FIR filter and time delay

Hi. I note that a suitably designed FIR filter with N coefficients can
have a time-delay of N/2 (or is it (N-1)/2?). I'm designing low pass
filters by the window method. In my application I'm taking whole sample
files and filtering after the fact. It is very important for me to be
able to say for any sample in my filtered sample what the original time
of that sample was in the original file. Hence it looks sensible for me
to reinterpret the coefficients of my FIR filter so that when producing
an output sample y[i], that my coefficients are applied to samples at
times x[i-N/2] to x[i+N/2], and to assume that all samples before the
start of my original sample or after its end are zero. Hence I get a
filtered version of my original file without any time shift. Is this
correct?

I haven't yet examined optimal/Remerez FIR filter design in great
detail. Is it easy and/or possible to predict the delay inherent in a
Remerez filter as it is for one designed by the windowing method? If
execution speed is not crucial, are there any other advantages in using
a FIR filter designed by the Remerez method compared to simply using a
FIR filter with a larger number of coefficients?

Thanks in anticipation,

Ross-c

Ross Clement (Email address invalid - do not use) wrote:
> Hi. I note that a suitably designed FIR filter with N coefficients can
> have a time-delay of N/2 (or is it (N-1)/2?). I'm designing low pass
> filters by the window method. In my application I'm taking whole sample
> files and filtering after the fact. It is very important for me to be
> able to say for any sample in my filtered sample what the original time
> of that sample was in the original file. Hence it looks sensible for me
> to reinterpret the coefficients of my FIR filter so that when producing
> an output sample y[i], that my coefficients are applied to samples at
> times x[i-N/2] to x[i+N/2], and to assume that all samples before the
> start of my original sample or after its end are zero. Hence I get a
> filtered version of my original file without any time shift. Is this
> correct?
>
> I haven't yet examined optimal/Remerez FIR filter design in great
> detail. Is it easy and/or possible to predict the delay inherent in a
> Remerez filter as it is for one designed by the windowing method? If
> execution speed is not crucial, are there any other advantages in using
> a FIR filter designed by the Remerez method compared to simply using a
> FIR filter with a larger number of coefficients?
>
> Thanks in anticipation,
>
> Ross-c

If you are using and/or have access to MATLAB along with the Signal
Processing Toolbox, then for your application I would recommend
filtering with the "filtfilt" command.  This command performs true
"Zero Phase" (i.e. no delay) filtering by using a forward-reverse
filtering implementation.  It works with any filter you design (FIR,
IIR, linear phase, etc.).

If you don't have access to MATLAB, then I would do a search on zero
phase filtering.

Alternatively, you can pad the data at the ends with intelligently
selected values and then take the sections of the filtered output which
correspond to your input.

Thanks. Looking at the page:

http://www-ccrma.stanford.edu/~jos/filters/Zero_Phase_Filters.html

It seems that my shifting of the coefficients of a FIR filter designed
by the windowing method produces a filter where h(n) = h(-n), which
fits the definition of a zero phase filter as per that page. The page
above doesn't describe zero phase as being produced by a
forwards-backwards procedure. Though, I can see how this would work and
be useful for filters such as IIR filters with, erm, "inconvenient"
phase responses. The phase effects of filtering backward would I
presume cancel out those of moving foward while the frequency response
would reinforce.

Since I'm using very large FIR filters (up to thousands of
coefficients), the caveat that phase oscillates between 0 and pi in the
stopband is unlikely to cause me any trouble as the response in the
stopband will be, I assume, tiny.

The page uses a Remez-designed FIR filter as an example and hence I
presume the coefficients of a Remez-designed filter are also symmetric
so that they can be shifted such that h(n) = h(-n).

I sort of have access to matlab. I have windows matlab but work on
linux and my uni doesn't seem set up for providing the linux version of
matlab. However, I much prefer to "do things myself". I have recently
moved "DSP-ward" from a different background recently and prefer to
work through things such as filter design "by hand" to make sure that I
have an understanding of what is going on. I don't mind writing my own
code then discarding it to use library code later on if the library
code is going to more numerically accurate than anything I'll ever
write, e.g. linear algebra code.

As a side question, would you recommend scilab? I've noted people
describing the documentation for the DSP toolbox for scilab being well
written and well worth reading for the educational side-effects.

Cheers,

Ross-c

Ross Clement (Email address invalid - do not use) wrote:
> Hi. I note that a suitably designed FIR filter with N coefficients can
> have a time-delay of N/2 (or is it (N-1)/2?).

All symmetric FIR filters with the same number of taps operating at the 
same sample rate have the same delay: (N-1)/2.

> I'm designing low pass
> filters by the window method. In my application I'm taking whole sample
> files and filtering after the fact. It is very important for me to be
> able to say for any sample in my filtered sample what the original time
> of that sample was in the original file.

Since the value of a filtered result depends on all of the values of the 
samples used to calculate it, that is a difficult philosophical as well 
as practical question.

> Hence it looks sensible for me
> to reinterpret the coefficients of my FIR filter so that when producing
> an output sample y[i], that my coefficients are applied to samples at
> times x[i-N/2] to x[i+N/2], and to assume that all samples before the
> start of my original sample or after its end are zero. Hence I get a
> filtered version of my original file without any time shift. Is this
> correct?

Sort of. It's as easy to subtract the delay of the filter from the index 
number of the finished sample. The result will be the same. Note that 
the output will be wrong when the filter isn't filled with valid data. 
All filters have start and end transients which become unimportant only 
with long inputs.

> I haven't yet examined optimal/Remerez FIR filter design in great
> detail. Is it easy and/or possible to predict the delay inherent in a
> Remerez filter as it is for one designed by the windowing method?

All symmetric FIR filters with the same number of taps operating at the 
same sample rate have the same delay: (N-1)/2.

> If
> execution speed is not crucial, are there any other advantages in using
> a FIR filter designed by the Remerez method compared to simply using a
> FIR filter with a larger number of coefficients?

Less delay?

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

Ross,

Looks like you are off to a good start.

For more info on forward-backwards (forward-reverse) zero-phase
filtering you can check out:
http://ccrma.stanford.edu/~jos/filters/Forward_Backwards_Filtering.html
http://www.mathworks.com/access/helpdesk/help/toolbox/signal/basic19a.html

I've never used Scilab, so I can't comment on that.

Thanks. I should have taken more note of your page on how to ask
questions on comp.dsp and checked out the properties of filters
designed by the Remez exchange method before posting.

The start and end transients are not too important for my application
as speech samples typically have "silence" at the beginning and end +/-
the occasional bit of breath noise.

Also, as for the philosophical question of the "time" of a sample in
the filtered output versus the original output... My thinking is as
follows and I would appreciate correction if it is wrong.

Imaging that there is a synthetic signal created by the addition of
sines (or cosines of course). Imagine that only one such sine wave has
a frequency of < 150hz, and all others have frequencies > 150hz. If the
full signal was filtered by a perfect (I do know this is impossible)
zero-phase lowpass filter with a cutoff of 150hz, then the single
relatively low frequency sine should pass through the filter unchanged
while all other sines are stopped completely. In this situation the
output would be a pure sine wave with observable minima, maxima, and
zero crossings. I would like the minima, maxima, and zero crossings of
the observed sine wave to be at the same times as the corresponding
events of the sine wave used when synthesising the original sound.
Therefore I could do things like cut out a portion of the original
signal which is one complete cycle of the 150hz component used when
synthesising the signal. If I have a known time-delay (e.g. 0) due to
filtering then I can cut out such a cycle so that the 150hz component
has a phase of 0 in my extracted cycle by observing the minima, maxima,
and zero crossings in the filtered signal and mapping the locations of
these onto the original signal. I now believe that the term "zero phase
filter[ing]" describes exactly what I want.

Cheers,

Ross-c

"Ross Clement (Email address invalid - do not use)" <clemenr@wmin.ac.uk> 
wrote in message 
news:1140014159.919792.202010@g43g2000cwa.googlegroups.com...
> Hi. I note that a suitably designed FIR filter with N coefficients can
> have a time-delay of N/2 (or is it (N-1)/2?). I'm designing low pass
> filters by the window method. In my application I'm taking whole sample
> files and filtering after the fact. It is very important for me to be
> able to say for any sample in my filtered sample what the original time
> of that sample was in the original file. Hence it looks sensible for me
> to reinterpret the coefficients of my FIR filter so that when producing
> an output sample y[i], that my coefficients are applied to samples at
> times x[i-N/2] to x[i+N/2], and to assume that all samples before the
> start of my original sample or after its end are zero. Hence I get a
> filtered version of my original file without any time shift. Is this
> correct?
>
> I haven't yet examined optimal/Remerez FIR filter design in great
> detail. Is it easy and/or possible to predict the delay inherent in a
> Remerez filter as it is for one designed by the windowing method? If
> execution speed is not crucial, are there any other advantages in using
> a FIR filter designed by the Remerez method compared to simply using a
> FIR filter with a larger number of coefficients?
>
> Thanks in anticipation,
>
> Ross-c

You have the right idea.  For post-processing filtering you can center the 
filter coefficients at t=0.  Doing that will cause no time shift in the 
output for a *linear phase* i.e. symmetric filter.  This requires that the 
filter be of odd length doesn't it?  That is, to line up with the samples 
being processed.

So, for L odd, the coefficients are re-registered in time from the typical:
0 to L-1 where there are L-1 intervals spanned - and with L odd, an equal 
number on either side of zero:

-(L-1)/2 to +(L-1)/2.

Whether you use a design program based on the Remez algorithm or some other 
method shouldn't matter much except filter transition widths and ripple. 
The length of the filter determines the compute load.  I wouldn't compare 
Remez-generated filters with filters having "a larger number of 
coefficients" (presumably done using the windowing method??) because there 
are too many variables to consider.  You could use a filter with "a larger 
number of coefficients" and still not get the performance you need.  How you 
get there is less important than the result.

Note:  If you want to do the filtering in the frequency domain, then shift 
the odd-length filter circularly so the center is the zeroeth (or first) 
term in the array and have (L-1)/2 coefficients at the high end of the 
array, 1 + (L-1)/2 at the bottom of the array and any zero padding in 
between.  Then the FFT will be purely real and even and the temporal 
registration of the IFFT of the result will match with the input as you 
desire.

So a filter like this:
[1 2 3 2 1 0 0 0 0 0 0 0 0 0 0 0]
Would be circularly shifted:
[3 2 1 0 0 0 0 0 0 0 0 0 0 0 1 2]
before computing its FFT.

Fred

"Ross Clement (Email address invalid - do not use)" <clemenr@wmin.ac.uk> 
wrote in message 
news:1140018951.443983.50100@g47g2000cwa.googlegroups.com...
> Thanks. Looking at the page:
>
> http://www-ccrma.stanford.edu/~jos/filters/Zero_Phase_Filters.html
>
>
> Since I'm using very large FIR filters (up to thousands of
> coefficients), the caveat that phase oscillates between 0 and pi in the
> stopband is unlikely to cause me any trouble as the response in the
> stopband will be, I assume, tiny.
>

Thousands of coefficients is pretty scary unless the sample rate is quite 
high also.
... there will be lots of temporal leakage and I don't think that's what you 
want for speech.

Oscillating phase?????
Maybe you mean that a long linear phase filter that has long delay and, 
thus, has a high phase slope and goes through 2pi frequently?  Well, the 
delay is flat and that's all you should care about.  If you make it zero 
delay then the phase is zero and the slope is zero and there is no "phase 
oscillation" of this sort.

Fred 

Ross, determining the delay of a filter is equivalent to finding the 
"center" of its impulse response.  For linear phase filters where the 
coefficients are the same on either side of the half length of the 
filter this has a pretty clear meaning.  For an impulse response of any 
other shape, what could one consider to be the center?

Since filters disperse each sample over time it doesn't have meaning in 
general to ask which output sample corresponds to the input sample.  I 
suppose one could define it to be the time where the impulse response 
peaks (which can be between samples) but there is no real basis for such 
a definition.

 > Hence it looks sensible for me
> to reinterpret the coefficients of my FIR filter so that when producing
> an output sample y[i], that my coefficients are applied to samples at
> times x[i-N/2] to x[i+N/2], and to assume that all samples before the
> start of my original sample or after its end are zero. Hence I get a
> filtered version of my original file without any time shift. Is this
> correct?

I'm not sure what you mean by re-interpret, but with that symmetry you 
are describing a linear phase filter.  Any filter designed to have 
linear phase will have a constant group delay of half the filter length 
(for even length filters) which you can make zero for batch processing 
by shifting the output back in time by that amount.  If you have the 
impulse response of a filter lacking this characteristic, there is a 
simple method for generating the linear phase filter which has the same 
magnitude response as the one you start with.  You take the FFT of the 
filter, zero the phase at all frequencies and then take the inverse FFT. 
  Finally you simply rotate the result of that by half its length.  It 
may be necessasary to zero pad the original to twice its length first 
but I don't remember for sure.


Bob
-- 

"Things should be described as simply as possible, but no simpler."

                                              A. Einstein

Thanks Bob.

As a folllow-up question, could I use that FFT method to do the
following? Imagine that I have a filter implementation, perhaps an IIR
filter or even an analogue filter. I then calculate or measure the
impulse response of the filter up to a "sufficient" length. I then take
the FFT of that, zero the phases, and then inverse FFT and shift to get
the coefficients of an FIR filter with approximately the same frequency
response as the original IIR or analogue filter, but zero phase? I have
no obvious application of this but am curious to know.

Cheers,

Ross-c