GATE EE 2022 | Continuous Time Periodic Signal Fourier Series Question 2 | Signals and Systems

GATE EE 2022

MCQ (Single Correct Answer)

-0.67

The discrete time Fourier series representation of a signal x[n] with period N is written as $$x[n] = \sum\nolimits_{k = 0}^{N - 1} {{a_k}{e^{j(2kn\pi /N)}}} $$. A discrete time periodic signal with period N = 3, has the non-zero Fourier series coefficients : a_$$-$$3 = 2 and a₄ = 1. The signal is

$$2 + 2{e^{ - \left( {j{{2\pi } \over 6}n} \right)}}\cos \left( {{{2\pi } \over 6}n} \right)$$

$$1 + 2{e^{\left( {j{{2\pi } \over 6}n} \right)}}\cos \left( {{{2\pi } \over 6}n} \right)$$

$$1 + 2{e^{\left( {j{{2\pi } \over 3}n} \right)}}\cos \left( {{{2\pi } \over 6}n} \right)$$

$$2 + 2{e^{\left( {j{{2\pi } \over 6}n} \right)}}\cos \left( {{{2\pi } \over 6}n} \right)$$

GATE EE 2017 Set 1

MCQ (Single Correct Answer)

-0.6

Let the signal $$$x\left(t\right)=\sum_{k=-\infty}^{+\infty}\left(-1\right)^k\delta\left(t-\frac k{2000}\right)$$$ be passed through an LTI system with frequency response $$H\left(\omega\right)$$, as given in the figure below GATE EE 2017 Set 1 Signals and Systems - Continuous Time Periodic Signal Fourier Series Question 18 English

GATE EE 2017 Set 1 Signals and Systems - Continuous Time Periodic Signal Fourier Series Question 18 English

The Fourier series representation of the output is given as

4000+4000cos(2000$$\mathrm\pi$$t)+4000cos(4000$$\mathrm\pi$$t)

2000+2000cos(2000$$\mathrm\pi$$t)+2000cos(4000$$\mathrm\pi$$t)

4000cos(2000$$\mathrm\pi$$t)

2000cos(2000$$\mathrm\pi$$t)

GATE EE 2015 Set 1

MCQ (Single Correct Answer)

-0.6

The signum function is given by $$$\mathrm{sgn}\left(\mathrm x\right)=\left\{\begin{array}{l}\frac{\mathrm x}{\left|\mathrm x\right|};\;\mathrm x\neq0\\0\;;\;\;\mathrm x=0\end{array}\right.$$$ The Fourier series expansion of sgn(cos(t)) has

only sine terms with all harmonics.

only cosine terms with all harmonics

only sine terms with even numbered harmonics.

only cosine terms with odd numbered harmonics.

GATE EE 2009

MCQ (Single Correct Answer)

-0.6

The Fourier Series coefficients, of a periodic signal $$x\left( t \right),$$ expressed as $$x\left( t \right) = \sum {_{k = - \infty }^\infty {a_k}{e^{j2\pi kt/T}}} $$ are given by
$${a_{ - 2}} = 2 - j1;\,\,{a_{ - 1}} = 0.5 + j0.2;\,\,{a_0} = j2;$$
$${a_1} = 0.5 - j0.2;\,\,{a_2} = 2 + j1;\,\,$$ and
$${a_k} = 0;$$ for $$|k|\,\, > 2.$$

Which of the following is true?

$$x(t)$$ has finite energy because only finitely many coefficients are non $$-$$ zero

$$x(t)$$ has zero average value because it is periodic

the imaginary part of $$x(t)$$ is constant

The real part of $$x(t)$$ is even