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FIR-ward Thinking, Continued

Are longer filters and more taps better? Or are we looking another example of “less can be more”

So, if a 1024 tap min phase FIR is effective to 150 Hz, a 1024 tap linear phase FIR will only be effective to 300 Hz.

To get back down to 150 Hz would require a 2048 tap filter, which would mean a doubling of the processing delay from about 10.7 ms to 21 ms.

For my example file, an 8196 tap FIR would be needed to flatten the entire pass band. The required processing delay is about 85 ms. Ouch!

At face value, the benefit of longer FIR filters appears to be an increase in frequency resolution, resulting in:

1) Operation to a lower frequency limit.
2) Finer filter detail across its bandwidth.

I’ve already shown that, at least for linear phase FIRs, operation to a lower frequency (more taps) increases the processing delay. This is not due to a lack processing power. It’s due to the interdependence of time and frequency. Low frequencies last a long time and occupy a lot of space. Digital processing can’t change that.

Proportional Benefits
There are 10 octaves in the audible spectrum, so each one occupies about 10 percent of the whole (log scale). Doubling the filter length, which doubles the processing delay, only buys you the ability to filter about 10 percent more of the spectrum.

Just like doubling the amplifier power to gain 3 dB of level, there is a point of diminishing returns where it’s just not worth it. With amplifier power the price is paid in dollars. With linear phase FIRs the price is paid in milliseconds. It could be said that the currency of digital processing is time. You can have some amazing things – if you can wait long enough.

The lowest two octaves (subwoofer range) fall well below the Schroder Frequency for most rooms, where the room modes have a profound effect on the seat-to-seat frequency response. Even if we could make the response “perfect” at a point in space with a linear phase FIR, it will be different at all other seats, so where is the benefit?

Is finer filter detail needed? A 1024 tap filter has data point frequency spacing of about 48 Hz. If that is insufficient for correcting the frequency response, then you are probably trying to equalize something that you shouldn’t, such as comb filtering caused by room reflections. The more “detail” there is in the frequency response, the more “position specific” is the time response, since the details are produced by sound reflecting from various surfaces.

Move the mic and the response changes dramatically. An FIR with very high detail will only be appropriate for “correcting” a single point in space. While this might be useful for acoustic echo cancellation for a specific microphone position in a conferencing system, it doesn’t provide a benefit for an audience area. Long FIRs don’t bring much to the picnic for live sound applications.

Do Higher Sample Rates Help?
That’s the word on the street. Unfortunately it doesn’t pan out. Referring back to Figure 6, the time spacing of the filter taps dt is 1/SR. Multiply this times the number of taps (N) to get the time span (T) of the filter. The frequency resolution (F) is 1/T.

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These are simple relationships, and they show that if you double the SR the frequency resolution of the filter is reduced by one-half. We made it worse! If you double the SR you now have 2x the number of samples to process, so you have to double the filter length to maintain the same frequency resolution.

Let’s go the other way. Halving the SR will increase the frequency resolution of the filter. But this comes at the expense of high frequency response, which extends to about SR/2. Nyquist-Shannon will not be denied.

None of this changes the required processing delay for a linear phase FIR, which is up to one-half the time span that the filter must influence. Higher (or lower) sample rates don’t change the time, frequency, or wavelength of the signal.

Conclusion
There is no doubt that technology will gift us with longer digital filters. Chips keep improving, and so do the products that use them. Many of us can remember when 16 bit/44.1 kHz “CD quality” audio was barely possible. Today it’s considered by many to be low resolution. FIRs will follow the same path.

But there are some road blocks to longer filters that have nothing to do with the technology. As I’ve shown, the major one is the processing delay, which is determined by time span that the filter must influence (see Figure 1).

In live sound systems, we can only tolerate so much delay. It’s a fuzzy limit, but most would agree that more than about 20 ms for filtering is a pretty long time to wait for a signal. This unavoidable processing delay adds to the latency of the other digital components in the chain, which can easily add another 10 ms or more. We simply have to give up on linear phase equalization at low frequencies due to the time premium, at least for live sound applications.

A combination of linear phase FIRs for HF equalization and minimum phase FIRs for LF equalization seems to be the way to go. These are called “mixed phase” filters, and in my opinion, they are the future. And while some are starting to consider IIR filters to be passé, there is no denying that they have the widest bandwidth, lowest processing delay, and use the least DSP horsepower of any digital filter. Please don’t take away my parametric equalizer blocks!

Having a limitation on the number of taps also means that we must give some thought to what we are trying to correct and why. It’s like only having one spoonful of peanut butter left in the jar. Use it wisely. How does my equalization affect the entire coverage area of the loudspeaker, not just a “sweet spot” where a measurement microphone is placed? The magic of linear phase FIR filters diminishes quickly when multiple seats are considered. They will likely continue to build auditoriums that way for the foreseeable future.

It’s better to have a well-considered application of a 1024 tap FIR than a sloppy application of a much longer filter. Just as human perception places a practical ceiling on the required sample rate and bit depth for analog-to-digital conversion, so do time, frequency, wavelength, and delay place a limitation on the benefits of longer FIR filters. In audio, there is always “pushback” on making anything larger, and digital filters are no exception. It’s important to let the pushback push back. This keeps things in balance and promotes proper thinking with regard to what really produces better sound versus what looks good on a spec sheet.

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It’s another example of “less can be more.”

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