lINEAR TIME INVARIEANT SYSTEM , IMPULSE RESPONSE

Unit –III
Linear Time-Invariant (LTI) system

Linear Time Invariant System(LTI)
• A system with linear and time invariant called LTI
system.
• Obeys principle of superposition and
homogeneity.
• Input and output characteristics do not change with
time.
y (t, T) = y(t-T)
• If input is delayed by T units then output also delayed
by T units.

LTI System properties
• Commutative property
• Distributive property
• Associative property
• System with and without Memory
• Invertibility
• Causality
• Stability
• Unit step response

Commutative property
x(t)∗h(t) = h(t)∗
x(t)

Distributive property
x(t) ∗[h1(t) + h2(t)] = x(t) ∗ h1(t) + x(t) ∗
h2(t)

Associative property
x(t) ∗[h1(t) * h2(t)] = [x(t) ∗ h1(t)] ∗
h2(t)

System with and without memory
Without memory system:
• Also called as static system.
• It depends on present inputs.
With memory system:
• Also called as dynamic system.
• It depends upon past and future inputs.

Invertibility
• A system is invertible if the input of the
system can be recovered from the output of
the system.
h(t) * h1(t) = δ(t)

∞
Stability:
• In a stable system, a bounded input results in a
bounded output.
∫−∞
|ℎ 𝞃|𝑑𝞃< ∞
Causality:
• A causal system depends only on the present and
past values of the input to the system.
Unit Step Response:
• It can be obtained by using convoluting unit step
input u(t) with impulse response h(t)
s(t)=h(t)*u(t)

Transfer Function of an LTI system

• Transfer function defined by Fourier or Laplace
transform.
• Ratio of Laplace transform of o/p signal to Laplace
transform of i/p signal when initial conditions are
zero.
𝐻𝑠
=
𝑦
𝑠
𝑥
𝑠
Or 𝐻𝑠 =
𝐿[ℎ(𝑡)]
• Impulse response is nothing but inverse Laplace
transform of transfer function.
h 𝑡 = 𝐿−1[𝐻(𝑠)]

• Ratio of Fourier transform of o/p signal to Fourier
transform of i/p signal when initial conditions
are zero.
𝐻ω
=
𝑦
ω
𝑥
ω
Or 𝐻ω =
𝐹[ℎ(𝑡)]
• Impulse response is nothing but inverse Laplace
transform of transfer function.
h 𝑡 = 𝐹−1[𝐻(ω)]

Write the condition for LTI system to be stable and causal.

Given the differential equation representation of the
system d2
y(t)/dt2
+2dy(t)/dt-3y(t)=2x(t). Examine the
frequency response.

Identify the differential equation relating the input and output a
CT system represented by ( ) =1 / [
𝐻 𝑗𝛺 ( )
𝑗𝞨 2
+8( )+1]
𝑗𝞨

Given the input x(t) =u(t) and h(t) = δ(t-1). Find
the response y(t).

List the properties for convolution integral.

The input - output relationship of the system is described as,
d2
y/dt2
+3dy/dt +2y=dx/dt. Find the system function H(s) of the system.

Summarize impulse response of an LTI system.

Given H(s) =1/[𝑠2
+2 +1]. Express the differential equation
𝑠
representation of the system.

Examine the Convolution of following
signals. x(t)= u(t) and h(t)= u(t) u(t-2)

lINEAR TIME INVARIEANT SYSTEM , IMPULSE RESPONSE

A stable LTI system is characterized by the differential equation
d2
y(t)/dt2
+ 4dy(t)/dt + 3y(t) = dx(t)/dt + 2x(t). Derive its frequency response
& impulse response using Fourier transform.

Identify the impulse response h(t) of the system given by the differential
equation d2
y(t)/dt2
+ 3dy(t)/dt + 2y(t) = x(t) with all initial conditions to be
zero.

Derive the output expression of the system described by the differential equation
d2
y(t)/dt2
+ 6dy(t)/dt +8y(t) = dx(t)/dt+ x(t), when the input signal is x(t) =u(t) and
the initial conditions are y(0+
)=1, dy(0+
)/dt=1

A system is described by the differential equation d2
y(t)/dt2
+6dy(t)/dt +
8y(t) = dx(t)/dt+ x(t). Evaluate the transfer function and the output signal y(t)
for x(t) = δ(t).

lINEAR TIME INVARIEANT SYSTEM , IMPULSE RESPONSE

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