Linear Infinite State Systems
Last time we
• Looked at systems with an infinite number of states (Reals N)
• Examined property of linearity
• Formulated (A, B, C, D) system definition
• Derived equation to calculate state response
Today we will
• Characterize linear discrete-time systems by their impulse
response
• Formulate zero-input output response using impulse response
• Compute impulse response for example systems
• See examples of FIR and IIR systems
EECS 20 Chapter 5 Part 2
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Zero-State and Zero-Input Responses
Last time, we came up with equations that give us s(n) and y(n):
n-1
s(n) = An s(0) +
y(n) = C An s(0) +
n-1
Σ An-1-m B x(m)
m=0
Σ C An-1-m B x(m) + D x(n)
m=0
We gave names to the terms in these equations:
Zero-input state response:
An s(0)
Zero-input output response:
C An s(0)
n-1
Zero-state state response:
Zero-state output response:
EECS 20 Chapter 5 Part 2
Σ An-1-m B x(m)
m=0
n-1
Σ
C An-1-m B x(m) + D x(n)
m=0
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Impulse Function
The general system response can also be described by its
response to a particular input signal called the impulse:
: Integers  {0, 1}
n  Integers,
(n) =
{
1 if n = 0
0 if n ≠ 0
This function is also known as the Kronecker Delta function.
EECS 20 Chapter 5 Part 2
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Impulse Response
When we apply the impulse function  as the input x to a SISO
system with an initial state of zero, we denote the resulting
output y by h.
This special output, h, is called the impulse response.
Using the formula for the zero-state output response,
n-1
y(n) =
Σ C An-1-m B x(m) + D x(n)
m=0
We see that when the  function is substituted for x,
h(0) = D
h(1) = CB
h(n) = CAn-1B for n > 0
EECS 20 Chapter 5 Part 2
h(2) = CAB
h(0) = D
h(3) = CA2B
h(n) = 0 for n < 0
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General Response Using Impulse Response
Looking again at the zero-state output response,
n-1
y(n) =
Σ C An-1-m B x(m) + D x(n)
m=0
we see that we can write a general output y(n) using the
impulse response h:
n-1
y(n) =
=
Σ h(n-m) x(m) + h(0) x(n)
m=0
n
Σ h(n-m) x(m)
Since h(n) = 0 for n < 0,
∞
y(n) =
Σ h(n-m) x(m)
m=-∞
m=0
EECS 20 Chapter 5 Part 2
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Convolution
This operation being performed on h and x is called convolution,
and it is denoted by 
∞
y(n) =
Σ h(n-m) x(m)

m=-∞
y=hx
The convolution operation is commutative:
hx=xh
∞
∞
m=-∞
k=-∞
Σ h(n-m) x(m) = Σ h(k) x(n-k)
The convolution operation has useful properties in the frequency
domain, covered in later chapters. That is one of the reasons we
use the impulse response characterization.
EECS 20 Chapter 5 Part 2
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Example: FIR
Consider the 7-day moving average:
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Σ x(n-k)
The impulse response has finite
duration (h(n) = 0 for all n > 6).
k=0
If a system has an impulse
Find the impulse response. response with finite duration, we
call it a finite impulse response
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(FIR) system.
h(n) = 1/7 Σ (n-k)
y(n) = 1/7
k=0
The (n-k) will equal 1 for k = n, 0 otherwise.
We can have k = n for n between 0 and 6.
h(n) = 1/7 for 0 ≤ n ≤ 6
h(n) = 0 otherwise
EECS 20 Chapter 5 Part 2
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Example: IIR
Consider the oscillator system, and find the impulse response:
cos ()
A=
sin ()
0
B=
1
-sin ()
cos ()
C=1 0
D=0
We showed that
h(n) = CAn-1B for n > 0
h(0) = D
We can also show that
An =
cos (n)
-sin (n)
sin (n)
cos (n)
h(n) = -sin( (n-1) ) for n>0
h(n) = 0 otherwise
EECS 20 Chapter 5 Part 2
h(n) = 0 for n < 0
The impulse response does
not die out; there is no N for
which h(n) = 0 for n>N.
We say these systems have
infinite impulse response
(IIR).
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Example: Unstable
Consider the bank interest system
s(n+1) = (1+r)s(n) + x(n)
y(n) = s(n)
Find the impulse response h.
Here, the impulse response has an interesting interpretation: it
is the yearly value of a bank account when $1 is deposited in
year 0, and nothing is deposited thereafter. We may show that
h(n) = (1+r)n
Here, the impulse response does not die out; it is an IIR system.
In addition, h(n) increases with n, without bound.
If a system produces an unbounded output for a bounded input,
we say the system is unstable.
EECS 20 Chapter 5 Part 2
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MIMO Systems
Since the impulse function  takes values in Reals, the impulse
response h is technically defined only for single-input systems.
We may define a matrix-valued function h with the same values
h(n) = CAn-1B for n > 0
h(0) = D
h(n) = 0 for n < 0
and the general output y will be given by the same convolution
∞
y(n) =
Σ h(n-m) x(m)
m=-∞

y=hx
So we can still have an h with the same properties for general
MIMO systems, but we won’t call it an impulse response.
EECS 20 Chapter 5 Part 2
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