# Manipulating the derivative of a sampled sound

Started by May 19, 2006
```Hi. This is just a thought exercise, which could easily be downright
silly, badly thought out, entirely pointless, or generally a waste of
time for any number of other reasons. But that never stopped me before.

It's possible to store sampled versions of waveforms, single samples
which are looped, long samples of acyclic sounds, or various
representations inbetween.

The standard method of storing a sample is to store the actual sample
values. An alternative method of storing a sample as delta modulation,
where the first actual sample values are stored, and then store for
each successive sample the difference from the previous sample. In
effect, the sample stored approximates (or, is a sampled version of?)
the differential of the stored sample. Given that we have the first
sample value, we can recover the original signal, which is effectively
an integration using the stored sample to solve for the constant of
integration.

It would also be possible to perform delta modulation on the delta
modulation, which would give a sampled approximation (or version) of
the second derivative. Provided that the first values of the sample and
the second derivative were stored, it would be possible to recover the
original signal.

I wondered if it would be possible to change the nature of the sound by
manipulating the second (or higher) derivatives directly. For a sine
wave, simply multiplying the derivative by a constant achieves nothing
useful because diff( a * sin( b * x ) ) dx = a * b * cos( b * x ). If
we then multiply the derivative by a new constant d, and integrate,
assuming an initial phase of 0, we get: a * d * sin( b * x ). Or, we've
multiplied the original signal by the same constant. Not very useful.
Same applies for the more general case a * sin( b * x + c ); Since any
periodic function can be expressed as a Fourier series, if this applies
to a sine wave, then by induction it will apply to any periodic
waveform. No?

Finally I get to the question. Are there any easy manipulations of the
derivatives (first, second, any) which will produce "interesting" and
potentially musically useful changes to sounds? What would happen if I
filtered the sampled second derivative, say with standard highpass,
lowpass, notch, etc., filters?

Cheers,

Ross-c

```
```Ross Clement (Email address invalid - do not use) wrote:
...
> I wondered if it would be possible to change the nature of the sound by
> manipulating the second (or higher) derivatives directly. For a sine
> wave, simply multiplying the derivative by a constant achieves nothing
> useful because diff( a * sin( b * x ) ) dx = a * b * cos( b * x ). If
> we then multiply the derivative by a new constant d, and integrate,
> assuming an initial phase of 0, we get: a * d * sin( b * x ). Or, we've
> multiplied the original signal by the same constant. Not very useful.
> Same applies for the more general case a * sin( b * x + c ); Since any
> periodic function can be expressed as a Fourier series, if this applies
> to a sine wave, then by induction it will apply to any periodic
> waveform. No?

Well... yes. Your argument is correct for sinusoidals. For broadband
signals, things become more involved (although it is based on your
idea). As far as I recall,

FT{d^n f/dt^n } = w^n FT{f(t)}

So basically, the more you differentiate, the more you enhance the
high-frequency parts of the spectrum.

However, it ought to be interesting to hear what effects one might
get during integration (n<0), and "fractional derivatives", where n is
a
non-integer number.

> Finally I get to the question. Are there any easy manipulations of the
> derivatives (first, second, any) which will produce "interesting" and
> potentially musically useful changes to sounds?

These are two questions in one. First, do these techniques
produce interesting results? I don't know. You have to try.
Second: are such techniques easy to implement? The non-integer n
integration/differentiation might not be, in time domain. Maybe you
are able to design some sort of approximation to an FIR filter
that does the job.

> What would happen if I
> filtered the sampled second derivative, say with standard highpass,
> lowpass, notch, etc., filters?

Don't have the faintest clue. You'll have to try.

It's a fascinating problem. I remember I enjoyed playing with these
sorts of techniques when I first learned image processing.

Rune

```
```Ross Clement (Email address invalid - do not use) wrote:
..
> The standard method of storing a sample is to store the actual sample
> values. An alternative method of storing a sample as delta modulation,
> where the first actual sample values are stored, and then store for
> each successive sample the difference from the previous sample. In
> effect, the sample stored approximates (or, is a sampled version of?)
> the differential of the stored sample.

It's an approximation. The difference filter has impulse response h_1 =
[1/2 -1/2] (scaled to have 0dB gain at Nyquist). The differentiator
filter is a antisymmetric acausal IIR filter with impulse response h_D
= [ ... -1/4, 1/3, -1/2, 1, 0, -1, 1/2, -1/3, 1/4, ... ] (ellipses
denote the obvious extension to infinity). You can get a better
approximation by windowing this impulse response.

> Given that we have the first
> sample value, we can recover the original signal, which is effectively
> an integration using the stored sample to solve for the constant of
> integration.

Yes, this perfectly reconstructs to original sequence. Even though the
first difference filter h_1 has a zero at DC (and therefore is not
invertible) you can reconstruct the original sequence by storing the
first sample and integrating from there on. Note that integration is an
instable filter, there will be numerical issues to consider in the
reconstruction process.

> It would also be possible to perform delta modulation on the delta
> modulation, which would give a sampled approximation (or version) of
> the second derivative. Provided that the first values of the sample and
> the second derivative were stored, it would be possible to recover the
> original signal.

The difference of the difference is called (wait for it) the second
difference. The n-th difference filter h_n is defined as the n-fold
convolution of the first difference, ie.

h_n = h_1^(*n).

Differencing is widely used in time series analysis to remove
polynomial trends (an n-th difference filter removes an n-th order
polynomial from the data - hence the need to store the n initial values
for the reconstruction).

> Finally I get to the question. Are there any easy manipulations of the
> derivatives (first, second, any) which will produce "interesting" and
> potentially musically useful changes to sounds? What would happen if I
> filtered the sampled second derivative, say with standard highpass,
> lowpass, notch, etc., filters?

For linear processing, there won't be much difference (see caveat
below), because linear processes commute. In other words, applying a
filter g to the differenced signal and then integrating is the same as
applying g to the original sequence:

h_n * g * h_n^-1 = h_n  * h_n^-1 * g = g

(provided you store the first sample - otherwise you loose the DC
content), where h_n and h_n^-1 denote differencing n times and
integrating n times (plus adding the integration constant).

The caveat is: since audio signals tend to contain less high-frequency
than low-frequency content, the differenced signal will tend to have
lower amplitude - this can be exploited for numerical benefit, as is
described by Keith here:

There exists other non-linear applications for the differencing filter
in audio: namely as a sidechain filter in digital audio dynamic range
compression units. The differencing removes low frequency content and
thus avoids low-frequency modulation of the compressor gain. Another
application could be to split the signal with the difference filter
into high- and low-pass bands, then do some non-linear processing to
the low-pass signal and finally add the processed low-pass signal to
the original high-pass signal. This attenuates aliasing due to
non-linear high-frequency distortion.

>
> Cheers,
>
> Ross-c

Regards,
Andor

```
```"Ross Clement (Email address invalid - do not use)" <clemenr@wmin.ac.uk>
wrote in message
> Hi. This is just a thought exercise, which could easily be downright
> silly, badly thought out, entirely pointless, or generally a waste of
> time for any number of other reasons. But that never stopped me before.
>
> It's possible to store sampled versions of waveforms, single samples
> which are looped, long samples of acyclic sounds, or various
> representations inbetween.
>
> The standard method of storing a sample is to store the actual sample
> values. An alternative method of storing a sample as delta modulation,
> where the first actual sample values are stored, and then store for
> each successive sample the difference from the previous sample. In
> effect, the sample stored approximates (or, is a sampled version of?)
> the differential of the stored sample. Given that we have the first
> sample value, we can recover the original signal, which is effectively
> an integration using the stored sample to solve for the constant of
> integration.
>
> It would also be possible to perform delta modulation on the delta
> modulation, which would give a sampled approximation (or version) of
> the second derivative. Provided that the first values of the sample and
> the second derivative were stored, it would be possible to recover the
> original signal.
>
> I wondered if it would be possible to change the nature of the sound by
> manipulating the second (or higher) derivatives directly. For a sine
> wave, simply multiplying the derivative by a constant achieves nothing
> useful because diff( a * sin( b * x ) ) dx = a * b * cos( b * x ). If
> we then multiply the derivative by a new constant d, and integrate,
> assuming an initial phase of 0, we get: a * d * sin( b * x ). Or, we've
> multiplied the original signal by the same constant. Not very useful.
> Same applies for the more general case a * sin( b * x + c ); Since any
> periodic function can be expressed as a Fourier series, if this applies
> to a sine wave, then by induction it will apply to any periodic
> waveform. No?
>
> Finally I get to the question. Are there any easy manipulations of the
> derivatives (first, second, any) which will produce "interesting" and
> potentially musically useful changes to sounds? What would happen if I
> filtered the sampled second derivative, say with standard highpass,
> lowpass, notch, etc., filters?
>
> Cheers,
>
> Ross-c
>
I would say anything with differentiation is too noisy. In control systems
we often talk of PID but the D term is not a real differentiator - only band
limited.

M.P

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```