Such filters are inherently unstable; we describe how they may be controlled, not only by the careful selection and constraining of parameters, but also by appropriate and idiomatic performance technique. We draw comparisons between these filters and the behaviour of acoustic instruments, which typically exhibit what we call excitable regions of sonic activity.
In general, any signal processing algorithm which is either time or input dependent may be classed as a non-linear system [Reid and Passin, 1992]. Such behaviour is normally avoided in digital filter algorithms because of the well-known problems of stability [Lynn and Fuerst, 1989]. They do however find a place in non-musical applications such as image processing [Embree and Kimble, 1991].
The problem with conventional synthesis techniques (from our point of view) is that they do not in general exhibit such bounded excitable regions -- a linear oscillator may easily be designed to span and exceed the human audible range. Only with the development of techniques of physical modelling drawing on aspects of chaos theory [Rodet, 1994][Mackenzie, 1995] has the essentially non-linear behaviour of musical instruments (from which much of their expression derives) begun to be realistically captured in synthesis.
Just as an instrumentalist first accepts, and then strives to exploit and transcend the so-called limitations of their instrument, so we in our researches are relatively unconcerned to use mathematical artifices to overcome all analogous limitations in a given filter; rather, we see such limitations as features of an instrument which an idiomatic performance technique can exploit musically. We give one example of this below.
2
X = X - C
new old
This can be shown to be stable for values of 0 < C <2.
In extending this formula to create musically useful waveforms we have added one or more delay elements, leading to a recurrence relation which combines a conventional linear filter with a non-linear delay:
2
X = a X + b X + d X - C
n n-1 n-2 n-L
With careful selection of coefficient values, a
wide range of decaying and sustained sounds, often of richly
time-varying spectral character, can be realized. Above all, these
oscillators exhibit Excitable Regions, outside which they will
either converge rapidly, or exceed the bounds of their numerical
representation (in effect, they will break). Within these regions,
they can be forced into unexpected modes of oscillation, in much the
same way that a wind instrument will generate multiphonics or squeak,
if played incorrectly or unconventionally.
We recast our oscillator to receive an input X(n):
2
Y = a Y + b Y + d Y + X - C
n n-1 n-2 n-L n
This can be described as a conventional linear Infinite Impulse
Response (IIR) filter combined with a non-linear delay length L,
plus the constant term.
2
Y = d Y + X - C
n n-L n
where, for stability, 0 < C <= 1, 0 < d < 1
(i.e. d and C are opposite in sign, but may be of
similar magnitude, |C| usually less than | d |)
The constant term C has the unavoidable side-effect of imposing a DC component, with starting transient, on the output. Although the filter exhibits useful behaviour without it, we have found that the most striking effects are dependent on its inclusion, despite the concomitant need for a DC removal filter on the output.
Although the filter is by its nature sensitive to the input, we have so far had no problems with stability so long as the input is constrained within the range -0.5 < X(n) < 0.5. This is however a maximum; the chaotic behaviour develops rapidly as the input rises above -12dB.
It is impossible to illustrate all the distinctive behaviours of this
filter; a fairly representative selection of results in the frequency
domain is shown;
in each case the spectrum of the output
given a chirp input is shown. We have given examples which fill
a large part of the spectrum, so that the filters' behaviour can be seen
clearly; practical filters would concentrate more of the activity in the
lower frequency ranges.
It will be seen that the filter exhibits distinctly comb-like behaviour as the delay L is increased. However, unlike the response of a linear comb filter, in which the peaks are regular in spacing, width and amplitude, this filter exhibits marked irregularities in all those aspects. Some responses look remarkably like formant regions, indicating that this filter has a distinctive set of personalities, and is therefore able to provide a range of `bodies' for our arithmetical instrument.
However, we consider the single most significant property of this filter to be the fact that, whereas an input sine wave will excite but one resonant peak of a linear comb filter, it will excite several in this case; the filter's Excitable Regions will be multiply activated in a moderately unpredictable way according to the characteristics of the input. Accordingly, we call this and its derivatives an ER filter.
The combination of the linear and non-linear elements can be interpreted in two equally useful ways.
Firstly, d and C can be taken together as a kind of modulation or `fractal' index (whose range must clearly reduced according to the gain of the linear filter), which, as it increases from zero, adds a comb-like irregularity to the linear band-pass response -- the ER filter can be `fractalized' dynamically. For stability the range of the index must be reduced according to the gain of the linear filter; putting it another way, we want to leave room for non-linear behaviour to be introduced by keeping the linear gain within conservative levels.
We note that in this way (by always commencing the fractal index from
zero) the startup transient described above can be obviated -- this is
therefore an idiomatic performance technique for this filter.
Appropriately for this interpretation, the delay L can be set at 3 for
the closest match between chaotic and linear responses
or it
can be set to any reasonable distance as for a comb filter.
Alternatively, the formula can be regarded as a non-linear comb filter with (typically) a low pass or low-frequency band-pass linear filter on the output, serving primarily to remove those extreme high-frequency components which will inevitably arise from the non-linear behaviour.
This further suggests a possible application of the ER filter as the basis of a non-linear reverberator. Our experiments with this are still at an early stage; we are not expecting to invite any qualitative comparisons with normal reverb (although with the interest in some commercial quarters for `retro' sounds such as plate and even spring reverbs we do not dismiss the thought entirely), rather we see this as a means to enhance the sense of the `size' or `presence' of our arithmetical instrument in a manner consistent with our overall approach.
2
Y = a Y + b Y + d Y + X - C
n n-1 n-M n-L n
where 2 <= M < L
In the absence of the non-linear components this amounts to a linear
recursive comb filter, which though unorthodox (the standard comb filter
includes a carefully matched non-recursive delay) seems worthy of a
deeper investigation in itself. In our present context we have noted
that introduction of the non-linear components does, somewhat to our
surprise, substantially preserve the linear frequency response.
The offset M is anti-symmetric in character; given L =
100, for example, the response where M = 10 is the inverse
of the response where M = 90
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