Nonlinear ElementsMany musical instrument models require nonlinear elements, such as signal bandwidth, it can cause aliasing in a discrete-time implementation. In the above examples, the nonlinearity also appears inside a feedback loop. This means the bandwidth expansion compounds over time, causing more and more aliasing.
The topic of nonlinear systems analysis is vast, in part because there is no single analytical approach which is comprehensive. The situation is somewhat analogous to an attempt to characterize ``all non-bacterial life''. As a result, the only practical approach is to identify useful classes of nonlinear systems which are amenable to certain kinds of analysis and characterization. In this chapter, we will look at certain classes of memoryless and passive nonlinear elements which are often used in digital waveguide modeling. It is important to keep in mind that a nonlinear element may not be characterized by its impulse response, frequency response, transfer function, or the like. These concepts are only defined, in general, for linear time-invariant systems. However, it is possible to generalize these notions for nonlinear systems using constructs such as Volterra series expansions . However, rather than getting involved with fairly general analysis tools, we will focus instead on approaching each class of nonlinear elements in the manner that best fits that class, with the main goal being to understand its audible effects on discrete-time signals. nonlinear element. Furthermore, many complex nonlinear systems can be broken down into a linear system containing a memoryless nonlinearity. Given a sampled input signal , the output of any memoryless nonlinearity can be written as
Clipping NonlinearityA simple example of a noninvertible (many-to-one) memoryless nonlinearity is the clipping nonlinearity, well known to anyone who records or synthesizes audio signals. In normalized form, the clipping nonlinearity is defined by
Since the clipping nonlinearity abruptly transitions from linear to hard-clipped in a non-invertible, heavily aliasing manner, it is usually desirable to use some form of soft-clipping before entering the hard-clipping range.
Arctangent NonlinearityA simple example of an invertible (one-to-one) memoryless nonlinearity is the arctangent mapping:
Cubic Soft ClipperIn §9.1.6, we used the cubic soft-clipper to simulate amplifier distortion:
Taylor series expansion:
where ``smooth'' means that derivatives of all orders must exist over the range of validity. Derivatives of all orders are obviously needed at by the above expansion, and for the expansion to be valid everywhere, the function must be smooth for all .
Spectrum of a Memoryless NonlinearitiesThe series expansion of a memoryless nonlinearity is a useful tool for quantifying the aliasing caused by that nonlinear mapping when introduced into the signal path of a discrete-time system.
Taylor series expansion such as Eq.(6.18), the simplest nonlinearity is clearly the square law nonlinearity:
series expansion of the arctangent nonlinearity is
soft-clipper, like any polynomial nonlinearity, is defined directly by its series expansion:
In the absence of hard-clipping ( ), bandwidth expansion is limited to a factor of three. This is the slowest aliasing rate obtainable for an odd nonlinearity. Note that smoothing the ``corner'' in the clipping nonlinearity can reduce the severe bandwidth expansion associated with hard-clipping.
nonlinearity in a stable feedback loop preserves stability provided the gain of the nonlinearity is less than one, i.e., . A simple proof for the case of a loop consisting of a continuous-time delay-line and memoryless-nonlinearity is as follows. The delay line can be interpreted as a waveguide model of an ideal string or acoustic pipe having wave impedance and a noninverting reflection at its midpoint. A memoryless nonlinearity is a special case of an arbitrary time-varying gain . By hypothesis, this gain has magnitude less than one. By routing the output of the delay line back to its input, the gain plays the role of a reflectance at the ``other end'' of the ideal string or acoustic pipe. We can imagine, for example, a terminating dashpot with randomly varying positive resistance . The set of all corresponds to the set of real reflection coefficients in the open interval . Thus, each instantaneous nonlinearity-gain corresponds to some instantaneously positive resistance . The whole system is therefore passive, even as changes arbitrarily (while remaining positive). (It is perhaps easier to ponder a charged capacitor terminated on a randomly varying resistor .) This proof method immediately extends to nonlinear feedback around any transfer function that can be interpreted as the reflectance of a passive physical system, i.e., any transfer function for which the gain is bounded by 1 at each frequency, viz., . The finite-sampling-rate case can be embedded in a passive infinite-sampling-rate case by replacing each sample with a constant pulse lasting seconds (in the delay line). The continuous-time memoryless nonlinearity is similarly a held version of the discrete-time case . Since the discrete-time case is a simple sampling of the (passive) continuous-time case, we are done.
nonlinear elements in a digital waveguide model:
- Verify that aliasing can be heard and sounds bad before working to get rid of it.
- Aliasing (bandwidth expansion) is reduced by smoothing ``corners'' in the nonlinearity.
- Consider an oversampling factor for nonlinear subsystems sufficient to accommodate the bandwidth expansion caused by the nonlinearity.
- Make sure there is adequate lowpass filtering in a feedback loop containing a nonlinearity.
- oversampling, and
- a lowpass filter to after the nonlinearity.
String Coupling Effects