Digital Signal Processing Reference
In-Depth Information
Fig. 13
An example of pure
frequency translation using
complex mixing
input spectrum
output spectrum
f
f
0
0
f
C
4.3
Frequency Translations and Filtering
4.3.1
Frequency Translations for Signals
One key operation in radio signal processing is the shifting of a signal spectrum
from one center-frequency to another. Conversions between baseband and bandpass
representations and I/Q modulation and demodulation (synchronous detection) are
special cases of this. The basis of all the frequency translations lies in multiplying a
signal with a complex exponential, generally referred to as complex or I/Q mixing.
This will indeed cause a pure frequency shift, i.e.,
e
j
ω
LO
t
y
(
t
)=
x
(
t
)
⇔
Y
(
f
)=
X
(
f
−
f
LO
)
(32)
where
denotes transforming between time and frequency domain. This forms
the basis, e.g., for all the linear modulations, and more generally for all frequency
translations. This is illustrated in frequency domain in Fig.
13
in the case where the
input signal is at baseband.
In general, since
⇔
e
j
ω
LO
t
x
(
t
)
=
x
I
(
t
)
cos
(
ω
LO
t
)
−
x
Q
(
t
)
sin
(
ω
LO
t
)
+
j
(
x
Q
(
t
)
cos
(
ω
LO
t
)+
x
I
(
t
)
sin
(
ω
LO
t
))
,
(33)
four real mixers and two adders are needed to implement a full complex mixer (full
case of real-valued input signal, only two mixers are needed.
Real mixing is obviously a special case of the previous complex one and results
in two frequency translations:
y
(
t
)=
x
(
t
)
cos
(
ω
LO
t
)
2
e
j
ω
LO
e
−
j
ω
LO
t
⇔
1
1
2
X
1
2
X
=
x
(
t
)
+
Y
(
f
)=
(
f
−
f
LO
)+
(
f
+
f
LO
)
(34)
Here, the original spectrum appears twice in the mixer output, the two replicas being
separatedby2
f
LO
in frequency. In receivers, this results in the so called image signal
or mirror-frequency problem since the signals from both
f
c
+
−
f
LO
will
appear at
f
c
after a real mixing stage. Thus if real mixing is used in the receiver,
the image signal or mirror-frequency band needs to be attenuated before the actual
f
LO
and
f
c
