Equalization using DFT

Hi,

I'm wondering, if there is no problem in equalization with DFT. Simply
window  the source signal, DFT, mutliply each frequency bin by a real value
defining it, and IDFT. This would even be zero-phase filtering wouldn't it?
Or am I missing something?

Thanks!
dmnc

On Nov 12, 11:26&#4294967295;am, "jungledmnc" <jungled...@gmail.com> wrote:
> Hi,
>
> I'm wondering, if there is no problem in equalization with DFT. Simply
> window &#4294967295;the source signal, DFT, mutliply each frequency bin by a real value
> defining it, and IDFT. This would even be zero-phase filtering wouldn't it?
> Or am I missing something?

what you're missing is the need to do overlap-adding or overlap-saving
with this.  and your "windowing" is really rectangular (i.e. no
windowing at all).  also, then the frequency response (what you use to
multiply each DFT frequency bin) must be such that the impulse
response corresponding to it is short enough, depending on the size of
the FFT and the "frame hop" distance between adjacent frames.

r b-j

>Hi,
>
>I'm wondering, if there is no problem in equalization with DFT. Simply
>window  the source signal, DFT, mutliply each frequency bin by a real
value
>defining it, and IDFT. This would even be zero-phase filtering wouldn't
it?
>Or am I missing something?
>
>Thanks!
>dmnc
>

Basically correct, but there are a ton of details.  Here's a link. 
Regards,
Steve

http://www.dspguide.com/ch18.htm

On Nov 12, 8:26 am, "jungledmnc" <jungled...@gmail.com> wrote:
> Hi,
>
> I'm wondering, if there is no problem in equalization with DFT. Simply
> window  the source signal, DFT, mutliply each frequency bin by a real value
> defining it, and IDFT. This would even be zero-phase filtering wouldn't it?
> Or am I missing something?
>
> Thanks!
> dmnc

There is a lot of relevant practical application literature in the
computer music analysis/synthesis field and the audio compression
field. In the audio compression literature, substitute your
equalization process for the coding and decoding portions.

Dale B. Dalrymple
http://dbdimages.com

Thanks guys. I forgot to mention the windowing and overlap-adding... but I
mean when you perform FFT correctly with all it needs and the only
modification is multiplying each frequency bin with desired frequency
response, then there is no problem?
If so, then why not to use it, since it is zero or linear phase (which one
btw?). And what's the advantage of using standard filters instead (e.g.
classical biquads) (speed of course, but that's all I can see)?
What if I for example cascade a few biquad filters, get its frequency
response for particular frequency bins, and use this FFT method instead?
Wouldn't it be cool way to make it zero/linear phase filter? I can also use
bidirectional filtering, but the requirements seem almost the same.


>also, then the frequency response (what you use to
>multiply each DFT frequency bin) must be such that the impulse
>response corresponding to it is short enough, depending on the size of
>the FFT and the "frame hop" distance between adjacent frames.

What you mean by this exactly? I think this cannot happen when I use
frequency response as described above, can it?

Thanks again!
dmnc

>
>Thanks guys. I forgot to mention the windowing and overlap-adding... but
I
>mean when you perform FFT correctly with all it needs and the only
>modification is multiplying each frequency bin with desired frequency
>response, then there is no problem?
>If so, then why not to use it, since it is zero or linear phase (which
one
>btw?). And what's the advantage of using standard filters instead (e.g.
>classical biquads) (speed of course, but that's all I can see)?
>What if I for example cascade a few biquad filters, get its frequency
>response for particular frequency bins, and use this FFT method instead?
>Wouldn't it be cool way to make it zero/linear phase filter? I can also
use
>bidirectional filtering, but the requirements seem almost the same.
>

FFT convolution (the technique you are describing) produces exactly the
same filtering results as an FIR filter carried out by standard
convolution. The advantage of FFT convolution is that it is very fast if
the kernel is long, say, greater than 50 points.  On the down side, the
code to get FFT convolution going is quite involved.  In comparision,
convolution is only a few lines of code.  

So your first question reduces to: "What are the advantages and
disadvantages of FIR (i.e., convolution) vs IIR (i.e., biquad or recursive)
filters?"  Here's a link.

http://www.dspguide.com/ch21.htm

A zero phase filter becomes a linear phase filter when the output is
shifted relative to the input.  For real time processing there is always a
delay, so the filter will be linear phase.  However, if the data are just
stored in arrays in a computer, it simply depends of how you stuff the
ouput data into the array, relative to the input array. 
Regards,
Steve







   

jungledmnc wrote:
> Thanks guys. I forgot to mention the windowing and overlap-adding... but I
> mean when you perform FFT correctly with all it needs and the only
> modification is multiplying each frequency bin with desired frequency
> response, then there is no problem?
> If so, then why not to use it, since it is zero or linear phase (which one
> btw?). And what's the advantage of using standard filters instead (e.g.
> classical biquads) (speed of course, but that's all I can see)?
> What if I for example cascade a few biquad filters, get its frequency
> response for particular frequency bins, and use this FFT method instead?
> Wouldn't it be cool way to make it zero/linear phase filter? I can also use
> bidirectional filtering, but the requirements seem almost the same.

Phase is part of a filter's effect. It is not generally possible to 
combine one filter's amplitude response with another's phase response.

>> also, then the frequency response (what you use to
>> multiply each DFT frequency bin) must be such that the impulse
>> response corresponding to it is short enough, depending on the size of
>> the FFT and the "frame hop" distance between adjacent frames.
> 
> What you mean by this exactly? I think this cannot happen when I use
> frequency response as described above, can it?

You need to get a few things straight:

    Every filter has an impulse response.

    The overlap-add method using Fourier transforms performs exactly the
    same convolution as the traditional transversal method of doing FIRs.

    The coefficients (tap weights, if you prefer) of a transversal FIR
    _are_ its impulse response. The impulse response of an overlap-add
    filter is the IFFT of the frequency-response shaping mask. Arbitrary
    masks can produce very long impulse responses, and the FFT has to be
    long enough to accommodate it.

    A tapered window is a sometimes useful adjunct to an FFT for
    frequency analysis. It is never needed and actually harmful when used
    in OL-A filtering.

You can't make a brick-wall filter simply by zeroing out the bins you 
don't want. The implied impulse response is far too long. Besides, you 
don't really control the response _between_ the bin centers, and they 
will bite you if the transitions are too abrupt. Moreover, the time 
response will ring like a fire bell.

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

>jungledmnc wrote:
>> Thanks guys. I forgot to mention the windowing and overlap-adding...
but I
>> mean when you perform FFT correctly with all it needs and the only
>> modification is multiplying each frequency bin with desired frequency
>> response, then there is no problem?
>> If so, then why not to use it, since it is zero or linear phase (which
one
>> btw?). And what's the advantage of using standard filters instead
(e.g.
>> classical biquads) (speed of course, but that's all I can see)?
>> What if I for example cascade a few biquad filters, get its frequency
>> response for particular frequency bins, and use this FFT method
instead?
>> Wouldn't it be cool way to make it zero/linear phase filter? I can also
use
>> bidirectional filtering, but the requirements seem almost the same.
>
>Phase is part of a filter's effect. It is not generally possible to 
>combine one filter's amplitude response with another's phase response.
>
>>> also, then the frequency response (what you use to
>>> multiply each DFT frequency bin) must be such that the impulse
>>> response corresponding to it is short enough, depending on the size
of
>>> the FFT and the "frame hop" distance between adjacent frames.
>> 
>> What you mean by this exactly? I think this cannot happen when I use
>> frequency response as described above, can it?
>
>You need to get a few things straight:
>
>    Every filter has an impulse response.
>
>    The overlap-add method using Fourier transforms performs exactly the
>    same convolution as the traditional transversal method of doing
FIRs.
>
>    The coefficients (tap weights, if you prefer) of a transversal FIR
>    _are_ its impulse response. The impulse response of an overlap-add
>    filter is the IFFT of the frequency-response shaping mask. Arbitrary
>    masks can produce very long impulse responses, and the FFT has to be
>    long enough to accommodate it.
>
>    A tapered window is a sometimes useful adjunct to an FFT for
>    frequency analysis. It is never needed and actually harmful when
used
>    in OL-A filtering.
>
>You can't make a brick-wall filter simply by zeroing out the bins you 
>don't want. The implied impulse response is far too long. Besides, you 
>don't really control the response _between_ the bin centers, and they 
>will bite you if the transitions are too abrupt. Moreover, the time 
>response will ring like a fire bell.
>
>Jerry
>-- 
>Engineering is the art of making what you want from things you can get.
>�����������������������������������������������������������������������
>
Isn't using a not rectangular window the same as using the window method
of FIR design?  Along those lines, having a brick wall filter would result
in a windowed sinc type filter correct?

SteveSmith wrote:

   ...

> FFT convolution (the technique you are describing) produces exactly the
> same filtering results as an FIR filter carried out by standard
> convolution. The advantage of FFT convolution is that it is very fast if
> the kernel is long, say, greater than 50 points.  On the down side, the
> code to get FFT convolution going is quite involved.  In comparision,
> convolution is only a few lines of code.  

dmnc's puzzlement stems from the equivalent FIR not being the one he 
imagines it to be. I'd like to think up an imaginative way to illustrate 
the difference.

   ...

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

jungledmnc wrote:
> Thanks guys. I forgot to mention the windowing and overlap-adding... but I
> mean when you perform FFT correctly with all it needs and the only
> modification is multiplying each frequency bin with desired frequency
> response, then there is no problem?
> If so, then why not to use it, since it is zero or linear phase (which one
> btw?). And what's the advantage of using standard filters instead (e.g.
> classical biquads) (speed of course, but that's all I can see)?
> What if I for example cascade a few biquad filters, get its frequency
> response for particular frequency bins, and use this FFT method instead?
> Wouldn't it be cool way to make it zero/linear phase filter? I can also use
> bidirectional filtering, but the requirements seem almost the same.
> 


Linear-phase EQ is widely used for audio mastering  (see e.g. Waves, 
Algorithmix, even the L-PP EQ in Apple's Logic Pro etc). Possibly all of 
them use FFTs, but I suspect  Algorithmix at least is doing some thing 
very esoteric as their EQ, while praised to the skies (used at 
astronomical sample rates for such things as SACD mastering), is 
vveerryy  slow, and uses huge impulse responses of over a second. 
Basically,  FFT-based EQ and convolution reverb use exactly the same 
algorithm, except for the length of the impulse responses. Reverb 
responses can be extremely long, and "partitioned convolution" of one 
form or another is used to get the latency low enough for real-time use. 
  The "trick" is simply to ensure the FFT is used to perform ~linear~ 
convolution of the source and the IR, which means a combination of 
zero-padding and plain 50% frame overlap (where the frame length is 
typically twice the raw length of the IR, usually rounded  up to a 
power-of-two size for FFT purposes). The other tricks are in designing 
the IR, which may be done in all manner of ways, from drawing of 
arbitrary shapes (with all the usual caveats) to use of standard FIR 
filter design tools such as Remez etc. The dspguide links posted earlier 
  in this thread illustrate  all that in detail.

The Voxengo Spline EQ is an example of a L-P EQ using almost-arbitrary 
drawing, showing  how closely the generated response matches the 
requested one:

http://www.voxengo.com/product/curveeq/


The reason audio people actually use IIR filters in audio most of the 
time is only partly because they are computationally cheap compared to 
FIRs (and easy to vary dynamically). Some analogue-modelled EQs are 
anything but cheap; but they endow the sound with that mythical 
"character" (no doubt thanks to the non-linear phase, and the odd 
sprinkle of dsp snake-oil) that recording and mixing engineers will get 
paid lots of money  to impart to a track. Bob Katz's mastering book is 
well worth a read on this topic (as on many others).


Richard Dobson