We can use all we've learned so far about quadrature signals by exploring the process of quadraturesampling. Quadrature sampling is the process of digitizing a continuous (analog) bandpass signal and downconverting its spectrum to be centered at zero Hz. Let's see how this popular process works by thinking of a continuous bandpass signal, of bandwidth B, centered about a carrier frequency of fc Hz as shown in Figure 817(a).
Figure 817. The "before and after" spectra of a quadraturesampled signal.
Our goal in quadrature sampling is to obtain a digitized version of the analog bandpass signal, but we want the digitized signal's discrete spectrum centered about zero Hz, not fc Hz as in Figure 817(b). That is, we want to mix a time signal with to perform complex downconversion. The frequency fs is the digitizer's sampling rate in samples/second. We show replicated spectra in Figure 817(b) to remind ourselves of this effect when A/D conversion takes place.
We can solve our sampling problem with the quadraturesampling block diagram (also known as I/Q demodulation) shown in Figure 818(a). That arrangement of two sinusoidal oscillators, with their relative 90o phase, is often called a quadrature oscillator. First we'll investigate the inphase (upper) path of the quadrature sampler. With the input analog xbp(t)'s spectrum shown in Figure 818(b), the spectral output of the top mixer is provided in Figure 818(c).
Figure 818. Quadraturesampling: (a) block diagram; (b) input spectrum; (c) inphase mixer output spectrum; (d) inphase filter output spectrum.
Those and terms in Figure 818 remind us, from Eq. (813), that the constituent complex exponentials comprise a real cosine duplicate and translate each part of Xbp(f)'s spectrum to produce the Xi(f) spectrum. There is a magnitude loss of a factor of 2 in Xi(f), but we're not concerned about that at this point. Figure 818(d) shows the output of the lowpass filter (LPF) in the inphase path.
Likewise, Figure 819 shows how we get the filtered continuous quadrature phase portion (bottom path) of our desired complex signal by mixing xbp(t) with –sin(2pfct). From Eq. (814) we know that the complex exponentials comprising the real –sin(2pfct) sinewave are and . The minus sign in the term accounts for the downconverted spectra in Xq(f) being 180o out of phase with the upconverted spectra.
Figure 819. Spectra within the quadrature phase (lower) signal path of the block diagram.
This depiction of quadrature sampling can be enhanced if we look at the situation from a threedimensional standpoint, as in Figure 820. There the +j factor rotates the "imaginaryonly" Q(f) by 90o, making it "realonly." This jQ(f) is then added to I(f) to yield the spectrum of a complex continuous signal x(t) = i(t) + jq(t). Applying this signal to two A/D converters gives our final desired discrete time samples of xc(n) = i(n) + jq(n) in Figure 818(a) having the spectrum shown in Figure 817(b).
Figure 820. Threedimensional view of combining the I(f) and Q(f) spectra to obtain the I(f) +jQ(f) spectra.
Some advantages of this quadraturesampling scheme are:
While the quadrature sampler in Figure 818(a) performed complex downconversion, it's easy to implement complex upconversion by merely conjugating the xc(n) sequence, effectively inverting xc(n)'s spectrum about zero Hz, as shown in Figure 821.
Figure 821. Using conjugation to control spectral orientation.
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