Topic: Digital Signal Processing
Let $Y(k)$ be the 5-point DFT of the sequence $y(n)=\{1~2~3~4~5\}$. What is the 5-point DFT of the sequence $Y(k)$?
- A. [15 -2.5+3.4j -2.5+0.81j -2.5-0.81 -2.5-3.4j]
- B. [1 5 4 3 2]
- C. [5 25 20 15 10]
- D. [5 4 3 2 1]
Answer: C. [5 25 20 15 10]
Explanation: Using the duality property of DFT, $\text{DFT}(\text{DFT}(y(n))) = N \cdot y(-n \bmod N)$. Here, $N=5$. The time-reversed sequence modulo 5 is $y(-n \bmod 5) = \{1, 5, 4, 3, 2\}$. Therefore, the DFT of $Y(k)$ is $5 \times \{1, 5, 4, 3, 2\} = \{5, 25, 20, 15, 10\}$.