ECE 308 -7
LTI Described by Difference Equations
Z. Aliyazicioglu
Electrical and Computer Engineering Department
Cal Poly Pomona
Discrete Time Systems Described by Difference Equations
Recursive and Nonrecursive Discrete-Time Systems
If a system output y(n) at time n depends on any number of past
output value y(n-1), y(n-2),… , it is called a recursive system.
Let’s have a DTS that gives the cumulative average
1 n
y ( n) = ∑ x(k )
n + 1 k =0
We can find y(n) more efficient by utilizing the output y(n-1).
1 n −1
y (n ) = ∑ x(k ) + x(n)
n + 1 k =0
1
= ( ny (n − 1) + x(n ) )
n +1
n 1
= y ( n − 1) + x( n )
n +1 n +1
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Recursive and Nonrecursive Discrete-Time Systems
x(n)
+ X
y(n)
1
n+1
X z-1
n
This system requires two multiplication, one addition, and one
memory location. This is a recursive system which means the
output at time n depends on any number of a past output values.
So, a recursive system has feed back output of the system into
the input. This feed back loop contains a delay element.
2 1
y (0) = x(0) y (2) = y (1) + x(2)
3 3
1 1
y (1) = y (0) + x(1) n0 1
2 2 y ( n0 ) = y ( n0 − 1) + x( n0 )
n0 + 1 n0 + 1
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Recursive and Nonrecursive Discrete-Time Systems
If y(n) depends only on the present and past input, it is called
nonrecursive.
For the causal FIR systems
M
y ( n) = ∑ h( k ) x ( n − k )
k =0
= h(0) x(n) + h(1) x(n − 1) + ... + h( M ) x(n − M )
= F [ x(n), x(n − 1),..., x( n − M ) ]
y(n)
x(n) F[y(n-1),..y(n-N),
x(n) F[x(n),x(n-1), y(n) x(n),…,x(n-M)]
…,x(n-M)]
z-1
Nonrecursive system Recursive system
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Constant-Coefficient Difference Equations
LTI Systems can be described by Constant-Coefficient
Difference Equations to represent the input-output relations
Let’s have a recursive system that is first-order difference
X(n)
y(n)
+
y ( n) = ay ( n − 1) + x ( n)
a
z-1
where is a constant and system is time invariant. We assume that
we have initial condition y(-1).
For , y(n) can be obtained
y (0) = ay (−1) + x(0) y (1) = ay (0) + x (1) = a 2 y (−1) + ax (0) + x(1)
y (n) = a n+1 y (−1) + a n x(0) + a n−1 x(1) + ... + x( n) y (n) = a n+1 y (−1) + a k x(n − k )
n
∑ k =0
n≥0
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Constant-Coefficient Difference Equations
The response y(n) of the system depends on
• initial condition y(-1) of the system and
• the system response to the input signal.
If the system is initially relaxed at time n=0, its memory should be
zero. So, y(-1)=0.
Then, system is at zero state and the corresponding output is
called zero-state response or forced response.
n
y zs (n) = ∑ a k x(n − k ) n≥0
k =0
If system is initially nonrelaxed (y(-1)≠0) and the input x(n)=0 for
all n.
The corresponding output is called zero-input response or natural
response.
y zi (n) = a n+1 y (−1) n≥0
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Constant-Coefficient Difference Equations
The total system response is
y (n) = yzs (n) + y zi (n)
General form of linear constant-coefficient difference equation is
N M
y (n) = − ∑ ak y (n − k ) + ∑ bk x (n − k )
k =1 k =0
N M
∑a
k =0
k y ( n − k ) = ∑ bk x(n − k )
k =0
In order to find y(n), we need to know initial conditions
y(n-1), y(-2),…,y(n-N) and the input x(n) for all n>=0.
Recursive system may be relaxed or non-relaxed, depending
on the initial condition.
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Constant-Coefficient Difference Equations
A system is linear if satisfy the following requirements:
1. The total response is equal to the sum of the zero-input and
zero-state responses
y (n) = y zs (n) + yzi (n)
2. The principles of superposition applies to the zero-state
response (zero-state linear)
3. The principles of superposition applies to the zero-input
response (zero-input linear)
If a system does not satisfy all three separate requirement,
system is called nonlinear.
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