On Sun, Jun 26, 2011 at 11:43:46AM +0200, Jörn Nettingsmeier wrote:
Right. And 'linear' here means 'without a constant term' - we don't
want our system to be a Hilbert transform for example.
> *group* *delay* is a *time* *delay* for a specific frequency. if you
Correct. It it the derivative of the phase response w.r.t. angular
frequency (minus that value if your convention is that a delay
corresponds to positive time).
Group delay actually tells us how the 'envelope' of a signal is
modified by nonlinear phase response, something we can easily hear
on any 'percussive' signals.
Let w = 2 * pi * f
Suppose you have some filter that has a non-linear phase
P(w) = a * w^2 (radians)
The corresponding phase delay is
D(w) = P(w) / w = a * w (seconds)
The group delay is
G(w) = dP(w)/dw = 2 * a * w (seconds)
Now if you have a relatively narrowband signal centered at
some frequency w1, e.g. a 'ping' with a gentle attack, then
it would appear to be delayed by 2 * a * w1, not a * w1,
because what we hear as delay is the delay on the envelope,
not on the 'cycles'.
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