Question

Fast Fourier Transform algorithms exploit:

• A. Four basic properties of phase factor
• B. Complex multiplications
• C. Indexing and addressing operations
• D. Symmetry and periodicity

Hint:

- The Fast Fourier Transform (FFT) algorithm significantly reduces computation time by exploiting the symmetry and periodicity of the twiddle factors in the Discrete Fourier Transform (DFT).
- This leads to an improved computational complexity of \( O(N \log_2 N) \) instead of \( O(N^2) \).

Answer 1

The Correct Option is D

Step 1: Understanding Fast Fourier Transform (FFT)
- The Fast Fourier Transform (FFT) is an algorithm to compute the Discrete Fourier Transform (DFT) efficiently.
- The computational complexity of DFT is \( O(N^2) \), while FFT reduces it to \( O(N \log_2 N) \).
Step 2: How FFT Works
- FFT exploits the properties of symmetry and periodicity in the twiddle factors: \[ W_N^k = e^{-j\frac{2\pi k}{N}} \] - By utilizing these properties, FFT reduces redundant calculations, making the transformation much faster.
Step 3: Evaluating the Options
- (A) Incorrect: The phase factor is used, but FFT primarily relies on symmetry and periodicity.
- (B) Incorrect: While FFT involves complex multiplications, its efficiency comes from reducing these computations.
- (C) Incorrect: Indexing and addressing operations play a role but are not the main concept exploited.
- (D) Correct: FFT exploits symmetry and periodicity to improve computational efficiency.