# anyone have an answer to this?

Started by April 7, 2008
```Hi there,

Just wondering if this question was answered...I know it is an old post,
but just taking a long shot here. Essentially I am getting the same
outcome...specifically, in matlab, I can generate an appropriate
correlation plot, and retrieve the correct delay with xcorr...as well as
using the TF method : max(ifft(fft(x1).*conj(fft(x2))), i.e. through
regular cross correlation.

However, when I try to retrieve phase correlation:
X1 = fft(x1);
X2 = fft(x2);
numerator = X1.*X2;
PhaseCorr = fftshift(ifft((numerator)./abs(numerator)));

I don't get the kroeneker delta that I should be getting in the correct
delay position. What am I doing wrong???

My signals (x1, and x2) are two sinusoids of freq f and delay T between
each other, and length N samples...they have been zero padded with N-1
extra zeroes.

Does someone have a solution to this probem.

```
```On Apr 7, 6:37&#2013266080;am, "bronx" <branko_blagoje...@hotmail.com> wrote:
> Hi there,
>
> Just wondering if this question was answered...I know it is an old post,
> but just taking a long shot here. Essentially I am getting the same
> outcome...specifically, in matlab, I can generate an appropriate
> correlation plot, and retrieve the correct delay with xcorr...as well as
> using the TF method : max(ifft(fft(x1).*conj(fft(x2))), i.e. through
> regular cross correlation.
>
> However, when I try to retrieve phase correlation:
> X1 = fft(x1);
> X2 = fft(x2);
> numerator = X1.*X2;
> PhaseCorr = fftshift(ifft((numerator)./abs(numerator)));
>
> I don't get the kroeneker delta that I should be getting in the correct
> delay position. What am I doing wrong???
>
> My signals (x1, and x2) are two sinusoids of freq f and delay T between
> each other, and length N samples...they have been zero padded with N-1
> extra zeroes.
>
> Does someone have a solution to this probem.

What exactly is "phase correlation"? Give me a description, not
equations.

Where do you expect the delta to be?

Why do you expect to get a delta function out of what you are doing?

Dirk
```
```>On Apr 7, 6:37=A0am, "bronx" <branko_blagoje...@hotmail.com> wrote:
>> Hi there,
>>
>> Just wondering if this question was answered...I know it is an old
post,
>> but just taking a long shot here. Essentially I am getting the same
>> outcome...specifically, in matlab, I can generate an appropriate
>> correlation plot, and retrieve the correct delay with xcorr...as well
as
>> using the TF method : max(ifft(fft(x1).*conj(fft(x2))), i.e. through
>> regular cross correlation.
>>
>> However, when I try to retrieve phase correlation:
>> X1 =3D fft(x1);
>> X2 =3D fft(x2);
>> numerator =3D X1.*X2;
>> PhaseCorr =3D fftshift(ifft((numerator)./abs(numerator)));
>>
>> I don't get the kroeneker delta that I should be getting in the
correct
>> delay position. What am I doing wrong???
>>
>> My signals (x1, and x2) are two sinusoids of freq f and delay T
between
>> each other, and length N samples...they have been zero padded with N-1
>> extra zeroes.
>>
>> Does someone have a solution to this probem.
>
>What exactly is "phase correlation"? Give me a description, not
>equations.
>
>Where do you expect the delta to be?
>
>Why do you expect to get a delta function out of what you are doing?
>
>Dirk
>

Hi,

sorry, I should have been more clear...I am trying to perform the
generalized cross correlation PHAT (phase transform). This is also
sometimes called normalized cross correlation. This is essentially a
regular cross correlation algorithm in the frequency domain, for two
signals, divided by the absolute value of the product of one signal and
the conjugate of the other. for example:

PHAT = X1.*conj(X2)./abs(X1.*conj(X2))

If the signals are just a delayed version of each other, then the inverse
fourier transform of the PHAT should provide a plot of the signal with a
kronecker delta in the location of the time delay between the two time
shifted signals. I am looking for this time delay.
More detail regarding the algorithm is found here.
http://en.wikipedia.org/wiki/Phase_correlation

My issue is that when i plot this in matlab using the following code to
retrieve the delay information using the following code:

(x1 and x2 are equivalent sinusoids (N samples in 20ms segment) delayed by
time T from one another)
time = 20ms;
numerator = X1.*X2;
PhaseCorr = fftshift(ifft((numerator)./abs(numerator)));
y = time*linspace(-1,1, 2*N-1);
plot(t,PhaseCorr);
[maxAmp maxIndex] = max(PhaseCorr);
Tretrieved = t(maxIndex);

I don't get a delta at time Tretrieved (which is the retrieved T delay) as
I should....When I plug in normal correlation (i.e. remove
'./abs(numerator)' in the code above, then I do get a maximum in the right
time interval T. Instead my maximum point under the PHAT algorithn (which
is not very large in amplitude) is closer to the origin rather than at
time T.
The problem with using the regular CC method is that it really gets messy
when there are multiple sinusoids in the signal, and multiple delays
between them. I read that weighting CC algorithms such as the PHAT, SCOT
and Hanan Thomson transform are typically used for the retrieval of a time
delay because they are more robust to noise, and signal overlay...and most
papers suggest the PHAT one in particular.

So, my question is whether I am doing something wrong, judging from the
code above...why am I getting the right delay for regular cross
correlation, but not for the PHAT algorithm...

Any suggestions??

```
```>>On Apr 7, 6:37=A0am, "bronx" <branko_blagoje...@hotmail.com> wrote:
>>> Hi there,
>>>
>>> Just wondering if this question was answered...I know it is an old
>post,
>>> but just taking a long shot here. Essentially I am getting the same
>>> outcome...specifically, in matlab, I can generate an appropriate
>>> correlation plot, and retrieve the correct delay with xcorr...as well
>as
>>> using the TF method : max(ifft(fft(x1).*conj(fft(x2))), i.e. through
>>> regular cross correlation.
>>>
>>> However, when I try to retrieve phase correlation:
>>> X1 =3D fft(x1);
>>> X2 =3D fft(x2);
>>> numerator =3D X1.*X2;
>>> PhaseCorr =3D fftshift(ifft((numerator)./abs(numerator)));
>>>
>>> I don't get the kroeneker delta that I should be getting in the
>correct
>>> delay position. What am I doing wrong???
>>>
>>> My signals (x1, and x2) are two sinusoids of freq f and delay T
>between
>>> each other, and length N samples...they have been zero padded with
N-1
>>> extra zeroes.
>>>
>>> Does someone have a solution to this probem.
>>
>>What exactly is "phase correlation"? Give me a description, not
>>equations.
>>
>>Where do you expect the delta to be?
>>
>>Why do you expect to get a delta function out of what you are doing?
>>
>>Dirk
>>
>
>Hi,
>
>sorry, I should have been more clear...I am trying to perform the
>generalized cross correlation PHAT (phase transform). This is also
>sometimes called normalized cross correlation. This is essentially a
>regular cross correlation algorithm in the frequency domain, for two
>signals, divided by the absolute value of the product of one signal and
>the conjugate of the other. for example:
>
>PHAT = X1.*conj(X2)./abs(X1.*conj(X2))
>
>If the signals are just a delayed version of each other, then the
inverse
>fourier transform of the PHAT should provide a plot of the signal with a
>kronecker delta in the location of the time delay between the two time
>shifted signals. I am looking for this time delay.
>More detail regarding the algorithm is found here.
>http://en.wikipedia.org/wiki/Phase_correlation
>
>My issue is that when i plot this in matlab using the following code to
>retrieve the delay information using the following code:
>
>(x1 and x2 are equivalent sinusoids (N samples in 20ms segment) delayed
by
>time T from one another)
>time = 20ms;
>numerator = X1.*X2;
>PhaseCorr = fftshift(ifft((numerator)./abs(numerator)));
>y = time*linspace(-1,1, 2*N-1);
>plot(t,PhaseCorr);
>[maxAmp maxIndex] = max(PhaseCorr);
>Tretrieved = t(maxIndex);
>
>I don't get a delta at time Tretrieved (which is the retrieved T delay)
as
>I should....When I plug in normal correlation (i.e. remove
>'./abs(numerator)' in the code above, then I do get a maximum in the
right
>time interval T. Instead my maximum point under the PHAT algorithn
(which
>is not very large in amplitude) is closer to the origin rather than at
>time T.
>The problem with using the regular CC method is that it really gets
messy
>when there are multiple sinusoids in the signal, and multiple delays
>between them. I read that weighting CC algorithms such as the PHAT, SCOT
>and Hanan Thomson transform are typically used for the retrieval of a
time
>delay because they are more robust to noise, and signal overlay...and
most
>papers suggest the PHAT one in particular.
>
>So, my question is whether I am doing something wrong, judging from the
>code above...why am I getting the right delay for regular cross
>correlation, but not for the PHAT algorithm...
>
>Any suggestions??
>
>
>
Sorry made a mistake in the code above...its actually:
numerator = X1.*conj(X2); %I didnt forget the conjugate in my actual code
```
```On Apr 7, 2:47&#2013266080;pm, "bronx" <branko_blagoje...@hotmail.com> wrote:
> >>On Apr 7, 6:37=A0am, "bronx" <branko_blagoje...@hotmail.com> wrote:
> >>> Hi there,
>
> >>> Just wondering if this question was answered...I know it is an old
> >post,
> >>> but just taking a long shot here. Essentially I am getting the same
> >>> outcome...specifically, in matlab, I can generate an appropriate
> >>> correlation plot, and retrieve the correct delay with xcorr...as well
> >as
> >>> using the TF method : max(ifft(fft(x1).*conj(fft(x2))), i.e. through
> >>> regular cross correlation.
>
> >>> However, when I try to retrieve phase correlation:
> >>> X1 =3D fft(x1);
> >>> X2 =3D fft(x2);
> >>> numerator =3D X1.*X2;
> >>> PhaseCorr =3D fftshift(ifft((numerator)./abs(numerator)));
>
> >>> I don't get the kroeneker delta that I should be getting in the
> >correct
> >>> delay position. What am I doing wrong???
>
> >>> My signals (x1, and x2) are two sinusoids of freq f and delay T
> >between
> >>> each other, and length N samples...they have been zero padded with
> N-1
> >>> extra zeroes.
>
> >>> Does someone have a solution to this probem.
>
> >>What exactly is "phase correlation"? Give me a description, not
> >>equations.
>
> >>Where do you expect the delta to be?
>
> >>Why do you expect to get a delta function out of what you are doing?
>
> >>Dirk
>
> >Hi,
>
> >sorry, I should have been more clear...I am trying to perform the
> >generalized cross correlation PHAT (phase transform). This is also
> >sometimes called normalized cross correlation. This is essentially a
> >regular cross correlation algorithm in the frequency domain, for two
> >signals, divided by the absolute value of the product of one signal and
> >the conjugate of the other. for example:
>
> >PHAT = X1.*conj(X2)./abs(X1.*conj(X2))
>
> >If the signals are just a delayed version of each other, then the
> inverse
> >fourier transform of the PHAT should provide a plot of the signal with a
> >kronecker delta in the location of the time delay between the two time
> >shifted signals. I am looking for this time delay.
> >More detail regarding the algorithm is found here.
> >http://en.wikipedia.org/wiki/Phase_correlation
>
> >My issue is that when i plot this in matlab using the following code to
> >retrieve the delay information using the following code:
>
> >(x1 and x2 are equivalent sinusoids (N samples in 20ms segment) delayed
> by
> >time T from one another)
> >time = 20ms;
> >numerator = X1.*X2;
> >PhaseCorr = fftshift(ifft((numerator)./abs(numerator)));
> >y = time*linspace(-1,1, 2*N-1);
> >plot(t,PhaseCorr);
> >[maxAmp maxIndex] = max(PhaseCorr);
> >Tretrieved = t(maxIndex);
>
> >I don't get a delta at time Tretrieved (which is the retrieved T delay)
> as
> >I should....When I plug in normal correlation (i.e. remove
> >'./abs(numerator)' in the code above, then I do get a maximum in the
> right
> >time interval T. Instead my maximum point under the PHAT algorithn
> (which
> >is not very large in amplitude) is closer to the origin rather than at
> >time T.
> >The problem with using the regular CC method is that it really gets
> messy
> >when there are multiple sinusoids in the signal, and multiple delays
> >between them. I read that weighting CC algorithms such as the PHAT, SCOT
> >and Hanan Thomson transform are typically used for the retrieval of a
> time
> >delay because they are more robust to noise, and signal overlay...and
> most
> >papers suggest the PHAT one in particular.
>
> >So, my question is whether I am doing something wrong, judging from the
> >code above...why am I getting the right delay for regular cross
> >correlation, but not for the PHAT algorithm...
>
> >Any suggestions??
>
> Sorry made a mistake in the code above...its actually:
> numerator = X1.*conj(X2); %I didnt forget the conjugate in my actual code- Hide quoted text -
>
> - Show quoted text -

You should post the equations for the frequencies, length of signals,
size of FFT, ...

I suspect this method is going to run into problems where there are a
lot of FFT bins where there is virtually no signal, such that the
phase is really based more on numerical noise than any signal
present.  In that case the phase difference between these bins would
not be dependent on the delay and would not contribute to your
result.  I don't know if you have this problem or not, but it would
somethng to check for.

Dirk
```
```>
>You should post the equations for the frequencies, length of signals,
>size of FFT, ...
>
>I suspect this method is going to run into problems where there are a
>lot of FFT bins where there is virtually no signal, such that the
>phase is really based more on numerical noise than any signal
>present.  In that case the phase difference between these bins would
>not be dependent on the delay and would not contribute to your
>result.  I don't know if you have this problem or not, but it would
>somethng to check for.
>
>Dirk
>

here is my code in detail

freq1 = 0.5e3;     %Frequency of incoming source in Hz
time = 20e-3;   %Total time of simulation in s
tau1 = 0.375e-3;     %time difference in arrival of 2 signals

N = pow2(11);
%t = (0:N-1)*time;
t = linspace(0,time,N);

% Signals arriving at input of microphones...with different phase

% MIC A
x1 = sin(2*pi*freq1*t);

% MIC B
x2 = sin(2*pi*freq1*(t + tau1));

figure(1);
plot(t, x1, "r");
hold on;
plot(t, x2, "g");

XR_CC_num = X1.*conj(X2);
XR_CC_den = abs(XR_CC_num);
XR = (XR_CC_num)./(XR_CC_den);
%Phase Correlation
PhaseCorr = fftshift((ifft(XR)));
%Cross Correlation
CrossCorr = fftshift(ifft(XR_CC_num));
CorrTime = time*linspace(-1,1, NPC);

figure(2);
%Plot Phase correlation in ms
plot(1e3*CorrTime,real(PhaseCorr),'r');

figure(3);
%Plot Cross Correlation in ms
plot(1e3*CorrTime,real(CrossCorr),'g');

%the location of the peak of each result is the time delay between signal
%%x1 and signal x2

let me know if you know why the peaks don't show the same time delay
index.

Thanks
```
```On Apr 7, 5:31&#2013266080;pm, "bronx" <branko_blagoje...@hotmail.com> wrote:
> >You should post the equations for the frequencies, length of signals,
> >size of FFT, ...
>
> >I suspect this method is going to run into problems where there are a
> >lot of FFT bins where there is virtually no signal, such that the
> >phase is really based more on numerical noise than any signal
> >present. &#2013266080;In that case the phase difference between these bins would
> >not be dependent on the delay and would not contribute to your
> >result. &#2013266080;I don't know if you have this problem or not, but it would
> >somethng to check for.
>
> >Dirk
>
> here is my code in detail
>
> freq1 = 0.5e3; &#2013266080; &#2013266080; %Frequency of incoming source in Hz
> time = 20e-3; &#2013266080; %Total time of simulation in s
> tau1 = 0.375e-3; &#2013266080; &#2013266080; %time difference in arrival of 2 signals
>
> N = pow2(11);
> %t = (0:N-1)*time;
> t = linspace(0,time,N);
>
> % Signals arriving at input of microphones...with different phase
>
> % MIC A
> x1 = sin(2*pi*freq1*t);
>
> % MIC B
> x2 = sin(2*pi*freq1*(t + tau1));
>
> figure(1);
> plot(t, x1, "r");
> hold on;
> plot(t, x2, "g");
>
>
> XR_CC_num = X1.*conj(X2);
> XR_CC_den = abs(XR_CC_num);
> XR = (XR_CC_num)./(XR_CC_den);
> %Phase Correlation
> PhaseCorr = fftshift((ifft(XR)));
> %Cross Correlation
> CrossCorr = fftshift(ifft(XR_CC_num));
> CorrTime = time*linspace(-1,1, NPC);
>
> figure(2);
> %Plot Phase correlation in ms
> plot(1e3*CorrTime,real(PhaseCorr),'r');
>
> figure(3);
> %Plot Cross Correlation in ms
> plot(1e3*CorrTime,real(CrossCorr),'g');
>
> %the location of the peak of each result is the time delay between signal
> %%x1 and signal x2
>
> let me know if you know why the peaks don't show the same time delay
> index.
>
> Thanks

the other.

Take a step back and find the difference in the phases for the two
FFTs.  Is the phase what you expect?

Dirk
```
```On Apr 8, 9:09&#2013266080;am, dbell <bellda2...@cox.net> wrote:
> On Apr 7, 5:31&#2013266080;pm, "bronx" <branko_blagoje...@hotmail.com> wrote:
>
>
>
>
>
> > >You should post the equations for the frequencies, length of signals,
> > >size of FFT, ...
>
> > >I suspect this method is going to run into problems where there are a
> > >lot of FFT bins where there is virtually no signal, such that the
> > >phase is really based more on numerical noise than any signal
> > >present. &#2013266080;In that case the phase difference between these bins would
> > >not be dependent on the delay and would not contribute to your
> > >result. &#2013266080;I don't know if you have this problem or not, but it would
> > >somethng to check for.
>
> > >Dirk
>
> > here is my code in detail
>
> > freq1 = 0.5e3; &#2013266080; &#2013266080; %Frequency of incoming source in Hz
> > time = 20e-3; &#2013266080; %Total time of simulation in s
> > tau1 = 0.375e-3; &#2013266080; &#2013266080; %time difference in arrival of 2 signals
>
> > N = pow2(11);
> > %t = (0:N-1)*time;
> > t = linspace(0,time,N);
>
> > % Signals arriving at input of microphones...with different phase
>
> > % MIC A
> > x1 = sin(2*pi*freq1*t);
>
> > % MIC B
> > x2 = sin(2*pi*freq1*(t + tau1));
>
> > figure(1);
> > plot(t, x1, "r");
> > hold on;
> > plot(t, x2, "g");
>
> > x1pad = [(x1) zeros(1,N-1)];
> > x2pad = [(x2) zeros(1,N-1)];
>
> > XR_CC_num = X1.*conj(X2);
> > XR_CC_den = abs(XR_CC_num);
> > XR = (XR_CC_num)./(XR_CC_den);
> > %Phase Correlation
> > PhaseCorr = fftshift((ifft(XR)));
> > %Cross Correlation
> > CrossCorr = fftshift(ifft(XR_CC_num));
> > CorrTime = time*linspace(-1,1, NPC);
>
> > figure(2);
> > %Plot Phase correlation in ms
> > plot(1e3*CorrTime,real(PhaseCorr),'r');
>
> > figure(3);
> > %Plot Cross Correlation in ms
> > plot(1e3*CorrTime,real(CrossCorr),'g');
>
> > %the location of the peak of each result is the time delay between signal
> > %%x1 and signal x2
>
> > let me know if you know why the peaks don't show the same time delay
> > index.
>
> > Thanks
>
> the other.
>
> Take a step back and find the difference in the phases for the two
> FFTs. &#2013266080;Is the phase what you expect?
>
> Dirk- Hide quoted text -
>
> - Show quoted text -

Sure they are!  Plot them and take a look.

Maurice Givens
```
```On Apr 7, 4:31&#2013266080;pm, "bronx" <branko_blagoje...@hotmail.com> wrote:
> >You should post the equations for the frequencies, length of signals,
> >size of FFT, ...
>
> >I suspect this method is going to run into problems where there are a
> >lot of FFT bins where there is virtually no signal, such that the
> >phase is really based more on numerical noise than any signal
> >present. &#2013266080;In that case the phase difference between these bins would
> >not be dependent on the delay and would not contribute to your
> >result. &#2013266080;I don't know if you have this problem or not, but it would
> >somethng to check for.
>
> >Dirk
>
> here is my code in detail
>
> freq1 = 0.5e3; &#2013266080; &#2013266080; %Frequency of incoming source in Hz
> time = 20e-3; &#2013266080; %Total time of simulation in s
> tau1 = 0.375e-3; &#2013266080; &#2013266080; %time difference in arrival of 2 signals
>
> N = pow2(11);
> %t = (0:N-1)*time;
> t = linspace(0,time,N);
>
> % Signals arriving at input of microphones...with different phase
>
> % MIC A
> x1 = sin(2*pi*freq1*t);
>
> % MIC B
> x2 = sin(2*pi*freq1*(t + tau1));
>
> figure(1);
> plot(t, x1, "r");
> hold on;
> plot(t, x2, "g");
>
>
> XR_CC_num = X1.*conj(X2);
> XR_CC_den = abs(XR_CC_num);
> XR = (XR_CC_num)./(XR_CC_den);
> %Phase Correlation
> PhaseCorr = fftshift((ifft(XR)));
> %Cross Correlation
> CrossCorr = fftshift(ifft(XR_CC_num));
> CorrTime = time*linspace(-1,1, NPC);
>
> figure(2);
> %Plot Phase correlation in ms
> plot(1e3*CorrTime,real(PhaseCorr),'r');
>
> figure(3);
> %Plot Cross Correlation in ms
> plot(1e3*CorrTime,real(CrossCorr),'g');
>
> %the location of the peak of each result is the time delay between signal
> %%x1 and signal x2
>
> let me know if you know why the peaks don't show the same time delay
> index.
>
> Thanks

x1 is a delayed version of x2 by tau1.  However, the question I think
you should be asking yourself is why you think the FFT of x1 would be
different than the FFT of x2.

Maurice Givens
```
```On Apr 8, 1:17&#2013266080;pm, maury <maury...@core.com> wrote:
> On Apr 8, 9:09&#2013266080;am, dbell <bellda2...@cox.net> wrote:
>
>
>
>
>
> > On Apr 7, 5:31&#2013266080;pm, "bronx" <branko_blagoje...@hotmail.com> wrote:
>
> > > >You should post the equations for the frequencies, length of signals,
> > > >size of FFT, ...
>
> > > >I suspect this method is going to run into problems where there are a
> > > >lot of FFT bins where there is virtually no signal, such that the
> > > >phase is really based more on numerical noise than any signal
> > > >present. &#2013266080;In that case the phase difference between these bins would
> > > >not be dependent on the delay and would not contribute to your
> > > >result. &#2013266080;I don't know if you have this problem or not, but it would
> > > >somethng to check for.
>
> > > >Dirk
>
> > > here is my code in detail
>
> > > freq1 = 0.5e3; &#2013266080; &#2013266080; %Frequency of incoming source in Hz
> > > time = 20e-3; &#2013266080; %Total time of simulation in s
> > > tau1 = 0.375e-3; &#2013266080; &#2013266080; %time difference in arrival of 2 signals
>
> > > N = pow2(11);
> > > %t = (0:N-1)*time;
> > > t = linspace(0,time,N);
>
> > > % Signals arriving at input of microphones...with different phase
>
> > > % MIC A
> > > x1 = sin(2*pi*freq1*t);
>
> > > % MIC B
> > > x2 = sin(2*pi*freq1*(t + tau1));
>
> > > figure(1);
> > > plot(t, x1, "r");
> > > hold on;
> > > plot(t, x2, "g");
>
> > > x1pad = [(x1) zeros(1,N-1)];
> > > x2pad = [(x2) zeros(1,N-1)];
>
> > > NPC = length(x1pad);
> > > X1 = fft(x1pad);
> > > X2 = fft(x2pad);
> > > XR_CC_num = X1.*conj(X2);
> > > XR_CC_den = abs(XR_CC_num);
> > > XR = (XR_CC_num)./(XR_CC_den);
> > > %Phase Correlation
> > > PhaseCorr = fftshift((ifft(XR)));
> > > %Cross Correlation
> > > CrossCorr = fftshift(ifft(XR_CC_num));
> > > CorrTime = time*linspace(-1,1, NPC);
>
> > > figure(2);
> > > %Plot Phase correlation in ms
> > > plot(1e3*CorrTime,real(PhaseCorr),'r');
>
> > > figure(3);
> > > %Plot Cross Correlation in ms
> > > plot(1e3*CorrTime,real(CrossCorr),'g');
>
> > > %the location of the peak of each result is the time delay between signal
> > > %%x1 and signal x2
>
> > > let me know if you know why the peaks don't show the same time delay
> > > index.
>
> > > Thanks
>
> > the other.
>
> > Take a step back and find the difference in the phases for the two
> > FFTs. &#2013266080;Is the phase what you expect?
>
> > Dirk- Hide quoted text -
>
> > - Show quoted text -
>
> Sure they are! &#2013266080;Plot them and take a look.
>
> Maurice Givens- Hide quoted text -
>
> - Show quoted text -