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Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by $${a_k}$$ , that is<br>$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}{e^{jk{{2\pi } \over T}t}}} .$$<br>The same function x(t) can also be considered as a periodic function with period T' = 40. Let b<sub>k</sub> be the Fourier series coefficients when period is taken as T'. If $$\sum\limits_{k = - \infty }^\infty {\left| {{a_k}} \right|} = 16,$$ then $$\sum\limits_{k = - \infty }^\infty {\left| {{b_k}} \right|} $$ is equal to
A
256
B
64
C
16
D
4
Correct Answer:
16
Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {a
k
} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):
1. The complex Fourier series coefficients of x(3t) are {a
k
} where k is integer valued.
2. The complex Fourier series coefficients of x(3f) are {3a
k
} where k is integer valued.
3. The fundamental angular frequency of x(3t) is 6π rad/s.
For the three statements above, which one of the following is correct?
A
Only 2 and 3 are true
B
Only 1 and 3 are true
C
Only 3 is true
D
Only 1 is true
For a function g(t), it is given that
$$\int\limits_{ - \infty }^{ + \infty } {g\left( t \right){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}} $$ for any real value $$\omega $$ . If $$y\left( t \right) = \int\limits_{ - \infty }^t {g\left( \tau \right)} d\tau ,\,{\rm{then}}\,\int\limits_{ - \infty }^{ + \infty } {y\left( t \right)} dt$$ is. . . . . . . .
A
0
B
-j
C
$$ - {j \over 2}$$
D
$${j \over 2}$$
If a + b + c + d = 4, then find the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$ + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$ + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$ + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$ is?
A
0
B
5
C
1
D
4
If a + b + c + d = 4, then the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$ + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$ + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$ + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$ is?
A
0
B
1
C
4
D
1 + abcd
A periodic signal x(t) has a trigonometric Fourier series expansion
$$x\left( t \right) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\,\cos \,n{\omega _0}t + {b_n}\sin \,n{\omega _0}t} \right)} $$
If $$x\left( t \right) = - x\left( { - t} \right) = - x\left( {{{t - \pi } \over {{\omega _0}}}} \right),$$ we can conclude that
A
a<sub>n</sub> are zero for all n and b<sub>n</sub> are zero for n even
B
a<sub>n</sub> are zero for all n and b<sub>n</sub> are zero for n odd
C
a<sub>n</sub> are zero for n even and b<sub>n</sub> are zero for n odd
D
a<sub>n</sub> are zero for n odd and b<sub>n</sub> are zero for n even
The function f(t) has the Fourier transform f(ω)
The Fourier transform of
$$g\left( t \right) = \left( {\int\limits_{ - \infty }^\infty {g\left( t \right){e^{ - j\omega }}dt} } \right)$$ is
A
2πf(-ω)
B
$${1 \over {2\pi }}f\left( \omega \right)$$
C
$${1 \over {2\pi }}f\left( { - \omega } \right)$$
D
None of these
Let the function
\[{\text{f}}\left( \theta \right) = \left| {\begin{array}{*{20}{c}} {\sin \theta }&{\cos \theta }&{\tan \theta } \\ {\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)} \\ {\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)} \end{array}} \right|\
A
<br>where \\
B
and \ denote the derivative of f with respect to \. Which of the following statements is/are TRUE?<br>I. There exists \ such that \<br>II. There exists \ such that\
C
<p><span>A.</span> l only
D
</span> ll only
The Fourier series representation of an impulse train denoted by
$$s\left( t \right) = \sum\limits_{n = - \infty }^\infty {\delta \left( {t - n{T_0}} \right)} \,{\rm{is}}\,{\rm{given}}\,{\rm{by}}$$
A
$${1 \over {{T_0}}}\sum\limits_{n = - \infty }^\infty {\exp \left( { - {{j2\pi nt} \over {{T_0}}}} \right)} $$
B
$${1 \over {{T_0}}}\sum\limits_{n = - \infty }^\infty {\exp } \left( { - {{j\pi nt} \over {{T_0}}}} \right)$$
C
$${1 \over {{T_0}}}\sum\limits_{n = - \infty }^\infty {\exp } \left( {{{j\pi nt} \over {{T_0}}}} \right)$$
D
$${1 \over {{T_0}}}\sum\limits_{n = - \infty }^\infty {\exp } \left( {{{j2\pi nt} \over {{T_0}}}} \right)$$
If {x} is a continuous, real valued random variable defined over the interval (-$$\infty $$, +$$\infty $$) and its occurrence is defined by the density function given as:
$${\text{f}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}$$ where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$
A
1
B
0.5
C
$$\pi $$
D
$$\frac{\pi }{2}$$
The value of the expression $$\frac{{{{\left( {a - b} \right)}^2}}}{{\left( {b - c} \right)\left( {c - a} \right)}} + $$ $$\frac{{{{\left( {b - c} \right)}^2}}}{{\left( {a - b} \right)\left( {c - a} \right)}} + $$ $$\frac{{{{\left( {c - a} \right)}^2}}}{{\left( {a - b} \right)\left( {b - c} \right)}}$$ = ?
A
0
B
3
C
$$\frac{1}{3}$$
D
2