Complex additions required in linear filtering using FFT algorithm

In linear filtering of a sequence using the Fast Fourier Transform (FFT) algorithm, the number of complex additions required can be determined based on the algorithm itself. The FFT algorithm reduces the number of computations required compared to the direct time-domain convolution method.

Complexity of FFT algorithm

The complexity of the FFT algorithm is determined by the number of complex multiplications and complex additions required. The number of complex multiplications is generally higher than the number of complex additions. However, for the purpose of this question, we are specifically concerned with the number of complex additions.

Number of complex additions

The number of complex additions required in the FFT algorithm can be calculated using the formula:

Number of complex additions = 2Nlog2N

Where N represents the length of the sequence being filtered.

Explanation of the formula

- Nlog2N: This term represents the number of complex additions required for the butterfly operations in the FFT algorithm. The butterfly operations involve combining the results of smaller DFTs to obtain the final DFT.

- 2N: This term represents the number of complex additions required for the final summation of the DFT values after the butterfly operations.

Comparison with the given options

Option a) (N/2)logN: This option is incorrect as it divides N by 2, which does not align with the formula for complex additions in the FFT algorithm.

Option b) 2Nlog2N: This option is correct as it matches the formula for the number of complex additions required in the FFT algorithm.

Option c) (N/2)log2N: This option is incorrect as it divides N by 2, which does not align with the formula for complex additions in the FFT algorithm.

Option d) Nlog2N: This option is incorrect as it does not include the factor of 2 in the formula for complex additions in the FFT algorithm.

Therefore, the correct answer is option b) 2Nlog2N, which represents the number of complex additions required in linear filtering of a sequence using the FFT algorithm.