Which of the following statements accurately describes the computational complexity of the Fast Fourier Transform (FFT) algorithm for computing the DFT of a sequence of length 𝑁?Select one:a.The computational complexity is 𝑂(𝑁log𝑁).b.The computational complexity is 𝑂(𝑁).c. The computational complexity depends on the specific properties of the input sequence and can vary.d.The computational complexity is 𝑂(𝑁2).
Question
Which of the following statements accurately describes the computational complexity of the Fast Fourier Transform (FFT) algorithm for computing the DFT of a sequence of length 𝑁?Select one:a.The computational complexity is 𝑂(𝑁log𝑁).b.The computational complexity is 𝑂(𝑁).c. The computational complexity depends on the specific properties of the input sequence and can vary.d.The computational complexity is 𝑂(𝑁2).
Solution
The correct answer is:
a. The computational complexity is 𝑂(𝑁log𝑁).
The Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse. The key idea is to decrease the computational complexity of calculating the DFT from 𝑂(𝑁2), which would be the case if we directly apply the definition of DFT, to 𝑂(𝑁log𝑁). This is achieved by recursively dividing the DFT of a sequence into smaller parts. This makes the FFT a much more efficient algorithm for computing the DFT.
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