# Reduced-Delay IIR Filters

This blog gives the results of a preliminary investigation of reduced-delay (reduced group delay) IIR filters based on my understanding of the concepts presented in a recent interesting blog by Steve Maslen [1].

## Development of a Reduced-Delay 2nd-Order IIR Filter

Maslen's development of a reduced-delay 2nd-order IIR filter begins with a traditional prototype filter, $H_{Trad}$, shown in Figure 1(a). The first modification to the prototype filter is to extract the b0 feedforward coefficient path from the filter's delay line as shown in the $H_{Mod}$ filter presented in Figure 1(b).

Figure 1. Equivalent 2nd-order IIR filters: (a) prototype traditional filter, $H_{Trad}$;
(b) modified filter, $H_{Mod}$.

As shown in Appendix A, the two filters' $H_{Trad}(z)$ and $H_{Mod}(z)$ z-domain transfer functions given in Figure 1 are identical. The final step in Reference [1]'s modifications is to shift $H_{Mod}$'s two feedforward coefficients upward in the delay line to produce the desired reduced-delay 2nd-order IIR filter, $H_{Red‑Del}$, shown in Figure 2.

Figure 2. Reduced-delay 2nd-order IIR filter.
The derivation of the $H_{Red‑Del}(z)$ transfer function in Figure 2 is given in Appendix B.

## 缩减延迟的2阶IIR滤波器的性能

[b,a] = butter(2, 0.25);
[b,a] = ellip(2, 2, 30, .25);
[b,a] = cheby1(2, .5, .25);
[b,a] = cheby2(2, 30, .25);

The first column of Figure 3 shows the z-plane pole/zero locations for the prototype traditional IIR filters, while the second column shows the z-plane pole/zero locations for the Figure 2 reduced-delay filters.

Figure 3. 2nd-order lowpass prototype filter and reduced-delay filter performance. First column is the z-plane of the prototype filter, the second column is the z-plane of the reduced-delay filter.

Because the prototype –to- reduced-delay filter conversion process modifies a prototype filter’s feedforward coefficients, notice how that process shifted the locations of the z-plane zeros for the reduced-delay filters and this affects their stopband attenuation.

In the third column of Figure 3 the dashed and solid curves show the frequency magnitude responses of the lowpass prototype traditional and Figure 2 reduced-delay filters respectively. Notice the reduced stopband attenuation of the reduced-delay filters’ solid curves in the third column.

In the forth column of Figure 3 the dashed and solid curves show the group delay plots of the lowpass prototype traditional and Figure 2 reduced-delay filters respectively. The ‘Delta Grp Del’ label above the fourth column’s plots give the group delay reduction of the reduced-delay filters at zero Hz (measured in samples).

## What We Should Learn From Figure 3

Regarding Reference [1]’s IIR filter group delay reduction process producing the Figure 2 filter, the main points of this blog are:

• The amount of group delay reduction in the passband of the Figure 2 reduced-delay filters is just less than one sample.

• Relative to the original prototype IIR filters, the stopband attenuation of the reduced-delay filters is significantly degraded.

• The amount of group delay reduction and the stopband attenuation of the reduced-delay filters depends on the design method of the original IIR filter being converted.

If this reduced-delay filter topic interests you, the PDF file associated with this blog presents the performance of 1st-, 3rd-, and 4th-order reduced-delay IIR lowpass filters.

## References

[1] Maslen, Steve, "Part 11. Using -ve Latency DSP to Cancel Unwanted Delays in Sampled-Data Filters/Controllers", Website: dsprelated.com, https://www.dsprelated.com/showarticle/1280.php
Appendix A: Proof of $H_{Mod}(z)$ = $H_{Trad}(z)$
Proving the equivalence of Figure 1's $H_{Trad}(z)$ and $H_{Mod}(z)$ transfer functions begins by expressing $H_{Mod}(z)$ as:

Putting the two terms in Eq. (A-1) over a common denominator yields:

Collecting the factors of z in Eq. (A-2) gives us:

Canceling the appropriate positive and negative terms in Eq. (A-3) gives us $H_{Mod}(z)$'s transfer function that is equal to the traditional IIR filter's $H_{Trad}(z)$ given as:

## Appendix B: Derivation of $H_{Red-Del}(z)$ Transfer Function

The derivation of the Figure 2’s $H_{Red-Del}(z)$ transfer function proceeds as follows:

Putting the two terms in Eq. (B-1) over a common denominator yields:

Collecting the factors of z in Eq. (B-2) gives us the desired 2nd-order $H_{Red-Del}(z)$ z-domain transfer function:

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