Flashcards on The Fast Fourier Transform (FFT)
The Fast Fourier Transform (FFT) Explained for Students
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Fast Fourier Transform (FFT)
26 cards
Card 1
Question: What common name is given to the Cooley–Tukey algorithm used for computing the DFT more efficiently?
Answer: Fast Fourier Transform (FFT).
Card 2
Question: Do the DFT and FFT produce different results?
Answer: No — DFT and FFT results do not differ; FFT computes the same result more efficiently.
Card 3
Question: How does FFT reduce the number of complex multiplications compared to the direct DFT computation?
Answer: FFT decreases the number of complex multiplications from N^2 to approximately (N/2)·log2(N).
Card 4
Question: What is the speed-up factor of FFT relative to direct DFT in terms of N?
Answer: Approximately 2N / log2(N).
Card 5
Question: Can the FFT algorithm be used to compute the inverse DFT?
Answer: Yes — a similar algorithm is used for the inverse DFT, called the inverse FFT (iFFT).
Card 6
Question: According to the slide examples, roughly how many multiplications does DFT require vs FFT for N=1024?
Answer: DFT requires about 1,048,576 multiplications; FFT requires about 5,120 multiplications (about 205:1 speed-up).
Card 7
Question: What happens in MATLAB/Python when the input signal length is a power of two?
Answer: The FFT algorithm is applied (i.e., a fast algorithm is used).
Card 8
Question: If the signal length given to FFT is shorter or longer than N, how is it handled?
Answer: If shorter, the signal is zero-padded to length N; if longer, the signal is truncated from the end to length N.
Card 9
Question: Give the basic MATLAB/Python commands for FFT and inverse FFT as shown.
Answer: MATLAB: X = fft(x); X = fft(x,N); x = ifft(X). Python (NumPy): X = numpy.fft.fft(x); X = numpy.fft.fft(x,N); x = numpy.fft.ifft(X).
Card 10
Question: What three FFT normalization variants exist in Python/NumPy?
Answer: "forward", "backward" (default), and "ortho" (orthogonal variant).