Waveshaping synthesis

As a metaphor to understand the fundamentals of waveshaping synthesis, imagine a note played on an electric stringed instrument connected to a vacuum-tube amplifier. If the volume knob of the amplifier is increased to its maximum, the vacuum-tubes will be saturated and the sound will clip and if the amplitude of the note is increased at its origin, before entering the amplifier, then the output will clip even more. If the note is a sinusoid, the louder the input, the more squared the output wave will be. If the note has a complex spectrum, then the output will be a signal blurred by distortion. 

Amplitude sensitivity is one of the key features of waveshaping s)mthesis. In the above example, the amount of amplification (or distortion) is proportional to the level of the input signal. The spectrum of the output therefore becomes richer as the level of the input is 

A particular family of polynomials called Chebyshev polynomials of the first kind has been widely used for specifying transfer functions for waveshaping synthesis. Chebyshev polynomials are represented as follows T fx) where k represents the order of the polynomial and X represents a sinusoid. Chebyshev polynomials have the useful property that when a cosine wave with amplitude equal to one is applied to Tj-fx), the resulting signal is a sinewave at the kth harmonic. For example, if a sinusoid of amplitude equal to one is applied to a transfer function given by the seventh-order Chebyshev polynomial, the result will be a sinusoid at seven times the frequency of the input. Chebyshev polynomials for Tj to Tjo are given in Appendix 1. 

Because each separate polynomial produces a particular harmonic of the input signal, a certain spectrum composed of various harmonics can be obtained by summing a weighted combination of Chebyshev polynomials, one for each desired harmonic (see Appendix 1 for 

Variations on the amplitude of the input signal can activate different portions of the waveshaper. Waveshaping synthesis is therefore very convenient to synthesise sounds with a considerable amount of time-varying spectral components. 

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Distorting the ramp a little would result in a distortion of the output wave. We can store any shape in the wavetable, resulting in a wide variety of possible outputs. Accessing a wavetable (often a simple waveform) with another (usually simple) waveform is called waveshaping synthesis. 

One common form of waveshaping synthesis, called Frequency Modulation (FM), uses sine waves for both input address and wavetable waveforms. 

To a large extent the best transfer functions for waveshaping synthesis are described using polynomials. The amplitude of the signal input to the waveshaper is represented by the variable x and the output is denoted by F(x), where F is the function and d are amplitude coefficients ... 

Whereas AM, FM and waveshaping are synthesis techniques inherited from the world of analog synthesis, Walsh synthesis is an inherently digital technique. Walsh synthesis works based upon square waves instead of sinewaves it creates other waveforms by combining square waves. Inasmuch digital systems work with only two numbers at their most fundamental level (i.e. 0 and 1), square waves are very straightforward to manipulate digitally. 

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