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Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {a<sub>k</sub>} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):<br>1. The complex Fourier series coefficients of x(3t) are {a<sub>k</sub>} where k is integer valued.<br>2. The complex Fourier series coefficients of x(3f) are {3a<sub>k</sub>} where k is integer valued.<br>3. The fundamental angular frequency of x(3t) is 6π rad/s.<br>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
Correct Answer:
Only 1 and 3 are true
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
$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}{e^{jk{{2\pi } \over T}t}}} .$$
The same function x(t) can also be considered as a periodic function with period T' = 40. Let b
k
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
The magnitude and phase of the complex Fourier series coefficient a
k
of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation: C is the set of complex number, R is the set of purely real numbers, and P is the set of purely imaginary numbers.
A
$$x\left( t \right) \in R$$
B
$$x\left( t \right) \in P$$
C
$$x\left( t \right) \in \left( {C - R} \right)$$
D
The information given is not sufficient to draw any conclusion about x(t)
One period (0, T) each of two periodic waveforms W
1
and W
2
are shown in the figure. The magnitudes of the nth Fourier series coefficients of W
1
and W
2
, for n ≥ 1, n is odd, are respectively proportional to
A
|n|<sup>-3</sup> and |n|<sup>-2</sup>
B
|n|<sup>-1</sup> and |n|<sup>-3</sup>
C
|n|<sup>-1</sup> and |n|<sup>-2</sup>
D
|n|<sup>-4</sup> and |n|<sup>-2</sup>
How do we represent a pairing of a periodic signal with its fourier series coefficients in case of continuous time fourier series?
A
x(t) ↔ Xn
B
x(t) ↔ Xn+1
C
x(t) ↔ X
D
x(n) ↔ Xn
The process of converting discrete-time continuous valued signal into discrete-time discrete valued (digital) signal is known as ____________
A
Sampling
B
Quantization
C
Coding
D
None of the mentioned
Which of the following statements is/are false regarding modern periodic table?
1. Elements in the modern periodic table are arranged in decreasing order of their atomic numbers.
2. Elements in the modern periodic table are arranged in increasing order of their atomic masses.
3. In the modern periodic table, isotopes are placed in adjacent groups.
4. Elements in modern periodic table are arranged in increasing order of their atomic numbers.
A
only 1
B
only 4
C
1, 2 and 4
D
1, 2 and 3
A continuous time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period T
s
. In which one of the following cases is the sampled signal periodic?
A
$$T = \sqrt 2 {T_s}$$
B
T = 1.2T<sub>s</sub>
C
Always
D
Never
Seven people A, B, C, D, E, F and G live on separate floors of a 7-floor building. Ground floor is numbered 1, first floor is numbered. 2 and so on until the topmost floor is numbered 7. Each one of these having a different cars-Cadillac, Ambassador, Fiat, Maruti, Mercedes, Bedford and Fargo but not necessarily in the same order. Only three people live above the floor on which A lives. Only one person lives between A and the one having a car Cadillac. F lives immediately below the one having a car Bedford. The one having a car Bedford lives on an even-numbered floor. Only three people live between the ones having a car Cadillac and Maruti. E lives immediately above C. E is not having a car Maruti. Only two people live between B and the one having a car Fargo. The one having a car Fargo lives below the floor on which B lives. The one having a car Fiat does not live immediately above D or immediately below B. D does not live immediately above or immediately below A. G does not have a car Ambassador. Question : How many people live between the floors on which D and the one having a car Bedford ?
A
One
B
Two
C
Three
D
Four
Each of the following problem has a question and two statements which are labeled 1. and. Use the data given in 1. and 2. to decide whether the statements are sufficient to answer the question.Select one of the following . a.Statements (1) ALONE is sufficient,but (2) ALONE is not sufficient. b.Statements (2) ALONE is sufficient,but (1) ALONE is not sufficient. c. Both statement TOGETHER are sufficient but statement alone is sufficient .How many minutes are there is time period T? (1) Time period T extends from 12.00 midnight to 12.00 noon.(2) Time period T is 2 hours and 180 seconds long.
A
A
B
B
C
C
D
D
Let x(t) and y(t) (with Fourier transforms X(f) and Y(f) respectively) be related as shown in the figure. Then Y(f) is
A
$$ - \frac{1}{2}X\left( {\frac{f}{2}} \right){e^{ - j2\pi f}}$$
B
$$ - \frac{1}{2}X\left( {\frac{f}{2}} \right){e^{j2\pi f}}$$
C
$$ - X\left( {\frac{f}{2}} \right){e^{j2\pi f}}$$
D
$$ - X\left( {\frac{f}{2}} \right){e^{ - j2\pi f}}$$
