On Mar 18, 2011, at 1:55 PM, Wen X wrote:
- when considering finite duration there is the uncertainty
principle, so
you always deal with a pack of frequencies rather than one
frequency, which
makes "latency" dependent on the content of that pack.
- however, using FT[th(t)]=j(FT[h(t)])', one can show that the
centroid of
time w.r.t. the impulse response equals the centroid of group delay
w.r.t.
frequency response.
so, when h(t)<0, does that cause negative time to be averaged into the
centroid calculation?
perhaps we should be looking for
+inf
integral{ t * |h(t)|^2 dt}
-inf
and normalize that with
+inf
integral{ |h(t)|^2 dt}
-inf
use |h(t)| or |h(t)|^2 instead of h(t) because h(t) is potentially
bipolar.
now i wonder if that integral above might be equal to
+inf
integral{ t_g(2*pi*f) * |H(j*2*pi*f)|^2 df}
-inf
where
H(s) is the Laplace transform of h(t) and
t_g(w) = -(d/dw)phi(w) is the group delay at angular frequency w.
is that a known fact?
i really don't know. but i would like to know.
(again, if the filter is causal then the centroid is nonnegative).
that's true, but i am not sure that the converse is true.
but what i want to make clear is that a causal filter *can* have
negative group delay over a wide range of frequencies. that's why i
referred to this old paper http://musicdsp.org/files/EQ-
Coefficients.pdf because it shows negative group delay for large
segments of frequency. and for the peaking EQ set to a negative peak
(in dB), the negative group delay is in the middle of the frequency
range surrounding the resonant frequency. i think, in that case, it
may very well be that the "average" group delay (depending on how it
is averaged) is negative. and the EQ filters are certainly causal.
This means if your filter is narrow band than the centroid of
group delay gives the latency in that band.
the latency of the *envelope* (or group). and inside that band.
If your filter is wide band than
I guess you can try evaluating group delay centroid from local
frequency
bands, but I'm not sure of a theory that supports doing so.
i am not aware of a theory that relates group delay to causality (even
though a relationship seems intuitive), but i would be happy to learn
such.
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
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