FFT with and without a low pass filtering 1

[es335]

When making a frequency measurement, one has to make sure that the Nyquist criterion is obeyed. But what if I have no frequency filtered data. Is it correct to make a FFT transformation and just look at the first half of the transformation? Is that the same as a low pass filtering?

 

[mcac]

No, because if the time domain signal contains frequency components beyond nyquist´s these components will appear reflected in the range (0-Nyquist). In order to eliminate this side effect you need to use an anti-aliasing filter.

 

[nbucska]

mcac is right !

The Nyquist-crit. frequency is the sampling rate. This is equivalent with chopping -- multiplication with _-_-_-.

This mixes down higher frequencies -- after mixing they can't be distinguished from original LF-s.


<nbucska@pcperipherals.com>

 

[bridog]

The only time I've seen an anit-aliasing filter intentionally left out is in some DSP demodulator designs. The reflection is taken advantage of to downconvert a modulated waveform at a known carrier frequency.

Bridog

 

[nbucska]

You can leave out AAF if there is no HF component which
would interfere with the intended measurement -- but
you better know what you are doing...

<nbucska@pcperipherals.com>

 

[bridog]

Absolutely. The modulated carrier is passed through a bandpass filter first in the receiver front end or an earlier IF stage. Its just cool to see the signal drop right into an A/D where the carrier frequency and sampling frequency are specifically related like an IF freq into a mixer. But like you said nbucska, you better know what you're doing. I've never tried it myself: just marveled at some SONAR and satellite designs...

But getting back to the thread, yes on needing the AAF for basic sampling, no on reading only the first half of the FFT.

 

[nbucska]

Based on 30+yrs of engineering experience, if you know
what you are doing, you can make shortcuts.

<nbucska@pcperipherals.com>

 

[MikeyP]

Interesting aside:

Deliberate aliasing is a fundamental part of zoom FFT analysis.

M

--
Dr Michael F Platten

 

[betkro]

es335, Let me see if I had well understood your question. By frequency measuring, I assume you mean “sampling”, because you associate to the Nyquist criterion.

Certainly, you are not obliged to obey this criterion to make a frequency measurement, just to faithfully recovering the waveform or, well understand of what it really happens if you sample at a different rate.

You’re are right by saying that sampling at a rate less than the bandwidth is equivalent to a filter. This is a pretty graphical explanation that is certainty mathematically based.

Filtering (antialiasing, transversal, Kalman, etc.) is another thing.

 

[mgopalan]

Some limited (and rather verbose) shared knowledge...

1) ideally your signal is sampled much faster than the frequency content of the data you are interested in. This is often but not always true.

2) If you did not use an AAF before you digitized AND you have high frequency content...then you are probably screwed. For example...If you sampled at 100Hz and you had a frequency component at 99Hz..then it would appear to be reflected and show up as a 1Hz signal. The problem is you cant tell whether it was a real 1Hz signal or a 99Hz signal...

3) Of course, if you sampled at 100Hz and you are pretty sure that you have no signal content above 50 Hz then you are safe..

4) The nyquist criterion is used basically to preserve frequency content. By itself it is not adequate to resolve waveform shape. This reflect back to 1) where you generally sample faster than the data you need.

5) So having said all of that...in an ideal case you do the following

a) Figure out the content you want to measure...say you want to &quot;properly&quot; digitize events that occur within a 100 Hz bandwidth. This means preserving Amplitude AND phase of all content below 100Hz.
b) Ok..so you pick an AAF whose 3dB is set so that it is FLAT AMPLITUDE and LINEAR PHASE to AT LEAST 100 Hz. Nothing is perfect so you use your engineering judgement and choose to have an AAF set so that the Amplitude transfer function is Flat to within 5% to 100Hz and the Phase is linear to within 5 degrees to 100 Hz. (this is not an &quot;or&quot; case...both must be satisfied...and you will find that you will often fail in one criterion long before you hit the second one)

c) So hypothetically you choose a 4th order Butterworth AAF. Its 3dB needs to be set at 107Hz so that its amplitude is 95% at 100Hz and phase is less than 5 degrees at 100 Hz. (these are not real numbers!!!again hypothetical)
c) Now, this filter has a 24 dB/Octave roll off.. Which means that if you apply the Nyquist criterion blindly and sample at 107*2Hz, you alias stuff....why? Because your filter hasnt rolled off until well above 107 Hz. 107Hz is the 3dB point!!!3dB is not adequate attenuation for most rigorous applications!!!
d) What do you consider rolled off? Depends on the SNR and Bit resolution you want. Lets say that you pick that 60dB is good enough for you (thats roughly a 10-bit A/D converter's resolution)...well at 24dB/octave and 60dB attenuation puts you at roughly 5.5*107Hz = 588.5 Hz.
e) Now, you can be SURE that above 588.5 Hz, you have NO frequency content that survives to be digitized.
f) NOW you apply nyquist and sample at twice of 588.5..This is about 1177.0Hz....
g) Now, if you have wideband input, you WILL have unwanted or unneccesary content between 100 and 588.5 Hz. Most cases you can ignore these, but digital filters work great to lop off this extra stuff didnt want anyway.but watch what these digital filters do to your pass band...no point in spending a lot of time acquiring great data and then screw it up with a shitty digital filter.

Having said all of that, if you are SURE that you didnt have high frequenct content before you digitized, you can run the FFT and look at the spectra. The FFT algorithm returns a mirrored waveform anyway (mirrored about the nyquist frequency)..most display programs just chop of the second half before they display it for you.

Now, there are LOTS of exceptions where you dont follow these rules. Examples have already been given.

I agree with nbuckska and bridog...If you are patient and know what you are doing, you can make real inferences from the data...but beware.

MG

All opinions are mine. The Facts I blame on physics.