Questions & Answers of Liner Time Invariant and Causal Systems

Consider a continuous-time system with input xt and output yt given by

yt=xtcost

This system is

Consider the following state-space representation of a linear time-invariant system.

$\overset.x\left(t\right)=\begin{bmatrix}1&0\\0&2\end{bmatrix}x\left(t\right),y\left(t\right)=c^Tx\left(t\right),\;c=\begin{bmatrix}1\\1\end{bmatrix}\;\mathrm{and}\;x\left(0\right)=\begin{bmatrix}1\\1\end{bmatrix}$

The value of yt for t=loge2 is __________.

The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where xt is the input signal. A signal zt is called eigen-signal of the system T , when T{z(t)}= y z(t), where γ is a complex number, in general, and is called an eigenvalue of T. Suppose the impulse response of the system T is real and even. Which of the following statements is TRUE

A moving average function is given by $y\left(t\right)=\;\frac1T\int\limits_{t-T}^tu\left(\zeta\right)\operatorname d\zeta$. If the input u is a sinusoidal signal of frequency 12THz, then in steady state, the output y will lag u (in degree) by ________.

The impulse response g(t) of a system, G, is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure(b)?

For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE?

The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables x and y. The integrationtime step is h.
xk+1-xkh=yk
yk+1-ykh=-xk
For this discrete-time system, which one of the following statements is TRUE?

xt is nonzero only for Tx<t<T'x , and similarly, yt is nonzero only for Ty<t<T'y. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE?

Consider an LTI system with transfer function

HS=1ss+4

If the input to the system is cos3t and the steady state output is Asin3t+α, then the value of A is

Consider an LTI system with impulse response ht=e-5tut. If the output of the system is yt=e-3tut-e-5tut then the input, xt, is given by

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

Let y[n] denote the convolution of h[n] and g[n], where h[n]=(1/2)n u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = 1/2, then g[1] equals

The input x(t) and output y(t) of a system are related as yt=-t xτcos3τdτ.The system is

Given two continuous time signals x(i)=e-t and y(i)=e-2t which exist for t>0, the convolution z(t)=x(t)*y(t) is

The system represented by the input-output relationship $\style{font-size:14px}{y\left(t\right)=\int\limits_{-\infty}^{5t}x\left(\tau\right)d\tau,t>0}$ is

Given the finite length input x[n] and the corresponding finite length output y[n] of an LTI system as shown below, the impulse response h[n] of the system is

A Linear Time Invariant system with an impulse response h(t) produces output y(t) when input x(t) is applied. When the input x(t-τ) is applied to a system with impulse response h(t-τ), the output will be

A cascade of 3 Linear Time Invariant systems is causal and unstable. From this, we conclude that

A signal e-αtsinωt is the input to a real Linear Time Invariant system. Given K and $\phi$ are constants, the output of the system will be of the form $\style{font-size:18px}{Ke^{-\beta t}\sin\left(\nu t+\phi\right)}$ where

The impulse response of a causal linear time-invariant system is given as h(t) . Now consider the following two statements :
Statement (I): Principle of superposition holds
Statement (II):h(t)=0 for t<0
Which one of the following statements is correct ?

A system with x(t) and output y(t) is defined by the input-output relation :

$y\left(t\right)=\int\limits_{-\infty}^{-2t}x\left(t\right)d\tau$

The system will be

A signal xt=sincαt where α  is a real constant sincx=sinπxπx is the input to a Linear Time Invariant system whose impulse response h(t)=sin(βt),where β is a real constant. If minα,βdenotes the minimum of α and β and similarly,maxα,β denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the output of the system ?

Let a signal $a_1\sin\left(\omega_1t+\phi_1\right)$ be applied to a stable linear time invariant system. Let the corresponding steady state output be represented as $a_2F\left(\omega_2t+\phi_2\right)$. Then which of the following statement is true?

If u(t), r(t) denote the unit step and unit ramp functions respectively and u(t)*r(t) their convolution, then the function u(t+1)*r(t-2) is given by