"Digital Synthesis of Complex Waveforms" – a waveshaping synthesizer hardware was designed and built using a microcomputer to provide facilities including frequency control over a four octave range, note duration control, envelope shaping and selection, dynamic spectrum generation, and voice selection and mixing.
Implementing Chebyshev polynomials of the first kind (a set of orthogonal polynomials), the synthesizer was designed to incorporate hardware dynamic waveshaping synthesis for reproduction of any static and dynamic harmonic spectra.
Chebyshev polynomials have the unique property of multiplying the frequency of a sine wave input. When fed a sine wave with amplitude 1.0, the output sine wave will have a frequency N times higher, with N being the degree of the polynomial. This makes Chebyshev polynomials act like frequency multipliers. For sine wave inputs less than 1.0 amplitude, the output contains a complex mixture of harmonics.
To achieve a target harmonic profile, multiple Chebyshev polynomials can be combined, each weighted by the desired amount of that harmonic. This allows precise tailoring of the harmonic signature for a chosen timbre. In summary, Chebyshev polynomials enable versatile harmonic shaping through frequency multiplication and distortion effects.
Performing the complex Chebyshev polynomial calculations at real-time audio rates can be computationally demanding. To optimize efficiency, the polynomial mappings are pre-computed and stored in a lookup table. During sound synthesis, the system simply retrieves the amplitude of the input sine wave and finds the corresponding polynomial result value in the table. This lookup approach drastically reduces computations required at run time. By leveraging pre-processing and table indexing, we can alleviate computational load while still harnessing the harmonic shaping capabilities of Chebyshev polynomials for audio signals. The lookup methodology streamlines audio rate polynomial evaluation, enabling efficient and flexible harmonic content modification in real-time sound production.
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| Waveform Synthesis |
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| | Excitation | |
| | Table | |
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| +------------------------+-----------------------+ |
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| v |
| +------------------------------------------------+ |
| | Shaping | |
| | Table | |
| +------------------------------------------------+ |
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| v |
| Synthesized waveform g(Θ) |
| where |
| g(Θ) = f(x) and |
| x = cos(Θ) |
+--------------------------------------------------------+
A waveshaping synthesizer consists of:
+---+---+---+---+---+---+
| x | x | x | x | x | x |
+---+---+---+---+---+---+
| | | | | |
v v v v v v
cos(wt) values in range
-1 <= x <= +1
A matrix containing values for a cosine wave over unit period defined as:
x = cos Θ with Θ = wt
where t is time, w the radian frequency and Θ the phase angle.
Thus x varies with time within the range -1 <= x <= +1 or the signed unit interval.
+---+---+---+---+---+---+
| f | f | f | f | f | f |
+---+---+---+---+---+---+
| | | | | |
v v v v v v
Shaping function values
in range [-1, +1]
A 1-D matrix containing values for a shaping function f with values within the signed unit interval.
+------------------------+
| For each x value: |
| f(x) = lookup(x) |
| in Shaping Table |
+------------------------+
|
v
Synthesized waveform g(Θ)
where
g(Θ) = f(x) and
x = cos(Θ)
To synthesize a steady state spectrum:
Successive values f(x) are computed by performing a table lookup in the Shaping Table of successive values of x pulled out of the Excitation Table.
The function f must meet the following conditions:
- f must take a number in the range (-1, +1) i.e. the signed unit interval.
- f must return a result in the signed unit interval.
An infinite range of waveforms including all symmetric waveshapes can be produced using this method.
If the resultant waveform is g(Θ) then g(Θ) = f(x)
but x = cos Θ
Θ = cos^-1(x)
hence f(x) = g(cos^-1(x))
To produce some desired waveshape g(Θ), f(x) can in principle be set as g(cos^-1(x))
The series of functions Tn known as Chebychev polynomials of the first kind can be shown to be very convenient wave-shaping functions.
The nth harmonic is produced when Tn is used as a shaping function.
Remembering that x = cos Θ,
Tn(x) = Tn(cos Θ)
= cos Θ -> the nth harmonic
The trigonometric expression above illustrates the convenience of employing Chebychev polynomials.
T0(x) = cos 0 = 1
T1(x) = cos Θ = x
A recursive formula for Tn + 1, with respect to Tn and Tn-1 can be derived thus:
cos u. cos v = (cos(u+v) + cos(u-v))/ 2
If u = n Θ and v = Θ
cos nΘ.cos Θ = (cos(nΘ+ Θ) + cos(cos nΘ - Θ))/ 2
= (cos(n+1)Θ + cos(n-1)Θ)/ 2
where cos Θ = x and cos nΘ = Tn (cos Θ)
Hence x.Tn(x) = (Tn+1 (x) + Tn-1 (x))/ 2
and Tn+1 (x) = 2x.Tn (x) - Tn-1 (x)
which is the required recursive formula to calculate successive Tn:
T0(x) = 1
T1(x) = x
T2(x) = 2x^2 - 1
T3(x) = 4x^3 - 3x
T4(x) = 8x^4 - 8x^2 + 1
etc.
To create any steady-state spectrum, a linear combination of appropriate weighted functions is required. Thus to synthesize a steady-state square wave spectrum:
f_square = (4/π).(T1(x) + (1/3).T3(x) + (1/5).T5(x) + ...)
Note that T0 represents a constant offset and may be disregarded for this purpose.