David Griesinger

Digital Reverberation is everywhere. In only eleven years it has gone from one of the first really cost-effective uses of digital technology to being indispensable to modern recording. This paper presents some of the origins of digital reverberation in the extensive earlier work at the British Broadcasting Corporation, Bell Laboratories, and elsewhere. It will discuss the advantages and disadvantages of some of the algorithms, and the hardware requirements of digital processors optimized for reverberation. IT CAN'T BE DONE Before discussing electronic reverberation it is worth pointing out why it is really not possible to electronically emulate natural reverberation. In practice there are two reasons, one of which, the complexity of natural impulse responses, merely makes the problem impracticably difficult. The other problem, the multitude of sources, makes it impossible. Once we have done our duty by pointing these problems out, we can continue with a good conscience to describe what we CAN do. IMPULSE RESPONSE OF NATURAL SPACES If we shoot a pistol at one point in a room, and record the sound pressure with microphone at another point we can capture an echogram, or the impulse response of the room. If we have a computer powerful enough to CONVOLVE recorded music with this measured impulse response we can duplicate the sound of the room exactly just as if we had a perfect loudspeaker at the position of the pistol. The process of convolution sounds more complicated than it is. First we digitize the impulse response and store it in the computer. To perform the convolution we take the first sample of the impulse response and multiply by the first sample of the music, and then add the product to the product of the second sample of the impulse response multiplied by the second sample of the music. We keep doing this until the impulse response has decayed enough that we needn't continue. The resulting sum is the first output sample. To find the second output sample we do the same thing, but this time multiplying the first sample of the impulse response by the SECOND sample of the music, and summing everything again shifted by one sample. Each sample of the output comes from a sum of a great many multiplications. If our sample rate is 44kHz, and the room has a 1 second reverb time, each output sample is the result of 44 thousand multiply and adds. To do these in real time requires a machine with 2000 mega instructions per second (MIPS) capacity. For a 2 second reverb time, we need twice this, or 4000 MIPS. For stereo we need 8000 MIPS. We could use FFT techniques to do the convolutions but this jumps ahead a bit. For the moment lets think about what happens if we simplify the impulse response so it contains only a few multiplies. Natural spaces have response functions of great complexity. If you record a pistol shot and play it back on an oscilloscope you get patterns similar to Figures 1 and 2. It is tempting to claim such an echograms can be characterized by just a few simple reflections. If the impulse response contains only 100 reflections we need only 100 multiply and adds -5 MIPS -a much more practical figure. Nearly all work in electronic reverberation demands such simplification. Does it work? Not really. A few prominent reflections are easy to identify in figure 1, but if we look more closely we find each reflection is not a simple spike but a more complicated shape. The spread in the individual reflections is due to the surface roughness and frequency dispersion of natural reflectors, a roughness which can easily extend over a time period of more than a millisecond. In good rooms the walls are bumpy -and who wants to record in bad rooms? In trying to reduce the problem to manageable size we have substituted a single 20 microsecond wide delay for the broader shape of a natural reflection. The difference in sound is large. Ironically the higher the bandwidth of the electronic reverberator the greater is the discrepancy, and the more unnatural the simple delay will sound. In addition there is a great deal of what looks like noise throughout the diagram, and at later times -see figure 2 -this noise begins to dominate the whole response. These wiggles are reflections from objects and surfaces of small size. These small reflections are vital to the perceived smoothness and diffusion in the reverberation. MULTIPLE SOURCES The impulse responses of rooms are chaotic, in that the impulse response depends strongly on the position of the source and the listener. Anytime you move either the microphone or the position of the sound source the impulse response changes dramatically, as does the timbre of any music convolved with this impulse response. Each musician will have their timbre modified by the room in a different way, and in addition each of them will make small movements to their instruments as they play. It is the superposition of all these changing impulse responses which we recognize as the sound of a good hall. To emulate it electronically we need a separate reverberation device for each instrument, each using an impulse response which changes slightly with time. It is this requirement for a great multitude of input channels which limits the performance of current reverberation devices most strongly. When' you have only one or two input channels the characteristic timbre of the reverberation device will always be audible, since the same timbre is applied to everything. We can do nothing about this problem, so lets ignore it for the moment. We should not complain, since the coloration which must result from a lack of multiple sources can mask colorations which arise from trying to generate reverberation with a practical processor. LETS DO IT ANYWAY -WITH RECIRCULA TION The process of convolution described above is a method of generating reverb which uses FIR or finite impulse response filters. FIR's are all limited by the number of delays you can sum in a practical processor. In digital filters involving long delay times or low frequencies it is usually better to use feedback or recirculation to get the response you want. Recirculated delays form IIR or infinite impulse response filters. Can we build a reverberator with IIR filters? The idea is promising, since such filters have inherently an exponentially decaying impulse response, and one multiply and one delay can account for many reflections. REVERBERATION WITH ACOUSTIC AND MAGNETIC DELAYS This is not a new problem. Recirculation of delayed sound has been used to emulate reverberation for many years. I will pick up the story with the work of Axon, Gilford, and Shorter at the BBC in the 1950's. They used acoustic delays (sound traveling in a pipe) and magnetic recorders to extensively study the reverberant behavior of recirculation. The simplest recirculator is a single delay with feedback. Figure 3 -from Schroeder (1962). If the attenuation (the coefficient of the multiply) is g, the attenuation in decibels each time through the loop is -20*log(g). Thus if the delay line has length t, the reverb time (the time it takes for a sound to decay 60dB) is T = t * 60/201og(l/g) = 3t/log(l/g) seconds. The impulse response and the tone response are given in the figure. The tone response is the familiar "comb filter", where the sharpness of the teeth depends on the attenuation g. If g is nearly one -to get a long reverb time the teeth are very sharp indeed. Input frequencies which fall on the peaks of the response will reverberate. All others decay away quickly. Any variation in the frequency response of the delay line will cause a large change in the reverb time of the peaks as a function of frequency. Axon et al point out that as a reverberator this system has two shortcomings. 1. The impulse response is not dense enough -individual echos can be heard as flutter, and 2. the timbre is very strongly colored by the fundamental resonance of the filter and its harmonics -which are the only frequencies which reverberate! If we make the delay longer the fundamental frequency becomes lower so the modal density (the number of comb teeth per octave ) becomes higher, but the time density of reflections becomes lower. We can add some more taps to the delay, while keeping the feedback confined only to the last Tap. Figure 4. This gives the impulse response shown. This arrangement is promising. We can increase the density of the impulse response simply by adding taps. Notice however that the impulse response has constant density with time once the pattern of taps is established it remains constant until the sound has decayed entirely. How about using two taps to derive the feedback? Figure 5. Note that now the impulse response becomes more dense as time progresses much more similar to the natural case. Unfortunately the amplitudes of the individual reflections fluctuate wildly. The same or worse fluctuation occurs in the tone response. In fact as the feedback is increased a few prominent frequencies will have very long reverb times, or even oscillate. Such a reverberator sounds very similar to a PA system running just below feedback. Axon and Gilford concluded that feedback should always come from a single tap, because the coloration and the stability of such as system is always predictable. Later work at the BBC (see Spring and Gilford) investigated the results of randomly switching or blending between two separate taps for feedback, with more natural results. Axon et al also tried putting several combs such as figure 4 in parallel, and this was the basis of a magnetic reverberation system built at the BBC. Several parallel comb filters of different lengths with multiple taps involving non commeasureate lengths have many advantages over simpler systems. The density of modes in the tone frequency response can be made large, as can the density of reflections in the impulse response. Again later experiments at the BBC with randomly varying the position of the delay taps resulted in significant improvements. Note that for any parallel system the number of reflections in any interval of time is proportional to the number of sections. Achieving twice the impulse density requires twice as much computation. We really would like to put reverberators in series. DIGITAL REVERBERATION Although the analysis of Axon and Gilford was insightful their hardware was limited and expensive. Electronic reverberation requires delay units with excellent signal to noise ratio, low distortion, and completely flat frequency response. These are properties of digital systems. It is no accident that the first digital audio product was a delay line, and the first real-time use of digital signal processing in audio was for reverberation. (see Blesser and Bader). M.R. Schroeder was the first to try making artificial reverberation through computer manipulation of digitized sounds. Schroeder's computer at Bell Labs was capable of working only in non-real time on relatively short samples of sound. However he made an exceedingly important contribution by recognizing the usefulness of IlR filters with all-pass or flat tone frequency response in the synthesis of reverberation. Schroeder pointed out that if you add a negative feed forward path around a comb filter the frequency response can be made flat. Figure 6. Note that although the steady state tone response is flat, the phase response, and the response to a rapidly varying signal such as music, is not flat at all. However all pass filters have a much smaller effect on timbre than a comb of similar delay and reverb time. The major advantage of all-pass filters is that they may be wired in SERIES. Ordinary comb filters placed in series sound very bad, since only tones which just happen to resonate in all of them will pass through. All pass filters do not have this problem -as many as you want can be placed in series and they will still pass all frequencies. The impulse response can become quite complex, since the number of impulses in any given time is multiplied in each section. Schroeder found that if sound were passed through two or more all pass filters in series after going through a network of comb filters in parallel a much denser impulse response results. Figure 7.