Audio computation signal processing
How sampling theorem used in music instruments?
Any function of time w.r.to amplitude, in the sense that having respected values takes
on the integers design should somehow determine the signals.Consider the following function,
f (n) = 1 ,
n = . . . , −1, 0, 1, . . .
Here f (t) = 1 for all real.It 0perfectly good function too,and nothing about the function’s values at the integer numbers it from the
simpler f (t) = 1.
How to form a digitization?
Audio signals are the secret sample code function. A function that is “possible to sample” should be one for that purpose use some number of values on non-integers
from its values on integers.
It is customary at this point in discussions of computer music to invoke the
famous Nyquist theorem.
How music functionality works from nyquist?
This states (roughly speaking) that if a function is a
finite or unodd infinite combination of normal sine signal none of whose angular frequencies
exceeds π, then, theoretically at least, it is fully determined by the function’s
values on the integers. One possible way of limits are higher- and higher-order polynomial interpolation.
The angular frequency π, called the Nyquist frequency, corresponds to R/2
cycles per second if R is the sample rate. The corresponding period is two
kind of samples. The Nyquist frequency is the best we can do in the sense that any
real sinusoid of higher frequency is equal, at the integers, to form lower than the Nyquist, and it is this lower frequency that will get
reconstructed by the ideal interpolation process.
How to make reconstruction?
For instance, a sinusoid with
angular frequency between π and 2π, say π + ω, can be written as
cos((π + ω)n + φ) = cos((π + ω)n + φ − 2πn)
= cos((ω − π)n + φ)
= cos((π − ω)n − φ)
for all integers n. (If n weren’t an integer the first step would fail.) So a sinusoid
with frequency between π and 2π is equal, on the integers at least, to one with
frequency between 0 and π; higher-frequency one you try to synthesize will come
out your speakers at the wrong frequency—specifically,
you will hear the unique frequency between 0 and π that the higher frequency lands on when reduced in the above way. 0 to ∞ is folded back and forth, in lengths of π, onto the
interval from 0 to π. The word aliasing means the same thing.shows that sinusoids of angular frequencies π/2 and 3π/2, for instance, can’t distinguished as digital audio signals.
Over all audio signal process
when, for instance, we’re computing values of a Fourier
series (Page 12), either as a wavetable or as a real-time signal, we had better
leave out any sinusoid in the sum whose frequency exceeds π. But the picture in
general is not this simple, since most techniques other than additive synthesis
don’t lead to neat, band-limited signals (ones whose components stop at some
limited frequency). For example, a sawtooth wave of frequency ω, of the form
put out by Pd’s phasor~ object but considered as a continuous function f (t),
f (t) =1
sin(2ωt) sin(3ωt)···sin(ωt) +2 π
which enjoys arbitrarily high frequencies; and moreover the hundredth partial
is only 40 dB weaker than the first one.