Digital Signal Processing Reference
In-Depth Information
(c) Compute
v
8. How would you
describe the transformation operated by the cascade on the input?
[
n
]
when
x
[
n
]=
cos
(
ω
0
n
)
for
ω
=
3
π/
0
7
8
l
g
r
,
y
i
d
.
,
©
,
L
s
(d) Compute
v
[
n
]
as before, with now
ω
=
0
Exercise 5.10: Analytic signals and modulation.
In this exercise we
explore a modulation-demodulation scheme commonly used in data trans-
mission systems. Consider two real sequences
x
,whichrepre-
sent
two
data streams that we want to transmit. Assume that their spectrum
is of lowpass type, i.e.
X
[
n
]
and
y
[
n
]
e
j
ω
)=
e
j
ω
)=
(
Y
(
0for
|
ω
|
>ω
c
. Consider further the
following derived signal:
c
[
n
]=
x
[
n
]+
jy
[
n
]
and the modulated signal:
e
j
ω
0
n
,
r
[
n
]=
c
[
n
]
ω
<ω
<π
−
ω
c
0
c
2andsketch
R
e
j
ω
)
forwhatevershapesyou
(a) Set
ω
c
=
π/
6,
ω
0
=
π/
(
e
j
ω
)
e
j
ω
)
choose for
X
(
,
Y
(
. Verify fromyour plot that
r
[
n
]
is an analytic
signal.
The signal
r
[
n
]
is called a
complex bandpass signal
.Ofcourseitcannotbe
transmitted as such, since it is complex. The transmitted signal is, instead,
Re
r
]
s
[
n
]=
[
n
This modulated signal is an example of Quadrature Amplitude Modulation
(QAM).
(b) Write out the expression for
s
.Nowyoucan
see the reason behind the term QAM, since we are modulating with
two carriers in quadrature (i.e. out of phase by 90 degrees).
[
n
]
in terms of
x
[
n
]
,
y
[
n
]
Now we want to recover
x
[
n
]
and
y
[
n
]
from
s
[
n
]
. To do so, follow these
steps:
j
h
]
=
(c) Show that
s
is the Hilbert filter.
In other words, we have recovered the analytic signal
r
[
n
]+
[
n
]
∗
s
[
n
r
[
n
]
,where
h
[
n
]
[
n
]
from its real
part only.
(d) Once you have
r
[
n
]
, show how to extract
x
[
n
]
and
y
[
n
]
.
