In Section 5.2 we introduced FIR filters with an averaging example, and that's where we first learned that the process of time-domain averaging performs low-pass filtering. In fact, successive time-domain outputs of an N-point averager are identical to the output of an (N–1)-tap FIR filter whose coefficients are all equal to 1/N, as shown in Figure 11-11.

Figure 11-11. An N-point averager depicted as an FIR filter.

The question we'll answer here is "What is the frequency magnitude response of a generic N-point averager?" We could evaluate Eq. (6-28), with all a(k) = 0, describing the frequency response of a generic N-stage FIR filter. In that expression, we'd have to set all the b(0) through b(N) coefficient values equal to 1/N and calculate HFIR(w)'s magnitude over the normalized radian frequency range of 0 w p. That range corresponds to an actual frequency range of 0 f fs/2 (where fs is the equivalent data sample rate in Hz). A simpler approach is to recall, from Section 5.2, that we can calculate the frequency response of an FIR filter by taking the DFT of the filter's coefficients. In doing so, we'd use an M-point FFT software routine to transform a sequence of N coefficients whose values are all equal to 1/N. Of course, M should be larger than N so that the sin(x)/x shape of the frequency response is noticeable. Following through on this by using a 128-point FFT routine, our N-point averager's frequency magnitude responses, for various values of N, are plotted in Figure 11-12. To make these curves more meaningful, the frequency axis is defined in terms of the sample rate fs in samples/s.

Figure 11-12. N-point averager's frequency magnitude response as a function of N.

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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

Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183 © 2008-2020.
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