> I offer below what I hope to be a nice justification for pulse
> width modulation that is independent of physical properties
> of the speaker. If it _is_ correct then it may be useful to
> provide limits to what one might expect in further development.
>
> Consider a pulse of finite duration dt and of height h, h<1.
> (A single scaled datum from an .au file for example). If you
> compare the Fourier coefficients of this pulse to that of a
> pulse of height 1 but duration (h)(dt) (e.g. a pulse width
> modulation) you will note that they are equal through
> the first order in dt. Unfortunately second order (error)
> terms are apparently always of the same sign and cannot be
> eliminated through some rearrangement of the PWM 'bits'.
> They are however about the same order of magnitude as the
> second order effects due to discretization of the original
> data (if my math is correct) [1].
>
> The only reason this has relevance here stems from the fact
> that the ear can be more accurately modeled by its response
> to a frequency distribution than to a sequence of amplitudes.
> In particular the relative phases of the frequency components
> are apparently unimportant, and the ear responds roughly
> logarithmicly (therefore weakly) to the amplitudes of the wave
> form, and consequently to the amplitudes of those components.
> (Please forgive me for neglecting masking effects.)
> Thus it is "legal" to compare only Fourier components as I have
> done and to some extent to neglect second order terms.
>
> The upshot of this is simply that not only can you "trade off"
> bit resolution for higher output frequencies (to 1st order),
> the ear also vastly-under samples the resulting waveform [2].
> In any case this means that the ear hears both waveforms, the
> original and the PWM, as roughly equal, even though in fact
> they may be quite different.
It's a lot easier than this. Yes, pulse width modulation creates
a Fourier series with highr order harmonics. The inductor in the
speaker coil simply cancels these frequencies out (an inductance
resists a dramatic change in current) thru integration.
You are left only with the low frequencies, which is what you're
looking for. The higher order harmonics don't get through.
You are right that the ear hears differenct frequencies in quite
special ways. This is the theory behind spectral encoding, which
is popular in compression methods like MPEG. It isn't especially
relevant in the PC speaker example.
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