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A finite duration discrete-time signal $x[n]$ is obtained by sampling the continuous-time signal $x\left ( t \right )=\cos\left ( 200\pi t \right )$ at sampling instants $t=n/400, n=0, 1, \dots ,7.$ The $8$-point discrete Fourier transform $\text{(DFT)}$ of $x[n]$ is defined as $$X\left [ k \right ]=\sum ^{7}_{n=0}x\left [ n \right ]e^{-j\frac{\pi kn}{4}}, k=0,1, \dots ,7.$$

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.
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