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A COMPLEX DOWN-CONVERSION EXAMPLE

We can use all we've learned so far about quadrature signals by exploring the process of quadrature-sampling. Quadrature sampling is the process of digitizing a continuous (analog) bandpass signal and down-converting 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 8-17(a).

Figure 8-17. The "before and after" spectra of a quadrature-sampled 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 8-17(b). That is, we want to mix a time signal with to perform complex down-conversion. The frequency fs is the digitizer's sampling rate in samples/second. We show replicated spectra in Figure 8-17(b) to remind ourselves of this effect when A/D conversion takes place.

We can solve our sampling problem with the quadrature-sampling block diagram (also known as I/Q demodulation) shown in Figure 8-18(a). That arrangement of two sinusoidal oscillators, with their relative 90o phase, is often called a quadrature oscillator. First we'll investigate the in-phase (upper) path of the quadrature sampler. With the input analog xbp(t)'s spectrum shown in Figure 8-18(b), the spectral output of the top mixer is provided in Figure 8-18(c).

Figure 8-18. Quadrature-sampling: (a) block diagram; (b) input spectrum; (c) in-phase mixer output spectrum; (d) in-phase filter output spectrum.

Those and terms in Figure 8-18 remind us, from Eq. (8-13), 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 8-18(d) shows the output of the lowpass filter (LPF) in the in-phase path.

Likewise, Figure 8-19 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. (8-14) we know that the complex exponentials comprising the real –sin(2pfct) sinewave are and . The minus sign in the term accounts for the down-converted spectra in |Xq(f)| being 180o out of phase with the up-converted spectra.

Figure 8-19. 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 three-dimensional standpoint, as in Figure 8-20. There the +j factor rotates the "imaginary-only" Q(f) by 90o, making it "real-only." 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 8-18(a) having the spectrum shown in Figure 8-17(b).

Figure 8-20. Three-dimensional view of combining the I(f) and Q(f) spectra to obtain the I(f) +jQ(f) spectra.

Some advantages of this quadrature-sampling scheme are:

  • Each A/D converter operates at half the sampling rate of standard real-signal sampling.
  • In many hardware implementations, operating at lower clock rates saves power.
  • For a given fs sampling rate, we can capture wider band analog signals.
  • Quadrature sequences make FFT processing more efficient due to a wider frequency range coverage.
  • Quadrature sampling also makes it easier to measure the instantaneous magnitude and phase of a signal during demodulation.
  • Knowing the instantaneous phase of signals enables coherent processing.

While the quadrature sampler in Figure 8-18(a) performed complex down-conversion, it's easy to implement complex up-conversion by merely conjugating the xc(n) sequence, effectively inverting xc(n)'s spectrum about zero Hz, as shown in Figure 8-21.

Figure 8-21. Using conjugation to control spectral orientation.

 
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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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