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WAVESHAPING
Waveshaping is in some ways a relation of modulation techniques such as frequency or phase modulation. Waveshaping can create quite dramatic sound transformations through the application of a very simple process. In FM (frequency modulation) modulation synthesis occurs between two oscillators, waveshaping is implemented using a single oscillator (usually a simple sine oscillator) and a so-called 'transfer function'. The transfer function transforms and shapes the incoming amplitude values using a simple look-up process: if the incoming value is x, the outgoing value becomes y. This can be written as a table with two columns. Here is a simple example:
| Incoming (x) Value | Outgoing (y) Value |
| -0.5 or lower
|
-1 |
| between -0.5 and 0.5
|
remain unchanged
|
| 0.5 or higher
|
1 |
Illustrating this in an x/y coordinate system results in the following graph:
Basic Implementation Model
Although Csound contains several opcodes for waveshaping, implementing waveshaping from first principles as Csound code is fairly straightforward. The x-axis is the amplitude of every single sample, which is in the range of -1 to +1. This number has to be used as index to a table which stores the transfer function. To create a table like the one above, you can use Csound's sub-routine GEN07. This statement will create a table of 4096 points with the desired shape:
giTrnsFnc ftgen 0, 0, 4096, -7, -0.5, 1024, -0.5, 2048, 0.5, 1024, 0.5
Now two problems must be solved. First, the index of the function table is not -1 to +1. Rather, it is either 0 to 4095 in the raw index mode, or 0 to 1 in the normalized mode. The simplest solution is to use the normalized index and scale the incoming amplitudes, so that an amplitude of -1 becomes an index of 0, and an amplitude of 1 becomes an index of 1:
aIndx = (aAmp + 1) / 2
The other problem stems from the difference in the accuracy of possible values in a sample and in a function table. Every single sample is encoded in a 32-bit floating point number in standard audio applications - or even in a 64-bit float in recent Csound. A table with 4096 points results in a 12-bit number, so you will have a serious loss of accuracy (= sound quality) if you use the table values directly. Here, the solution is to use an interpolating table reader. The opcode tablei (instead of table) does this job. This opcode then needs an extra point in the table for interpolating, so we give 4097 as the table size instead of 4096.
This is the code for simple waveshaping using our transfer function which has been discussed previously:
EXAMPLE 04E01_Simple_waveshaping.csd
<CsoundSynthesizer>
<CsOptions>
-odac
</CsOptions>
<CsInstruments>
sr = 44100
ksmps = 32
nchnls = 2
0dbfs = 1
giTrnsFnc ftgen 0, 0, 4097, -7, -0.5, 1024, -0.5, 2048, 0.5, 1024, 0.5
giSine ftgen 0, 0, 1024, 10, 1
instr 1
aAmp poscil 1, 400, giSine
aIndx = (aAmp + 1) / 2
aWavShp tablei aIndx, giTrnsFnc, 1
outs aWavShp, aWavShp
endin
</CsInstruments>
<CsScore>
i 1 0 10
</CsScore>
</CsoundSynthesizer>
Powershape
The powershape opcode performs waveshaping by simply raising all samples to the power of a user given exponent. Its main innovation is that the polarity of samples within the negative domain will be retained. It simply performs the power function on absolute values (negative values made positive) and then reinstates the minus sign if required. It also normalises the input signal between -1 and 1 before shaping and then rescales the output by the inverse of whatever multiple was required to normalise the input. This ensures useful results but does require that the user states the maximum amplitude value expected in the opcode declaration and thereafter abide by that limit. The exponent, which the opcode refers to as 'shape amount', can be varied at k-rate thereby facilitating the creation of dynamic spectra upon a constant spectrum input.
If we consider the simplest possible input - a sine wave - a shape amount of '1' will produce no change. (Raising any value to the power of 1 leaves that value unchanged.)
A shaping amount of 2.5 will visibly 'squeeze' the waveform as values less than 1 become increasingly biased towards the zero axis.
Much higher values will narrow the positive and negative peaks further. Below is the waveform resulting from a shaping amount of 50.
Shape amounts less than 1 (but greater than zero) will give the opposite effect of drawing values closer to -1 or 1. The waveform resulting from a shaping amount of 0.5 shown below is noticeably more rounded than the sine wave input.
Reducing shape amount even closer to zero will start to show squaring of the waveform. The result of a shape amount of 0.1 is shown below.
The sonograms of the five examples shown above are as shown below:
As power (shape amount) is increased from 1 through 2.5 to 50, it can be observed how harmonic partials are added. It is worth noting also that when the power exponent is 50 the strength of the fundamental has waned somewhat. What is not clear from the sonogram is that the partials present are only the odd numbered ones. As the power exponent is reduced below 1 through 0.5 and finally 0.1, odd numbered harmonic partials again appear but this time the strength of the fundamental remains constant. It can also be observed that aliasing is becoming a problem as evidenced by the vertical artifacts in the sonograms for 0.5 and in particularl 0.1. This is a significant concern when using waveshaping techniques. Raising the sampling rate can provide additional headroom before aliasing manifests but ultimately subtlety in waveshaping's use is paramount.
Distort
The distort opcode, authored by Csound's original creator Barry Vercoe, was originally part of the Extended Csound project but was introduced into Canonical Csound in version 5. It waveshapes an input signal according to a transfer function provided by the user using a function table. At first glance this may seem to offer little more than what we have already demonstrated is easily possible from first principles but it offers a number of additional features that enhance its usability. The input signal first has soft-knee compression applied before being mapped through the transfer function. Input gain is also provided via the 'distortion amount' input argument and this provides dynamic control of the waveshaping transformation. The result of using compression means that spectrally the results are better behaved than is typical with waveshaping. A common transfer function would be the hyperbolic tangent (tanh) function. Csound now possesses an GEN routine GENtanh for the creation of tanh functions:
GENtanh f # time size "tanh" start end rescale
By adjusting the 'start' and 'end' values we can modify the shape of the tanh transfer function and therefore the aggressiveness of the waveshaping. ('Start' and 'end' values should be the same absolute values and negative and positive respectively if we want the function to pass through the origin from the lower left quadrant to the upper right quadrant.)
Start and end values of -1 and 1 will produce a gentle 's' curve.
This represents only a very slight deviation from a straight line function from (-1,-1) to (1,1) - which would produce no distortion - therefore the effects of the above used as a transfer function will be extremely subtle.
Start and end points of -10 and 10 will produce a much more dramatic curve and more dramatic waveshaping:
f 1 0 1024 "tanh" -5 5 0
Note that the GEN routine's p7 argument for rescaling is set to zero ensuring that the function only ever extends from -1 and 1. The values provided for 'start' and 'end' only alter the shape.
In the following test example a sine wave at 200 hz is waveshaped using distort and the tanh function shown above.
EXAMPLE 04E02_Distort_1.csd
<CsoundSynthesizer> <CsOptions> -dm0 -odac </CsOptions> <CsInstruments> sr = 44100 ksmps =32 nchnls = 1 0dbfs = 1 giSine ftgen 1,0,1025,10,1 ; sine function giTanh ftgen 2,0,257,"tanh",-10,10,0 ; tanh function instr 1 aSig poscil 1, 200, giSine ; a sine wave kAmt line 0, p3, 1 ; rising distortion amount aDst distort aSig, kAmt, giTanh ; distort the sine tone out aDst*0.1 endin </CsInstruments> <CsScore> i 1 0 4 </CsScore> </CsoundSynthesizer>
The resulting sonogram looks like this:
As the distort amount is raised from zero to 1 it can be seen from the sonogram how upper partials emerge and gain in strength. Only the odd numbered partials are produced therefore over the fundemental at 200 hz partials are present at 600, 1000, 1400 hz and so on. If we want to restore the even numbered partials we can simultaneously waveshape a sine at 400 hz, one octave above the fundemental as in the next example.
EXAMPLE 04E03_Distort_2.csd
<CsoundSynthesizer> <CsOptions> -dm0 -odac </CsOptions> <CsInstruments> sr = 44100 ksmps =32 nchnls = 1 0dbfs = 1 giSine ftgen 1,0,1025,10,1 giTanh ftgen 2,0,257,"tanh",-10,10,0 instr 1 kAmt line 0, p3, 1 ; rising distortion amount aSig poscil 1, 200, giSine ; a sine aSig2 poscil kAmt*0.8,400,giSine ; a sine an octave above aDst distort aSig+aSig2, kAmt, giTanh ; distort a mixture of the two sines out aDst*0.1 endin </CsInstruments> <CsScore> i 1 0 4 </CsScore> </CsoundSynthesizer>
The higher of the two sines is faded in using the distortion amount control so that when distortion amount if zero we will be left with only the fundamental. The sonogram looks like this:
What we hear this time is something close a sawtooth waveform with a rising low-pass filter. The higher of the two input sines at 400 hz will produce overtones at 1200, 2000, 2800... thereby filling in the missing partials.
REFERENCES
Distortion Synthesis - a tutorial with Csound examples by Victor Lazzarini http://www.csounds.com/journal/issue11/distortionSynthesis.html