How many complex multiplications are need to be performed for each FFT algorithm?Select one:a. (N/2)log2Nb. Nlog2Nc. None of the mentionedd. (N/2)logN

To determine how many complex multiplications are needed for each Fast Fourier Transform (FFT) algorithm, let’s go through the problem step by step.

1. Break Down the Problem

To analyze the number of complex multiplications in the FFT algorithm, we need to understand:

• The general process of the FFT.
• The logarithmic complexity of the algorithm.
• The contributions of each stage in the FFT to the total number of multiplications.

2. Relevant Concepts

The FFT uses a divide-and-conquer approach to compute the Discrete Fourier Transform (DFT):

• The complexity of the FFT is typically $O(N \log N)$.
• Each stage of the FFT consists of butterfly operations that use complex multiplication.

3. Analysis and Detail

• In each of the $\log_2 N$ stages, we perform $N/2$ pairwise computations (butterflies).
• Each butterfly operation involves one complex multiplication (and potentially additional additions).
• Therefore, the total number of complex multiplications can be estimated as follows:

$\text{Total Complex Multiplications} = \frac{N}{2} \cdot \log_2 N$

4. Verify and Summarize

The above analysis summarizes that for the FFT, the number of complex multiplications needed varies as: $\frac{N}{2} \cdot \log_2 N$ This matches option (a).

Final Answer

The number of complex multiplications needed for each FFT algorithm is $\frac{N}{2} \log_2 N$. Thus, the correct answer is (a) $\frac{N}{2} \log_2 N$.