Question

The number of additions required to compute N-point DFT using radix-2 FFT is given by:

• A. \( N \log_2 N \)
• B. \( (N - 1) \log_2 N \)
• C. \( (N / 2) \log_2 N \)
• D. \( 4N \log_2 N \)

Hint:

- The radix-2 FFT algorithm reduces the complexity of computing DFT from \( O(N^2) \) to \( O(N \log_2 N) \). - The number of additions required follows the formula: \[ N \log_2 N \]

Answer 1

The Correct Option is A

Step 1: Understanding Radix-2 FFT Algorithm
- The Discrete Fourier Transform (DFT) has a direct computation complexity of \( O(N^2) \).
- The Fast Fourier Transform (FFT) significantly reduces this to \( O(N \log_2 N) \).
- The radix-2 FFT is an efficient way to compute N-point DFT using a divide-and-conquer approach.
Step 2: Addition Complexity in Radix-2 FFT
- In each stage of the radix-2 FFT, \( N \) additions are required.
- There are \( \log_2 N \) stages in the computation.
- Thus, the total number of additions required is: \[ N \log_2 N \]
Step 3: Evaluating the Options - (A) Correct: The total number of additions required in the radix-2 FFT is \( N \log_2 N \). - (B) Incorrect: \( (N - 1) \log_2 N \) is not the correct expression for addition complexity. - (C) Incorrect: \( (N / 2) \log_2 N \) underestimates the number of required additions. - (D) Incorrect: \( 4N \log_2 N \) is incorrect and overestimates the number of required additions.