Csound

# CONVOLUTION

Convolution is a mathematical procedure whereby one function is modified by another. Applied to audio, one of these functions might be a sound file or a stream of live audio whilst the other will be, what is referred to as, an impulse response file; this could actually just be another shorter sound file. The longer sound file or live audio stream will be modified by the impulse response so that the sound file will be imbued with certain qualities of the impulse response. It is important to be aware that convolution is a far from trivial process and that realtime performance may be a frequent consideration. Effectively every sample in the sound file to be processed will be multiplied in turn by every sample contained within the impulse response file. Therefore, for a 1 second impulse response at a sampling frequency of 44100 hertz, each and every sample of the input sound file or sound stream will undergo 44100 multiplication operations. Expanding upon this even further, for 1 second's worth of a convolution procedure this will result in 44100 x 44100 (or 1,944,810,000) multiplications. This should provide some insight into the processing demands of a convolution procedure and also draw attention to the efficiency cost of using longer impulse response files.

The most common application of convolution in audio processing is reverberation but convolution is equally adept at, for example, imitating the filtering and time smearing characteristics of vintage microphones, valve amplifiers and speakers. It is also used sometimes to create more unusual special effects. The strength of convolution based reverbs is that they implement acoustic imitations of actual spaces based upon 'recordings' of those spaces. All the quirks and nuances of the original space will be retained. Reverberation algorithms based upon networks of comb and allpass filters create only idealised reverb responses imitating spaces that don't actually exist. The impulse response is a little like a 'fingerprint' of the space. It is perhaps easier to manipulate characteristics such as reverb time and high frequency diffusion (i.e. lowpass filtering) of the reverb effect when using a Schroeder derived algorithm using comb and allpass filters but most of these modification are still possible, if not immediately apparent, when implementing reverb using convolution. The quality of a convolution reverb is largely dependent upon the quality of the impulse response used. An impulse response recording is typically achieved by recording the reverberant tail that follows a burst of white noise. People often employ techniques such as bursting balloons to achieve something approaching a short burst of noise. Crucially the impulse sound should not excessively favour any particular frequency or exhibit any sort of resonance. More modern techniques employ a sine wave sweep through all the audible frequencies when recording an impulse response. Recorded results using this technique will normally require further processing in order to provide a usable impulse response file and this approach will normally be beyond the means of a beginner.

Many commercial, often expensive, implementations of convolution exist both in the form of software and hardware but fortunately Csound provides easy access to convolution for free. Csound currently lists six different opcodes for convolution, convolve (convle), cross2, dconv, ftconv, ftmorf and pconvolve. convolve (convle) and dconv are earlier implementations and are less suited to realtime operation, cross2 relates to FFT-based cross synthesis and ftmorf is used to morph between similar sized function table and is less related to what has been discussed so far, therefore in this chapter we shall focus upon just two opcodes, pconvolve and ftconv.

## pconvolve

pconvolve is perhaps the easiest of Csound's convolution opcodes to use and the most useful in a realtime application. It uses the uniformly partitioned (hence the 'p') overlap-save algorithm which permits convolution with very little delay (latency) in the output signal. The impulse response file that it uses is referenced directly, i.e. it does not have to be previously loaded into a function table, and multichannel files are permitted. The impulse response file can be any standard sound file acceptable to Csound and does not need to be pre-analysed as is required by convolve. Convolution procedures through their very nature introduce a delay in the output signal but pconvolve minimises this using the algorithm mentioned above. It will still introduce some delay but we can control this using the opcode's 'ipartitionsize' input argument. What value we give this will require some consideration and perhaps some experimentation as choosing a high partition size will result in excessively long delays (only an issue in realtime work) whereas very low partition sizes demand more from the CPU and too low a size may result in buffer under-runs and interrupted realtime audio. Bear in mind still that realtime CPU performance will depend heavily on the length of the impulse file. The partition size argument is actually an optional argument and if omitted it will default to whatever the software buffer size is as defined by the -b command line flag. If we specify the partition size explicitly however, we can use this information to delay the input audio (after it has been used by pconvolve) so that it can be realigned in time with the latency affected audio output from pconvolve - this will be essential in creating a 'wet/dry' mix in a reverb effect. Partition size is defined in sample frames therefore if we specify a partition size of 512, the delay resulting from the convolution procedure will be 512/sr (sample rate).

In the following example a monophonic drum loop sample undergoes processing through a convolution reverb implemented using pconvolve which in turn uses two different impulse files. The first file is a more conventional reverb impulse file taken in a stairwell whereas the second is a recording of the resonance created by striking a terracota bowl sharply. If you wish to use the three sound files I have used in creating this example the mono input sound file is here and the two stereo sound files used as impulse responses are here and here. You can, of course, replace them with ones of your own but remain mindful of mono/stereo/multichannel integrity.

#### EXAMPLE 05H01_pconvolve.csd

```<CsoundSynthesizer>

<CsOptions>
-odac
</CsOptions>

<CsInstruments>

sr     =  44100
ksmps  =  512
nchnls =  2
0dbfs  =  1

gasig init 0

instr 1 ; sound file player
gasig           diskin2   p4,1,0,1
endin

instr 2 ; convolution reverb
; Define partion size.
; Larger values require less CPU but result in more latency.
; Smaller values produce lower latency but may cause -
; - realtime performance issues
ipartitionsize  =         256
ar1,ar2         pconvolve gasig, p4,ipartitionsize
; create a delayed version of the input signal that will sync -
; - with convolution output
; create a dry/wet mix
outs      aMixL,aMixR
gasig           =         0
endin

</CsInstruments>

<CsScore>
; instr 1. sound file player
;    p4=input soundfile
; instr 2. convolution reverb
;    p4=impulse response file
;    p5=dry/wet mix (0 - 1)

i 1 0 8.6 "loop.wav"
i 2 0 10 "Stairwell.wav" 0.3

i 1 10 8.6 "loop.wav"
i 2 10 10 "Dish.wav" 0.8
e
</CsScore>

</CsoundSynthesizer>
```

## ftconv

ftconv (abbreviated from 'function table convolution) is perhaps slightly more complicated to use than pconvolve but offers additional options. The fact that ftconv utilises an impulse response that we must first store in a function table rather than directly referencing a sound file stored on disk means that we have the option of performing transformations upon the audio stored in the function table before it is employed by ftconv for convolution. This example begins just as the previous example: a mono drum loop sample is convolved first with a typical reverb impulse response and then with an impulse response derived from a terracotta bowl. After twenty seconds the contents of the function tables containing the two impulse responses are reversed by calling a UDO (instrument 3) and the convolution procedure is repeated, this time with a 'backwards reverb' effect. When the reversed version is performed the dry signal is delayed further before being sent to the speakers so that it appears that the reverb impulse sound occurs at the culmination of the reverb build-up. This additional delay is switched on or off via p6 from the score. As with pconvolve, ftconv performs the convolution process in overlapping partitions to minimise latency. Again we can minimise the size of these partitions and therefore the latency but at the cost of CPU efficiency. ftconv's documentation refers to this partition size as 'iplen' (partition length). ftconv offers further facilities to work with multichannel files beyond stereo. When doing this it is suggested that you use GEN52 which is designed for this purpose. GEN01 seems to work fine, at least up to stereo, provided that you do not defer the table size definition (size=0). With ftconv we can specify the actual length of the impulse response - it will probably be shorter than the power-of-2 sized function table used to store it - and this action will improve realtime efficiency. This optional argument is defined in sample frames and defaults to the size of the impulse response function table.

#### EXAMPLE 05H02_ftconv.csd

```<CsoundSynthesizer>

<CsOptions>
-odac
</CsOptions>

<CsInstruments>

sr     =  44100
ksmps  =  512
nchnls =  2
0dbfs  =  1

; impulse responses stored as stereo GEN01 function tables
giStairwell     ftgen   1,0,131072,1,"Stairwell.wav",0,0,0
giDish          ftgen   2,0,131072,1,"Dish.wav",0,0,0

gasig init 0

; reverse function table UDO
opcode tab_reverse,0,i
ifn             xin
iTabLen         =               ftlen(ifn)
iTableBuffer    ftgentmp        0,0,-iTabLen,-2, 0
icount          =               0
loop:
ival            table           iTabLen-icount-1, ifn
tableiw         ival,icount,iTableBuffer
loop_lt         icount,1,iTabLen,loop
icount          =               0
loop2:
ival            table           icount,iTableBuffer
tableiw         ival,icount,ifn
loop_lt         icount,1,iTabLen,loop2
endop

instr 3 ; reverse the contents of a function table
tab_reverse p4
endin

instr 1 ; sound file player
gasig           diskin2   p4,1,0,1
endin

instr 2 ; convolution reverb
; buffer length
iplen   =       1024
; derive the length of the impulse response
iirlen  =       nsamp(p4)
ar1,ar2 ftconv  gasig, p4, iplen,0, iirlen
; delay compensation. Add extra delay if reverse reverb is used.
adel            delay     gasig,(iplen/sr) + ((iirlen/sr)*p6)
; create a dry/wet mix
outs      aMixL,aMixR
gasig           =         0
endin

</CsInstruments>

<CsScore>
; instr 1. sound file player
;    p4=input soundfile
; instr 2. convolution reverb
;    p4=impulse response file
;    p5=dry/wet mix (0 - 1)
;    p6=reverse reverb switch (0=off,1=on)
; instr 3. reverse table contents
;    p4=function table number

; 'stairwell' impulse response
i 1 0 8.5 "loop.wav"
i 2 0 10 1 0.3 0

; 'dish' impulse response
i 1 10 8.5 "loop.wav"
i 2 10 10 2 0.8 0

; reverse the impulse responses
i 3 20 0 1
i 3 20 0 2

; 'stairwell' impulse response (reversed)
i 1 21 8.5 "loop.wav"
i 2 21 10 1 0.5 1

; 'dish' impulse response (reversed)
i 1 31 8.5 "loop.wav"
i 2 31 10 2 0.5 1

e
</CsScore>

</CsoundSynthesizer
```

Suggested avenues for further exploration with ftconv could be applying envelopes to, filtering and time stretching and compressing the function table stored impulse files before use in convolution.

The impulse responses I have used here are admittedly of rather low quality and whilst it is always recommended to maintain as high standards of sound quality as possible the user should not feel restricted from exploring the sound transformation possibilities possible form whatever source material they may have lying around. Many commercial convolution algorithms demand a proprietary impulse response format inevitably limiting the user to using the impulse responses provided by the software manufacturers but with Csound we have the freedom to use any sound we like.