Signals and Systems

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 discrete sequence $x[...

A time-limited waveform g(x) is specified as follows: g(x) = { -k, -π < x ≤ 0 +k, 0 < x ≤ π 0, otherwise A new waveform f(x) is constructed from g(x) as follows: f(x) = Σ_{m=-∞}^{∞...

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 δ(t) be the continuous-time unit impulse function. The va...

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} |x(t)|^2 dt = $ ____...

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be $x_1(t) = \begin{cases} 1, & t \in [0,1] \\ 2-t, & t \in [1,2] \\ 0, & \text{otherwise} \end{cases}$ and $x_2(t) = \begin{cases...

Suppose signal y(t) is obtained by the time-reversal of signal x(t), i.e., y(t) = x(−t), −∞ < t < ∞. Which one of the following options is always true for the convolution of x(t) a...

If u(t) is the unit step function, then the region of convergence (ROC) of the Laplace transform of the signal x(t) = e^(t^2) [u(t − 1) – u(t − 10)] is

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 is

The inverse Laplace transform of H(s) = $\frac{s+3}{s^2+2s+1}$ for t ≥ 0 is

Consider the system with following input-output relation $y[n] = (1+(-1)^n) x[n]$ where, $x[n]$ is the input and $y[n]$ is the output. The system is

A continuous-time input signal $x(t)$ is an eigenfunction of an LTI system, if the output is

The output y(t) of the following system is to be sampled, so as to reconstruct it from its samples uniquely. The required minimum sampling rate is

The Laplace transform of f(t)=$$2\sqrt{t/\mathrm\pi}$$ is $$s^{-3/2}$$. The Laplace transform of g(t)=$$\sqrt{1/\mathrm{πt}}$$ is

The unilateral Laplace transform of f(t) is $$\frac1{s^2\;+\;s\;+\;1}$$. The unilateral Laplace transform of tf(t) is

Consider the function $$F\left( s \right) = {5 \over {s\left( {{s^2} + 3s + 2} \right)}}$$ Where $$F(s)$$ is the Laplace transform of the function $$f(t).$$ The initial value of $$...

The Laplace transform of $$\left(t^2\;-\;2t\right)u\left(t\;-\;1\right)$$ is