On Jul 13, 2011, at 9:29 AM, Olli Niemitalo wrote:
On Sat, Jul 9, 2011 at 10:53 PM, robert bristow-johnson
r...@audioimagination.com wrote:
On Dec 7, 2010, at 5:27 AM, Olli Niemitalo wrote:
[I] chose that the ratio a(t)/a(-t) [...] should be preserved
by preserved, do you mean constant over all t?
Constant over all r.
i think i figgered that out after hitting the Send button.
what is the fundamental reason for preserving a(t)/a(-t) ?
I'm thinking outside your application of automatic finding of splice
points. Think of crossfades between clips in a multi-track sample
editor. For a cross-fade in which one signal is faded in using a
volume envelope that is a time-reverse of the volume envelope using
which the other signal is faded out, a(t)/a(-t) describes by what
proportions the two signals are mixed at each t. The fundamental
reason then is that I think it is a rather good description of the
shape of the fade, to a user, as it will describe how the second
signal swallows the first by time.
okay, i get it.
so instead of expressing the crossfade envelope as
a(t) = e(t) + o(t)
i think we could describe it as a constant-voltage crossfade (those
used for splicing perfectly correlated snippets) bumped up a little by
an overall loudness function. an envelope acting on the envelope.
and, as you correctly observed, for constant-voltage crossfades, the
even component is always
e(t) = 1/2
so, pulling another couple of letters outa the alfabet, we can
represent the crossfade function as
a(t) = e(t) + o(t) = g(t)*( 1/2 + p(t) )
where
g(-t) = g(t) is even
and
p(-t) = -p(t) is odd
g(t) = 1 for constant-voltage crossfades, when r=1.
for constant-power crossfades, r=0, we know that g(0) = sqrt(2) 1
the shape p(t) is preserved for different values of r and we want to
solve for g(t) given a specified correlation value r and a given
shape family p(t). indeed
a(t)/a(-t) = (1/2 + p(t))/(1/2 - p(t))
and remains preserved over r if p(t) remains unchanged.
p(t) can be spec'd initially exactly like o(t) (linear crossfade,
Hann, Flattened Hann, or whatever odd function your heart desires). i
think it should be easy to solve for g(t). we know that
e(t) = 1/2 * g(t)
o(t) = g(t) * p(t)
and recall the result
e(t) = sqrt( (1/2)/(1+r) - (1-r)/(1+r)*(o(t))^2 )
which comes from
(1+r)*( e(t) )^2 + (1-r)*( o(t) )^2 = 1/2
so
(1+r)*( 1/2*g(t) )^2 + (1-r)*( g(t)*p(t) )^2 = 1/2
( g(t) )^2 * ( (1+r)/4 + (1-r)*(p(t))^2 ) = 1/2
and picking the positive square root for g(t) yields
g(t) = 1/sqrt( (1+r)/2 + 2*(1-r)*(p(t))^2 )
might this result match what you have? (assemble a(t) from g(t) and
p(t) just as we had previously from e(t) and o(t).)
remember that p(t) is odd so p(0)=0 so when
r=1 --- g(t) = 1 (constant-voltage crossfade)
and
r=0 --- g(0) = sqrt(2)(constant-power crossfade)
The user might choose one shape
for a particular crossfade. Then, depending on the correlation between
the superimposed signals, an appropriate symmetrical volume envelope
could be applied to the mixed signal to ensure that there is no peak
or dip in the contour of the mixed signal. Because the envelope is
symmetrical, applying it preserves a(t)/a(-t). It can also be
incorporated directly into a(t).
All that is not so far off from the application you describe.
but i don't think it is necessary to deal with lags where Rxx(tau)
0. why
splice a waveform to another part of the same waveform that has
opposite
polarity? that would create an even a bigger glitch.
Splicing at quiet regions with negative correlation can give a smaller
glitch than splicing at louder regions with positive correlation.
okay. i would still like to hunt for a splice displacement around
that quiet region that would have correlation better than zero. and,
if both x(t) and y(t) have no DC, it should be possible to find
something.
This
applies particularly to rhythmic material like drum loops, where the
time lag between the splice points is constrained, and it may make
most sense to look for quiet spots. However, if it's already so quiet
in there, I don't know how much it matters what you use for a
cross-fade.
Apart from it's so quiet it doesn't matter, I can think of one other
objection against using cross-fades tailored for r 0: For example,
let's imagine that our signal is white noise generated from a Gaussian
distribution, and we are dealing with given splice points for which
Rxx(tau) 0 (slightly).
but you should also be able to find a tau where Rxx(tau) is slightly
greater than zero because Rxx(tau) should be DC free (if x(t) is DC
free). if it were true noise, it should not be far from zero so you
would likely use the r=0 crossfade function.
Now, while the samples of the signal were
generated independently, there is by accident a bit of
