Filtering In Frequency Domain v Time Domain

A low pass time domain filter has a roll off.  A low pass Fourier
transform frequency filter can eliminate all the frequencies above a
cutoff frequency 100% without attenuating any of the lower frequencies
at all.

If the goal is to determine a phase angle between two signals, is
there any advantage in low pass filtering in the time domain before
frequency filtering in the frequency domain?


Bret Cahill

On Sat, 5 Mar 2011 09:49:41 -0800 (PST), Bret Cahill
<BretCahill@peoplepc.com> wrote:

>A low pass time domain filter has a roll off.  A low pass Fourier
>transform frequency filter can eliminate all the frequencies above a
>cutoff frequency 100% without attenuating any of the lower frequencies
>at all.
>
>If the goal is to determine a phase angle between two signals, is
>there any advantage in low pass filtering in the time domain before
>frequency filtering in the frequency domain?
>
>
>Bret Cahill
>
>

A filter is a filter. Which domain (and there are more than two) you
prefer to evaluate it in is your choice.

John

On 3/5/2011 9:49 AM, Bret Cahill wrote:
> A low pass time domain filter has a roll off.  A low pass Fourier
> transform frequency filter can eliminate all the frequencies above a
> cutoff frequency 100% without attenuating any of the lower frequencies
> at all.
>
> If the goal is to determine a phase angle between two signals, is
> there any advantage in low pass filtering in the time domain before
> frequency filtering in the frequency domain?
>
>
> Bret Cahill

Bret,

Well...........
 From an operational point of view, of course "filter can eliminate all 
the frequencies above a cutoff frequency 100%" by multiplying (perhaps 
actually replacing) with zeros.

But, you have to ask yourself what this does in the time domain.

Any flat-zero segment in one domain will create an "infinite" sequence 
in the transform.  This is what happens when we zero-pad a time sequence 
in order to have higher frequency resolution or to implement circular 
convolution.  This doesn't mean, necessarily, that doing something like 
this is "bad".  You just have to know a bit about what you're doing.

One of my favorite applications of doing just that is this:
- The objective is temporal interpolation of a sequence - which is 
basically an increase of the sample rate with some filtering involved.
- One simple way is to intersperse a bunch of regular, zero-valued 
samples in time and then pass the new padded sequence through a lowpass 
filter in order to get nonzero values where those added zeros were place.
- Another way is to rather "assume" the new sample rate and plot the new 
frequency content.  This will look like the original spectrum but 
repeated as many times as the interpolation factor "I".  THEN, we 
lowpass filter (by multiplying in frequency) to remove all of the energy 
bumps at fsOLD, 2fsOLD .... up to but not including the fsNEW.
That suggests the same process you describe.

The real trick here is to not create any "sharp edges" when multiplying 
by a contiguous sequence of zeros.  If you do, there will be stuff 
spread out in time just the transform dual of spectral spreading when we 
use a rectangular window in time.

So, one approach in this interpolation case is to do this.
- Assume we know little or nothing of the signal spectrum because, after 
all, we're building an interpolation "system".  So, if we just multiply 
by a sequence of zeros, we have no idea what anomalies we're creating.
- Do the interpolation in stages.  In fact, the first stage would be to 
repeat the spectral sequence to only 2fsOLD and then lowpass filter the 
data with an actual LPF - so that the spectral energy is fairly well 
limited to 1/4 the fsNEW1 or 1/2 fsOLD.
Only now can you know that the spectral data at fsNEW/4 to 3fsNEW/4 will 
be "nearly zero".
- Now you can pad the spectral sequence with zeros symmetrically about 
fs/2 - effectively increasing the value of fs or N ... whichever way you 
like to think of it.  This padding operation is exactly like what we do 
in the time domain when we want higher N but have limited data - for 
whatever reason.  NOTE: This padding is exactly the same as looking at 
the repeated spectrum up to I*fsOLD and then multiplying that sequence 
with zeros from fsOLD/2 to fsNEW-fsOLD/2.

In effect, the edges of the zero sequence will fall on frequencies where 
there is little energy (due to the prefiltering).  And so, there will be 
assuredly little temporal spreading caused by doing this.

So this is an example of doing what you want but taking the precaution 
of pre-filtering before multplying by a sequence of zeros. It's the same 
sort of thing as doing windowing in time to reduce the effects of 
rectangular  windowing - done in advance of zero padding in time.  But, 
in this case our "window" in frequency has the shape of an actual 
lowpass filter - which is a "rectangular window with smoothed edges and 
long (nonzero) stopband tails"

So, that's a long-winded caveat.  Now on to your question:

First of all, "time domain" filtering and "frequency domain filtering" 
are two different animals.  You can't do temporal IIR filtering in 
frequency unless it's a special case made to look like a FIR.  So that's 
a limitation of sorts.  And, you are forcing yourself to do block or 
finite sequence oriented processing.  If your objective is to deal with 
streaming data then maybe time domain processing is better - or you will 
have to deal with how to combine the blocks.

The advantage of frequency domain filtering is that it can save a LOT of 
compute time if the (FIR) filter has any appreciable length.

As long as you pay attention to the sequence lengths and what that means 
in your system then I see no reason to do time domain filtering *before* 
frequency domain filtering.  Also remember that frequency domain 
filtering is equivalent to circular convolution in time and the temporal 
sequences have to account for that (be zero padded) to avoid overlap in 
time.

Fred

On 3/5/2011 11:22 AM, Fred Marshall wrote:
> This padding is exactly the same as looking at the repeated spectrum up
> to I*fsOLD and then multiplying that sequence with zeros from fsOLD/2 to
> fsNEW-fsOLD/2.

Or, one might chose to have the zeros run from fsOLD to fsNEW-fsOLD so 
that the stopband of the lowpass filter are included, rather than 
forcing those values to zero as well.

One might also imagine that avoiding sharp edges caused by multiplying 
by a sequence of zeros would be helped if the value and perhaps the 
first derivative of the original lowpassed sequence were to be zero at 
the edges of the "split point" .. the fs/2 point at this stage.  So, if 
the requirements are particularly stiff then that might be a 
consideration.  Again, similar to tapered time domain windowing ideas.

Fred

Every filter has an impulse response. When modifying the DFT of a signal, then inverse DFTing the result, the effective transforms must be longer than the impulse response of the modifying function. (You can use long transforms or an overlap method.) How long is the impulse response of the brick-wall function you proposed?

Jerry
-- 
Engineering is the art of making what you want from things you can get.

> > This padding is exactly the same as looking at the repeated spectrum up
> > to I*fsOLD and then multiplying that sequence with zeros from fsOLD/2 to
> > fsNEW-fsOLD/2.
>
> Or, one might chose to have the zeros run from fsOLD to fsNEW-fsOLD so
> that the stopband of the lowpass filter are included, rather than
> forcing those values to zero as well.
>
> One might also imagine that avoiding sharp edges caused by multiplying
> by a sequence of zeros would be helped if the value and perhaps the
> first derivative of the original lowpassed sequence were to be zero at
> the edges of the "split point" .. the fs/2 point at this stage. �So, if
> the requirements are particularly stiff then that might be a
> consideration. �Again, similar to tapered time domain windowing ideas.

What is interesting is how much time you can spend playing around with
these transforms.

Does a 10 degree lag phase angle in the reference introduce amplitude
errors in match filtering?

You can find out on Excel.


Bret Cahill

> > > This padding is exactly the same as looking at the repeated spectrum up
> > > to I*fsOLD and then multiplying that sequence with zeros from fsOLD/2 to
> > > fsNEW-fsOLD/2.
>
> > Or, one might chose to have the zeros run from fsOLD to fsNEW-fsOLD so
> > that the stopband of the lowpass filter are included, rather than
> > forcing those values to zero as well.
>
> > One might also imagine that avoiding sharp edges caused by multiplying
> > by a sequence of zeros would be helped if the value and perhaps the
> > first derivative of the original lowpassed sequence were to be zero at
> > the edges of the "split point" .. the fs/2 point at this stage. �So, if
> > the requirements are particularly stiff then that might be a
> > consideration. �Again, similar to tapered time domain windowing ideas.
>
> What is interesting is how much time you can spend playing around with
> these transforms.
>
> Does a 10 degree lag phase angle in the reference introduce amplitude
> errors in match filtering?
>
> You can find out on Excel.

Unlike phase sensitive filtering where the phase must be known to get
the amplitude the phase of the reference must be irrelevant with match
filtering.

I'll check it out to make sure.


Bret Cahill

Please clue me in here. What is "phase-sensitive filtering"? What kind of filter (aside from possibly a lock-in amplifier) needs a reference? 

Jerry
-- 
Engineering is the art of making what you want from things you can get.

On 3/5/2011 9:49 AM, Bret Cahill wrote:
> If the goal is to determine a phase angle between two signals,

Bret,

What are you trying to do?

If there are 2 sinusoids, at different frequencies, then there is no 
"phase angle" between them - although there may be a delay between them.

If there are 2 sinusoids at the same frequency then, in the context of a 
single DFT, they will add and appear to be one sinusoid.  But, maybe the 
system is different.

If they happen at different times then there are other system 
considerations - as in "where is the time reference?"

Fred

> > If the goal is to determine a phase angle between two signals,
>
> Bret,
>
> What are you trying to do?

Match filter with an out of phase reference.  For awhile I thought it
wouldn't matter and in fact the absolute value in the frequency domain
is the same for two signals that are identical except for a phase
angle..

But after trying it out on Excel it seems the phase angle must still
be calculated from the real and imaginary components, the difference
in the angles between the signals.

The noise will probably require some kind of average of the phase
angles.

An iterative technique may be necessary.  First you get an estimate of
the phase angle from the noisy signal and reference then you phase
adjust the reference then you match filter then you recheck to see if
the phase angle is converging.

> If there are 2 sinusoids, at different frequencies, then there is no
> "phase angle" between them - although there may be a delay between them.

Assume just one fundamental frequency.

> If there are 2 sinusoids at the same frequency then, in the context of a
> single DFT, they will add and appear to be one sinusoid. =A0But, maybe th=
e
> system is different.

> If they happen at different times then there are other system
> considerations - as in "where is the time reference?"

The SNR may be as high as 20 so hopefully one cycle will be enough.
In that case only 90% of the noise needs to be eliminated.


Bret Cahill