Reducing IIR Filter Computational Workload

This blog describes a straightforward method to significantly reduce the number of necessary multiplies per input sample of traditional IIR lowpass and highpass digital filters.

Reducing IIR Filter Computations Using Dual-Path Allpass Filters

We can improve the computational speed of a lowpass or highpass IIR filter by converting that filter into a dual-path filter consisting of allpass filters as shown in Figure 1.

                FIGURE 1. IIR filter –to– dual-path allpass filters conversion.

The dual A₀(z) and A₁(z) filter paths in Figure 1 are cascades of simple 1st- and 2nd-order allpass filters. (Allpass filters are a specialized class of IIR filters whose frequency magnitude responses are constant over the full frequency range from zero to the f_s input data sample rate [1].)

Figure 2 shows an example of this standard IIR –to– parallel allpass filter conversion process, where our original IIR filter is a 5th-order lowpass filter.

         FIGURE 2. Example of "standard" IIR filter –to– parallel-path allpass filter conversion.

The original IIR and the dual-path filters in Figure 2 have identical frequency magnitude responses. A key point of this blog is that the filter on the left side of Figure 2 requires 11 multiplies per input sample whereas the parallel filter on the right side only requires five multiplies per input sample.

Details of the allpass sections within the 5th-order dual-path filter on the right side of Figure 2 are given in Figure 3. The c_k coefficients are real-valued scalars.

     FIGURE 3. 1st- and 2nd-order allpass sections and their z-domain transfer functions.

We see in Figure 3 that H₁(z) and H₂(z) are indeed allpass filters because their numerator coefficients are in reverse order from their denominator coefficients.

At first glance the notion of implementing a lowpass or highpass IIR filter using allpass filters seems impossible. An explanation of how this is possible is found in Reference [2], graciously supplied online by its author DSP guru fred harris.

As additional examples, we show the result of our standard IIR –to– dual-path allpass conversion for standard 3rd-, 7th-, and 9th-order IIR filters in Figure 4.

              FIGURE 4. Dual-path allpass IIR filters: (a) 3rd- and 7th-order; (b) 9th-order.

Computational Savings When Using the Dual-Path Allpass Filters

Table 1 compares the computational workloads of traditional Direct Form I IIR filters relative to the dual-path IIR filters in Figures 3 and 4. The bottom row of Table 1 illustrates the computational advantage of the dual-path allpass filters.

To keep this blog from being too lengthy, the PDF file associated with this blog provides:

References

[1] S. Mitra, Digital filter lecture slides (allpass filter discussion starts at slide# 23).

http://www.emba.uvm.edu/~gmirchan/classes/EE275/Handouts_Ed4/Ch07(4e)Handouts/Ch7(1)Handouts_4e.pdf

[2] f. harris, "A Most Efficient Digital Filter: The Two-Path Recursive All-Pass Filter", Streamlining Digital Signal Processing, A Tricks of the Trade Guidebook (IEEE Press/Wiley, 2007), Chapter 9, pp. 85-104. Available at:

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