Let's Write A Reverb

Here's a simple feedback loop:

The input signal is sent through a delay, which then goes to the output - but it's also fed back into the delay input. This means the signal travels round in a loop, producing a series of echoes from a single input pulse.

There are a few variations on this, like where in the loop we take the output from, or where the gain is placed. The essential part is the feedback which produces an infinitely-looping output, with a gain to make it decay over time.

Now let's look at a multi-channel version of the same loop:

An important property here is that each channel gets delayed by a different amount, so they each produce a different echo pattern. If we mix all the output channels together, the result still has repeating patterns, but it's more complex than the single-channel version:

Adding a mixing matrix

That more complex sound is good - we don't want identifiable repeating patterns in our reverb. But in the loop above, each channel only repeats its own echoes. Could we get an even more complex result if they picked up each other's echoes as well?

We can achieve this by putting a mixing matrix inside the feedback loop. This is an N \times N orthogonal matrix, where N is the number of channels.

Matrices, and orthogonal matrices

If you aren't familiar with matrices, for this article, we can just say: a matrix is a function which takes in multiple inputs, and adds those together in different ways to produce multiple outputs:

A simple example would be a mid-side encoder, which takes two input channels, and returns them in a linear combination:

For a mixing matrix, we treat each simultaneous set of input samples (from the N channels) as the inputs to the function, and use that to generate one sample for each output channel.

Orthogonal Matrix

An orthogonal matrix is a matrix where the total energy of the output always equals the total energy of the input.

You don't need to know how to construct these, we're going to use some off-the-shelf orthogonal mixing matrices. However, this property where the output energy matches the input energy is going to be really useful later.

Because of this cross-mixing, as the signal travels around the loop, the echoes appear more and more often, giving an increasingly chaotic result:

The tail end of that is even starting to sound like a reverb, right?

Caution

At this point, it's tempting to mix the channels as much as possible, or put a bunch of other things inside the feedback loop to speed up diffusion.

There are some interesting designs here - and they work, but they often require careful tuning. As well as finding a good compromise for the delay time, using too much inter-channel mixing can lock the delays together so they act like a cohesive unit, which isn't great for longer tails.

Designing a feedback loop to produce a diffuse, long-lasting sound with no strong resonances is hard - but we don't have to do it!

Even a basic feedback loop handles the "long-lasting" part nicely. Rather than trying to get the feedback loop to also produce a diffuse sound, we're going to separate the two concerns:

If the diffuser does its job well, we shouldn't really need to build up echo density in the feedback loop.

Now we just need to design a really good diffuser.

Down-mixing

At some point in our final effect, we need to mix these multi-channel signals back down to a mono (or stereo) output.

When you do this downmixing, the system as a whole stops being an allpass (even if you're just mixing down to the same number of channels as you started with).

This is fine as long as your phase-shifts don't have regular patterns which could produce an identifiable "comb" timbre. The diffuser design we've explored here gives a nicely irregular phase-response, particularly for larger numbers of channels (N).

Because the Hadamard matrix has mixed things already, we can just take the top 1 or 2 output channels to get a good result - this is what we did for the examples above.

In our final reverb design, the diffuser's multi-channel output will head straight into the multi-channel feedback loop, so we don't downmix until later.

Choosing delay times

Once you've decided what range of delay times you want for a particular diffusion step, you could pick each channel's delay time randomly. However, if you divide the range into equal segments, and use one sub-range for each channel, they end up approximately evenly-distributed while still being randomised.

So: we have a multi-channel diffuser, and we have a multi-channel feedback-delay loop. All we have to do is duplicate our input, and mix down our output, and we're done!

Here's our combined design:

The input and output don't have to be mono - just choose something appropriate when converting to/from N channels, and you can have a stereo (or more) reverb.