GATE EC 2005 Set 1 Q2 — Continuous-time Signals

Two analog signals $x_{1}(t)$ and $x_{2}(t)(t$ in second $)$, are sampled at a rate $F_{s}=40 \mathrm{~Hz}$, where $$x_{1}(t)=\cos (20 \pi t), t \geq 0, \text{ and } x_{2}(t)=\cos (100 \pi t), t \geq 0$$ The first ten samples (starting from $t=0$ ) are considered for the analysis. Which of the following statements is TRUE?

The continuous time signal $x(t)$ is real, periodic with period $T$ and satisfies the Dirichlet conditions. The Fourier series representation of $x(t)=\sum_{-\infty}^{\infty} a_{n} e^{j\left(\frac{2 \pi n t}{T}\right)}$ and $x(t)$ satisfies the following: $x\left(t-\frac{T}{2}\right)=-x(t)$. For any integer $m$, which of the following options is …

Consider a real signal $x(t),-\infty < t < \infty$, such that $x(t) = 0\quad\text{for}\quad t <0, x(t) =2 \quad\text{for} \quad0 \leq t < 1\quad \text{and}\quad x(t) = 0\quad \text{for} \quad t \geq 1$. Let $E[x(t)]=\int_{-\infty}^{\infty}[x(t)]^{2} d t$. Which of the following options correctly represents the ratio, …

Consider a real baseband signal $x(t)=e^{-2 t}$, for $t$ (in seconds) $\geq 0$. If $99 \%$ of energy of $x(t)$ lies within $B \mathrm{~Hz}$, then which of the following options is TRUE for the value of ${B}?$

Let $x_{1}(t)=\cos (2 \pi n t)$ and $x_{2}(t)=2 \sin (4 \pi n t)$ represent two sinusoids for a positive integer $n$ and $-\infty < t < \infty$. Which of the following statements about $x_{1}(t)$ and $x_{2}(t)$ is/are valid?

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega)=\int_{-\infty}^{\infty} f(t) \exp (-j \: \omega t) d t$ exists. Which one of the following statements is always TRUE?