SNR estimation using a windowed FFT

Hi,

What is the best way to estimate the SNR of a sinusoidal signal in
white noise using an FFT? For the best SNR estimate, is it better to
do one long FFT or to break up into smaller FFTs to average the noise
power? Also does windowing have an effect on the measured SNR?

Thanks,

Rizwan

On Jul 13, 7:44 pm, reah...@gmail.com wrote:
> Hi,
>
> What is the best way to estimate the SNR of a sinusoidal signal in
> white noise using an FFT? For the best SNR estimate, is it better to
> do one long FFT or to break up into smaller FFTs to average the noise
> power? Also does windowing have an effect on the measured SNR?
>
> Thanks,
>
> Rizwan

Is the frequency of the sinusoid known?

For your estimation problem, recall that due to
Parseval's relations, power estimate of white
noise can be done either in frequency or in
time.  The issue is to isolating the effect of
the sinusoid.

Julius

On Jul 14, 5:17 pm, julius <juli...@gmail.com> wrote:
> On Jul 13, 7:44 pm, reah...@gmail.com wrote:
>
> > Hi,
>
> > What is the best way to estimate the SNR of a sinusoidal signal in
> > white noise using an FFT? For the best SNR estimate, is it better to
> > do one long FFT or to break up into smaller FFTs to average the noise
> > power? Also does windowing have an effect on the measured SNR?
>
> > Thanks,
>
> > Rizwan
>
> Is the frequency of the sinusoid known?
>
> For your estimation problem, recall that due to
> Parseval's relations, power estimate of white
> noise can be done either in frequency or in
> time.  The issue is to isolating the effect of
> the sinusoid.
>
> Julius

Yes, the frequency is known (the experiment is injecting a tone to
measure receiver Noise figure).  What I am worried about is the effect
of the windowing on the noise / signal power.  Without windowing,
noise power of the FFT output rises linearly with N while signal power
rises as N^2 (so SNR goes up by N).  However, with windowing,

noise power should go up by sum from n = 0 to N-1 of w[n]^2

while signal power goes up by ( sum from n = 0 to N-1 of w[n] ) ^2


for a hamming window, the resulting SNR is ~1.75 dB less than for a
rectangular window.  Is this correct or am I missing something here?

Thanks,

Rizwan

On Jul 15, 12:42 am, reah...@gmail.com wrote:
> On Jul 14, 5:17 pm, julius <juli...@gmail.com> wrote:
>
>
>
> > On Jul 13, 7:44 pm, reah...@gmail.com wrote:
>
> > > Hi,
>
> > > What is the best way to estimate the SNR of a sinusoidal signal in
> > > white noise using an FFT? For the best SNR estimate, is it better to
> > > do one long FFT or to break up into smaller FFTs to average the noise
> > > power? Also does windowing have an effect on the measured SNR?
>
> > > Thanks,
>
> > > Rizwan
>
> > Is the frequency of the sinusoid known?
>
> > For your estimation problem, recall that due to
> > Parseval's relations, power estimate of white
> > noise can be done either in frequency or in
> > time.  The issue is to isolating the effect of
> > the sinusoid.
>
> > Julius
>
> Yes, the frequency is known (the experiment is injecting a tone to
> measure receiver Noise figure).  What I am worried about is the effect
> of the windowing on the noise / signal power.  Without windowing,
> noise power of the FFT output rises linearly with N while signal power
> rises as N^2 (so SNR goes up by N).  However, with windowing,
>
> noise power should go up by sum from n = 0 to N-1 of w[n]^2
>
> while signal power goes up by ( sum from n = 0 to N-1 of w[n] ) ^2
>
> for a hamming window, the resulting SNR is ~1.75 dB less than for a
> rectangular window.  Is this correct or am I missing something here?
>
> Thanks,
>
> Rizwan

For a pure sinusoidal signal the amount of loss will depend on if the
tone falls exactly on FFT bin or not. You will suffer the coherent
loss when the tone fall exactly on an FFT bin, and the scalloping loss
when it is exactly half way.

You may want to look at a Flat Top type window - see the B&K tech.
manuals. You still have sidelobes and other considerations.

Cheers,
David

On Jul 14, 9:42 pm, reah...@gmail.com wrote:
> On Jul 14, 5:17 pm, julius <juli...@gmail.com> wrote:
>
>
>
> > On Jul 13, 7:44 pm, reah...@gmail.com wrote:
>
> > > Hi,
>
> > > What is the best way to estimate the SNR of a sinusoidal signal in
> > > white noise using an FFT? For the best SNR estimate, is it better to
> > > do one long FFT or to break up into smaller FFTs to average the noise
> > > power? Also does windowing have an effect on the measured SNR?
>
> > > Thanks,
>
> > > Rizwan
>
> > Is the frequency of the sinusoid known?
>
> > For your estimation problem, recall that due to
> > Parseval's relations, power estimate of white
> > noise can be done either in frequency or in
> > time.  The issue is to isolating the effect of
> > the sinusoid.
>
> > Julius
>
> Yes, the frequency is known (the experiment is injecting a tone to
> measure receiver Noise figure).  What I am worried about is the effect
> of the windowing on the noise / signal power.  Without windowing,
> noise power of the FFT output rises linearly with N while signal power
> rises as N^2 (so SNR goes up by N).  However, with windowing,
>
> noise power should go up by sum from n = 0 to N-1 of w[n]^2
>
> while signal power goes up by ( sum from n = 0 to N-1 of w[n] ) ^2
>
> for a hamming window, the resulting SNR is ~1.75 dB less than for a
> rectangular window.  Is this correct or am I missing something here?

The second moment of a Hamming window is less than
that of the same width rectangular window.  This is
related to why a Hamming window (or any of the "hump
shaped" windows) appears to act like a shorter or
narrower rectangular window in terms of signal power
and noise power, which as you found, results in a
lower SNR.

However, if your sinusoidal signal is not centered
in an fft bin, then it is much easier to subtract
out most of the signal from the combined S+N fft
result given a Hamming windowed fft, since the
transform of a Hamming window is much more localized
than a complex sinc.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M