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ISRO Scientist EC 2016 Official Paper

Option 2 : 1

CT 3: Building Materials

2962

10 Questions
20 Marks
12 Mins

**Concept:**

The power spectral density is basically the Fourier transform of the autocorrelation function of the power signal, i.e.

\({S_x}\left( f \right) = F.T.\left\{ {{R_x}\left( \tau \right)} \right\}\)

**Analysis:**

Given, the autocorrelation function of the random signal X(t) as:

\({R_X}\left( \tau \right) = {e^{ - 2\left| \tau \right|}}\)

So, its power spectral density is obtained as:

\({S_X}\left( f \right) = {\cal F}\left\{ {{R_X}\left( \tau \right)} \right\}\)

\( = \mathop \smallint \limits_{ - \infty }^\infty {R_X}\left( \tau \right){e^{ - j2\pi f\tau }}d\tau \)

\( = \mathop \smallint \limits_{ - \infty }^0 {e^{2\tau }}{e^{ - j2\pi f\tau }}d\tau + \mathop \smallint \limits_0^\infty {e^{ - 2\tau }}{e^{ - j2\pi f\tau }}d\tau \)

\(= \frac{1}{{2\left( {1 - j\pi f} \right)}} + \frac{1}{{2\left( {1 + j\pi f} \right)}}\)

\(= \frac{1}{{{1} + {\pi ^2}{f^2}}}\)

The peak value of PSD is at f = 0,

S_{X}(0) = 1