For a vector xฬ… = [x[0], x[1],......x[7]] the 8-point discrete Fourier transform (DFT) is denoted by Xฬ… = DFT (xฬ…) = X[0], X[1], ...., X[7]], where

\(\rm X[k]=\sum_{n=0}^{7}x[n]\ exp\left(-j\frac{2\pi}{8}nk\right) \space \)

Here j = √-1, if Xฬ… = [1, 0, 0, 0, 2, 0, 0, 0] and yฬ… = (DFT (xฬ…)), then the value of y[0] is ________ (rounded off to one decimal place).

This question was previously asked in
GATE EC 2022 Official Paper
View all GATE EC Papers >

Answer (Detailed Solution Below) 8

Free
GATE EC 2023: Full Mock Test
3.3 K Users
65 Questions 100 Marks 180 Mins

Detailed Solution

Download Solution PDF

Concept:

By duality property

 \(x(n)\xrightarrow [ ] {DFT}\ \ \xrightarrow [] {DFT} Nx(-k)\space \)

Calculation:

Given 

Xฬ…  = DFT x(n) =  X[k] = [1, 0, 0, 0, 2, 0, 0, 0] 

\(\rm X[k]=\sum_{n=0}^{7}x[n]\ exp\left(-j\frac{2\pi}{8}nk\right) \space \)

Therefore we hane to find 

yฬ… = (DFT (xฬ… )) = DFT (DFT x(n))

By duality property

\(x(n)\xrightarrow [ ] {DFT}\ \ \xrightarrow [] {DFT} Nx(-k)\space \) 

DFT (DFT x(n)) = yฬ…  = N x(-k)

X(k) = [1, 0, 0, 0, 2, 0, 0, 0] 

X(-k) = [1, 0, 0, 0, 2, 0, 0, 0] 

y(n) =  N X(-k) = 8 [1, 0, 0, 0, 2, 0, 0, 0] 

y(0) = 8

Latest GATE EC Updates

Last updated on Jan 8, 2025

-> The GATE EC Call Letter has been released on 7th January 2025.

-> The GATE EC 2025 Exam will be held on 15th February 2025.

-> The mode of the GATE EC exam will be a Computer-Based test of 100 marks. 

-> Candidates preparing for the exam can refer to the GATE EC Previous Year Papers to improve their preparation and increase the chances of selection. 

-> Candidates must attempt the GATE EC Mock tests

More Definition of Discrete Fourier Transform (DFT) Questions

More Discrete Fourier Transform (DFT) and Discrete Fourier Series (DFS) Questions

Hot Links๏ผš teen patti joy official teen patti win teen patti master update teen patti master gold teen patti bodhi