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

To solve this problem, we need to calculate the number of complex additions required for 16 direct computations in the Discrete Fourier Transform (DFT).

1. Understanding the Concepts:

- Discrete Fourier Transform (DFT): The DFT of a sequence \( x(n) \) of length \( N \) is given by the formula:

\[ X(k) = \sum_{n=0}^{N-1} x(n) e^{-j \frac{2\pi}{N} kn} \]

For each \( X(k) \), the summation involves \( N \) terms, and since the DFT is applied for each frequency component \( k \), the number of operations can be computed.

2. Number of Complex Additions:

For a DFT of length \( N \), the computation of each \( X(k) \) requires \( N-1 \) complex additions (since the sum involves \( N \) terms). Thus, for \( N = 16 \), the number of additions per frequency component is:

\[ N - 1 = 16 - 1 = 15 \text{ complex additions per } X(k) \]

Since there are 16 frequency components (from \( k = 0 \) to \( k = 15 \)), the total number of complex additions for all computations is:

\[ 15 \times 16 = 240 \text{ complex additions} \]

Final Answer:

The number of complex additions needed for 16 direct computations in the DFT is 240.