Hi all,

I wrote this posting within another topic on this group already, but
since there was no response it possibly just was in the wrong place,
and also people might have stopped reading in the middle because it
started another topic in an already existing discussion. I would like
to ask it again, because I'm sure someone will know about this.

It is about specifying spectra as power spectra or power spectral
density results. Assume a system which emits a number of tones, i.e.
single frequencies within the spectrum. There is also broadband-like
noise in the system, i.e. some pink-shaped noise, and broadband noise
with white characteristics, e.g. from fans or background noise.
Assume further I could measure the sources in sequential order, i.e.
just noise A, then A+B, then A+B+C and so on. I would like to compare
the sources, and specify e.g. the SNR of the different sources with
respect to the lowest (white) noise.

I could show it in a frequency domain plot as power spectrum or
magnitude spectrum, but depending on the number of FFT-coefficients,
the broadband noise will be raised or lowered, while the single-tone
signals will keep their values. This is why for broadband noise one
will typically use the power spectral density, which gives a definite
measure, independend of the FFT used. Nevertheless, defining power
spectral density (V^2/Hz vs. Hz) for the single tones not only feels
somehow weird, it also for these signals gives results which will be
lower or higher depending on the number of FFT coefficients.

My question is: how do I state a comparison value like SNR for such a
case (mix of different broadband noises and single tones), and how can
I show it graphically in a way that makes sense: Power spectrum or
power spectral density or...?

Any thoughts are most welcome. I didn't found a solution which solves
all the questions raised. 

Sincerely

stereo

stereo skrev:
> Hi all,
>
> I wrote this posting within another topic on this group already, but
> since there was no response it possibly just was in the wrong place,
> and also people might have stopped reading in the middle because it
> started another topic in an already existing discussion. I would like
> to ask it again, because I'm sure someone will know about this.
>
> It is about specifying spectra as power spectra or power spectral
> density results. Assume a system which emits a number of tones, i.e.
> single frequencies within the spectrum. There is also broadband-like
> noise in the system, i.e. some pink-shaped noise, and broadband noise
> with white characteristics, e.g. from fans or background noise.
> Assume further I could measure the sources in sequential order, i.e.
> just noise A, then A+B, then A+B+C and so on. I would like to compare
> the sources, and specify e.g. the SNR of the different sources with
> respect to the lowest (white) noise.
>
> I could show it in a frequency domain plot as power spectrum or
> magnitude spectrum, but depending on the number of FFT-coefficients,
> the broadband noise will be raised or lowered, while the single-tone
> signals will keep their values. This is why for broadband noise one
> will typically use the power spectral density, which gives a definite
> measure, independend of the FFT used.

Yep.

> Nevertheless, defining power
> spectral density (V^2/Hz vs. Hz) for the single tones not only feels
> somehow weird,

Maybe, but it is still correct. The DFT can only measure the
sinusoidal with some degree of certainty. It can not measure
the exact amplitude of an infinitely narrow sinusoidal. The
windowing because of a finite record-length sees to that.

> it also for these signals gives results which will be
> lower or higher depending on the number of FFT coefficients.

Wrong. Most FFT implementations skip it for varuious practical
reasons, but formally you need to include a 1/sqrt(N) in the FFT.
You have to include it yourself, if you want to use the FFT for
power spectrum computations.

> My question is: how do I state a comparison value like SNR for such a
> case (mix of different broadband noises and single tones), and how can
> I show it graphically in a way that makes sense: Power spectrum or
> power spectral density or...?

Power spectral density is the way to go.

Rune

> My question is: how do I state a comparison value like SNR for such a
> case (mix of different broadband noises and single tones), and how can
> I show it graphically in a way that makes sense: Power spectrum or
> power spectral density or...?
>
>


this is a great question and the issue come up often in RF measurements
as well...


I suggest you define your  SNR as the ratio of  Signal power which is
the power of the CW tone(s) ...to ...noise power as measured in (or
integrated over) a specified bandwidth.  The sensible BW to use to
measure and specify the noise depends upon the application.  For RF TV
channels it can be 6 MHz or 4.5 MHz.  For FM radio channel 200 kHz.
For AM radio channel 10 kHz etc.   For audio it might be 20 Hz to 20
kHz and it might or might not be "A weighted"   You have to specify and
measure the noise over the BW that matters to the specific application.

As far as graphical representation...

I suggest you use PS rather than PSD.  Use a PS but in order to remove
the ambiguity, you must specify the measurement BW.    i.e.  here is a
PS of the output AS MEASURED IN A 100 HZ BE (for example).   The
measurement BW is simply the resolution BW of the spectrum analyzer or
the resolution of the FFT.  Note that PSD is exactly the same as above
except that the resolution BW = 1 Hz by the definition of density.
Using PSD with a 1 Hz resolution is non-intuitive to many people
because the res BW is so narrow (1 Hz) it makes the noise look so low.
  So I suggest using a res BW that is appropriate for the instrument
being used and including that in the description of the graphical
display to remove the ambiguity.

Mark