Fourier Transforms, Frequency Response and Correlation

Question 1
The exponential Fourier series representation of a continuous-time periodic signal x(t) is defined as
x\left ( t \right )=\sum_{k=-\infty }^{\infty }a_{k}e^{jk\omega _{0}t}
where \omega _0 is the fundamental angular frequency of x(t) and the coefficients of the series are a_{k}. The following information is given about x(t) and a_{k}.

I. x(t) is real and even, having a fundamental period of 6
II. The average value of x(t) is 2
III. a_{k}=\left\{\begin{matrix} k, & 1\leq k\leq 3\\ 0,& k> 3 \end{matrix}\right.

The average power of the signal x(t) (rounded off to one decimal place) is ____________
A
14
B
45
C
63
D
32
GATE EC 2021   Signals and Systems
Question 1 Explanation: 
1. x(t) is real and even so a_{k} is also real and even a_{k}=a_{-k}
2. Average of x(t) is 2 i.e., a_{0}=2.
\begin{array}{llll} 3. \;x(t) \rightarrow a_{k}=k &1 \leq k \leq 3 \quad& a_{1}=1 \quad& a_{-1}=1 \\ \quad \quad \quad \quad \;\;\;\;\;\;\;\;0 & k>3 & a_{2}=2 & a_{-2}=2 \\ && a_{3}=3 & a_{-3}=3 \end{array}
4. \quad T_{0}=6
Parsval's Power Theorem
\begin{aligned} \frac{1}{T} \int_{0}^{T}\left|x(t)^{2}\right| d t &=\sum_{n=-\infty}^{+\infty}\left|a_{k}\right|^{2} \\ P(t) &=\sum_{n=-\infty}^{+\infty}\left|a_{k}\right|^{2}=\left|a_{-3}\right|^{2}+\left|a_{-2}\right|^{2}+\left|a_{-1}\right|^{2}+\left|a_{0}\right|^{2}+\left|a_{1}\right|^{2}+\left|a_{2}\right|^{2}+\left|a_{3}\right|^{2} \\ &=2\left|a_{1}\right|^{2}+2\left|a_{2}\right|^{2}+2\left|a_{3}\right|^{2}+\left|a_{0}\right|^{2} \\ &=2 \times 1^{2}+2 \times(2)^{2}+2(3)^{2}+(2)^{2} \\ &=2+8+18+4 \\ P_{x(t)} &=32 \end{aligned}
Question 2
X(\omega ) is the Fourier transform of x(t) shown below. The value of \int_{-\infty }^{\infty }|X(\omega )|^2 d\omega (rounded off to two decimal places) is ______.
A
58.61
B
42.45
C
26.12
D
78.92
GATE EC 2020   Signals and Systems
Question 2 Explanation: 
\begin{aligned} \int_{-\infty }^{\infty }\left | X(\omega ) \right |^{2}d\omega &=2\pi \int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt \\ &=2\pi \int_{-\infty }^{\infty }\left | y(t) \right |^{2}\\ &=2\times 2\pi \int_{-2}^{0}\left | y(t) \right |^{2}dt\\ &=2\times 2\pi\left [ \int_{-2}^{-1}(t+2)^{2}dt+\int_{-1}^{0}(2t+3)^{2}dt \right ]\\&=4\pi \left [\left \{ \frac{(t+2)^{3}}{3} \right \} _{-1}^{-2}+\left \{ \frac{(2t+3)^3}{3\times 2} \right \}_{0}^{-1} \right ]\\&=4\pi \left [ \frac{1-0}{3}+\frac{3^{3}-1}{6} \right ]\\&=4\pi \left [ \frac{1}{3}+\frac{26}{6} \right ]\\&= 4\pi \times \left [ \frac{1}{3}+\frac{26}{6} \right ] \\ &=4\pi \times \left [ \frac{1}{3}+\frac{13}{3} \right ]\\&=4\pi \times \frac{14}{3} \\&=\frac{56\pi }{3}=58.61\end{aligned}
Question 3
The input 4sinc(2t) is fed to a Hilbert transformer to obtain y(t), as shown in the figure below:

Here sinc(x)=\frac{sin(\pi x)}{\pi x}. The value (accurate to two decimal places) of \int_{-\infty }^{\infty } |y(t)|^{2}dt is _______.
A
7
B
8
C
9
D
10
GATE EC 2018   Signals and Systems
Question 3 Explanation: 
Hilbert transform does not alter the amplitude spectrum of the signal.
\begin{aligned} \text{So, }\quad \int_{-\infty}^{\infty}|y(t)|^{2} d t&=\int_{-\infty}^{\infty}|x(t)|^{2} d t=\int_{-\infty}^{\infty}|X(f)|^{2} d f \\ x(t)&=4 \sin \mathrm{c}(2 t) \\ \text{sinc}(t) &\stackrel{CTFT}{\longleftrightarrow} \text{rect}(f) \\ 4 \text{sinc}(2 t) &\stackrel{CTFT}{\longleftrightarrow}+\frac{4}{2} \text{rect}\left(\frac{f}{2}\right)=2 \text{rect}\left(\frac{f}{2}\right) \end{aligned}\\ \begin{aligned} \int_{-\infty}^{\infty}|X(f)|^{2} d f &=2 \times(2)^{2}=8 \\ \text { So. } \int_{-\infty}^{\infty}|y(t)|^{2} d t &=8 \end{aligned}

Question 4
Consider an LTI system with magnitude response
|H(f)|=\left\{\begin{matrix} 1-\frac{|f|}{20} & |f|\leq 20\\ 0 & |f|> 20 \end{matrix}\right.
And phase response Arg {H(f)}=-2f
If the input to the system is
x(t)=8cos(20\pi t+\frac{\pi }{4})+16sin(40\pi t+\frac{\pi }{8})+24cos(80\pi t+\frac{\pi }{16})
Then the average power of the output signal y(t) is _________.
A
6
B
7
C
8
D
9
GATE EC 2017-SET-2   Signals and Systems
Question 4 Explanation: 
Since,

\begin{aligned} \text{So,}\quad y(t)&=\frac{1}{2} \times 8 \cos (20 \pi t+\phi) \\ \text{So,}\quad &=4 \cos (20 \pi t+\phi) ; \phi=\frac{\pi}{4}-20^{\circ} \\ P_{y}&=\frac{4^{2}}{2}=8 \mathrm{W} \end{aligned}
Question 5
A continuous time signal x(t)=4cos(200\pi t)+8cos(400 \pi t), where t is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response

h(t)=\left\{\begin{matrix} \frac{2sin(300\pi t)}{\pi t} & t\neq 0\\ 600 & t=0 \end{matrix}\right.

Let y(t) be the output of this filter. The maximum value of |y(t)| is ________.
A
8.0
B
6.5
C
7.3
D
8.9
GATE EC 2017-SET-1   Signals and Systems
Question 5 Explanation: 
x(t)=4 \cos 200 \pi t+8 \cos 400 \pi t



\begin{aligned} \text{So.}\quad y(t)&=8 \cos 200 \pi t \\ \therefore \quad|y(t)|_{\max }&=8 \end{aligned}
Question 6
If the signal x(t)=\frac{sin(t)}{\pi t}*\frac{sin(t)}{\pi t} with * denoting the convolution operation, then x(t) is equal to
A
\frac{sin(t)}{\pi t}
B
\frac{sin(2t)}{2\pi t}
C
\frac{2sin(t)}{\pi t}
D
(\frac{sin(t)}{\pi t})^{2}
GATE EC 2016-SET-3   Signals and Systems
Question 6 Explanation: 




\begin{aligned} x(t) &=x_{1}(t) * x_{1}(t) \\ X(\omega) &=X_{1}(\omega) \cdot X_{1}(\omega) \end{aligned}


\therefore \quad x(t)=\frac{\sin t}{\pi t}
Question 7
The energy of the signal x(t)=\frac{sin(4\pi t)}{4\pi t} is ________
A
0
B
0.25
C
0.5
D
0.75
GATE EC 2016-SET-2   Signals and Systems
Question 7 Explanation: 




\begin{aligned} \text { Energy, } E_{x(i)} &=\frac{1}{2 \pi} \int_{-\infty}^{\infty}|X(\omega)|^{2} d \omega \\ &=\frac{1}{2 \pi} \int_{-4 \pi}^{4 \pi}\left(\frac{1}{4}\right)^{2} d \omega \\ &=\frac{1}{2 \pi} \times \frac{1}{16}[8 \pi]=\frac{1}{4}=0.25 \mathrm{J} \end{aligned}
Question 8
A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form

\sum_{k=1}^{3}a_{k}cos(k\omega _{0}t), where a_{k}\neq 0, \omega _{0}\neq 0.

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?
A
\sum_{k=1}^{3}b_{k}cos(k\omega _{0}t+\varnothing _{k}) , where b_{k}\neq a_{k}, \forall k
B
\sum_{k=1}^{4}b_{k}cos(k\omega _{0}t+\varnothing _{k}), where b_{k}\neq 0, \forall k
C
\sum_{k=1}^{3}a_{k} cos(k\omega _{0}t+\varnothing _{k})
D
\sum_{k=1}^{2}a_{k} cos(k\omega _{0}t+\varnothing _{k})
GATE EC 2016-SET-1   Signals and Systems
Question 8 Explanation: 


When a sinusoidal input is given to LTI system, the output is also a sinusoid with change in magnitude and the phase shift offered by LTI system.
Question 9
Consider the function g(t)=e^{-t}sin(2\pi t)u(t) where u(t) is the unit step function. The area under g(t) is _______.
A
0.15
B
0.34
C
0.5
D
1
GATE EC 2015-SET-3   Signals and Systems
Question 9 Explanation: 
\begin{aligned} g(t) &=e^{-t} \sin (2 \pi t) u(t) \\ \text { Area } &=\int_{-\infty}^{\infty} g(t) d t \\ &=\int_{-\infty}^{\infty} \sin (2 \pi t) u(t) e^{-t} d t=\left.G(s)\right|_{s=0} \\ G(s) &=\frac{2 \pi}{(s+1)^{2}+4 \pi^{2}} \\ \text { Area } &=\left.\frac{2 \pi}{(s+1)^{2}+4 \pi^{2}}\right|_{S=0}=0.155 \end{aligned}
Question 10
A real-valued signal x(t) limited to the frequency band |f|\leq \frac{W}{2} is passed through a linear time invariant system whose frequency response is
H(f)=\left\{\begin{matrix} e^{-j4\pi f}, &|f|\leq \frac{W}{2} \\ 0, & |f|> \frac{W}{2} \end{matrix}\right.
The output of the system is
A
x(t+4)
B
x(t-4)
C
x(t+2)
D
x(t-2)
GATE EC 2014-SET-4   Signals and Systems
Question 10 Explanation: 
\begin{aligned} \text{Given:} \quad H(f)&=\left\{\begin{array}{cc}e^{-j 4 \pi t}, & |f| \leq \frac{\omega}{2} \\ 0, & |f|>\frac{\omega}{2}\end{array}\right. \\ \text{So, } \quad|H(t)|&=1 \\ H(f)&=e^{-j 2.2 \pi t} \\ H(\omega)&=e^{j 2 \omega} &|f| \leq \frac{\omega}{2} \\ h(t)&=\delta(t-2)\\ \text{As we know that}\\ y(t)&=x(t) * h(t)\\ \text{or}\quad y(t)&=X(\omega) \cdot H(\omega) \\ y(t)&=x(t) * \delta(t-2) \\ y(t)&=x(t-2) \end{aligned}
There are 10 questions to complete.