If we implement a linear phase FIR digital filter using the standard structure in Figure 13-16(a), there's a way to reduce the number of multipliers when the filter has an odd number of taps. Let's look at the top of Figure 13-16(a) where the five-tap FIR filter coefficients are h(0) through h(4) and the y(n) output is


Figure 13-16. Conventional and simplified structures of an FIR filter: (a) with an odd number of taps; (b) with an even number of taps.

If the FIR filter's coefficients are symmetrical we can reduce the number of necessary multipliers. That is, if h(4) = h(0), and h(3) = h(1), we can implement Eq. (13-62) by

Equation 13-63

where only three multiplications are necessary as shown at the bottom of Figure 13-16(a). In our five-tap filter case, we've eliminated two multipliers at the expense of implementing two additional adders. This minimum-multiplier structure is called a "folded" FIR filter.

In the general case of symmetrical-coefficient FIR filters with S taps, we can trade (S–1)/2 multipliers for (S–1)/2 adders when S is an odd number. So in the case of an odd number of taps, we need only perform (S–1)/2 + 1 multiplications for each filter output sample. For an even number of symmetrical taps as shown in Figure 13-16(b), the savings afforded by this technique reduces the necessary number of multiplications to S/2.

As of this writing, typical programmable-DSP chips cannot take advantage of the folded FIR filter structure because it requires a single addition before each multiply and accumulate operation.

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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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