•
In general, every time sequence
x
[
n
] can be expressed as
=
+
 +
 +…= = ∞

xn
x0δn x1δn 1 x2δn 2
k
0
xkδn k
•
If the system is linear and time invariant (LTI), then we have
δ
[
n
] →
h
[
n
]
(impulse response)
δ
[
n
–
k
] →
h
[
n
–
k
]
(shifting)
x
[
k
]
δ
[
n
–
k
] →
x
[
k
]
h
[
n
–
k
]
(homogeneity)
= ∞
 → = ∞

k
0
xkδn k
k
0
xkhn k
(additivity)
•
Discretetime convolution equation
:
=
=
∞
[

]
yn
k
0
xkh n k
•
A system is said to be causal
if its current output depends only on its current and past inputs. If the output
depends on future inputs, the system is said to be noncausal
. A system is causal if and only if
h
[
n
] = 0
for
n
< 0
.
•
A digital system is called a finiteimpulseresponse (FIR)
system or filter if its impulse response has a
finite number of nonzero entries. A causal FIR filter
is said to have length
N
if
h
[
N
– 1] ≠ 0
and
h
[
n
] = 0
for all
n
>
N
– 1
. If
N
= 1
, or
h
[0] =
a
≠ 0
and
h
[
n
] = 0
for
n
≠ 0
, then this reduces to
y
[
n
] =
ax
[
n
]
where
a
is a constant. (Multiplier).
•
A digital system is called an infiniteimpulseresponse (IIR)
filter if its impulse response has infinitely
many nonzero values.
•
Discretetime difference equation
(Delayed form):
+

+…+
+

=
+

+…+
+

a1yn
a2yn 1
aN
1yn N
b1xn
b2xn 1
bM
1xn M
where
a
i
and
b
i
are real constants, and
M
and
N
are positive integers. With
a
1
≠ 0
is the most general form of difference
equations to describe LTIL (lumped LTI) and causal systems. If
a
N
+1
≠ 0
and
b
M
+1
≠ 0
, the difference equation is said to
have order
max(
M, N
)
.
•
Nyquist Frequency Range:
(
π
/
T
,
π
/
T
]
•
Nonrecursive difference equation
:
=
+
 +…+
+
[ 
]
yn
b1xn b2xn 1
bM 1x n M
•
Recursive difference equation
:
=
 …
+

+
+
 +…+
+
[ 
]
yn
a2yn 1
aN 1yn N b1xn b2xn 1
bM 1x n M
•
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 Fall '10
 ChiTsongChen
 LTI system theory, Impulse response, Finite impulse response, Infinite impulse response

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