engineering recuitment GATE ECE 2023-24 Test Series Signals and Systems Discrete Fourier Transform (DFT) and Discrete Fourier Series (DFS) Definition of Discrete Fourier Transform (DFT)
A finite duration discrete-time signal x[n] is obtained by sampling the continuous-time signal x(t) = cos (200πt) at sampling instants t = n/400, n = 0, 1, …, 7. The 8-point discrete Fourier transform (DFT) of x[n] is defined as:
\(X\left[ k \right] = \mathop \sum \limits_{n = 0}^7 x\left[ n \right]{e^{ - j\frac{{\pi kn}}{4}}},\;k = 0,\;1,\; \ldots ,\;7.\)
Which one of the following statements is TRUE?1
All X[k] are non-zero
2
Only X[4] is non-zero
3
Only X[2] and X[6] are non-zero
4
Only X[3] and X[5] are non-zero