Question

How many complex additions are needed for 16 direct computations in discrete Fourier transform?

• A. \( \text{240} \)
• B. \( \text{56} \)
• C. \( \text{756} \)
• D. \( \text{32} \)

Hint:

For a direct computation of an $N$-point DFT:

• Number of complex multiplications: $N^2$
• Number of complex additions: $N(N-1)$

These formulas are crucial for understanding the computational complexity of the DFT and the efficiency gains offered by the Fast Fourier Transform (FFT) algorithms. Always remember that direct computation is computationally intensive for large $N$.

Answer 1

The Correct Option is A

The Discrete Fourier Transform (DFT) of a sequence $x[n]$ of length $N$ is given by: $$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j 2\pi k n / N} \quad \text{for } k = 0, 1, \ldots, N-1$$ For each value of $k$, we need to compute the sum of $N$ terms.
Each term $x[n] e^{-j 2\pi k n / N}$ involves one complex multiplication.
The sum of $N$ terms involves $N-1$ complex additions.Since there are $N$ values of $k$ (from $0$ to $N-1$), the total number of complex additions required for direct computation of DFT is $N \times (N-1)$.
In this problem, $N = 16$.Therefore, the number of complex additions needed is $16 \times (16-1)$.
Number of complex additions $= 16 \times 15$.Number of complex additions $= 240$.