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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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The given continuous-time signal is $x(t) = \cos(200\pi t)$. This signal is sampled at instants $t = n/400, n = 0, 1, \ldots ,7$ to obtain the discrete-time signal x[n].

The discrete-time signal (x[n]) can be written as:

$x[n] = \cos(200\pi \cdot n/400) = \cos(\pi/2 \cdot n)$

The 8-point DFT of (x[n]) is defined as:

$X[k] = \sum_{n=0}^{7} x[n]e^{-j2\pi nk/8}, k=0,1,\ldots,7$

Substituting (x[n]) into the equation, we get:

$X[k] = \sum_{n=0}^{7} \cos(\pi/2 \cdot n) \cdot e^{-j2\pi nk/8}$

We can calculate the DFT for each (k) value. However, we know that $\cos(n \cdot \pi/2))$ is 0 for odd (n) and 1 for even (n). Therefore, the non-zero terms in the sum are for even (n) (0, 2, 4, 6).

For these terms, $e^{-j2\pi nk/8}$ equals 1 when (k) is a multiple of 4. Therefore, the only non-zero terms in the DFT are when (k) equals 0 or 4.

So, the correct answer is: Only X[0] and X[4] are non-zero. However, this option is not given in the choices. There seems to be a mistake in the question or the provided options.

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The correct answer is "Only $X[2]$ and $X[6]$ are non-zero."

Here's why:

Given that the signal x(t) = cos(200πt) is sampled at t = n/400 for n = 0, 1, …, 7, the discrete-time signal x[n] can be written as cos(πn/2). The 8-point discrete Fourier transform (DFT) of x[n] is given by X[k] = Σx[n]*e^(-jπkn/4) for k = 0, 1, …, 7.

The terms with n = 2 and n = 6 are the only ones that produce non-zero results because cos(πn/2) = 0 for all odd n. Hence, the DFT X[k] will be non-zero only for X[2] and X[6]. For all other values of k, X[k] will be zero. Therefore, the statement "Only X[2] and X[6] are non-zero" is TRUE.

This question is typical of discrete signal processing problems you might encounter in a course like GATE ECE. It requires understanding of both continuous and discrete signals, as well as Fourier transforms.

This answer is given by OpenAI

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