286
Technique 3—Nonlinear Functions
T
0
(
x
) = 1
T
1
(
x
) =
x
T
2
(
x
) = 2
x
2
−
1
T
3
(
x
) = 4
x
3
−
3
x
T
4
(
x
) = 8
x
4
−
8
x
2
+ 1
T
5
(
x
) = 16
x
5
−
20
x
3
+ 5
x
T
6
(
x
) = 32
x
6
−
48
x
4
+ 18
x
2
−
1
T
7
(
x
) = 64
x
7
−
112
x
5
+ 56
x
3
−
7
x
T
8
(
x
) = 128
x
8
−
256
x
6
+ 160
x
4
−
32
x
2
+ 1
T
9
(
x
) = 256
x
9
−
576
x
7
+ 432
x
5
−
120
x
3
+ 9
x
Figure 19.5
The first ten Chebyshev polynomials.
Figure 19.6
Doubling.
For sound design they offer a great shortcut in the synthesis
method of waveshaping. If a pure sinusoidal wave with frequency
f
is applied, the result is a harmonically shifted version at
nf
for
T
n
. The amplitude of the new harmonic can be made to depend
on the input amplitude too. Let’s ignore
T
0
and
T
1
since one has
a range that’s a constant and the other is the identity (which
gives the input signal). But we can look at the first useful one,
T
2
, which is a frequency doubler. For practical purposes we can
ignore the multiplier and offset and reduce the analysis to that
of
x
2
.
time
frequency
400
1.000
801
0.560
Figure 19.7
The second Chebyshev polynomial
T
2
creates a 2
f
component from
f
.
We already looked at the properties of squaring a signal in the context of
envelope curves. When a sine wave is squared the result is a new sine wave at
19.2 Chebyshev Polynomials
287
twice the frequency raised above zero. A way of looking at this is that since
x
2
is the same as
x
×
x
then we are multiplying, or modulating, a signal with
itself. In the frequency domain this gives us a sum and difference. The sum will
be
x
+
x
= 2
x
, making twice the original frequency. The difference,
x
−
x
= 0,
gives a frequency of zero, or a DC offset. Another clue is to remember that
the squares of negative numbers are positive, so for a bipolar input we will
only get a unipolar output, and so the output must always be above zero. The
patch of figure 19.6 shows a squared sine added to an unmodified copy. A
object removes the DC offset, so the signal sits around zero as seen in the
time domain graph of figure 19.7. In the right-hand graph we see a spectrum
snapshot showing the new harmonic at twice the input frequency.
Figure 19.8
A third harmonic.
We can extend this principle to get the second, third,
and higher harmonics. For the first few Chebyshev polyno-
mials it’s not too difficult to implement them directly using
basic arithmetic objects. Figure 19.8 shows a patch for
T
3
using a few multiply operations. Chebyshev polynomials
are alternately odd and even functions. Only the functions
that are odd contain the original frequency; those that are
even produce the first harmonic instead. This one is odd
because it implements 4
x
3
+ 3
x
. In this example we can
blend between the fundamental and second harmonic by
varying the amplitude, so there is no need to explicitly mix
in a copy of the driving sinusoid if we need it. To demon-
strate the rapid growth of complexity, one more example
for
T
4
is given in figure 19.9.
Keypoint
Chebyshe
v
polynomials can be used to add specific harmonics.
Figure 19.9
Chebyshev
T
4
.
It’s an even function since
T
4
(
x
) = 8
x
4
−
8
x
2
+ 1
contains only even coefficients (plus a constant we can
ignore). Notice that every other coefficient term is sub-
tracted or added to the previous. This causes some har-
monics to be created out of phase and cancel with others.
Because of this the output amplitude is always within a
normalised range. In fact Chebyshev polynomials are spe-
cial, carefully constructed cases of more general rules that
let us predict a spectrum from any polynomial function.
By combining polynomials we can theoretically produce
any spectra at a much lower cost than using oscillators
additively. Furthermore, they can then be factored and
simplified to produce a single polynomial (perhaps with
a great many terms) that will produce the spectrum we
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