Complex Multiplications in FFT Algorithm

The Fast Fourier Transform (FFT) is a widely used algorithm for computing the Discrete Fourier Transform (DFT) of a sequence or signal. The FFT algorithm computes the DFT by dividing the input sequence into smaller sub-sequences and recursively computing their DFTs.

The key operations in the FFT algorithm are complex additions and multiplications. The number of complex multiplications required by the FFT algorithm depends on the size of the input sequence.

Formula for Complex Multiplications

The formula for the number of complex multiplications required by the FFT algorithm is given by:

(N/2)log2N,

where N is the size of the input sequence.

Explanation of Options

a) (N/2)logN: This option is almost correct, but the base of the logarithm should be 2, not N. Hence, this option is incorrect.

b) Nlog2N: This option is incorrect because it only considers the number of complex additions, not the number of complex multiplications.

c) (N/2)log2N: This option is correct because it considers both the number of complex additions and multiplications. The FFT algorithm requires (N/2) complex additions and (N/2)log2N complex multiplications.

d) None of the mentioned: This option is incorrect because option c is the correct answer.

Conclusion

Thus, the correct answer to the question is option c, which states that (N/2)log2N complex multiplications are required by the FFT algorithm for each input sequence of size N.