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From: ChrisM on 17 Jun 2010 17:23
Hi,

I'm rather new to signal processing and I would like to deepen my
understanding of spectral densities...

From the book 'Spectral Analysis of Signals' by Stoica and Moses I learned
that I must interpret my signals as one realisation of a random sequence to
account for the probabilistic character of noisy signals.

This works fine for me in the time domain:
I understand that in the time domain I will get (slightly) different
measurements each time I run my experiment due to sensor noise. As I know
the sensor specifications I can determine the confidence interval around
the measured values in which the 'real' value may be found (with a certain
probability).

But how does this work for the Power Spectral Density (PSD)?
Is there something like a confidence interval for the values of the PSD at
a certain frequency?
How can I describe the accuracy of the PSD of a signal?


From: Rune Allnor on 18 Jun 2010 10:12
On 17 Jun, 23:23, "ChrisM" <merkl(a)n_o_s_p_a_m.mytum.de> wrote:
> Hi,
>
> I'm rather new to signal processing and I would like to deepen my
> understanding of spectral densities...
>
> From the book 'Spectral Analysis of Signals' by Stoica and Moses I learned
> that I must interpret my signals as one realisation of a random sequence to
> account for the probabilistic character of noisy signals.
>
> This works fine for me in the time domain:
> I understand that in the time domain I will get (slightly) different
> measurements each time I run my experiment due to sensor noise. As I know
> the sensor specifications I can determine the confidence interval around
> the measured values in which the 'real' value may be found (with a certain
> probability).
>
> But how does this work for the Power Spectral Density (PSD)?
> Is there something like a confidence interval for the values of the PSD at
> a certain frequency?
> How can I describe the accuracy of the PSD of a signal?

It depends entirely on your estimator for the PSD.
The naive estimator, the periodogram, has a variance
on the order of

Var[P[k]] = P[k]^2

where P[k] is the k'th coefficient of the periodogram.
Not awfully useful, as the coefficients of the periodogram
don't tell you much more than that you measured *something*.
Which you - presumably - already knew.

Rune
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