Question

Which of the following is true regarding the number of computations required to compute an \( N \)-point DFT?

• A. \( N^2 \) complex multiplications and \( N(N-1) \) complex additions
• B. \( N^2 \) complex additions and \( N(N-1) \) complex multiplications
• C. \( N^2 \) complex multiplications and \( N(N+1) \) complex additions
• D. \( N^2 \) complex additions and \( N(N+1) \) complex multiplications

Hint:

For an \( N \)-point DFT, the computational complexity is: - \( N^2 \) complex multiplications. - \( N(N-1) \) complex additions. For FFT, this is reduced to \( O(N \log N) \).

Answer 1

The Correct Option is A

Step 1: The Discrete Fourier Transform (DFT) is computed using the formula: \[ X(k) = \sum_{n=0}^{N-1} x(n) W_N^{kn}, \quad k = 0,1,\dots,N-1 \] where \( W_N = e^{-j2\pi/N} \) is the twiddle factor. 
Step 2: Computational Complexity: - Each output \( X(k) \) requires summing over \( N \) terms. 
- Each term involves one complex multiplication. 
- Since there are \( N \) outputs, the total number of complex multiplications is: \[ N^2 \] 
- Each summation involves \( (N-1) \) complex additions. 
- Since there are \( N \) outputs, the total number of complex additions is: \[ N(N-1) \] 
Step 3: Evaluating options: 
- (A) Correct: \( N^2 \) complex multiplications and \( N(N-1) \) complex additions. 
- (B) Incorrect: Complex additions and multiplications are swapped. 
- (C) Incorrect: Addition count is incorrectly given as \( N(N+1) \). 
- (D) Incorrect: Both operations are incorrectly counted.