In this section, we present a crude (but simple to implement) complex signal envelope detection scheme. By envelope detection, we mean estimating the instantaneous magnitude of a complex signal xc(n). The process is straightforward: we sum the absolute values of a complex signal's real and imaginary parts, and apply that sum to a simple first-order lowpass IIR filter to obtain an envelope signal E(n) as shown in Figure 13-78(a). The filter's feedback coefficient a is in the range of 0 to 1. (That lowpass filter is our exponential averager discussed in Section 11.5, which some DSP folks call a leaky integrator.) The E(n) sequence is proportional to the desired instantaneous magnitude of xc(n), or


Figure 13-78. Envelope detection: (a) block diagram; (b) |xr(n)|+|xi(n)| adder output, and E(n) for a = 0.4; (c) E(n) for a = 0.7 and a = 0.9.

To gauge the envelope detector's performance, consider a sampled version of an amplitude modulated sinusoid such as the xr(n) in Figure 9-7(a) from which a sampled analytic (complex) xc(n) can been generated. If xc(n) is applied to our envelope detection process, the processing results are shown in Figure 13-78(b) and 13-78(c), where the solid curves represent E(n) and the dashed curves are the true magnitude of xc(n). Notice how the amount of smoothing of the E(n) fluctuations depends on the value of a.

Sequence xr(n) must be used to generate a complex analytic xc(n) sequence (using one of the methods discussed in Sections 9.4 and 9.5) upon which this envelope detector scheme can be applied. The advantage of this envelope detection process is that, of course, no squaring or square root computations in Eq. (13-133), nor the |xr(n)| and |xi(n)| comparisons in the vector magnitude approximations in Section 13.2, need be performed.

Whether this envelope approximation technique yields sufficiently accurate results is for the user to decide. Its accuracy may be below the requirements of most AM (amplitude modulation) detection requirements, but the process may well be useful for estimating signal magnitude in automatic gain control (AGC) or energy detection applications.

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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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