Linear Time-Invariant (LTI) Systems
Dr. Ray Kwok
SJSU
Fall 2013
The unit impulse function
The unit impulse function AKA Dirac delta function is given as:
 Very large, t = 0
otherwise
0,
δ (t )
δ (t ) = 
1
K
− 2 −1
The value at t = 0 is very large, and δ ( t ) = 0 for
The duration is very short.
The area is one.
K
0
1
2
3
t
t =/ 0
The unit impulse function is a mathematical model to represent signals that are
highly localized in time.
Dr. Ray Kwok
LTI Systems
2
Systems, Networks, and Circuits
Network theory is mainly concerned with network topology (interconnections of
component)
System theory is mainly concerned with input - output relationship.
Electrical system is divided to:
Linear system described by set of linear equations.
nonlinear system described by set of nonlinear equations.
And into:
Time-invariant
time-varying system
And into:
Passive system
Active system
Also
Lumped system
Distributed system
And……
Dr. Ray Kwok
LTI Systems
3
Mathematical model
To aid analyzing and designing systems, mathematical models are formulated.
A mathematical model describes the behavior of physical system or device in
terms of a set of mathematical equations, with schematic diagram of the device
connection and the symbols of its component and numerical values.
Several models for continuous systems and techniques for system design and
analysis by analytical methods are proposed.
A system or transform maps one signal that is called input signal x(t) into another
signal which is called the output signal or response y(t) :
y ( t ) = T { x ( t )}
System block diagram: (the operator)
Dr. Ray Kwok
LTI Systems
4
Properties and characteristics of continuous systems
Properties and characteristics of continuous systems:
Linearity
Dr. Ray Kwok
Homogeneity
Additivity
Time-invariance
Causality
stability
x ( t ) = input
y ( t ) = output
( excitation )
( response )
LTI Systems
5
Linear system
A system is linear if and only if it satisfies the principle of homogeneity and the
principle of additvity.
The principle of homogeneity.
If an input x1(t) applied to a linear system produces the output y1(t), when a
scaled input signal by constant C is applied to the linear system, x2 ( t ) = Cx1 ( t )
the output y2 ( t ) = Cy1 ( t ) results.
Dr. Ray Kwok
x1 ( t )
LINEAR
SYSTEM
y1 ( t )
Cx1 ( t )
LINEAR
SYSTEM
Cy1 ( t )
LTI Systems
6
Linear system
The principle of additivity.
If an input x1(t) applied to a linear system produces the output y1(t), and x2 (t)
produces y2(t) , when a new input x1(t) + x2(t) is applied to the linear system, the
output y1(t) + y2(t) results.
x1 ( t )
y1 ( t )
x2 ( t )
y2 ( t )
y1 ( t ) + y2 ( t )
x1 ( t ) + x2 ( t )
Dr. Ray Kwok
LTI Systems
7
Linear system
The principle of superposition.
If an input x1(t) applied to a linear system produces the output y1(t), and x2 (t)
produces y2(t) , when a new input Ax1(t) +B x2(t) is applied to the linear system,
the output Ay1(t) + By2(t) results.
x1 ( t )
y1 ( t )
x2 ( t )
y2 ( t )
Ay1 ( t ) + By2 ( t )
Ax1 ( t ) + Bx2 ( t )
Dr. Ray Kwok
LTI Systems
8
Time invariance
If an input x1(t) applied to a system produces the output y1(t), when a timeshifted version of input x2(t) = x1(t - to) is applied to the linear system, the
output y2(t) = y1(t - to) results for arbitrary x1(t) and to and for all t , then the
system is said to be a time invariant system.
Loosely speaking, the system parameter do not change with time.
The same input applied at different times will produce outputs that are identical
in shape and size but shifted in time
y1 ( t )
x1 ( t )
x2 ( t ) = x1 ( t − t0 )
Dr. Ray Kwok
y2 ( t ) = y1 ( t − t0 )
LTI Systems
9
Causality
A causal system is a non-anticipative system in that the output does not precede,
or anticipate, the input.
i.e. The system’s output depends only on the past and current input, but not on
the future inputs.
All nature, physics operate under this principal – called causality (cause and
effect).
The impulse response of a causal system must be 0 for all t < 0. [ i.e. the
impulse input is δ(t) ].
Dr. Ray Kwok
LTI Systems
10
Memoryless system and system with memory
A system has memory if its output signal depends on past or/and future values of
its input signal. System with a memory is also called dynamic system.
A system is memoryless if the output signal depends only on the present value of
the input signal. A memoryless system is also called instantaneous system.
The resistor is an example of a memoryless system:
v ( t ) = Ri ( t )
While the capacitor has memory:
t
v ( t ) = v ( t0 ) + ∫ i (τ ) dτ
t0
Dr. Ray Kwok
LTI Systems
11
Stability
System stability can be defined from several points of view.
The bounded input bounded output (BIBO) criterion
A system is bounded-input bounded-output (BIBO) stable if for any bounded input
defined by x
x ≤ k1
The corresponding output y is also bounded defined by
y ≤ k2
where k2 and k2 are finite real numbers.
Dr. Ray Kwok
LTI Systems
12
System invertibility and inverse systems
A system is said to be invertible if distinct inputs lead to distinct output. An inverse
system is a system that when is cascaded with duplicate of itself yields an output
equal to the input of cascaded system
x (t )
Dr. Ray Kwok
y (t )
LTI Systems
x (t )
13
Thevenin’s Theorem
A linear two-terminal circuit can be replaced by an equivalent circuit consisting of
a voltage source in series with a resistor where Vth is the open-circuit voltage at
the terminals and is the input or equivalent resistance at the terminals when the
independent sources are turned off.
Norton’s Theorem
A linear two-terminal circuit can be replaced by an equivalent circuit consisting of
a current source in parallel with a resistor, where Is is the short-circuit current
through the terminals and R is the input or equivalent resistance at the terminals
when the independent sources are turned off.
VTh
Dr. Ray Kwok
LTI Systems
14
Operation for linear systems
Operation for linear systems
Continuous systems
Laplace transform, (s)
Convolution integral (time)
Correlation integral, (time)
Fourier series and transform
DFT approximation
Convolution (frequency, f )
Correlation (frequency, f )
Discrete systems
Dr. Ray Kwok
z- transform,
Convolution sum (time)
Correlation sum, (time)
Discrete time Fourier transform (DTFT) implemented by FFT
Discrete Fourier transform (DFT) implemented by Fast Fourier transform (FFT)
Convolution (frequency, θ )
Correlation (frequency, θ )
LTI Systems
15
Models for continuous and discrete systems
Models for contiguous and discrete systems
Continuous systems
Differential equation, DE’s
Transfer function, H(s)
Frequency response, H(jω)
State differential equations
Unit impulse response, h(t)
Signal flowgraph or block diagram
Discrete systems
Difference equation, DE’s
Transfer function, H(z)
Frequency response, H ( e jθ )
State difference equations
Unit sample response, h[n]
Signal flowgraph or block diagram
Dr. Ray Kwok
LTI Systems
16
Linear differential equation
The output y(t) and input x(t) of a LTIC system are related by a linear
differential equation with constant-coefficient of the form
d n y (t )
d n−1 y ( t )
dx
d mx
an
+ an−1
+ L + a0 y = b0 x ( t ) + b1 ( t ) + L + bm m
dx n
dx n−1
dt
dt
The right hand side terms are often lumped together and called forcing function
dx
d mx
as:
f ( t ) = b0 x ( t ) + b1
dt
( t ) + L + bm
dt m
n −1
with initial condition y ( 0 ) ,L, y .( 0 )
The complete response is of the form y t = yh 0 t + y f 0 t
where yh 0 t the homogenous response is the solution to the differential
equation with f ( t ) = 0 and contain n arbitrary constant.
and y f 0 ( t ) the forced response is that one particular solution to the differential
equation that contains no part of the yh 0 ( t ) .
n −1
The n arbitrary constant may be found by applying the values of y ( 0 ) ,L, y ( 0 )
()
()
()
()
to it.
Dr. Ray Kwok
LTI Systems
17
Linear differential equation
The homogenous response yh 0 ( t ) also called natural response, free response,
or complementary response.
And the transient response is y f 0 ( t )
The terms forced, particular integral, final and steady state are used
interchangeably.
A more concise representation using the finite summation
∂ k y ( t ) k =m ∂ k x ( t )
ak
= ∑ bk
∑
k
∂t
∂t k
k =0
k =0
k =n
Note: the notation is restricted to the practical situation where the number of the
derivatives of the output is greater than or equal to the number of input derivative,
that is n ≥ m .
The order of a differential equation is the order the highest derivative of the output
function that appears in the equation.
Dr. Ray Kwok
LTI Systems
18
The characteristic equation
The characteristic equation (CE) of the system is found by substituting a trial
solution y ( t ) = Ce st into the homogenous differential equation. k =n ∂ k y ( t )
∑a
k
k =0
(
) + a s ( Ce ) + L + a s ( Ce ) = 0
1
n
Since Ce st can not be zero (this corresponds to the trivial solution y
it can be factored out. The remaining terms must satisfy the algebraic
equation a s 0 + a s1 + L + a s n = 0
0
=0
Obtained by setting to zero terms involving the input and its derivative, with the
result:
0
st
1
st
n
st
a0 s Ce
∂t k
1
(t ) = 0
n
This result is known as the characteristic equation and can also be written as
k =n
∑a s
k
k
=0
k =0
Dr. Ray Kwok
LTI Systems
19
The characteristic equation
The n roots of the equation
k =n
∑a s
k
k
=0
k =0
s1 = r1 , s2 = r2 L sn = rn
are called characteristic roots. The characteristic equation can be written in factored
k =n
form:
k
∑a s
k
= an ( s − r1 )( s − r2 )L( s − rn ) = 0
k =0
where r1 , r2 ,L , rn , may be real or complex. If the DE has real coefficient, complex
roots must appear in conjugate pair. The solution of the homogenous DE
∂k y (t )
ak
=0
∑
k
∂t
k =0
k =n
for the given set of initial condition
y (t )
Dr. Ray Kwok
t =0
, ∂y ( t ) / ∂t
t =0
, ∂ n−1 y ( t ) / ∂t n−1
LTI Systems
t =0
20
The characteristic equation
The solution of the homogenous DE
k =n
∑a s
k
k
=0
k =0
For the given set of initial condition
y (t )
t =0
, ∂y ( t ) / ∂t
t =0
, ∂ n−1 y ( t ) / ∂t n−1
t =0
is called the initial condition (IC) solution and for simple (non-repeating) roots is of
the form
rnt
r1t
r2t
yIC ( t ) = C1e + C2e + L + Cne
or in compact form
n
yIC ( t ) = ∑ Ck e rk t
k =1
where the Ck for k = 1, 2,L n are coefficient that must be determined in order
to satisfy the given set of initial condition and rk for k = 1, 2,L n , are the
characteristic roots.
Dr. Ray Kwok
LTI Systems
21
The CE with multiple roots
If the CE contain multiple roots indicated by the factor
form
sk t
q −1 sk t
q −2 sk t
C1t e + C2t
( s − sk )
q
terms of the
e + L + Cq e
will appear in the initial condition
Dr. Ray Kwok
LTI Systems
22
Linear differential equation
Example:
Given the first order differential equation
Suggested solution
Substituting to original differential equation
dy
+ py = q ⇒
dx
p and q = C
y = yh 0 + y f 0
d ( yh 0 + y f 0 )
dx
+ p ( yh 0 + y f 0 ) = q
To find the homogenous solution, set forced function to zero:
The yh 0 ( t ) is
dyh 0
+ pyh 0 = 0
dx
yh 0 = Ae − px
Dr. Ray Kwok
LTI Systems
23
Distortionless transmission
What are the conditions for distortionless transmission?
By distortionless transmission, we mean that output signal of system is an exact
replica of the input signal except for two minor modifications.
1 – A possible scaling of amplitude
2 – A constant time delay
A signal x(t) is transmitted through a system without distortion if the output signal
is defined by y(t)
x (t )
y (t )
y ( t ) = k x ( t − t0 )
Dr. Ray Kwok
LTI Systems
24
Distortionless transmission
Here the constant k accounts for a change in amplitude and
accounts for a delay in transmission:
t0
constant
Y ( jω ) = k X ( jω ) e − j ω t0
The frequency response of a distortionless system is:
H ( jω ) =
Y ( jω )
= k e − j ω t0
X ( jω )
Correspondingly, the impulse response of the system is
h(t ) = kδ (t − to )
Dr. Ray Kwok
LTI Systems
25
Condition for CT LTI distortionless transmission
Conditions for distortionless transmission of CT LTI system
with transfer function H ( jω ) is
1) The magnitude response H(jω ) must be constant for all frequencies of interest.
H(jω ) = k
2) For the same frequencies of interest ∠ H ( jω ) is linear in frequency with
slope − t0 and intercept of zero.
∠ H ( jω ) =- ω t0
where t0 accounts for a delay in transmission through the system.
Dr. Ray Kwok
LTI Systems
26
Condition for CT LTI distortionless transmission
For DT LTI system with transfer function
1 − The magnitude response H(e jω ) is constant
for all frequencies of interest
H(e jω ) = k
2 − For the same frequencies of interest, the phase
response ∠ H (e jω ) is linear in frequency
∠ H (e jω ) = − n0 ω
where n0 accounts for an integer delay in transmission
through the system.
Dr. Ray Kwok
LTI Systems
27
Frequency Response
Frequency response for distortionless transmission through a linear time-invariant
system
H ( jω )
K
0
ω
arg H ( jω )
Slope = -t0
ω
a) Magnitude response
b) Phase response
Dr. Ray Kwok
LTI Systems
28