Digital Signal Processing Reference
In-Depth Information
1
1
l
g
r
,
y
i
d
.
,
©
,
L
s
0
0
-
π
0
π
-
π
0
π
Figure 12.18
Magnitude spectrum of an analytic signal
x
[
n
]
.
X
(
e
j
ω
)
(left) and
X
∗
(
(right).
e
−j
ω
)
Since
x
[
n
]
is analytic, by definition
X
(
e
j
ω
)=
0for
−
π
≤
ω<
0,
X
∗
(
e
−j
ω
)=
0
e
j
ω
)
does not overlap with
X
∗
(
e
−j
ω
)
for 0
<ω
≤
π
and
X
(
(Fig. 12.18). We can
therefore use (12.21) to write:
2
X
r
(
e
j
ω
)
for 0
≤
ω
≤
π
e
j
ω
)=
X
(
(12.23)
0
for
−
π<ω<
0
Now,
x
r
is a real sequence and therefore its Fourier transformis conjugate-
symmetric, i.e.
X
r
[
n
]
X
∗
r
(
e
j
ω
)=
(
e
−j
ω
)
;asaconsequence
0
for 0
≤
ω
≤
π
X
∗
(
e
−j
ω
)=
(12.24)
2
X
r
(
e
j
ω
)
for
−
π<ω<
0
By using (12.23) and (12.24) in (12.22) we finally obtain:
−jX
r
(
e
j
ω
)
for 0
≤
ω
≤
π
e
j
ω
)=
X
i
(
(12.25)
e
j
ω
)
+
jX
r
(
for
−
π<ω<
0
e
j
ω
)
which is the product of
X
r
with the frequency response of a Hilbert
filter (Sect. 5.6). In the time domain this means that the imaginary part of an
analytic signal can be retrieved from the real part only via the convolution:
(
x
i
[
n
]=
h
[
n
]
∗
x
r
[
n
]
At the demodulator,
s
ˆ
[
n
]=
s
[
n
]
is nothing but the real part of
c
[
n
]
and
therefore the analytic bandpass signal is simply
j
h
]
c
ˆ
[
n
]=ˆ
s
[
n
]+
[
n
]
∗
ˆ
s
[
n
In practice, the Hilbert filter is approximated with a causal, 2
L
1-tap type
III FIR, so that the structure used in demodulation is that of Figure 12.19.
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