GATE 2023 EE – Question 50

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Numerical answer

The discrete-time Fourier transform of a signal $x[n]$ is $X(\Omega)=(1+\cos\Omega)e^{-j\Omega}$. Consider that $x_p[n]$ is a periodic signal of period $N=5$ such that

$x_p[n]=x[n]$, for $n=0,1,2$

$\ \ \ \ \ \ \ \ =0$, for $n=3,4$

Note that $x_p[n]=\sum_{k=0}^{N-1}a_ke^{j\frac{2\pi}Nkn}$. The magnitude of the Fourier series coefficient $a_3$ is ________________ (Round off to 3 decimal places).

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Correct answer: 0.03 to 0.05

Explanation

$X(\Omega)=e^{-j\Omega}+\tfrac12e^{-j2\Omega}\cdot$ ... expanding $(1+\cos\Omega)e^{-j\Omega}=\tfrac12+e^{-j\Omega}+\tfrac12e^{-j2\Omega}$ gives $x[n]=\tfrac12\delta[n]+\delta[n-1]+\tfrac12\delta[n-2]$, so $x_p[0..4]=0.5,\,1,\,0.5,\,0,\,0$. Then $a_3=\dfrac15\left(0.5+e^{-j6\pi/5}+0.5e^{-j12\pi/5}\right)=\dfrac15(-0.1545+j0.1125)$, with magnitude $\dfrac{0.1911}{5}=0.038$.