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GATE 2015 EC – Question 53

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Multiple choice

Two sequences $[a, b, c]$ and $[A, B, C]$ are related as,

$$\begin{bmatrix} A \\ B \\ C \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & W_3^{-1} & W_3^{-2} \\ 1 & W_3^{-2} & W_3^{-4} \end{bmatrix}\begin{bmatrix} a \\ b \\ c \end{bmatrix} \text{ where } W_3 = e^{j\frac{2\pi}{3}}.$$

If another sequence $[p, q, r]$ is derived as,

$$\begin{bmatrix} p \\ q \\ r \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & W_3^1 & W_3^2 \\ 1 & W_3^2 & W_3^4 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 \\ 0 & W_3^2 & 0 \\ 0 & 0 & W_3^4 \end{bmatrix}\begin{bmatrix} A/3 \\ B/3 \\ C/3 \end{bmatrix},$$

then the relationship between the sequences $[p, q, r]$ and $[a, b, c]$ is

  1. $[p, q, r] = [b, a, c]$
  2. $[p, q, r] = [b, c, a]$
  3. $[p, q, r] = [c, a, b]$
  4. $[p, q, r] = [c, b, a]$

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Show answer and explanation

Correct answer: (C) $[p, q, r] = [c, a, b]$

Explanation

The first relation is a 3-point DFT and the second is an inverse DFT of $A$, $B$, $C$ after multiplying each by the phase factor $W_3^{2k}$. A phase factor $W_3^{-mk}$ in the frequency domain corresponds to a circular shift by $m$ in time. Here the factor $W_3^{2k} = W_3^{-k}$ shifts the sequence circularly by one position, so $[p, q, r] = [c, a, b]$.