Global Positioning System Reference
In-Depth Information
x
(
n
)
x
(
n
−
)
x
(
n
−
)
1
2
Delay
Delay
a
1
a
2
1
y
(
n
)
FIGURE 1.10. Simple linear system with input-output relation
y
(
n
)
=
x
(
n
)
+
a
1
x
(
n
−
1
)
+
a
2
x
(
n
−
)
2
.
1.4
Linear Time-Invariant Systems
Let us first consider a continuous-time, linear, time-invariant (LTI) system char-
acterized by an impulse response
h
(
t
)
, which is defined to be the response
y
(
t
)
from the LTI system to a unit impulse
δ(
t
)
.Thatis,
h
(
t
)
≡
y
(
t
)
when
x
(
t
)
=
δ(
t
).
The response to the input
x
(
t
)
is found by convolving
x
(
t
)
with
h
(
t
)
in the time
domain:
∞
(
)
=
(
)
∗
(
)
=
(λ)
(
−
λ)
λ.
y
t
x
t
h
t
h
x
t
d
(1.19)
−∞
Convolution is denoted by the
∗
.Since
h
(
t
)
=
0for
t
<
0 for causal systems, we
can also write
y
(
t
)
as
∞
y
(
t
)
=
x
(λ)
h
(
t
−
λ)
d
λ.
−∞
e
j
2
π
ft
. From (1.19) we
Define the continuous-time exponential signal
x
(
t
)
=
then have
∞
e
j
2
π
ft
e
j
2
π
f
(
t
−
λ)
d
y
(
t
)
=
h
(
t
)
∗
x
(
t
)
=
h
(
t
)
∗
=
h
(λ)
λ
−∞
e
j
2
π
ft
∞
−∞
e
−
j
2
π
f
λ
d
=
h
(λ)
λ
.
(1.20)
The expression in brackets is the Fourier transform of
h
(
t
)
, which we denote
H
(
.
Now define the discrete-time exponential sequence
x
f
)
e
j
ω
n
.
We can then characterize a discrete-time LTI system by its frequency response to
x
e
j
2
π
fn
(
)
=
=
n
(
)
. By means of the convolution sum formula, a discrete version of (1.20), we
obtain the response
n
∞
e
−
j
ω
k
e
j
ω
n
∞
e
j
ω
k
h
y
(
n
)
=
(
n
−
k
)
=
h
(
k
)
.
(1.21)
k
=−∞
k
=−∞
The expression in parentheses is the discrete-time Fourier transform of the im-
pulse response
h
(
n
)
, which we denote
H
(ω)
.
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