Recent questions in Signals and Systems

Consider the infinite-length, discrete-time sequence $x[n]=0.9^{|n|}$, where $n$ is an integer. The region of convergence of its $Z$-transform $X(Z)$ is given by:(Note: $...

Let $x_{c}(t)$ be any continuous-time periodic signal with period $T$. It is sampled uniformly with a sampling period $T_{S}$ where $T_{S} \neq T$, resulting in the discr...

A time-limited waveform $\mathrm{g}(x)$ is specified as follows:\[g(x)=\left\{\begin{aligned}-k, && -\pi < x \leq 0 \\+k, && 0 < x \leq \pi \\ 0 , && \text { otherwise ...

Consider a discrete-time linear time-invariant (LTI) system $\boldsymbol{S}$, where$$ y[n]=\boldsymbol{S}\{x[n]\} $$Let$$ \boldsymbol{S}\{\delta[n]\}=\left\{\begin{array}...

Consider a continuous-time signal$$ x(t)=-t^{2}\{u(t+4)-u(t-4)\}$$where $u(t)$ is the continuous-time unit step function. Let $\delta(t)$ be the continuous-time unit impu...

A continuous time periodic signal $x(t)$ is$ x(t)=1+2 \cos 2 \pi t+2 \cos 4 \pi t+2 \cos 6 \pi t$If $T$ is the period of $x(t)$, then $\frac{1}{T} \int_{0}^{T} \mid x(t)\...

Let continuous-time signals $x_{1}(t)$ and $x_{2}(t)$ be$$x_{1}(t)=\begin{cases} 1, & t \in[0,1] \\ 2-t, & t \in[1,2] \\ 0, & \text { otherwise } \end{cases} \quad \text ...

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system $\boldsymbol{S}$. The output is observed to be the contin...

An ideal low pass filter has frequency response given by$H(j \omega)=\begin{cases}1, & \mid \omega \mid \leq 200 \pi \\ 0, & \text { otherwise }\end{cases}$Let $h(t)$ be ...

Let $X(\omega)$ be the Fourier transform of the signal\[x(t)=e^{-t^{4}} \cos t, \quad-\infty < t < \infty.\]The value of the derivative of $X(\omega)$ at $\omega=0$ is (r...

​​​The input $x(t)$ and the output $y(t)$ of a system are related as\[y(t)=e^{-t} \int_{-\infty}^{t} e^{\tau} x(\tau) d \tau, \quad-\infty < t < \infty. \]The system isno...

​​​Consider the discrete-time systems $T_{1}$ and $T_{2}$ defined as follows:\[\begin{array}{c}\left\{T_{1} x\right\}[n]=x[0]+x +\cdots+x[n] \\\left\{T_{2} x\right\}[n]=x...

​​​If the $Z$-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the $Z$ transform of the signal $y[n]=x[2 n]$ is$Y(z)=X\left(z^{2}\right)$$Y(z)=\...

​​​​​Consider the function $f(t)=(\max (0, t))^2$ for $-\infty<t<\infty$, where $\max (a, b)$ denotes the maximum of $a$ and $b$. Which of the following statements is/are...

If the energy of a continuous-time signal $x(t)$ is $\text{E}$ and the energy of the signal $2 x(2 t-1)$ is $c E$, then $c$ is ___________ (rounded off to $1$ decimal pla...

A continuous-time system that is initially at rest is described by $$ \frac{d y(t)}{d t}+3 y(t)=2 x(t),$$ where$x(t)$ is the input voltage and $y(t)$ is the output voltag...

The Fourier transform $X(\omega)$ of the signal $x(t)$ is given by$$\begin{aligned}X(\omega) & =1, \text { for }|\omega|<W_o \\& =0, \text { for }|\omega|>W_0\end{aligned...

The $\text{Z}$-transform of a discrete signal $\text{x[n]}$ is$\text{X(z)}=\frac{4 z}{\left(z-\frac{1}{5}\right)\left(z-\frac{2}{3}\right)(z-3)}$ with $\text{ROC=R}$ Whic...

Which of the following statement(s) is/are true? If an $\text{LTI}$ system is causal, it is stableA discrete time $\text{LTI}$ system is causal if and only if its respons...