# Understanding Natural And Digital Reverb

In the digital domain, it’s easy to realize that each sample point of the response can be viewed as a discrete echo of the impulse. Since, ideally, the impulse is a single non-zero sample, it’s not a stretch to realize that a series of samples—a sound played in the room—would be the sum of the responses of each individual sample at their respective times (this is called superposition).

In other words, if we have a digitized impulse response, we can easily add that exact room characteristic to any digitized dry sound. Multiplying each point of the impulse response by the amplitude of a sample yields the room’s response to that sample; we simply do that for each sample of the sound that we want to “place” into that room. This yields a bunch—as many as we have samples—of overlapping responses that we simply add together.

Easy. But extremely expensive computationally. Each sample of the input is multiplied individually by each sample of the impulse response, and added to the mix. If we have n samples to process, and the impulse response is m samples long, we need to perform n+m multiplications and additions. So, if the impulse response is three seconds (a big room), and we need to process one minute of music, we need to do about 350 trillion multiplications and the same number of additions (assuming a 44.1KHz sampling rate).

This may be acceptable if you want to let your computer crunch the numbers for a day before you can hear the result, but it’s clearly not usable for real-time effects. Too bad, because its promising in several aspects. In particular, you can accurately mimic any room in the world if you have its impulse response, and you can easily generate your own artificial impulse responses to invent your own “rooms” (for instance, a simple decaying noise sequence gives a smooth reverb, though one with much personality).

Actually, there’s a way to handle this more practically. We’ve been talking about time-domain processing here, and the process of multiplying the two sampled signals is called “convolution.” While convolution in the time domain requires many operations, the equivalent in the frequency domain requires drastically reduced computation (convolution in the time domain is equivalent to multiplication in the frequency domain). I won’t elaborate here, but you can check out Bill Gardner’s article, “Efficient Convolution Without Input/Output Delay” for a promising approach. (I haven’t tried his technique, but I hope to give it a shot when I have time.)

## A Practical Approach To Digital Reverb

The digital reverbs we all know and love take a different approach. Basically, they use multiple delays and feedback to built up a dense series of echoes that dies out over time. The functional building blocks are well known; it’s variations and how they are stacked together that give an digital reverb units its characteristic sound.

The simplest approach would be a single delay with part of the signal fed back into the delay, creating a repeating echo that fades out (the feedback gain must be less than 1). Mixing in similar delays of different sizes would increase the echo density and get closer to reverberation. For instance, using different delay lengths based on prime numbers would ensure that each echo fell between other echoes, enhancing density.

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In practice, this simple arrangement doesn’t work very well. It takes too many of these hard echoes to make a smooth wall of reverb. Also, the simple feedback is the recipe for a comb filter, resulting in frequency cancellations that can mimic room effects, but can also yield ringing and instability. While useful, these comb filters alone don’t give a satisfying reverb effect.

Comb filter reverb element

By feeding forward (inverted) as well as back, we fill in the frequency cancellations, making the system an all-pass filter. All-Pass filters give us the echoes as before, but a smoother frequency response. They have the effect of frequency-dependent delay, smearing the harmonics of the input signal and getting closer to a true reverb sound. Combinations of these comb and all-pass recirculating delays—in series, parallel, and even nested—and other elements, such as filtering in the feedback path to simulate high-frequency absorption, result in the final product.

All-Pass filter reverb element

I’ll stop here, because there are many readily available texts on the subject and this is just an introduction. Personally, I found enough information for my own experiments in “Musical Applications of Microprocessors” by Hal Chamberlin, and Bill Gardner’s works on the subject.

Read and comment on the original article here.

Nigel Redmon is a musician, electrical and software engineer, and independent developer, specializing in digital audio signal processing applications. He has developed products for Line 6, Equator Audio, Alesis, Oberheim, and others.

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