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GATE EC 2015 Set 1 — Question 53

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MCQ+2 / -0.67MediumDFT & FFTDiscrete-Time SignalsSignals & Systems

Signals & Systems → Discrete-Time Signals → DFT & FFT

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Two sequences [a,b,c][a, b, c] and [A,B,C][A, B, C] are related as,(ABC)=(1111W3−1W3−21W3−2W3−4)(abc) where W3=ej2π3.\begin{pmatrix} A \\ B \\ C \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & W_3^{-1} & W_3^{-2} \\ 1 & W_3^{-2} & W_3^{-4} \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} \text{ where } W_3 = e^{j\frac{2\pi}{3}}.If another sequence [p,q,r][p, q, r] is derived as,(pqr)=(1111W31W321W32W34)(1000W32000W34)(A/3B/3C/3),\begin{pmatrix} p \\ q \\ r \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & W_3^{1} & W_3^{2} \\ 1 & W_3^{2} & W_3^{4} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & W_3^{2} & 0 \\ 0 & 0 & W_3^{4} \end{pmatrix} \begin{pmatrix} A/3 \\ B/3 \\ C/3 \end{pmatrix},then the relationship between the sequences [p,q,r][p,q,r] and [a,b,c][a, b, c] is
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