T-61.140 Signal Processing Systems
Spring 2003
Basic Properties LTI Systems
Linear TimeTime-Invariant
Systems
• The Commutative Property
• The Distributive Property
• The Associative Property
Tik-61.140 / Chapter 2
 Olli Simula
Convolution
y[ n] =
2
The Commutative Property
∞
x[ n] * h[ n] = h[ n] * x[ n]
∑ x[k ]h[ n − k ] = x[n ] * h[n ]
k =−∞
• Let r=n-k or k=n-r; substituting to convolution sum:
h[n]
x[n]
y[n]
x[n ] * h[ n ] =
∞
∑x[ k ]h[n − k ] =
k = −∞
+∞
∞
y (t ) = ∫− ∞ x(τ ) h(t − τ )dτ = x (t ) * h (t )
Tik-61.140 / Chapter 2
 Olli Simula
∑x[n − r ]h[r ] = h[ n] * x[ n]
r = −∞
3
 Olli Simula
The Commutative Property
x[ n ]
h[n]
y[n ]
h[n]
x[n]
y[n]
Chapter 2 / O. Simula
Tik-61.140 / Chapter 2
4
The Distributive Property
x[n ]* (h1[ n] + h2 [n ]) =
= x[n ] * h1 [n] + x[ n] * h2 [ n]
• The output of an LTI system with input x[n] and
unit impulse response h[n] is identical to the output
of an LTI system with input h[n] and unit impulse
response x[n]
 Olli Simula
Tik-61.140 / Chapter 2
5
• The distributive property has a useful interpretation
in terms of system interconnections
=> PARALLEL INTERCONNECTION
 Olli Simula
Tik-61.140 / Chapter 2
6
1
T-61.140 Signal Processing Systems
Spring 2003
The Distributive Property
h1[ n ] + h2 [ n ]
x[ n ]
The Associative Property
x[n ]* (h1 [n] * h2 [n]) =
y[n ]
= ( x[ n] * h1[n ]) * h2 [ n]
y1 [n ]
h1[ n ]
x[n]
• As a consequence of associative property the
following expression is unambiguous
y[n ]
+
h2 [n]
y[n] = x[n ]* h1[n ] * h2 [n ]
y 2 [ n]
Tik-61.140 / Chapter 2
 Olli Simula
7
 Olli Simula
Tik-61.140 / Chapter 2
8
The Associative Property
The Associative Property
y[n] = x[n ]* (h1 [n ]* h2 [n ])
• The associative property can be interpreted as
x [n]
h1 [ n ] * h2 [ n ]
y[n ]
y[n ] = ( x[n ] * h1 [n ])* h2 [ n ] = y1[n ] * h2 [ n ]
=> SERIES (OR CASCADE)
INTERCONNECTION OF SYSTEMS
y1 [ n ]
x [n ]
h1 [ n ]
h2 [ n ]
y [n ]
Tik-61.140 / Chapter 2
 Olli Simula
9
 Olli Simula
The Associative and Commutative
Property
h2 [ n ] * h1 [ n ]
y [n ]
y[ n] = (x[n] * h2 [n])* h1[ n] = y2 [n] * h1 [n]
x [n ]
 Olli Simula
Chapter 2 / O. Simula
h2 [ n ]
y 2 [n ]
h1[ n ]
Tik-61.140 / Chapter 2
10
The Properties of Cascade
Connection of Systems
y[ n] = x[n ] * (h1[ n ] * h2 [ n] ) = x[ n ] * (h2 [ n] * h1[ n] )
x[ n ]
Tik-61.140 / Chapter 2
• The order of the systems in cascade can
be interchanged
• The intermediate signal values, wi[n],
between the systems are different
y [n ]
11
 Olli Simula
Tik-61.140 / Chapter 2
12
2
T-61.140 Signal Processing Systems
Spring 2003
The Cascade Connection of
Systems
The Cascade Connection of
Systems
y1 [n ]
x [ n]
h1 [ n ]
x[n ]
h1 [ n ] * h 2 [ n ]
y[ n ]
x [ n]
h 2 [ n ] * h1 [ n ]
y[n ]
x [ n]
y2[n ]
h2 [ n]
y[ n ]
h1 [ n ]
Tik-61.140 / Chapter 2
 Olli Simula
• The properties of the cascade system depend on
the sequential order of cascaded blocks
• The behavior of discrete-time systems with finite
wordlength is sensitive to signal values, wi[n],
between the blocks
• What is the optimal sequential order of
cascaded blocks ?
y[ n ]
h 2[ n ]
13
Tik-61.140 / Chapter 2
 Olli Simula
Stability for LTI Systems
14
Stability for LTI Systems
• Consider an input x[n] that is bounded in magnitude
Definition of stability:
|x[n]| < B for all n
A system is stable if and only if
every bounded input produces
a bounded output
• The output is given by the convolution sum
∞
∑
| y [ n ] |=
h [ k ]x [n − k ]
k = −∞
| y [ n ] |≤
Bounded-input bounded-output (BIBO) stability
∞
∑
| h [ k ] || x [ n − k ] |
k = −∞
Tik-61.140 / Chapter 2
 Olli Simula
15
Stability for LTI Systems
• The unit step response is often used to
characterize LTI systems
∞
∑
| h[ k ] |
for
all n
s[n ] = u[ n] * h[ n]
k = −∞
• The output y[n] is bounded if the the impulse response
is absolutely summable
=
h[ k ] < ∞
=
A SUFFICIENT CONDITION FOR STABILITY !
=
∞
∑
k = −∞
 Olli Simula
Chapter 2 / O. Simula
Tik-61.140 / Chapter 2
16
The Unit Step Response
• For bounded input x[n-k]<B
| y [ n ] |≤ B
Tik-61.140 / Chapter 2
 Olli Simula
17
 Olli Simula
∞
∑ u[k]h[n − k ]
k = −∞
∞
∑ h[k]u [n − k ]
k = −∞
n
∑ h[k ]
k = −∞
Tik-61.140 / Chapter 2
18
3
T-61.140 Signal Processing Systems
Spring 2003
Invertible Systems
s[n ]
h[n]
Inverse
Accumulator
Accumulator
s[n ] =
n
∑ h[k ]
Causal LTI Systems
h[n ]
h[n] = s[n] − s[n −1]
k = −∞
• The step response of a DT LTI system is the
running sum its impulse response and the impulse
response of a DT LTI system is the first difference
of its step response
Tik-61.140 / Chapter 2
 Olli Simula
19
• Continuous-time systems:
Linear constant-coefficient differential
equations are used to describe a wide variety
of systems and physical phenomena
• Discrete-time systems:
Linear constant-coefficient difference
equations are used to describe the sequential
behavior of many different proceses
Linear Constant-Coefficient
Difference Equations
∑
k=0
ak y[n − k ] =
M
∑
k=0
20
Linear Constant-Coefficient
Difference Equations
• Nth-order constant-coefficient difference equation
N
Tik-61.140 / Chapter 2
 Olli Simula
• Rearranging and solving for y[n]
b k x[ n − k ]
y[n] =
• The condition for initial rest:
1  M
 ∑ b k x[ n − k ] −
a 0  k=0
N
∑
k =1

a k y[ n − k ]

If x[n]=0 for n < n0 , then y[n]=0 for n < n0
• The output y[n] at time n is expressed in terms of
previous values of the input and output
With initial rest the system is LTI and causal
• Auxiliary conditions: In order to calculate y[n] we
need to know y[n-1],…, y[n-N]
Tik-61.140 / Chapter 2
 Olli Simula
21
 Olli Simula
Linear Constant-Coefficient
Difference Equations
y[n] =
1  M
 ∑ b k x[ n − k ] −
a 0  k=0
N
∑
k =1
Tik-61.140 / Chapter 2
22
Linear Constant-Coefficient
Difference Equations
• In a special case when N=0 the difference equation
reduces to

a k y[ n − k ]

 M b

y [ n ] =  ∑ k x[ n − k ] =
 k=0 a 0

• Recursive equation:
M
∑
h [ k ]x [ n − k ]
k =0
• Nonrecursive equation:
The output is specified recursively in terms of
the input and previous outputs
Previously computed output values are not needed to
compute the present value of the output
Auxiliary conditions are not needed !
 Olli Simula
Chapter 2 / O. Simula
Tik-61.140 / Chapter 2
23
 Olli Simula
Tik-61.140 / Chapter 2
24
4
T-61.140 Signal Processing Systems
Spring 2003
Linear Constant-Coefficient
Difference Equations
Linear Constant-Coefficient
Difference Equations
• Consider the difference equation
• The impulse response corresponding to the
nonrecursive system is
bn
,

h[ n ] =  a0
 0,

1
y [ n − 1] = x [ n ]
2
1
y [n ] = x [n ] +
y [ n − 1]
2
y [n ] −
0 ≤ n ≤ M
• The impulse response is obtained as a response to x[n]=δ[n]
otherwise
n= 0:
The system specified by the nonrecursive equation is
often called a Finite Impulse Response (FIR) system
Tik-61.140 / Chapter 2
 Olli Simula
25
n =1 :
1
1
y [− 1 ] = 1 +
y [− 1 ] = 1
2
2
1
1
1
y [1 ] = δ [1 ] +
y[0 ] = 0 +
y[ 0 ] =
2
2
2
y [0 ] = δ [ 0 ] +
Tik-61.140 / Chapter 2
 Olli Simula
Impulse Response of a First-Order
System
26
Linear Constant-Coefficient
Difference Equations
• The impulse response can be written as
 1  n

h [ n ] =   2  ,
 0,

• Difference equations with N>1 are recursive and result in
an impulse responseof infinite length
n ≥ 0
The systems specified by recursive equations are
called Infinite Impulse Response (IIR) systems
n < 0
n
 1 
h[ n ] =   u [n ]
 2 
 Olli Simula
• In general, recursive difference equations will be used in
describing and analyzing discrete-time systems that are
linear, time-invariant, and causal, and consequently the
assumption of initial rest will usually be made
Tik-61.140 / Chapter 2
27
Block Diagram Representation of
Systems
Tik-61.140 / Chapter 2
 Olli Simula
First–Order Difference Equation
Basic elements:
y [n ] + ay [n − 1 ] = bx [ n ]
x1[ n ]
• Addition:
• Multiplication:
• Unit delay:
 Olli Simula
Chapter 2 / O. Simula
28
x 2 [ n]
xx[n
[n]]
x[ n ]
Tik-61.140 / Chapter 2
a
D
y [ n ] = bx [ n ] − ay [ n − 1]
x1 [n ] + x 2 [ n ]
+
x[n]
a
ax
ax[n
[n]]
b
y[n]
+
−a
D
y[ n − 1]
• Block diagram representation of a causal discrete-time
system (a first-order IIR digital filter)
x[n − 1]
29
 Olli Simula
Tik-61.140 / Chapter 2
30
5
T-61.140 Signal Processing Systems
Spring 2003
Linear Constant-Coefficient
Differential Equations
Linear Constant-Coefficient
Differential Equations
• Consider a first order differential equation
dy( t )
+ 2 y (t ) = x (t )
dt
where y(t) denotes the output of the system and x(t)
is the input
• Differential equations provide implicit
specification of the system, i.e., the relationship
between the input and output
Tik-61.140 / Chapter 2
 Olli Simula
31
• In order to obtain an explicit solution,
the differential equation must be solved
• More information is needed than that provided by
the equation alone, i.e.,
auxiliary conditions must be specified
=>
y (t)= yp (t)+ yh(t)
33
Auxiliary conditions must be specified:
• Different choices of auxiliary conditions leadto
different relationships between the input and
output
• For the most part , the condition of initial rest will
be used for systems described by differential
equations,
e.g., x(t)=0 for t<0, the condition for initial rest
implies the initial condition y(0)=0
Chapter 2 / O. Simula
34
Linear Constant-Coefficient
Differential Equations
• Under the condition of initial rest the system is
linear time-invariant (LTI) and causal
• The condition of initial rest does not specify a zero
initial condition at a fixed point of time, but rather
adjusts this point in time so that the response is
zero until the input becomes nonzero
• For example, if x(t)=0 for t <t0 for a causal LTI
system described a differential equation, then
y(t)=0 for t <t0 and the initial condition y(t0 )=0
would be usedto solve the output for t>t0
Tik-61.140 / Chapter 2
Tik-61.140 / Chapter 2
 Olli Simula
Linear Constant-Coefficient
Differential Equations
 Olli Simula
32
Linear Constant-Coefficient
Differential Equations
• The response to an input x(t) will generally consist
of the sum of
– a particular solution, yp (t), to the differential
equation - a signal of the same form as the input
- i.e., the forced response, and
– a homogeneous solution, yh(t), - a solution to the
differential equation with the input set to zero i.e., the natural response
Tik-61.140 / Chapter 2
Tik-61.140 / Chapter 2
 Olli Simula
Linear Constant-Coefficient
Differential Equations
 Olli Simula
A differential equation describes a constraint
between the input and output of the system,
but to characterize the system completely
auxiliary conditions must be specified
35
• A general Nth-order linear constant-coefficient differential
equation is given by
N
∑a
k= 0
k
d k y ( t)
=
dt k
M
∑b
k= 0
k
d k x ( t)
dt k
where the order refers to the highest derivative of y(t)
• In the casewhenN=0
k
y ( t) =
1
a0
d x( t )
M
∑b
k= 0
k
dt k
y(t) is an explicit function of x(t) and its derivatives
 Olli Simula
Tik-61.140 / Chapter 2
36
6
T-61.140 Signal Processing Systems
Spring 2003
Linear Constant-Coefficient
Differential Equations
Linear Constant-Coefficient
Differential Equations
• For N>1, the output is specified implicitly by the input
• The solution of the equation consists of two parts:
– a particular solution and
– a solution of the homogeneous differential equation
• The solutions to the homogeneous differential equation
d k y (t )
N
∑a
k= 0
k
dt k
d N − 1y ( t 0 )
dy ( t0 )
= ... =
=0
dt
dt N −1
• Under the condition of initial rest, the system is causal
and LTI
y ( t0 ) =
=0
are referred to as natural responses of the system
 Olli Simula
Chapter 2 / O. Simula
Tik-61.140 / Chapter 2
• In order to determine the input-output relationship of the
system completely, auxiliary conditions must be identified
• Different choices of auxiliary conditions result in different
input-output relationships
• The condition of initial rest: If x(t)=0 for t<t0,
it is assumed that y(t)=0 for t<t0 and, therefore,
the response for t>t0 can be calculated from
37
 Olli Simula
Tik-61.140 / Chapter 2
38
7