Help comparing two FFT approaches

[RobertMacy]

This is in essence what FFTs do behind the scene for decimation in time
and since you don't care about the phases if you are going to average
the power spectra you can study detail in the lower frequency band more
effectively this way (with some risk of aliassing).


The only question here is whether you can get to the same result faster.
There is no free lunch but sometimes you can get a discount...

I only reference the term 'energy' because there in lies the limits of
reception.

Signals are fast enough to reside INSIDE the smallest length, actually AT
LEAST 5 cycles appear within the shortest packet length. Do not expect
anything lower frequency. No signals are faster than 89% of sampling
Nyquist rate. Phase AND magnitude of received signal actually contain the
information being sought! So, MUST preserve.

 

[RobertMacy]

If all the operations are linear, the average of a sum is the same as
the sum of the average. I think you're referring to coherent vs.
incoherent averaging, i.e. averaging power spectra vs taking the power
spectrum of the average. That's different, for sure. I don't think the
OP has told us which he's actually contemplating.

Cheers

Phil Hobbs

Each signal is measured for amplitude AND phase.

 

[Martin Brown]

I only reference the term 'energy' because there in lies the limits of
reception.

But 'energy' has a very well defined meaning in spectrum analysis.

Signals are fast enough to reside INSIDE the smallest length, actually
AT LEAST 5 cycles appear within the shortest packet length. Do not
expect anything lower frequency. No signals are faster than 89% of
sampling Nyquist rate. Phase AND magnitude of received signal actually
contain the information being sought! So, MUST preserve.

If you have a known target carrier frequency then synchronous detection
is always superior the only question is by how much. There might be some
advantage in oversampling a little bit more. It would be very neat if
you could choose your transform length so that an exact multiple of the
main component(s) target frequency fit into it.

The longer time t the chunk of data you FFT at once the better your
frequency discrimination df - only you can decide how much is enough.

 

[RobertMacy]

If you have a known target carrier frequency then synchronous detection
is always superior the only question is by how much. There might be some
advantage in oversampling a little bit more. It would be very neat if
you could choose your transform length so that an exact multiple of the
main component(s) target frequency fit into it.

The longer time t the chunk of data you FFT at once the better your
frequency discrimination df - only you can decide how much is enough.

Martin, thank you for the exact URL's to those papers!

Sadly, the way I go through groups often accidentally deletes postings I
WANT to read! Luckily, there is a way to regain a deleted posting, but
until this morning didn't find much of this thread [especially valuable
postings with URL's, too!]

 

[o pere o]

Think average sum versus sum of averages.
If SNR is sufficiently high then both are equvalent.
At low SNR, average of sum is more accurate.

Vladimir Vassilevsky
DSP and Mixed Signal Designs
www.abvolt.com

Years ago from memory, simulating signals in white noise; there was NO
difference [within numerical accuracies 10^-15 ] between the two
techniques. However, you have me curious to go back and do the
experiments again, this time with 'normal' S/N and with extremely low
S/N. just to see if the simulations predict a difference. I predict
there will be NO difference in the simulation results, BECAUSE the
results were identical except for numerical accuracies. It would seem
that any potential differences related to the original S/N would have
shown up. ...just thinking out loud BEFORE doing all that work.

In the ACTUAL data when comparing the two techniques:
The average of a series of short packet langths caused the higher
frequency signals to completely disappear from the spectral display.
The very long packet length the higher frequency signals were very
discernible.

That seems like the opposite of your statement that average of the sum
of short lengths would yield better results.

As in my other response: a longer FFT gives you better frequency
resolution. This means that you will get information for many more
frequencies. But for those frequencies that do appear with both, the
long and the short one, the result is exactly the same:

Here, for simplicity, compare a 4 point FFT with 2x 2 point FFTs with
octava/matlab

x=rand(1,4);
x1=x(1:2);
x2=x(3:4);

fft(x1)+fft(x2) -> 2.83254 f=0
0.11009 f=fs/2

fft(x) -> 2.83254 + 0.00000i f=0
-0.17807 - 0.24707i f=fs/4
0.11009 + 0.00000i f=fs/2
-0.17807 + 0.24707i f=-fs/4

I can't see how you could loose information for high (or any)
frequencies. Perhaps you were averaging FFTs for non-consecutive samples
of the signal? This could make some periodicities disappear.

Pere

 

[RobertMacy]

Here, for simplicity, compare a 4 point FFT with 2x 2 point FFTs with
octava/matlab

x=rand(1,4);
x1=x(1:2);
x2=x(3:4);

fft(x1)+fft(x2) -> 2.83254 f=0
0.11009 f=fs/2

fft(x) -> 2.83254 + 0.00000i f=0
-0.17807 - 0.24707i f=fs/4
0.11009 + 0.00000i f=fs/2
-0.17807 + 0.24707i f=-fs/4

I can't see how you could loose information for high (or any)
frequencies. Perhaps you were averaging FFTs for non-consecutive samples
of the signal? This could make some periodicities disappear.

Pere

My comparison was a bit longer, but came to the same conclusion.

Never thought of having any gaps in the sampling.

Don't think it's happening but don't KNOW it's not happening. Thanks for
the heads up on that potential flaw.