Digital Signal Processing Reference
In-Depth Information
exp(
j
w
LO
t
)
x
(
t
)
y
(
t
)
y
I
(
t
)
x
I
(
t
)
sin(
w
LO
t
)
cos(
w
LO
t
)
x
Q
(
t
)
y
Q
(
t
)
Fig. 14
Illustration of complex mixing (complex signal multiplication) in terms of complex
signals (
upper
) and parallel real signals (
lower
)
C
t
j
e
f
f
f
C
f
C
Re[
.
]
input
output
Fig. 15
Principal structure of I/Q modulation using complex signal notations
mixer stage. This is the case, e.g., in the classical superheterodyne receiver. Similar
effects have to be taken into consideration also in transmitters, meaning that the
unwanted spectral replica produced by real mixing needs to be attenuated.
Linear I/Q modulation methods are basically just a special case of complex
mixing. Given a complex message signal
x
(
t
)=
x
I
(
t
)+
jx
Q
(
t
)
,itisfirstcomplex-
e
j
ω
c
t
, after which only the real part is actually transmitted. This
modulated as
x
(
t
)
can be written as
)=
ℜ
x
e
j
ω
C
t
=
y
(
t
(
t
)
x
I
(
t
)
cos
(
ω
c
t
)
−
x
Q
(
t
)
sin
(
ω
c
t
)
1
2
x
1
2
x
∗
(
e
j
ω
C
t
e
−
j
ω
C
t
=
(
t
)
+
t
)
(35)
While physical implementations build on the middle expression where
x
I
(
t
)
and
x
Q
(
are modulated onto two orthogonal (cosine and sine) carriers, the complex
models are very handy e.g. from spectral analysis point of view. Notice that both
terms or spectral components (at
t
)
+
f
c
and
−
f
c
) contain all the original information
(i.e.,
x
). This overall process, also termed lowpass-to-bandpass transformation, is
(
t
)
