Fractional Fourier Transform

Hello

The Fourier Transform decomposes a function of time (a signal) into the
frequencies, and it's inverse transform means that a time domain signal
can be constructed as a sum of trigonometric sinusoidal functions.
I am now also trying to understand and visualize the Fractional Fourier
Transform (FrFT). 

I am actually new in this topic and have found some usefull articles
regarding FrFT on the web but they are all explaning the maths behind
FrFT, but I would like to get some visualization that FrFT can be
interpreted as Fourier or a point of view that is localized through a line
(time-frequency) where the signal has both time and frequency components
in a space.

Is there a way that we can interpret the Fractional Fourier Transform as
the Fourier Transform? A time domain signal decomposition into frequencies
and composition from the frequencies?

Thanks in advances!
---------------------------------------
Posted through http://www.DSPRelated.com

On Wed, 05 Oct 2016 07:09:22 -0500, runinrainy wrote:

> Hello
> 
> The Fourier Transform decomposes a function of time (a signal) into the
> frequencies, and it's inverse transform means that a time domain signal
> can be constructed as a sum of trigonometric sinusoidal functions.
> I am now also trying to understand and visualize the Fractional Fourier
> Transform (FrFT).
> 
> I am actually new in this topic and have found some usefull articles
> regarding FrFT on the web but they are all explaning the maths behind
> FrFT, but I would like to get some visualization that FrFT can be
> interpreted as Fourier or a point of view that is localized through a
> line (time-frequency) where the signal has both time and frequency
> components in a space.
> 
> Is there a way that we can interpret the Fractional Fourier Transform as
> the Fourier Transform? A time domain signal decomposition into
> frequencies and composition from the frequencies?
> 
> Thanks in advances!
> ---------------------------------------
> Posted through http://www.DSPRelated.com

Have you looked at the references cited in the Wikipedia article on the 
FrFT?  Some of the titles looked like they might be practice-oriented.  
Actually, there's a graphic in that article that looks intriguing.

Or search on "Fractional fourier transform applications".

I suspect that what you'll find will be applications that live on the 
edges of a number of different applications areas -- certainly in 
communications we've worked hard to arrange the world into nice tidy 
frequency bands, thus putting ourselves outside of a domain that demands 
the use of the FrFT.

-- 
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work!  See my website if you're interested
http://www.wescottdesign.com

>Hello
>
>The Fourier Transform decomposes a function of time (a signal) into the
>frequencies, and it's inverse transform means that a time domain signal
>can be constructed as a sum of trigonometric sinusoidal functions.

[...snip...]

>Is there a way that we can interpret the Fractional Fourier Transform as
>the Fourier Transform? A time domain signal decomposition into
frequencies
>and composition from the frequencies?
>
>Thanks in advances!
>---------------------------------------
>Posted through http://www.DSPRelated.com

I am afraid that you are working under an idealized misconception of the
Fourier Transform.  The only cases for which it works the way you describe
are when the signal is a repeating waveform and the analysis frame is an
integer multiple of the waveform length.  If the analysis frame is
infinitely long, any mix of infinitely long pure tones will also work as
you describe.

For cases with a finite analysis frame, for any tone with a frequency that
is not an integer multiple of the frame size, the "constituent
frequencies" you get from the FT do *NOT* represent frequencies of the
tone, they represent the "building blocks" to reconstruct that signal
within the analysis frame.  If you extend the inverse beyond the limits of
the frame you will not get an extension of such a tone, you will get a
repeat of the analysis frame.

The study of window functions is largely an exercise to make the FT behave
more like the idealized version you describe.  In common parlance, this is
known as "reducing leakage", which implies that for a tone of a given
frequency, integer or not, the impact in the "frequency space" should be
localized around the closest bins corresponding to the frequency.

Hope this helps.

Ced
---------------------------------------
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You are kind of forgetting the quadrature oscillator that was posted on this forum earlier in the year. It is so fast that it actually makes an alternative to the traditional FFT algorithm.  After all all you have to do is multiply and integrate your signal with a multiple of those oscillators each at a different frequency.  I'm doing that for a PSK31 decoder I am writing at the moment.

>On Wed, 05 Oct 2016 07:09:22 -0500, runinrainy wrote:
>
>> Hello
>> 
>> The Fourier Transform decomposes a function of time (a signal) into
the
>> frequencies, and it's inverse transform means that a time domain
signal
>> can be constructed as a sum of trigonometric sinusoidal functions.
>> I am now also trying to understand and visualize the Fractional
Fourier
>> Transform (FrFT).
>> 
>> I am actually new in this topic and have found some usefull articles
>> regarding FrFT on the web but they are all explaning the maths behind
>> FrFT, but I would like to get some visualization that FrFT can be
>> interpreted as Fourier or a point of view that is localized through a
>> line (time-frequency) where the signal has both time and frequency
>> components in a space.
>> 
>> Is there a way that we can interpret the Fractional Fourier Transform
as
>> the Fourier Transform? A time domain signal decomposition into
>> frequencies and composition from the frequencies?
>> 
>> Thanks in advances!
>> ---------------------------------------
>> Posted through http://www.DSPRelated.com
>
>Have you looked at the references cited in the Wikipedia article on the 
>FrFT?  Some of the titles looked like they might be practice-oriented.  
>Actually, there's a graphic in that article that looks intriguing.
>
>Or search on "Fractional fourier transform applications".
>
>I suspect that what you'll find will be applications that live on the 
>edges of a number of different applications areas -- certainly in 
>communications we've worked hard to arrange the world into nice tidy 
>frequency bands, thus putting ourselves outside of a domain that demands

>the use of the FrFT.
>
>-- 
>Tim Wescott
>Control systems, embedded software and circuit design
>I'm looking for work!  See my website if you're interested
>http://www.wescottdesign.com

Thanks you Tim. I made several search on the web and also looked at
multiple articles in IEEE, and finally understand and now can visualize
the FrFT. It is basically the generalization of the classical Fourier
Transform and decomposes a time domain signal into chirp signals, means
that the frequency of the signal is changed linearly in time.
---------------------------------------
Posted through http://www.DSPRelated.com

>>Hello
>>
>>The Fourier Transform decomposes a function of time (a signal) into the
>>frequencies, and it's inverse transform means that a time domain signal
>>can be constructed as a sum of trigonometric sinusoidal functions.
>
>[...snip...]
>
>>Is there a way that we can interpret the Fractional Fourier Transform
as
>>the Fourier Transform? A time domain signal decomposition into
>frequencies
>>and composition from the frequencies?
>>
>>Thanks in advances!
>>---------------------------------------
>>Posted through http://www.DSPRelated.com
>
>I am afraid that you are working under an idealized misconception of the
>Fourier Transform.  The only cases for which it works the way you
describe
>are when the signal is a repeating waveform and the analysis frame is an
>integer multiple of the waveform length.  If the analysis frame is
>infinitely long, any mix of infinitely long pure tones will also work as
>you describe.
>
>For cases with a finite analysis frame, for any tone with a frequency
that
>is not an integer multiple of the frame size, the "constituent
>frequencies" you get from the FT do *NOT* represent frequencies of the
>tone, they represent the "building blocks" to reconstruct that signal
>within the analysis frame.  If you extend the inverse beyond the limits
of
>the frame you will not get an extension of such a tone, you will get a
>repeat of the analysis frame.
>
>The study of window functions is largely an exercise to make the FT
behave
>more like the idealized version you describe.  In common parlance, this
is
>known as "reducing leakage", which implies that for a tone of a given
>frequency, integer or not, the impact in the "frequency space" should be
>localized around the closest bins corresponding to the frequency.
>
>Hope this helps.
>
>Ced
>---------------------------------------
>Posted through http://www.DSPRelated.com

Thanks Ced, for your explanation.
---------------------------------------
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