There's a specialized FIR filter that's proved useful in decimation applications[21–25]. Called a half-band FIR filter, its frequency response is symmetrical about the fs/4 point as shown in Figure 5-33(a). As such, the sum of fpass and fstop is fs/2. This symmetry has the beautiful property that the time-domain FIR impulse response has every other filter coefficient being zero, except at the peak. This enables us to avoid approximately half the number of multiplications when implementing this kind of filter. By way of example, Figure 5-33(b) shows the coefficients for a 31-tap half-band filter where Df was defined to be approximately fs/32 using the Remez Exchange method. (To preserve symmetry, the parameters dp and ds were specified to be equal to each other.)

Figure 5-33. Half-band FIR filter: (a) frequency magnitude response [transition region centered at fs/4]; (b) 31-tap filter coefficients; 7-tap half-band filter structure.

Notice how the alternating h(k) coefficients are zero, so we perform 17 multiplications per output sample instead of the expected 31 multiplications. For an S-tap half-band FIR filter, we'll only need perform (S + 1)/2 + 1 multiplications per output sample.[] Be careful though; there's a restriction on the number of coefficients. To build linear phase half-band FIR filters, S + 1 must be an integer multiple of four. The structure of a simple seven-coefficient half-band filter is shown in Figure 5-33(c), with the h(1) and h(5) multipliers absent.

[] Section 13.7 shows a technique to further reduce the number of necessary multiplies for linear phase tapped delay line FIR filters including half-band filters.

Prev don't be afraid of buying books Next

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

Flylib.com © 2008-2020.
If you may any questions please contact us: flylib@qtcs.net