Continuous Time Periodic Signal Fourier Series · Signals and Systems · GATE EE
Marks 1
A continuous time periodic signal $x(t)$ is
$$ x(t)=1+2 \cos 2 \pi t+2 \cos 4 \pi t+2 \cos 6 \pi t $$
If $T$ is the period of $x(t)$, then $\frac{1}{T} \int_0^T|x(t)|^2 d t=$________(round off to the nearest integer).
The Fourier transform $$X(\omega)$$ of the signal $$x(t)$$ is given by
$$X(\omega ) = 1$$, for $$|\omega | < {W_0}$$
$$ = 0$$, for $$|\omega | > {W_0}$$
Which one of the following statements is true?
Here, $$\left\lfloor t\right\rfloor$$ represents the largest integer less than or equal to t and $$\left\lceil t\right\rceil$$ denotes the smallest integer greater than or equal to t. The coefficient of the second harmonic component of the Fourier series representing g(t) is _________.
$$\omega = 2\pi \left( {2k} \right)/T;\,\,k = 1,2,........$$ Also, no sine terms are present. Then $$x(t)$$ satisfies the equation
Marks 2
A time-limited waveform $g(x)$ is specified as follows:
$$ g(x)=\left\{\begin{array}{cc} -k, & -\pi A new waveform $f(x)$ is constructed from $g(x)$ as follows: $$ f(x)=\sum_{m=-\infty}^{\infty} g(x+2 \pi n), \text { for all } x \in R $$ The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of $f(x)$ is $\frac{2}{3 \pi}$. What is the value of $k$ ?
The discrete time Fourier series representation of a signal x[n] with period N is written as $$x[n] = \sum\nolimits_{k = 0}^{N - 1} {{a_k}{e^{j(2kn\pi /N)}}} $$. A discrete time periodic signal with period N = 3, has the non-zero Fourier series coefficients : a$$-$$3 = 2 and a4 = 1. The signal is
The Fourier series representation of the output is given as$${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\left( t \right) = \left\{ {\matrix{ {1, - {\raise0.5ex\hbox{$\scriptstyle T$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} < t \le {\raise0.5ex\hbox{$\scriptstyle {3T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}} \cr { - 1,{\raise0.5ex\hbox{$\scriptstyle {3T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} < t \le {\raise0.5ex\hbox{$\scriptstyle {7T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}},\,\,\,} \cr { - x\left( {t + T} \right)} \cr } } \right.$$ Which among the following gives the fundamental Fourier term of $$x(t)$$?
Marks 5
$$(a)$$$$\,\,\,\,\,\,\,\,$$ the dc component of $$V,$$
$$(b)$$$$\,\,\,\,\,\,\,\,$$ the amplitude of the fundamental component of $$V,$$ and
$$(c)$$$$\,\,\,\,\,\,\,\,$$ the $$rms$$ value of the ac part of $$V$$
