbered samples of x(n), then such an FFT algorithm is known as decimationintime algorithm.
Explanation: Let us consider the computation of the N=2v point DFT by the divide and conquer approach. We select M=N/2 and L=2. This selection results in a split of N point data sequence into two N/2 point data sequences f_{1}(n) and f_{2}(n) corresponding to the even numbered and odd numbered samples of x(n), respectively, that is
f_{1}(n)=x(2n)
f_{2}(n)=x(2n+1) ,n=0,1,2…N/21
Thus f_{1}(n) and f_{2}(n) are obtained by decimating x(n) by a factor of
2, and hence the resulting FFT algorithm is called a decimationintime algorithm.
If we split the N point data sequence into two N/2 point data sequences f_{1}(n) and f_{2}(n) corresponding to the even numbered and odd numbered samples of x(n) and F_{1}(k) and F_{2}(k) are the N/2 point DFTs of f_{1}(k) and f_{2}(k) respectively, then what is the N/2 point DFT X(k) of x(n)?
Explanation: From the question, it is given that
f_{1}(n)=x(2n)
f_{2}(n)=x(2n+1) ,n=0,1,2…N/21
If X(k) is the N/2 point DFT of the sequence x(n), then what is the value of X(k+N/2)?
Explanation: We know that, X(k) = F_{1}(k)+W_{N}^{k} F_{2}(k)
We know that F_{1}(k) and F_{2}(k) are periodic, with period N/2, we have F_{1}(k+N/2)= F_{1}(k) and F_{2}(k+N/2)= F_{2}(k). In addition, the factor W_{N}^{k+N/2}= W_{N}^{k}.
Thus we get, X(k+N/2)= F_{1}(k) W_{N}^{k} F_{2}(k).
How many complex multiplications are required to compute X(k)?
Explanation: We observe that the direct computation of F_{1}(k) requires (N/2)^{2} complex multiplications. The same applies to the computation of F_{2}(k). Furthermore, there are N/2 additional complex multiplications required to compute W_{N}^{k}. Hence it requires N(N+1)/2 complex multiplications to compute X(k).
The total number of complex multiplications required to compute N point DFT by radix2 FFT is:
Explanation: The decimation of the data sequence should be repeated again and again until the resulting sequences are reduced to one point sequences. For N=2^{v}, this decimation can be performed v=log_{2}N times. Thus the total number of complex multiplications is reduced to (N/2)log_{2}N.
The total number of complex additions required to compute N point DFT by radix2 FFT is:
Explanation: The decimation of the data sequence should be repeated again and again until the resulting sequences are reduced to one point sequences. For N=2^{v}, this decimation can be performed v=log_{2}N times. Thus the total number of complex additions is reduced to Nlog_{2}N.
The following butterfly diagram is used in the computation of:
Explanation: The above given diagram is the basic butterfly computation in the decimationintime FFT algorithm.
For a decimationintime FFT algorithm, which of the following is true?
Explanation: In decimationintime FFT algorithm, the input is taken in bit reversal order and the output is obtained in the order.
The following butterfly diagram is used in the computation of:
Explanation: The above given diagram is the basic butterfly computation in the decimationinfrequency FFT algorithm.
For a decimationintime FFT algorithm, which of the following is true?
Explanation: In decimationinfrequency FFT algorithm, the input is taken in order and the output is obtained in the bit reversal order.
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