Digital Signal Processing Reference
In-Depth Information
One additional key property is obtained from the transfer function interpretation
of modulated complex filters. For
H
N
n
z
−
n
, we can write
(
z
)=
∑
0
h
(
n
)
=
n
=
0
h
(
n
)
e
j
ω
0
n
z
−
n
N
n
=
0
h
(
n
)
z
−
1
e
j
ω
0
n
N
=
=
H
(
z
)
|
z
−
1
(37)
z
−
1
e
j
ω
0
←
This means that the modulated filter can also be implemented by simply replacing
the unit delays (
z
−
1
elements) of the original filter with generalized elements
z
−
1
e
j
ω
0
. Thus implementing frequency translations is very straight-forward also for
IIR type filters.
We illustrate the modulated FIR filter characteristics with a design example
where analytic bandpass filter is obtained through complex modulation. Target is to
have passband at 0
and the filter length is 50. Equiripple (Remez) design
is used, and the lowpass prototype is an ordinary LPF with passband
.
6
π
...
0
.
8
π
−
0
.
1
π
...
0
.
1
π
.
Then complex modulation with
e
j
0
.
7π
n
is deployed. The results are illustrated in
After learning that we can generally build complex (analytic) bandpass filters,
it's also easy to devise an alternative strategy, other than the classical scheme
with complex down-conversion and lowpass filtering, for I/Q demodulation. This
bandpass filter after which complex downconversion takes place. Notice that in this
scheme the complex bandpass filter creates already complex output signal and thus
a true complex mixer is required (4 muls and 2 adds). This structure has, however,
some benefits e.g. from analysis point of view, and it is also very suitable for digital
I/Q demodulation combined with decimation/down-sampling since the complex
filter output is free from negative frequencies.
Additional good example of applying complex signal processing tools in radio
transceivers is, e.g., a dual-carrier or dual-channel receiver in which the RF front-
end implements wideband I/Q downconversion of the received signal such that the
two interesting carriers are located at positive and negative (small) intermediate
frequencies (IFs) after the analog front-end. The signal is then sampled and the
two carriers are demodulated in parallel in the digital front-end to baseband for
there are two possibilities how to implement the carrier separation and demodulation
in the digital front-end: (i) complex digital bandpass filters centered at positive
and negative IFs, respectively, followed by complex digital downconversions or (ii)
complex digital downconversions from positive and negative IFs to baseband (in
parallel) and real digital lowpass filtering for both signals. In practice, this is also
accompanied with sample rate adaptation (decimation).
