# DSP Tricks: An odd way to build a simplified FIR filter structure

November 17, 2009

If you implement a linear phase Finite Impulse Response (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.

 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.

Let's look at the top of Figure 13-16(a) above where the five-tap FIR filter coefficients are h(0) through h(4) and the y(n) output is

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

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.

However, remember not all programmable-DSP chips can take advantage of the folded FIR filter structure because it requires a single addition before each multiply and accumulate operation.

Used with the permission of the publisher, Prentice Hall, this on-going series of articles is based on copyrighted material from "Understanding Digital Signal Processing, Second Edition" by Richard G. Lyons. The book can be purchased on line.

Richard Lyons is a consulting systems engineer and lecturer with Besser Associates. As a lecturer with Besser and an instructor for the University of California Santa Cruz Extension, Lyons has delivered digitasl signal processing seminars and training course at technical conferences as well at companies such as Motorola, Freescale, Lockheed Martin, Texas Instruments, Conexant, Northrop Grumman, Lucent, Nokia, Qualcomm, Honeywell, National Semiconductor, General Dynamics and Infinion.

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