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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 90o 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 90o 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 xbp(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.

Figure 8-22. Quadrature sampling with digital mixing method.

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 fc = fs/4, the cosine and sine oscillator outputs are the repetitive four-element cos(pn/2) = 1,0,–1,0, and –sin(pn/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.)

Figure 8-23. Spectra of quadrature sampling with digital mixing within the in-phase (upper) signal path.

With all its advantages, you should have noticed one drawback of this quadrature sampling with digital mixing process: the fs sampling rate must be four times the signal's fc center frequency. In practice, 4fc 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 modd = 5 to set the fs sampling rate at 40 MHz. This forces one of the replicated spectra of the sampled |X(m)| to be centered at fs/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 fs/4 (10 MHz) and digital lowpass filtering.

Figure 8-24. Bandpass sampling effects used to reduce the sampling rate of quadrature sampling with digital mixing: (a) analog input signal spectrum; (b) A/D converter spectrum.

Section 13.1 provides a clever trick for reducing the computational workload of the lowpass filters in Figure 8-22, when this fs/4 down-conversion method is used with decimation by two.