Questions & Answers of Linear Time-Invariant (LTI) Systems

Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems (ut denotes the unit-step function)?

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form

$\sum_{k=1}^3a_k\;\cos\left(k\omega_0t\right),\;\mathrm{where}\;a_k\neq0\;,\omega_0\neq0.$

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

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       P:1-e-t/T
Y: Unit step    Q:t-T1-e-t/T
Z: Ramp   R:e-t/T

If the signal xt=sintπt*sintπt with * ; denoting the convolution operation, then x(t) is equal to

The result of the convolution x (–t) * δ(–t – to) is

The impulse response of an LTI system can be obtained by

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input x[n], the response y[n] is

The complex envelope of the bandpass signal xt=-2sinπt/5πt/5sin(πt-π4),centered about f=12Hz, is

A continuous, linear time-invariant filter has an impulse response h(t) described by

ht=3for 0t30otherwise

When a constant input of value 5 is applied to this filter, the steady state output is_____.

For a function g(t), it is given that -+gte-jwtdt=ωe-2ω2 for any real value ω.
If yt=-tgtdτ, , then -tytdt  is

Consider a discrete-time signal

xn=nfor 0n100otherwise.

If y[n] is the convolution of x[n] with itself, the value of y[4] is _________.

The input -3e2tut, where ut is the unit step function, is applied to a system with transfer function s-2s+3. If the initial value of the output is −2, then the value of the output at steady state is _______.

The sequence xn=0.5nun, where un is the unit step sequence, is convolved with itself to obtain yn. Then n=-+yn is _______.

The DFT of a vector abcd is the vector αβγδ . Consider the product

pqrs=abcdabcddabccdabbcda

The DFT of the vector pqrs is a scaled version of

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

A system is defined by its impulse response h(n) = 2n u(n − 2). The system is

The impulse response ht of a linear time-invariant continuous time system is described by ht=expαtut+expβtu-t, where ut denotes the unit step function, and α and β are real constants. This system is stable if

A discrete time linear shift-invariant system has an impulse response h[n] with h[0]=1, h[1]=-1. h[2]=-2, and zero otherwise. The system is given an input sequence x[n] with x[0]=x[2]=1, and zero otherwise. The number of nonzero samples in the outp ut sequence y[n], and the value of y[2] are, respectively