GATE Questions & Answers of Fourier Series Representation of Continuous Periodic Signals

•  (A)  $f\left(x\right)=a_0+\sum _{k=1}^{\infty }{a}_k\cos \left(kx\right)$
•  (B)  $f\left(x\right)=\sum _{k=1}^{\infty }{b}_k\sin \left(kx\right)$
•  (C)  $f\left(x\right)=a_0+\sum _{k=1}^{\infty }{a}_{2k}\cos \left(kx\right)$
•  (D)  $f\left(x\right)=\sum _{k=0}^{\infty }{a}_{2k+1}\sin \left(2k+1\right)x$

The signum function is given by

$sgn\left(x\right)=\left\{\begin{array}{l}\frac{x}{\left|x\right|};x\ne 0\\ 0; x=0\end{array}\right.$

The Fourier series expansion of $sgn\left(\cos\left(t\right)\right)$ has

•  (A)  only sine terms with all harmonics.
•  (B)  only cosine terms with all harmonics.
•  (C)  only sine terms with even numbered harmonics.
•  (D)  only cosine terms with odd numbered harmonics.

Let $g:[0,\infty )\to :[0,\infty )$ be a function defined by $g\left(x\right)=x-\left[x\right]$, where $\left[x\right]$ represents the integer part of x. (That is, it is the largest integer which is less than or equal to x). The value of the constant term in the Fourier series expansion of $g\left(x\right)$ is _______

The fourier series expansion $\style{font-size:18px}{f\left(t\right)=a_0+\sum\limits_{n=1}^\infty a_n\cos n\omega t+b_n\sin n\omega t}$ of the periodic signal shown below will contain the following nonzero terms

•  (A) a₀ and b_n,n = 1,3,5, ...∞
•  (B) a₀ and a_n,n = 1,2,3, ...∞
•  (C) a₀, a_n and b_n,n = 1,2,3, ...∞
•  (D) a₀ and a_n n = 1,3,5, ..∞

The second harmonic component of the periodic waveform given in the figure has an amplitude of

•  (A) 0
•  (B) 1
•  (C) 2/ π
•  (D) $\sqrt{5}$

The Fourier Series coefficient, of a periodic signal x(t), expressed as $\style{font-size:18px}{x\left(t\right)={\textstyle\sum\limits_{k=-\infty}^\infty}a_ke^{\mathrm j2\mathrm{pkt}/\mathrm T}}$ are given by a_-2 = 2 - j1; a_-1 = 0.5 + j0.2; a₀ = j2; a₁ = 0.5 + j0.2; a₂ = 2 + j1; and $a_k=0;for \left|k\right|>2$. Which of the following is true?

•  (A) x(t) has finite energy becuase only finitely many coefficints are none-zero
•  (B) x(t) has zero average value besause it is periodic
•  (C) The imaginary part of x(t) is constant
•  (D) The real part of x(t) is even

Let x(t) be a periodic signal with time period T,Let y(t)=x(t-t₀)+x(t+t₀) for some t₀.The Fourier Series coefficients of y(t) are denoted by b_k. if b_k=0 for all odd k,then t₀ can be equal to

•  (A) T/8
•  (B) T/4
•  (C) T/2
•  (D) 2T

A signal x(t) is given by $x\left(t\right)=\left\{\begin{array}{lc}1,& -T/4<t\le 3T/4\\ -1,& 3T/4<t\le 7T/4\\ -x\left(t+T\right)& \end{array}\right.$

Which among the following gives the fundamental Fourier term of x(t)?

•  (A) $\frac{4}{\pi }\cos \left(\frac{\pi t}{T}-\frac{\pi }{4}\right)$
•  (B) $\frac{\pi }{4}\cos \left(\frac{\pi t}{2T}+\frac{\pi }{4}\right)$
•  (C) $\frac{4}{\pi }\sin \left(\frac{\pi t}{T}-\frac{\pi }{4}\right)$
•  (D) $\frac{\pi }{4}\sin \left(\frac{\pi t}{2T}+\frac{\pi }{4}\right)$