FFT White Noise Spectrum

Hi,

I have a simple question regarding the relationship between the RMS
voltage and frequency spectrum of Gaussian distributed white noise. I
would like to be able to relate the RMS value of the time domain signal
to the magnitude I see on the spectrum analyzer. As I understand it,
for white noise, the energy in the signal will be smeared over the pass
band of the spectrum analyzer. There will be equal energy in each
frequency component and as the number of FFT points increases then the
noise voltage of each frequency bin will reduce. How do I convert the
voltage values my spectrum analyzer is giving me to the RMS voltage of
the time domain signal?

Regards

James Lancaster

jameslancaster7371@mailinator.com wrote:
> Hi,
> 
> I have a simple question regarding the relationship between the RMS
> voltage and frequency spectrum of Gaussian distributed white noise. I
> would like to be able to relate the RMS value of the time domain signal
> to the magnitude I see on the spectrum analyzer. As I understand it,
> for white noise, the energy in the signal will be smeared over the pass
> band of the spectrum analyzer. There will be equal energy in each
> frequency component and as the number of FFT points increases then the
> noise voltage of each frequency bin will reduce. How do I convert the
> voltage values my spectrum analyzer is giving me to the RMS voltage of
> the time domain signal?
> 
> Regards
> 
> James Lancaster
> 

One way is to set RBW < noise BW and put the SA into "marker noise" 
mode. The marker will give you the noise density in dBm/Hz. To that 
measurement you can add 10log10(BW) to get noise power in dBm. From 
there it is straightforward to get Vrms.

John

jameslancaster7371@mailinator.com skrev:
> Hi,
>
> I have a simple question regarding the relationship between the RMS
> voltage and frequency spectrum of Gaussian distributed white noise. I
> would like to be able to relate the RMS value of the time domain signal
> to the magnitude I see on the spectrum analyzer. As I understand it,
> for white noise, the energy in the signal will be smeared over the pass
> band of the spectrum analyzer.

Yes.

> There will be equal energy in each
> frequency component and as the number of FFT points increases then the
> noise voltage of each frequency bin will reduce.

If the noise is perfectly white acording to the theory, yes.
In practice, there may be deviations, but leave that for now.

> How do I convert the
> voltage values my spectrum analyzer is giving me to the RMS voltage of
> the time domain signal?

Formally (and if you have access to the data) you can sum the
energy density over the bandwidth of the spectrum analyzer
and relate that to the energy in a frame in time domain, and
derive an RMS value from that.

I am not sure if that's a very practical approach, though.

Rune

I do have access to the data values. Basically I'm generating a 1VRMS
noise signal band limited to 110 KHz and performing a 4096 point FFT.
My spectrum analyzer is a PC based oscilloscope and is giving me values
in dB's relative to 1/sqrt(2) Volts, the spectrum is flat and has an
averaged value of -30dB. Am I correct in thinking that in order to
calculate the energy density I covert these values to volts, square
them and divide by the resolution bandwidth? If I then multiply this
value by the bandwidth of my measurement I will have the total energy
in my noisy signal?

On 17 Nov, 14:39, "Rune Allnor" <all...@tele.ntnu.no> wrote:
> jameslancaster7...@mailinator.com skrev:
>
> > Hi,
>
> > I have a simple question regarding the relationship between the RMS
> > voltage and frequency spectrum of Gaussian distributed white noise. I
> > would like to be able to relate the RMS value of the time domain signal
> > to the magnitude I see on the spectrum analyzer. As I understand it,
> > for white noise, the energy in the signal will be smeared over the pass
> > band of the spectrum analyzer.Yes.
>
> > There will be equal energy in each
> > frequency component and as the number of FFT points increases then the
> > noise voltage of each frequency bin will reduce.If the noise is perfectly white acording to the theory, yes.
> In practice, there may be deviations, but leave that for now.
>
> > How do I convert the
> > voltage values my spectrum analyzer is giving me to the RMS voltage of
> > the time domain signal?Formally (and if you have access to the data) you can sum the
> energy density over the bandwidth of the spectrum analyzer
> and relate that to the energy in a frame in time domain, and
> derive an RMS value from that.
>
> I am not sure if that's a very practical approach, though.
> 
> Rune

jameslancaster7371@mailinator.com skrev:
> I do have access to the data values. Basically I'm generating a 1VRMS
> noise signal band limited to 110 KHz and performing a 4096 point FFT.
> My spectrum analyzer is a PC based oscilloscope and is giving me values
> in dB's relative to 1/sqrt(2) Volts, the spectrum is flat and has an
> averaged value of -30dB. Am I correct in thinking that in order to
> calculate the energy density I covert these values to volts, square
> them and divide by the resolution bandwidth?

I don't know. Chances are that the numbers you get actually
is the energy density.

> If I then multiply this
> value by the bandwidth of my measurement I will have the total energy
> in my noisy signal?

Yes, once you have obtained the energy density numbers.
Be aware, though, that this is an estimate that will be
off if the noise spectrum is not flat.

Rune

jameslancaster7371@mailinator.com wrote:
> How do I convert the
> voltage values my spectrum analyzer is giving me to the RMS voltage of
> the time domain signal?

Unless your spectrum analyzer is not correctly normalized, it makes no 
difference whether you calculate the RMS value in the time domain or in 
the frequency domain. The FFT will not affect the total energy. The RMS 
value is simply the sum of squares of all (nonzero) values - either of 
the samples in the time domain or of the amplitudes in the frequency domain.

In case of really white noise (which has no gaussian profile by 
definition) there is little motivation to prefer either method. But in 
real applications it might be helpful to discard some frequency channels 
because they only contain unwanted noise.


Marcel