Convolution

The Laplace Transform of the signal $x(t) = u(t - 2) * (tu(t))$ is given by which of the following expressions? [" * " represents convolution operator]

A continuous time signal x(t) = 2 cos(8πt + π/3) is sampled at a rate of 15 Hz. The sampled signal x_s(t) when passed through an LTI system with impulse response h(t) = (sin 2πt /...

A causal and stable LTI system with impulse response h(t) produces an output y(t) for an input signal x(t) . A signal x(0.5t) is applied to another causal and stable LTI system wit...

Let an input x[n] having discrete time Fourier transform $X(e^{j\Omega}) = 1 - e^{-j\Omega} + 2e^{-3j\Omega}$ be passed through an LTI system. The frequency response of the LTI sys...

Let x(t) = 10 cos(10.5Wt) be passed through an LTI system having impulse response h(t) = π (sin Wt / πt)^2 cos 10Wt. The output of the system is

Let x1(t) = u(t + 1.5) - u(t - 1.5) and x2(t) is shown in the figure below. For y(t) = x1 (t) * x2 (t), the $\int_{-\infty}^{\infty}y(t)dt$ is ________ (rounded off to the nearest...

Let x 1 (t) = e $$-$$t u(t) and x 2 (t) = u(t) $$-$$ u(t $$-$$ 2), where u( . ) denotes the unit step function. If y(t) denotes the convolution of x 1 (t) and x 2 (t), then $$\math...

For a unit step input $u[n]$, a discreate-time $L T I$ system produces an output signal $(2 \delta[n+1]+\delta[n]+\delta[n-1])$. Let $y[n]$ be the output of the system for an input...

A binary baseband digital communication system employs the signal $$$p\left( t \right) = \left\{ {\matrix{ {{1 \over {\sqrt {{T_s}} }},} & {0 \le t \le {T_s}} \cr {0,} & {otherwise...

If the signal x(t) = $${{\sin (t)} \over {\pi t}}*{{\sin (t)} \over {\pi t}}$$ with * denoting the convolution operation, then x(t) is equal to

The result of the convolution $$x\left( { - t} \right) * \delta \left( { - t - {t_0}} \right)$$ is

Consider a continuous-time signal defined as $$x(t) = \left( {{{\sin \,(\pi t/2)} \over {(\pi t/2)}}} \right)*\sum\limits_{n = - \infty }^\infty {\delta (t - 10n)} $$ Where ' * ' d...

A continuous, linear time - invariant fiilter has an impulse response h(t) described by $$h\left( t \right) = \left\{ {\matrix{ {3\,for\,0 \le t \le 3} \cr {0\,otherwise} \cr } } \...

The sequence x $$\left[ n \right]$$ = $${0.5^n}$$ u[n], where u$$\left[ n \right]$$ is the unit step sequence, is convolved with itself to obtain y $$\left[ n \right]$$ . Then $$\s...

Consider a discrete-time signal $$x\left[ n \right] = \left\{ {\matrix{ {n\,\,for\,\,0 \le n \le 10} \cr {0\,\,otherwise} \cr } } \right.$$ If $$y\left[ n \right]$$ is the convolut...

Two system with impulse responses h 1 (t) and h 2 (t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

Let g(t) = $${e^{ - \pi {t^2}}}$$, and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is

The impulse response of a system is h(t) = t u(t). For an input u(t - 1), the output is

Let $$y\left[ n \right]$$ denote the convolution of $$h\left[ n \right]$$ and $$g\left[ n \right]$$, where $$h\left[ n \right]$$ $$ = \,{\left( {1/2} \right)^2}\,\,u\left[ n \right...

An input x(t) = exp( -2t) u(t) + $$\delta $$(t-6) is applied to an LTI system with impulse response h(t) = u(t). The output is

Two discrete time systems with impulse responses $${h_1}\left[ n \right]\, = \delta \left[ {n - 1} \right]$$ and $${h_2}\left[ n \right]\, = \delta \left[ {n - 2} \right]$$ are con...

A discrete time linear shift - invariant system has an impulse response $$h\left[ n \right]$$ with $$h\left[ 0 \right]$$ $$ = 1,\,\,h\left[ 1 \right]\,\, = - 1,\,\,h\left[ 2 \right...

Let $$g\left( t \right){\mkern 1mu} {\mkern 1mu} \,\,\,\,\,{\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} p\left( t \right){}^ * p\left( t \right)$$ where $$ * $$ denotes convolution and...

Let g(t) = p(t) * p(t), where * denotes convolution and p(t) = u(t) - (t-1) with u(t) being the unit step function. The impulse response of filter matched to the singal s(t) = g(t)...

Convolution of x(t + 5) with impulse function $$\delta \left( {t\, - \,7} \right)$$ is equal to

The transfer function of a system is given by $$H\left( s \right) = {1 \over {{s^2}\left( {s - 2} \right)}}$$. The impulse response of the system is

A linear time invariant system has an impulse response e 2t , t > 0. If the initial conditions are zero and the input is e 3t , the output for t > 0 is

Let u(t) be the unit step function. Which of the waveforms in Fig.(a) -(d) corresponds to the convolution of $$\left[ {u\left( t \right)\, - \,u\left( {t\, - \,1} \right)} \right]$...

A linear time invariant system has an impulse response $${e^{2t}},\,\,t\, > \,0.$$ If the initial conditions are zero and the input is $${e^{3t}}$$, the output for $$t\, > \,0$$ is

The Nyquist sampling frequency (in Hz) of a signal given by $$16 \times {10^{4\,}}\,\sin {c^2}(400t)*{10^6}\,\sin {c^3}(100t)$$ is

The unit impulse response of a linear time invariant system is the unit step function u(t). For t>0, the response of the system ot an excitation e -at u(t), a > 0 will be

Consider a rectangular pulse g(t) existing between $$t = \, - {T \over 2}\,and\,{T \over 2}$$. Find and sketch the pulse obtained by convolving g(t) with itself. The Fourier transf...

The ACF of a rectangular pulse of duration T is

The unit impulse response of a linear time invariant system is the unit step function u(t). For t>0, the response of the system to an excitation e -at u(t), a>0 will be

A rectangular pulse of duration T is applied to a filter matched to this input. The output of the filter is a

An input signal A exp $$\left( { - \alpha \,t} \right)$$ u(t) with $$\alpha > 0$$ is applied to a causal filter, the impulse response of which is A exp $$\,( - \alpha \,\,t)$$. Det...

A system having a unit impulse response $$h\left( n \right)$$ = $$u\left( n \right)$$ is excited by a signal $$x\left( n \right)$$ $$ = \,{\alpha ^n}\,\,u\left( n \right).\,$$ Dete...

A signal v(t)= [1+ m(t) ] cos $$({\omega _c}t)$$ is detected using a square law detector, having the characteristic $${v_0}(t) = {v^2}(t)$$. If the Fourier transform of m(t) is con...

Match each of the items, A, B and C, with an appropriate item from 1, 2, 3, 4 and 5 A. Fourier transform of a Gaussian function B. Convolution of a rectangular pulse with itself C....

Let h(t) be the impulse response of a linear time invariant system. Then the response of the system for any input u(t) is

Sketch the waveform (with properly marked axes) at the output of a matched filter matched for a signal s(t), of duration T, given by $$s(t) = \left\{ {\matrix{ {A\,\,\,\,for} & {0...

The voltage across an impedance in a network is V(s) = Z(s) I(s), where V(s), Z(s) and $${\rm I}$$(s) are the Laplace Transforms of the corresponding time functions V(t), z(t) and...

A rectangular pulse existing between t = 0 and t = T and having an amplitude A is to be received by a matched filter (a) Draw the impulse response of the matched filter labeling al...