The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?

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

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)

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

Consider with respect to HusserI's concept of phenomena.
1. Phenomena is a substantial unity that has real properties, real parts and real changes.
2. Phenomena is no substantial unity, that has no real parts and no real changes.
3. Phenomena has no nature, but only essence.
4. Phenomena have no essence, but only nature.

A

Only (1) is true

B

Only (2) and (3) are true

C

Only (1) and (3) are true

D

Only (1), (3) and (4) are true

Half Range Fourier Series contains either sine or cosine terms.

A

True

B

False

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 Fourier series of the function,
\[\begin{array}{*{20}{c}} {{\text{f}}\left( {\text{x}} \right) = 0,}&{ - \pi The convergence of the above Fourier series at x = 0 gives

A

$$\sum\limits_{{\text{n}} = 1}^\infty {\frac{1}{{{{\text{n}}^2}}} = \frac{{{\pi ^2}}}{6}} $$

B

$$\sum\limits_{{\text{n}} = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{{\text{n}} + 1}}}}{{{{\text{n}}^2}}} = \frac{{{\pi ^2}}}{{12}}} $$

C

$$\sum\limits_{{\text{n}} = 1}^\infty {\frac{1}{{{{\left( {{\text{2n}} - 1} \right)}^2}}} = \frac{{{\pi ^2}}}{8}} $$

D

$$\sum\limits_{{\text{n}} = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{{\text{n}} + 1}}}}{{{\text{2n}} - 1}} = \frac{\pi }{4}} $$

Computing the Fourier transform of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v), Fourier transformed function of f(x, y) an input image of size M*N, and H(u, v), the filter used for implementing Laplacian in frequency domain. This dual relationship is expressed as Fourier transform pair notation given by_____________ F(u,v) b) ∇2 f(x,y)↔-F(u,v) c) ∇2 f(x,y)↔-F(u,v) d) ∇2 f(x,y)↔F(u,v)

A

∇2 f(x,y)↔F(u,v)

B

(u –M/2)2+ (v –N/2)2

C

(u –M/2)2+ (v –N/2)2

D

(u –M/2)2+ (v –N/2)2

One period (0, T) each of two periodic waveforms W₁ and W₂ are shown in the figure. The magnitudes of the nth Fourier series coefficients of W₁ and W₂, for n ≥ 1, n is odd, are respectively proportional to

A

|n|-3 and |n|-2

B

|n|-1 and |n|-3

C

|n|-1 and |n|-2

D

|n|-4 and |n|-2