Recent questions in 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...

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}^{\inf...

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\q...

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 op...

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 sta...

Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $\left[-\omega_{c}, \omega_{c}\right]$, where $\ome...

Let $f(t)$ be a periodic signal with fundamental period $T_{0}>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$.The Fourier series expansions of $f(t)$ and $y...

Let $m(t)$ be a strictly band-limited signal with bandwidth $B$ and energy $E$. Assuming $\omega_{0}=10 B$, the energy in the $m(t) \cos \omega_{0} t$ is$\frac{E}{4}$$\f...

The Fourier transform $X(\omega)$ of $x(t)=e^{-t^{2}}$ isNote: $\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi}$$\sqrt{\pi} e^{\frac{\omega^{2}}{2}}$$\frac{e^{-\frac{\o...

For a real signal, which of the following is/are valid power spectral density/densities?$\text{S}_X(\omega)=\frac{2}{9+\omega^2}$$\text{S}_X(\omega)=e^{-\omega^2} \cos ^2...

Let $x(t)=10 \cos (10.5 W t)$ be passed through an LTI system having impulse response $h(t)=\pi\left(\frac{\sin W t}{\pi t}\right)^{2} \cos 10 Wt$. The output of the sys...

Let $\mathrm{x}_1(\mathrm{t})$ and $\mathrm{x}_2(\mathrm{t})$ be two band-limited signals having bandwidth $B=4 \pi \times 10^3 \; \mathrm{rad} / \mathrm{s}$ each. In the...

Let $\mathrm{x}_1(\mathrm{t})=\mathrm{u}(\mathrm{t}+1.5)-\mathrm{u}(\mathrm{t}-1.5)$ and $\mathrm{x}_2(\mathrm{t})$ is shown in the figure below. For $\mathrm{y}(\mathrm{...

Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems.$x(t)=\sin (t), y(t)=\cos (t),$$x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$Which of these...