The general formula for the number of operations required for the Cooley-Tukey Fast Fourier Transform (FFT) algorithm is indeed O(N*log2(N)), where N is the number of samples. This formula provides an estimate of the number of operations required by the algorithm, but it is not an exact calculation.

In practice, the actual number of operations required by the FFT algorithm may be larger or smaller than the estimate provided by the formula, depending on the specific implementation of the algorithm and the hardware used.

For example, the Cooley-Tukey algorithm requires more operations when the number of samples is not a power of 2, which may be the case for N = 256. In this case, some implementations may use additional operations to pad the signal with zeros to make the number of samples a power of 2, which could increase the number of operations required.

Furthermore, the actual number of operations required may also depend on the specific hardware used for the computation. Modern processors may use various optimizations and hardware accelerators to speed up FFT computations, which may result in a different number of operations than predicted by the formula.

Therefore, it is difficult to give a minimum number of samples required to satisfy the O(N*log2(N)) formula. In practice, the formula provides a rough estimate of the number of operations required, and the actual number may depend on various factors, including the specific implementation of the algorithm and the hardware used for the computation.

Do you count a complex number multiplication as one?

The number of multiplications required for the Fast Fourier Transform (FFT) algorithm depends on the size of the input sequence. For an input sequence of size N, the number of multiplications required by the Cooley-Tukey FFT algorithm is approximately N log2 N. This makes the FFT algorithm much more efficient than a naive implementation of the Discrete Fourier Transform (DFT), which requires O(N^2) multiplications. The use of FFT has revolutionized many areas of science and engineering where signal processing is involved, such as digital signal processing, image processing, and data compression.

No of Calculations required in Fast Fourier Transform-FFT is N/2log2N for Complex Multiplications and N log2N Complex Additions.

The number of multiplications in the Fast Fourier Transform (FFT) depends on the size of the input sequence. For an input sequence of size N, the number of multiplications required by the Cooley-Tukey FFT algorithm is approximately N log2 N.

For example, if the input sequence has a size of 1024, then the number of multiplications required by the Cooley-Tukey FFT algorithm is approximately 1024 * log2(1024) = 1048576.

The number of multiplications in the FFT can be reduced by using other algorithms, such as the Bluestein FFT or the split-radix FFT. However, these algorithms are more complex to implement, and they may not be faster for all input sizes.

In general, the number of multiplications in the FFT is a function of the algorithm used, the size of the input sequence, and the hardware platform on which the FFT is implemented.