There are many digital communications applications where a real signal is centered at one fourth the sample rate, or fs/4. This condition makes quadrature downconversion particularly simple. (See Sections 8.9 and 13.1.) In the event that you'd like to generate an interpolated (increased sample rate) version of the bandpass signal but maintain its fs/4 center frequency, there's an efficient way to do so[37]. Suppose we want to interpolate by a factor of two so the output sample rate is twice the input sample rate, fs-out = 2fs-in. In this case the process is: quadrature downconversion by fs-in/4, interpolation factor of two, quadrature upconversion by fs-out/4, and then take only the real part of the complex upconverted sequence. The implementation of this scheme is shown at the top of Figure 13-36.

Figure 13-36. Bandpass signal interpolation scheme, and spectra.

The sequences applied to the first multiplier in the top signal path are the real x(n) input and the repeating mixing sequence 1,0,–1,0. That mixing sequence is the real (or in-phase) part of the complex exponential

needed for quadrature downconversion by fs/4. Likewise, the repeating mixing sequence 0,–1,0,1 applied to the first multiplier in the bottom path is the imaginary (or quadrature phase) part of the complex downconversion exponential . The 2 symbol means insert one zero-valued sample between each signal at the A nodes. The final subtraction to obtain y(n) is how we extract the real part of the complex sequence at Node D. (That is, we're extracting the real part of the product of the complex signal at Node C times ej2p(1/4).) The spectra at various nodes of this process are shown at the bottom of Figure 13-35. The shaded spectra indicate true spectral components, while the white spectra represent spectral replications. Of course, the same lowpass filter must be used in both processing paths to maintain the proper time delay and orthogonal phase relationships.

There are several additional issues worth considering regarding this interpolation process[38]. If the amplitude loss, inherent in interpolation, of a factor of two is bothersome, we can make the final mixing sequences 2,0,–2,0, and 0,2,0,–2 to compensate for that loss. Because there are so many zeros in the sequences at Node B (three/fourths of the samples), we should consider those efficient polyphase filters for the lowpass filtering. Finally, if it's sensible in your implementation, consider replacing the final adder with a multiplexer (because alternate samples of the sequences at Node D are zeros). In this case, the mixing sequence in the bottom path would be changed to 0,–1,0,1.

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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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