ISRO EC 2023 Q.5: solution and answer

Official answer: DSource: ISRO's revised and final answer key of 16 February 2024

1. The trigonometric Fourier series of a periodic signal of period $T$ (with ${\omega }_0=2\pi /T$) is $x(t)=a_0+{\sum }_{n=1}^{\infty }\left(a_n\cos n{\omega }_{0}t+b_n\sin n{\omega }_{0}t\right)$, where $a_0=\frac{1}{T}{\int }_{T}x(t)dt$, $a_n=\frac{2}{T}{\int }_{T}x(t)\cos n{\omega }_{0}tdt$ and $b_n=\frac{2}{T}{\int }_{T}x(t)\sin n{\omega }_{0}tdt$.

2. Even function, $x(-t)=x(t)$: the product $x(t)\sin n{\omega }_{0}t$ is (even) $\times$ (odd) = odd, so its integral over a period symmetric about $t=0$ is zero. Hence $b_n=0$ for all $n$. The terms left are $a_0$ (the DC term) and the cosine terms. So an even function has only a DC term and cosine terms. This makes (d) true and (a) false.

3. Odd function, $x(-t)=-x(t)$: the product $x(t)\cos n{\omega }_{0}t$ is (odd) $\times$ (even) = odd, so $a_n=0$. Also $x(t)$ itself is odd, so $a_0=\frac{1}{T}{\int }_{T}x(t)dt=0$ (zero average).

4. So an odd function has only sine terms and no DC term. Option (b) is false because it names cosine terms, and (c) is false because it includes a DC term.

5. Only (d) is a true statement.