8.9. AN ALTERNATE
DOWN-CONVERSION
METHOD
The quadrature sampling method of complex down-conversion in Figure 8-18(a) works
perfectly
on paper, but it's difficult to maintain the necessary exact 90
o
phase relationships with high frequency, or wideband, signals in practice. One- or two-degree phase errors are common in the laboratory. Ideally, we'd need perfectly phase-matched coax cables, two
oscillators
exactly 90
o
out of phase, two ideal mixers with identical behavior and no DC output component, two analog lowpass filters with identical magnitude and phase characteristics, and two A/D converters with exactly identical performance. (Sadly, no such electronic
components
are available to us.) Fortunately, there's an easier-to-implement quadrature sampling method[5].
Consider the process shown in Figure 8-22, where the analog
x
bp
(
t
) signal is initially digitized with the follow-on mixing and filtering being performed digitally. This
quadrature sampling with digital mixing
method mitigates the problems with the Figure 8-18(a) quadrature sampling method and eliminates one of the A/D converters.
We use Figure 8-23 to show the spectra of the in-phase
path
of this quadrature sampling with digital mixing process. Notice the similarity between the continuous
I
(
f
) in Figure 8-18(d) and the discrete
I
(
m
) in Figure 8-23(d). A sweet feature of this process is that with
f
c
=
f
s
/4, the cosine and sine oscillator outputs are the repetitive four-element cos(
p
n
/2) = 1,0,–1,0, and –sin(
p
n
/2) = 0,–1,0,1, sequences, respectively. (See Section 13.1 for details of these special mixing sequences.) No actual mixers (or multiplies) are needed to down-convert our desired spectra to zero Hz! After lowpass filtering, the
i
(
n
) and
q
(
n
) sequences are typically decimated by a factor of two to reduce the data rate for following processing. (Decimation is a topic covered in Section 10.1.)
With all its advantages, you should have noticed one drawback of this quadrature sampling with digital mixing process: the
f
s
sampling rate must be four times the signal's
f
c
center frequency. In practice, 4
f
c
could be an unpleasantly high value. Happily, we can take advantage of the effects of bandpass sampling to reduce our sample rate. Here's an example: consider a real analog signal whose center frequency is 50 MHz, as shown in Figure 8-24(a). Rather than sampling this signal at 200 MHz, we perform bandpass sampling, and use Eq. (2-13) with
m
odd
= 5 to set the
f
s
sampling rate at 40 MHz. This forces one of the replicated spectra of the sampled
X
(
m
) to be centered at
f
s
/4, as shown in Figure 8-24(b), which is what we wanted. The A/D converter
x
(
n
) output is now ready for complex down-conversion by
f
s
/4 (10 MHz) and digital lowpass filtering.
Section 13.1 provides a clever trick for reducing the computational workload of the lowpass filters in Figure 8-22, when this
f
s
/4 down-conversion method is used with decimation by two.
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