GATE EC 2017 Set 1 — Question 31

Let the original signal be $x(t)$ with fundamental period $T_1 = 1$ s.
The fundamental frequency is $f_1 = 1/T_1 = 1$ Hz.
The fundamental angular frequency is $\omega_1 = 2\pi f_1 = 2\pi$ rad/s.
The Fourier series representation of $x(t)$ is given by:$$x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j k \omega_1 t} = \sum_{k=-\infty}^{\infty} a_k e^{j k 2\pi t}$$where $\{a_k\}$ are the complex Fourier series coefficients.Now, consider the time-scaled signal $y(t) = x(3t)$. This is a time compression by a factor of 3.Analysis of Period and Frequency (Statement III):
The new fundamental period $T_2$ is given by $T_2 = T_1 / 3 = 1/3$ s.
The new fundamental frequency is $f_2 = 1/T_2 = 3$ Hz.
The new fundamental angular frequency is $\omega_2 = 2\pi f_2 = 2\pi(3) = 6\pi$ rad/s.
Therefore, Statement III is true.Analysis of Fourier Coefficients (Statements I and II):
Let's find the Fourier series for $y(t) = x(3t)$. We substitute $3t$ for $t$ in the Fourier series expansion of $x(t)$:$$y(t) = x(3t) = \sum_{k=-\infty}^{\infty} a_k e^{j k \omega_1 (3t)} = \sum_{k=-\infty}^{\infty} a_k e^{j k (3\omega_1) t}$$The Fourier series of $y(t)$ is also defined with respect to its own fundamental frequency $\omega_2 = 3\omega_1$:$$y(t) = \sum_{k=-\infty}^{\infty} b_k e^{j k \omega_2 t} = \sum_{k=-\infty}^{\infty} b_k e^{j k (3\omega_1) t}$$where $\{b_k\}$ are the Fourier coefficients of $y(t)$.
By comparing the two expressions for $y(t)$, we can see that the coefficients must be equal for each harmonic index $k$.$$b_k = a_k$$This is a standard property of the Fourier series: time scaling a signal $x(t)$ to $x(\alpha t)$ does not change the values of the Fourier series coefficients, it only changes the fundamental frequency to $\alpha \omega_1$.Therefore, the complex Fourier series coefficients of $x(3t)$ are $\{a_k\}$.
Statement I is true.
Statement II is false.Conclusion:
Statements I and III are true.
Thus, the correct option is (B).