On 2015-11-05, robert bristow-johnson wrote:
I think I was slightly off when I said that the units of psd are
power per unit frequency -- since the whole signal has infinite
power,
no, i don't think so.
Me neither. Power is already by definition energy per unit time. Even if
an infinitely long signal often (not always) has infinite energy, *any*
practical signal you would be dealing with has finite power over its
entire length. And in fact physical signals can't have even infinite
length or energy, so even if you go crazy and conceptually integrate
everything from the beginning of the time to the End Times, it's
reasonable to model any realistic signal as being globally, not just
locally, square integrable (the square integral from -inf to +inf being
the total energy).
the units really need to be power per unit frequency per unit time,
which (confusingly) is the same thing as power.
the signal has infinite energy because it goes on (with power greater
than some positive lower-bound) forever. but it's not infinite power
unless it's something like "true" white noise (which has infinite
bandwidth).
Precisely. So what probably trips some up here is "power per unit time".
That's nonsensical. What we need is power per unit frequency, i.e.
energy per unit time per unit frequency.
what comes out of a random number generator (a good one) is white only
up to Nyquist. not infinite-bandwidth white noise.
And, as a matter of fact, if you go to the kind of distributional stuff
we talked about a while back, you can even deal with real white noise to
a degree. That's because you can do local integration in the Fourier
domain, where white noise has unity norm.
The same argument then explains why a signal which has infinite length
and infinite energy in the time domain is absolutely no problem for the
kind of analysis we're talking about here: STFT analysis already makes
your stuff local in time, so as long as the signal is of finite power,
you'll get sensible local results, even if the signal is globally
speaking of infinite energy (say like an ideal sinusoid).
The only real kink is that when you localise your analysis, you're
bringing in an extra degree of freedom: what precisely do the length and
shape of your windowing/apodization function do to the results of the
analysis. In spectral analysis work, that then mostly revolves around
energy concentration within an FFT band, and specral spillover to the
adjacent ones. Sometimes statistical estimation theory, and what phase
does over successive windows.
so the integral, from -Nyquist to +Nyquist of the PSD must equal the
variance, as derived from the p.d.f. and that value also has to be the
zeroth-lag value of the autocorrelation.
Yes, and that by definition. If you have to deal with DC, then you have
to separate autocorrelation from autocovariance. Also in that case the
DC part spills over asymmetrically after windowing, because essentially
it will be AM modulated by the window, and will alias upwards across the
zero frequency point.
This could be another reason why some special scaling is needed as
compared to a finite-length FFT.
really, the only scaling would be that comparing the Fourier integral
(with truncated and finite limts) to a Riemann summation (which could
be expressed as the DFT).
As I understand it, scaling is mostly necessary because of numerical
concerns. I mean...
When you do longer STFT's, the implicit filter represented by each bin
grows narrower and more selective. In other words, more and more
resonant. If it then so happens that you hit a sinusoid right in the
middle of the passband, a growing analysis window leads to an unlimited
amount of power gathered on that coefficient. After all, the continuous
time Fourier transform of a sinusoid is a Dirac distribution, and with a
growing analysis window you'll approach that -- the series doesn't
converge in the normal but only the weak sense, so that your STFT bin
blows up. So there's a tradeoff between headroom and noise floor, here.
Though, I could be talking about a different scaling problem than you
folks. I did jump into the fray pretty late. :)
--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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