The Laplace transform of the causal periodic square wave of period T shown in the figure below is

A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to t = 0). Match the excitation signals X, Y, Z with the corresponding time responses for t ≥ 0:

X: Impulse                 

Y: Unit step         

Z: Ramp         

The response of the system  to the unit input u(t) is y(t). The value of at is _____________.

Let the signal  outside the interval  where  and  are finite. Furthermore, . The region of convergence (ROC) of the signal’s bilateral Laplace transform F(s) is

By performing cascading and/or summing/differencing operations using transfer function blocks  and, one CANNOT realize a transfer function of the form

Input x(t) and output y(t) of an LTI system are related by the differential equation . If the system is neither causal nor stable, the impulse response h(t) of the system is

Let  with , where u(t) is unit step function. If the bilateral Laplace transform of x(t) is  

Then the value of β is __________.

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. . When y(0)=1 and u(t) is a unit step function, y(t) is

The input , where u(t) is the unit step function, is applied to a system with transfer function  .If the initial value of output is -2, then the value of the steady state is ___________.

Let h(t) denote the impulse response of a causal system with transfer function .

Consider the following three statements.

S1: The system is stable.

S2:   is independent of t for t>0.

S3: A non-causal system with the same transfer function is stable.

For the above system,

A stable linear time invariant (LTI) system has a transfer function  . To make this system causal it needs to be cascaded with another LTI system having a transfer function . A correct choice for  among the following options is

A causal LTI system has zero initial conditions and impulse response h(t). Its input x(t) and output y(t) are related through the linear constant-coefficient differential equation

 .         

Let another signal g(t) be defined as

 .

If G(s) is the Laplace transform of g(t), then the number of poles of G(s) is _______.

Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is

A system is described by the differential equation. Let x(t) be a rectangular pulse given by . Assuming that  and  at , the Laplace transform of y(t) is

A system described by a linear, constant coefficient, ordinary, first order differential equation has an exact solution given by y(t) for , when the forcing function is x(t) and the initial condition is y(0). If one wishes to modify the system so that the solution becomes  for  , we need to

A system with transfer function  is excited by sin (ωt).

The steady-state output of the system is zero at

The differential equation  describes a system with an input x(t) and an output y(t). The system, which is initially relaxed, is excited by a unit step input.

The output y(t) can be represented by the waveform

If the unit step response of a network is .

Then its unit impulse response is

An input  is applied to an LTI system with impulse response.

The output is

If  then the initial and final values of f (t) are respectively

A system with the transfer function  has an output  for the input signal . Then, the system parameter ‘p’ is

A continuous time LTI system is described by .

Assuming Zero initial conditions, the response y(t) of the above system for the input  is given by

The impulse response h(t) of a linear time-invariant continuous time system is described by , where u(t) denotes the unit step function, and  and  are real constants. This system is stable if

A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at s = -2 and s = -4, and one simple zero at s = -1. A unit step u(t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

If the Laplace transform of a signal y(t) is  then its final value is

The frequency response of a linear, time-invariant-system is given by. The step response of the system is

The unit-step response of a system starting from rest is given by  for

The transfer function of the system is:

In what range should Re(s) remain so that the Laplace transform of the function  exists?

A system described by the following differential equation  is initially at rest.

For input x(t) =2u(t), the output y(t) is

A causal system having the transfer function  is excited with 10u(t). The time at which the output reaches 99% of its steady state value is

The Laplace transform of i(t) is given by As, the value of i(t) tends to  

The Laplace transform of a continuous-time signal x(t) is . If the Fourier transform of this signal exists, then x(t) is

The transfer function of a system is given by. The impulse response of the system is: (* denotes convolution, and U(t) is unit step function)

Given that , , .  is

A linear time invariant system has an impulse response, . If the initial conditions are zero and the input is, the output for  is

For the linear, time invariant system whose block diagram is shown in Figure (a), with input x(t) and output y(t),

(a) Find the transfer function.

(b) For the step response of the system [i.e. find y(t) when x(t) is a unit step function and the initial conditions are zero]

(c) Find y(t), if x(t) is as shown in Figure (b), and the initial conditions are zero.

Figure (a)

Figure (b)

If , then  is equal to

If , then the value of

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

The Laplace Transform of $\scriptstyle e^{\alpha t} \cos(\alpha t)$ is equal to

The inverse Laplace transform of the function  is

If  then  and  are given by

[Note: ‘L’ stand for ‘Laplace’ Transform of’]

Match the following

(A) Fourier transform of a Gaussian function

(B) Convolution of a rectangular pulse with itself

(C) Current through an inductor for a step input voltage  

(1) Gaussian function

(2) Rectangular pulse

(3) Triangular pulse

(4) Ramp function

(5) Zero

The Laplace transform of a unit ramp function starting at , is

If G(s) is a stable transfer function, then  is always a stable transfer function
(True=1,False=0)

If  then  is given by

The voltage across an impedance in a network is , where ,  are the Laplace transforms of the corresponding time function v(t), z(t) and i(t). The voltage v(t) is: