Fourier Transform unique?

Can two diffrent signals have the same fourier tranform in both magnitude
and phase but the time domain signal be diffrent?

On Sep 5, 11:25 am, "westocl" <cwest...@hotmail.com> wrote:
> Can two diffrent signals have the same fourier tranform in both magnitude
> and phase but the time domain signal be diffrent?

The answer depends upon how mathematically accurate you want to be.
For practical purposes, the time domain signal has to be the same.
Mathematically, you would say that the signals may at most be
different on a set of measure zero.

Dilip.

westocl wrote:
> Can two diffrent signals have the same fourier tranform in both
> magnitude and phase but the time domain signal be diffrent?

Just say "no".

Except for anomalous cases, the Fourier Transform is a 1 to 1 
transformation.  That means that any unique sequence transforms to a 
corresponding unique sequence and the latter inverse transforms accordingly.

Fred

>westocl wrote:
>> Can two diffrent signals have the same fourier tranform in both
>> magnitude and phase but the time domain signal be diffrent?
>
>Just say "no".
>
>Except for anomalous cases, the Fourier Transform is a 1 to 1 
>transformation.  That means that any unique sequence transforms to a 
>corresponding unique sequence and the latter inverse transforms
accordingly.
>
>Fred
>
>
>
are these signals anomalous?  I may have sampled them wrong. but they seem
to have a strong sinusoidal group delay and the same FFT.
signal 1
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signal b
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westocl wrote:
> are these signals anomalous?  I may have sampled them wrong. but they
> seem to have a strong sinusoidal group delay and the same FFT.

By my calculations, they don't have FFTs that look even slightly 
similar.....
Now, had you asked about their *magnitudes* then that's a different matter 
but not what we were talking about.  They aren't equal but are much 
closer....  but "close" isn't what we were talking about either.

You may have sampled *something* wrong but these sequences are what they 
are.  So *they* aren't "wrong" they just "are".

Fred 

Dilip Warrier wrote:
> On Sep 5, 11:25 am, "westocl" <cwest...@hotmail.com> wrote:

>>Can two diffrent signals have the same fourier tranform in both magnitude
>>and phase but the time domain signal be diffrent?

For the Fourier series, you can consider aliasing, which allows
different signals to have the same transform.  Though it is
actually the sampling, and not the transform, that causes
the problem.

> The answer depends upon how mathematically accurate you want to be.
> For practical purposes, the time domain signal has to be the same.
> Mathematically, you would say that the signals may at most be
> different on a set of measure zero.

For the continuous transform, any signal with a non-infinite
(non delta function) discontinuity at a point can have the
same transform.  Consider, for example,

   f(x)=0  x>0
   f(0)=1
   f(x)=0  x<0

The transform will be indistinguishable from zero.
Any signal with such added or subtracted will also be
have its transform unchanged.

There are also signals like sin(exp(x)) or sin(1/x)
that may not transform very well.

-- glen

>westocl wrote:
>> are these signals anomalous?  I may have sampled them wrong. but they
>> seem to have a strong sinusoidal group delay and the same FFT.
>
>By my calculations, they don't have FFTs that look even slightly 
>similar.....
>Now, had you asked about their *magnitudes* then that's a different
matter 
>but not what we were talking about.  They aren't equal but are much 
>closer....  but "close" isn't what we were talking about either.
>
>You may have sampled *something* wrong but these sequences are what they

>are.  So *they* aren't "wrong" they just "are".
>
>Fred 
>
>
>

I looked at the signals and found the same thing.  The magnitudes are
close, but not identical, but the phases are quite different.  
Regards,
Steve

Fred Marshall wrote:
> westocl wrote:
>> are these signals anomalous?  I may have sampled them wrong. but they
>> seem to have a strong sinusoidal group delay and the same FFT.
>
> By my calculations, they don't have FFTs that look even slightly
> similar.....
> Now, had you asked about their *magnitudes* then that's a different
> matter but not what we were talking about.  They aren't equal but are
> much closer....  but "close" isn't what we were talking about either.
>
> You may have sampled *something* wrong but these sequences are what
> they are.  So *they* aren't "wrong" they just "are".
>
> Fred

Oh... you did ask about magnitude AND phase... which I interpreted to mean 
Real part and Imaginary part as they should map 1:1.  But, phase doesn't map 
1:1 but many:1 so we talk about "unwrapping" phase.  Yet, I believe 
sequences must map 1:1.  Somebody more inclined that I might help out 
here....

Fred 

>Fred Marshall wrote:
>> westocl wrote:
>>> are these signals anomalous?  I may have sampled them wrong. but they
>>> seem to have a strong sinusoidal group delay and the same FFT.
>>
>> By my calculations, they don't have FFTs that look even slightly
>> similar.....
>> Now, had you asked about their *magnitudes* then that's a different
>> matter but not what we were talking about.  They aren't equal but are
>> much closer....  but "close" isn't what we were talking about either.
>>
>> You may have sampled *something* wrong but these sequences are what
>> they are.  So *they* aren't "wrong" they just "are".
>>
>> Fred
>
>Oh... you did ask about magnitude AND phase... which I interpreted to
mean 
>Real part and Imaginary part as they should map 1:1.  But, phase doesn't
map 
>1:1 but many:1 so we talk about "unwrapping" phase.  Yet, I believe 
>sequences must map 1:1.  Somebody more inclined that I might help out 
>here....
>
>Fred 
>
>

Hi Fred,
The "many:1" property of the phase doesn't stop the Fourier Transform from
being a 1:1 mapping.  Say you have a point in the frequency domain with a
phase of P.  This is ambiguous with a phase of P + 2 pi, P + 4 pi, and so
on.  However, in the time domain this corresponds to shifting a cosine wave
by 2 pi, 4 pi, etc., which leave it unchanged.  So P in the frequency
domain (with its 2 pi ambiguity) is a 1:1 match with a single waveform in
the time domain (with its 2 pi ambiguity).  
Regards,
Steve 

>Fred Marshall wrote:
>> westocl wrote:
>>> are these signals anomalous?  I may have sampled them wrong. but they
>>> seem to have a strong sinusoidal group delay and the same FFT.
>>
>> By my calculations, they don't have FFTs that look even slightly
>> similar.....
>> Now, had you asked about their *magnitudes* then that's a different
>> matter but not what we were talking about.  They aren't equal but are
>> much closer....  but "close" isn't what we were talking about either.
>>
>> You may have sampled *something* wrong but these sequences are what
>> they are.  So *they* aren't "wrong" they just "are".
>>
>> Fred
>
>Oh... you did ask about magnitude AND phase... which I interpreted to
mean 
>Real part and Imaginary part as they should map 1:1.  But, phase doesn't
map 
>1:1 but many:1 so we talk about "unwrapping" phase.  Yet, I believe 
>sequences must map 1:1.  Somebody more inclined that I might help out 
>here....
>
>Fred 
>
>

Hi Fred,
The "many:1" property of the phase doesn't stop the Fourier Transform from
being a 1:1 mapping.  Say you have a point in the frequency domain with a
phase of P.  This is ambiguous with a phase of P + 2 pi, P + 4 pi, and so
on.  However, in the time domain this corresponds to shifting a cosine wave
by 2 pi, 4 pi, etc., which leave it unchanged.  So P in the frequency
domain (with its 2 pi ambiguity) is a 1:1 match with a single waveform in
the time domain (with its 2 pi ambiguity).  
Regards,
Steve