[Martin Brown]

On 03/03/2011 13:25, Arny Krueger wrote:
"Randy wrote in message
m
On 03/02/2011 09:37 PM, David Nebenzahl wrote:
On 3/2/2011 5:09 PM Dick Pierce spake thus:

Arny Krueger wrote:

"Sergey wrote in message

Music is _NOT_ a pure sinusoidal waves and there are
other things like attack, shape etc.

Actually, every music wave form can be entirely
represented as a collection of enveloped sine wave.

Not even "enveloped" sine waves: simply sine waves.

Yes. Just what is an "enveloped" sine wave anyway, pray
tell?

Amplitude-modulated.

Right.

I would like to hear how one simulates say piano notes by means of just
linear mixing of continuous sine waves.

A quick handwaving version of how you do it is to choose a suitable
mixture of sinewaves centred around the fundamental and its harmonics
with the phases tweaked to have a sharp attack and a slow decay around
the peak envelope position and to cancel elsewhere. It would not be an
efficient representation of a piece of music but it could be done.

A more detailed explanation is that a multiplication of the signal in
the time domain is a convolution in the frequency domain (and vice
versa). That is amplitude modulation of a simple continuous carrier wave
with or without harmonics can be achieved in the frequency domain by
convolving with the Fourier transform of the envelope shape you want to
impose. Shannons sampling theorem for a band limited function and the
fact that the Fourier transform preserves all information allows a
formal proof.

ISTR in the late 70's there was an infamous near unplayable direct
mastered vinyl record of the 1812 which featured on the cover an
electron micrograph of the offending groove. It destroyed expensive
styluses as well as playing through very few times before failing.

Whilst you can produce an unplayable CD with laser readback it never
damages the playback device though it might damage the speakers.

Regards,
Martin Brown