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The total number of complex additions required to compute N point DFT by radix-2 FFT is:
- a)(N/2)log2N
- b)Nlog2N
- c)(N/2)logN
- d)None of the mentioned
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The total number of complex additions required to compute N point DFT ...
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=2v, this decimation can be performed v=log2N times. Thus the total number of complex additions is reduced to Nlog2N.
Most Upvoted Answer
The total number of complex additions required to compute N point DFT ...
To understand why the correct answer is option 'B', let's break down the steps involved in computing the N-point DFT using radix-2 FFT.
1. Radix-2 FFT algorithm:
- The radix-2 FFT algorithm divides the input sequence into two halves, each containing N/2 points.
- It then recursively computes the DFT of the even-indexed points and odd-indexed points separately.
- Finally, it combines the results to obtain the DFT of the entire sequence.
2. Number of stages:
- In the radix-2 FFT algorithm, the input sequence is divided into two halves at each stage.
- The number of stages required to compute the N-point DFT is log2(N), as each stage reduces the problem size by half.
3. Number of butterfly operations:
- At each stage, there are N/2 butterfly operations.
- A butterfly operation computes the sum and difference of two complex numbers.
- Each butterfly operation requires two complex additions and two complex multiplications.
4. Total number of complex additions:
- At each stage, there are N/2 butterfly operations, which means N/2 complex additions.
- Since there are log2(N) stages, the total number of complex additions is (N/2) * log2(N).
Therefore, the correct answer is option 'B': Nlog2N. This represents the total number of complex additions required to compute the N-point DFT using radix-2 FFT.
1. Radix-2 FFT algorithm:
- The radix-2 FFT algorithm divides the input sequence into two halves, each containing N/2 points.
- It then recursively computes the DFT of the even-indexed points and odd-indexed points separately.
- Finally, it combines the results to obtain the DFT of the entire sequence.
2. Number of stages:
- In the radix-2 FFT algorithm, the input sequence is divided into two halves at each stage.
- The number of stages required to compute the N-point DFT is log2(N), as each stage reduces the problem size by half.
3. Number of butterfly operations:
- At each stage, there are N/2 butterfly operations.
- A butterfly operation computes the sum and difference of two complex numbers.
- Each butterfly operation requires two complex additions and two complex multiplications.
4. Total number of complex additions:
- At each stage, there are N/2 butterfly operations, which means N/2 complex additions.
- Since there are log2(N) stages, the total number of complex additions is (N/2) * log2(N).
Therefore, the correct answer is option 'B': Nlog2N. This represents the total number of complex additions required to compute the N-point DFT using radix-2 FFT.
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