Complex Multiplications in Radix-2 FFT

In order to understand why the total number of complex multiplications required to compute the N-point DFT (Discrete Fourier Transform) by radix-2 FFT (Fast Fourier Transform) is (N/2)log2N, let's first understand the concept of radix-2 FFT.

Radix-2 FFT is a widely used algorithm for fast computation of the DFT. It takes advantage of the divide-and-conquer approach by recursively decomposing the DFT into smaller DFTs. The algorithm operates on complex-valued data and performs various butterfly operations to combine the results.

Number of Butterfly Operations

In radix-2 FFT, each stage of the algorithm consists of log2N butterfly operations, where N is the number of input points. At each stage, the algorithm combines the results of the previous stage in a specific way using complex multiplications and additions.

Each butterfly operation involves two complex multiplications and two complex additions. Therefore, the total number of complex multiplications required in each stage is (2^(stage number)). For example, in the first stage, there are 2^(stage number) = 2^1 = 2 complex multiplications.

Total Number of Stages

The total number of stages in radix-2 FFT is log2N, where N is the number of input points. This is because each stage reduces the number of points by a factor of 2.

Total Number of Complex Multiplications

To calculate the total number of complex multiplications required to compute the N-point DFT by radix-2 FFT, we sum up the number of complex multiplications in each stage.

In the first stage, there are 2 complex multiplications. In the second stage, there are 4 complex multiplications. In the third stage, there are 8 complex multiplications, and so on. The number of complex multiplications in each stage follows a geometric progression with a common ratio of 2.

Using the formula for the sum of a geometric progression, the total number of complex multiplications can be calculated as:

2 + 4 + 8 + ... + (2^(log2N))

This can be simplified as:

2 * (1 - (2^(log2N)))/(1 - 2)

Simplifying further, we get:

2 * (1 - N)/(1 - 2) = (N/2) * (1 - N)

Therefore, the total number of complex multiplications required to compute the N-point DFT by radix-2 FFT is (N/2)log2N.

Hence, the correct answer is option A: (N/2)log2N.