chebyshevpoly — Efficiently evaluates the sum of Chebyshev polynomials of arbitrary order.

The *chebyshevpoly* opcode calculates the value of a polynomial expression with a single a-rate input variable that is made up of a linear combination of the first N Chebyshev polynomials of the first kind. Each Chebyshev polynomial, Tn(x), is weighted by a k-rate coefficient, *kn*, so that the opcode is calculating a sum of any number of terms in the form *kn*Tn(x)*. Thus, the *chebyshevpoly* opcode allows for the waveshaping of an audio signal with a *dynamic* transfer function that gives precise control over the harmonic content of the output.

*ain* -- the input signal used as the independent variable of the Chebyshev polynomials ("x").

*aout* -- the output signal ("y").

*k0, k1, k2, ...* -- k-rate multipliers for each Chebyshev polynomial.

This opcode is very useful for dynamic waveshaping of an audio signal. Traditional waveshaping techniques utilize a lookup table for the transfer function -- usually a sum of Chebyshev polynomials. When a sine wave at full-scale amplitude is used as an index to read the table, the precise harmonic spectrum as defined by the weights of the Chebyshev polynomials is produced. A dynamic spectrum is acheived by varying the amplitude of the input sine wave, but this produces a non-linear change in the spectrum.

By directly calculating the Chebyshev polynomials, the *chebyshevpoly* opcode allows more control over the spectrum and the number of harmonic partials added to the input can be varied with time. The value of each *kn* coefficient directly controls the amplitude of the nth harmonic partial if the input *ain* is a sine wave with amplitude = 1.0. This makes *chebyshevpoly* an efficient additive synthesis engine for N partials that requires only one oscillator instead of N oscillators. The amplitude or waveform of the input signal can also be changed for different waveshaping effects.

If we consider the input parameter *ain* to be "x" and the output *aout* to be "y", then the *chebyshevpoly* opcode calculates the following equation:

y = k0*T0(x) + k1*T1(x) + k2*T2(x) + k3*T3(x) + ...

where the Tn(x) are defined by the recurrence relation

T0(x) = 1,

T1(x) = x,

Tn(x) = 2x*T[n-1](x) - T[n-2](x)

More information about Chebyshev polynomials can be found on Wikipedia at
*http://en.wikipedia.org/wiki/Chebyshev_polynomial*

Here is an example of the chebyshevpoly opcode. It uses the file *chebyshevpoly.csd*.

**Example 103. Example of the chebyshevpoly opcode.**

See the sections *Real-time Audio* and *Command Line Flags* for more information on using command line flags.

<CsoundSynthesizer> <CsOptions> ; Select audio/midi flags here according to platform -odac ;;;RT audio out ;-iadc ;;;uncomment -iadc if RT audio input is needed too ; For Non-realtime ouput leave only the line below: ; -o chebyshevpoly.wav -W ;;; for file output any platform </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 2 ; time-varying mixture of first six harmonics instr 1 ; According to the GEN13 manual entry, ; the pattern + - - + + - - for the signs of ; the chebyshev coefficients has nice properties. ; these six lines control the relative powers of the harmonics k1 line 1.0, p3, 0.0 k2 line -0.5, p3, 0.0 k3 line -0.333, p3, -1.0 k4 line 0.0, p3, 0.5 k5 line 0.0, p3, 0.7 k6 line 0.0, p3, -1.0 ; play the sine wave at a frequency of 256 Hz with amplitude = 1.0 ax oscili 1, 256, 1 ; waveshape it ay chebyshevpoly ax, 0, k1, k2, k3, k4, k5, k6 ; avoid clicks, scale final amplitude, and output adeclick linseg 0.0, 0.05, 1.0, p3 - 0.1, 1.0, 0.05, 0.0 outs ay * adeclick * 10000, ay * adeclick * 10000 endin </CsInstruments> <CsScore> f1 0 32768 10 1 ; a sine wave i1 0 5 e </CsScore> </CsoundSynthesizer>