Question 1 |

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(t)=\sum_{k=-\infty }^{\infty }a_{k}e^{jk\frac{2\pi }{T}t}

The same function x(t) can also be considered function with period {T=40}'. Let b_{k} be the fourier series cofficients when period is taken as {T}'. If \sum_{k=-\infty}^{\infty}|a_{k}|=16 ,then \sum_{k=-\infty}^{\infty}|a_{k}| is equal to

x(t)=\sum_{k=-\infty }^{\infty }a_{k}e^{jk\frac{2\pi }{T}t}

The same function x(t) can also be considered function with period {T=40}'. Let b_{k} be the fourier series cofficients when period is taken as {T}'. If \sum_{k=-\infty}^{\infty}|a_{k}|=16 ,then \sum_{k=-\infty}^{\infty}|a_{k}| is equal to

256 | |

64 | |

16 | |

4 |

Question 1 Explanation:

Change in only time period or frequency does not change in the value of Fourier series coefficients

So,

\begin{aligned} b_{k} &=a_{k} \\ \sum_{k=-\infty}^{\infty}\left|b_{k}\right| &=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|=16 \end{aligned}

So,

\begin{aligned} b_{k} &=a_{k} \\ \sum_{k=-\infty}^{\infty}\left|b_{k}\right| &=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|=16 \end{aligned}

Question 2 |

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):

I. The complex Fourier series coefficients of x(3t) are \{a_{k}\} where k is integer valued

II. The complex Fourier series coefficients of x(3t) are \{3{a_{k}}\} where k is integer valued

III. The fundamental angular frequency of x(3t) is 6\pi rad/s

For the three statements above, which one of the following is correct?

I. The complex Fourier series coefficients of x(3t) are \{a_{k}\} where k is integer valued

II. The complex Fourier series coefficients of x(3t) are \{3{a_{k}}\} where k is integer valued

III. The fundamental angular frequency of x(3t) is 6\pi rad/s

For the three statements above, which one of the following is correct?

only II and III are true | |

only I and III are true | |

only III is true | |

only I is true |

Question 2 Explanation:

Initially T = 1\text{sec}, so \omega_{0} = 2\pi \text{rad/sec}. when
is compressed by 3, frequency will expand by
same factor but there is no change in values of
a_{k}.

So, both statement I and Ill are correct

So, both statement I and Ill are correct

Question 3 |

A periodic signal x(t) has a trigonometric Fourier series expansion

x(t)=a_{0}+\sum_{n=1}^{\infty}(a_{n}cosn\omega_{0}t)

If x(t)=-x-(t)=-x(t-\pi/\omega_{0}), we can conclude that

x(t)=a_{0}+\sum_{n=1}^{\infty}(a_{n}cosn\omega_{0}t)

If x(t)=-x-(t)=-x(t-\pi/\omega_{0}), we can conclude that

a_{n} are zero for all n and b_{n} are zero for n even | |

a_{n} are zero for all n and b_{n} are zero for n odd | |

a_{n} are zero for n even and b_{n} are zero for n odd | |

a_{n} are zero for n odd and b_{n} are zero for n even |

Question 3 Explanation:

Signal has odd and half wave symmetries.So at
a_{n} are zero and b_{n} are zero to n even.

Question 4 |

The magnitude and phase of the complex Fourier series coefficients 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 numbers, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

x(t)\in R | |

x(t)\in P | |

x(t)\in (C-R) | |

the information given is not sufficient to draw any conclusion about x(t) |

Question 4 Explanation:

\begin{aligned} \left|a_{k}\right|&-\text{ even symmetry} \\ \angle a_{k}&-\text{ Odd symmetry} &(\pi \text{ can be } -\pi) \end{aligned}

\Rightarrow x(t) is real.

\Rightarrow x(t) is real.

Question 5 |

Consider the periodic square wave in the figure shown.

The ratio of the power in the 7^{th} harmonic to the power in the 5^{th} harmonic for this waveform is closest in value to ______.

The ratio of the power in the 7^{th} harmonic to the power in the 5^{th} harmonic for this waveform is closest in value to ______.

0.51 | |

0.25 | |

0.75 | |

1 |

Question 5 Explanation:

\begin{aligned} X(\omega) &=\sum_{K=-\infty}^{\infty} 2 \pi a_{k} \delta\left(0-K \omega\right) \\ a_{K} &=\frac{1}{2 \pi} \int_{0}^{2} x(t) e^{-j K \omega_{0} t} d t \end{aligned}

\begin{aligned} &=\frac{1}{2 \pi}\left\{\int_{0}^{1} 1 \cdot e^{-j K \omega_{o} t} d \omega-\int_{1}^{2} 1 \cdot e^{-j K \omega_{o} t} d \omega\right\}\\ &=\frac{1}{2\pi}\left[\frac{1}{-j K \omega_{o}}[e^{-j K \omega_{o}}-1]-\frac{1}{-j K \omega_{o}}[e^{(-j 2K \omega_{o})}-e^{(-j 2K \omega_{o})}] \right ]\\ &=\frac{1}{2\pi}\left[\frac{e^{-j K \omega_{o}l2}}{j K \omega_{o}}.2j\sin\frac{K\omega_{o}}{2}-\frac{1}{j K \omega_{o}}e^{-\frac{j K \omega_{o}3}{2}}\left(2j\sin\frac{K\omega_{0}}{2} \right ) \right ]\\ &=\frac{1}{\pi}\left[\frac{\sin \frac{K \omega_{o}}{2}}{K \omega_{o}}\left(e^{-j K \omega_{0}}\right)\left(e^{\frac{j K \omega_{0}}{2}}-e^{-\frac{K_{m_{2}}}{2}}\right)\right] \\ &=\frac{1}{\pi}\left(\frac{\sin K \omega_{0} / 2}{K \omega_{o}}\right) e^{-j K \omega_{0}} \times 2 j \sin \left(\frac{\left.K_{\left(0_{0}\right.}\right)}{2}\right) \\ &=\frac{1}{\pi^{2}} \frac{\left(\sin K \frac{\pi}{2}\right)^{2}}{K} 2 j \times \cos K \pi \end{aligned}\\

7^{\text {th }} harmonic power =\left(2 \pi a_{7}\right)^{2}

5^{\text {th }} harmonic power =\left(2 \pi a_{5}\right)^{2}

Ratio =\left(\frac{a_{7}}{a_{5}}\right)^{2}=\left(\frac{5}{7}\right)^{2}=\frac{25}{49}=0.51

There are 5 questions to complete.