An Introduction to the Mathematics of Impulse Responses

There is nothing in this article that is required reading to use an IR to make your acoustic guitar pickup sound more like a mic, but if you are curious...

IR stands for Impulse Response. A very complex filter (equalizer) compared to what we are used to with sliders and knobs can be made with digital computer logic and (for the subclass of linear, time invariant, and finite response) they are programmed with their "IR".

Electric guitarists love their distorting (nonlinear) tube amps and IRs are largely useless in reproducing distortion. But electric guitarist also like the frequency response of vintage speaker drivers and cabinets. A filter programmed with an IR can model any of those cabinets. Not really that surprising if you can hear the difference between a D-18 and a D-28. Cabinet IRs are plentiful for purchase and free download on the Internet.

For us acoustic guitarists, products like Aura and ToneDexter use an IR programmed filter to make your pickup sound more like your mic. Is that problem linear, time invariant, and finite? It is close enough for most piezo pickups (skip to the end for a discussion of the exceptions). One problem is loading and playing back the IR in performance and the other is creating that IR in the first place (Aura library, ToneDexter training, etc.). This website is addressing the problem of sourcing IRs to allow Aura like results using an inexpensive IR loader pedal.

It might be easy to imagine sampling a signal and feeding back some of the delayed signal into the live signal. A delay pedal is an example. If you set the delay pedal to feedback the delayed signal back into the delay line input, you’ve got the possibility for something that could make a signal last forever. The first of these two possibilities, no feedback of the delay output into the delay input, is an example of a very simple Finite Impulse Response (FIR) filter. The other is an example of an Infinite Impulse Response (IIR) Filter. The length of a FIR filter that might make a good reverb is typically very long with compromised SNR and frequency response (to keep the memory to store the echo reasonably priced). The FIR filter we will be discussing here is very short and very high fidelity. It is being used as an EQ and will not be long enough for the more familiar time effects such as reverb or delay.

All these samples going into delay lines are examples of things you can do in what is called the “Time Domain”. But you are very familiar with the tone controls on your equipment. They are examples of things where the controls we use to manipulate them are labeled in the “Frequency Domain” (bass, treble, etc.). A French mathematician, Fourier, invented a transform which can be used to move mathematical formulas (representations of something you are interested in) between these two domains. His actual goal, like how a slide rule uses logarithms to turn multiplication into addition, was to simplify all sorts of hard stuff which in his case was heat transfer.

An impulse is a signal infinitely high in magnitude and infinitely short in duration. Of course, something like this is impossible in the real world, but close enough is good enough. The Fourier transform of an impulse in the time domain becomes "1" in the frequency domain (very cool and simplifying). If you put something like an impulse into your electric guitar amp speaker and sample the speaker output, what comes out is the impulse response. Transform that IR to the frequency domain and you’ve got something like the tone control settings to duplicate that cabinet’s frequency response. In the frequency domain EQ filters simplify into multiplication and that the transform of an impulse is a simple “1” means all that is left is the equalizer’s programming. Multiplication in the frequency domain becomes something called convolution in the time domain which is something you can do easily with a FIR filter (a FIR filter is really a convolution machine). But how to program that filter? It can be easily mathematically derived from that impulse response. It is as simple as the impulse response is the programming of that FIR filter. IR and FIR filter coefficients have become interchangeable ways to refer to the same thing in the music industry’s lexicon.

If you record your acoustic guitar pickup and a mic pointed at the guitar, you now have two time domain responses of your playing. Taking the digital version of the Fourier transform (usually referred to as an FFT, or fast Fourier transform) and transform them to frequency domain, divide the mic by the pickup, reverse the transformation, and you’ve got the programming for your acoustic guitar IR pedal. To make this sound good it is much more work than this simplified view (and much harder than cabinet modeling). This process is often referred to as “training”. There is a lot of art to choosing what parts of the original recording to use and how to post process.

Since the IR is only about programming a very complicated equalizer with a fixed setting, there are systems with which it will be less effective or not work at all. An example of a system that is not compatible with an IR is the Baggs Session VTC. Because the Session distorts the pickup output (intentionally nonlinear) it generates frequency content that was never in the original guitar pickup’s output and will be absent from the mic recording. The training process outlined above will fail. Magnetic pickups not placed adjacent to the bridge have a different frequency response as you play up the neck. A fixed filter can not deal effectively with this changing pickup output.