Fourier Transform Question

Suppose I have a signal x(t) with Fourier Transform X(jw). It is known
that if one scales the time axis by a factor K, then the frequency
axis is scaled by 1/K. I am curious about non-linear scalings of the
time axis. Is there a way to derive the Fourier Transform of
x(exp(t)), given that one knows the Fourier Transform of x(t)??


Bob Adams

Liz wrote:

> Suppose I have a signal x(t) with Fourier Transform X(jw). It is known
> that if one scales the time axis by a factor K, then the frequency
> axis is scaled by 1/K. I am curious about non-linear scalings of the
> time axis. Is there a way to derive the Fourier Transform of
> x(exp(t)), given that one knows the Fourier Transform of x(t)??
> 
> 
> Bob Adams

If I were a sadistic professor and you were an undergrad I'd reply to 
that by saying "hey, do you want to get a Doctorate?"

I don't think there'd be a general solution to that, but I'm willing to 
be surprised.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

On 2004-08-02 21:39:41 +0200, robert.w.adams@verizon.net (Liz) said:

> Suppose I have a signal x(t) with Fourier Transform X(jw). It is known
> that if one scales the time axis by a factor K, then the frequency
> axis is scaled by 1/K. I am curious about non-linear scalings of the
> time axis. Is there a way to derive the Fourier Transform of
> x(exp(t)), given that one knows the Fourier Transform of x(t)??
> 
> 
> Bob Adams

You could use non-uniform sampling (commonly called "warping") to 
achieve exponential spacing. In fact this is quite common, for example 
to get a constant-Q behaviour out of a constant bandwidth transform. 
Look for the keywords "warping", "Mellin transform" or "warped Fourier 
transform" to get started.
-- 
Stephan M. Bernsee
http://www.dspdimension.com

In article <9dec5a83.0408021139.4da826f8@posting.google.com>,
Liz <robert.w.adams@verizon.net> wrote:
>Suppose I have a signal x(t) with Fourier Transform X(jw). It is known
>that if one scales the time axis by a factor K, then the frequency
>axis is scaled by 1/K. I am curious about non-linear scalings of the
>time axis. Is there a way to derive the Fourier Transform of
>x(exp(t)), given that one knows the Fourier Transform of x(t)??

With the change of variables exp(t)=s, and assuming convergence 

int_{-infinity}^infinity exp(-ikt) x(exp(t)) dt
= int_0^infinity s^(-ik-1) x(s) ds

I'd like to write this as 

int_{-infinity}^infinity G(w) X(w) dw

where G(w) is the Fourier transform of Heaviside(s) s^(-ik-1).
That doesn't quite work when k is real because it's too singular at s=0, 
but you can either replace k by k+i epsilon or Heaviside(s) by
Heaviside(s-epsilon) where epsilon>0, and take the limit as epsilon->0.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2

israel@math.ubc.ca (Robert Israel) wrote in message news:<cemnk7$btr$1@nntp.itservices.ubc.ca>...
> In article <9dec5a83.0408021139.4da826f8@posting.google.com>,
> Liz <robert.w.adams@verizon.net> wrote:
> >Suppose I have a signal x(t) with Fourier Transform X(jw). It is known
> >that if one scales the time axis by a factor K, then the frequency
> >axis is scaled by 1/K. I am curious about non-linear scalings of the
> >time axis. Is there a way to derive the Fourier Transform of
> >x(exp(t)), given that one knows the Fourier Transform of x(t)??
> 
> With the change of variables exp(t)=s, and assuming convergence 
> 
> int_{-infinity}^infinity exp(-ikt) x(exp(t)) dt
> = int_0^infinity s^(-ik-1) x(s) ds
> 
> I'd like to write this as 
> 
> int_{-infinity}^infinity G(w) X(w) dw
> 
> where G(w) is the Fourier transform of Heaviside(s) s^(-ik-1).
> That doesn't quite work when k is real because it's too singular at s=0, 
> but you can either replace k by k+i epsilon or Heaviside(s) by
> Heaviside(s-epsilon) where epsilon>0, and take the limit as epsilon->0.
> 
> Robert Israel                                israel@math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel 
> University of British Columbia            
> Vancouver, BC, Canada V6T 1Z2


Thanks!

I followed OK until;

>I'd like to write this as 
> 
> int_{-infinity}^infinity G(w) X(w) dw


How did you make this step?


Regards


Bob Adams

israel@math.ubc.ca (Robert Israel) wrote in message news:<cemnk7$btr$1@nntp.itservices.ubc.ca>...
> In article <9dec5a83.0408021139.4da826f8@posting.google.com>,
> Liz <robert.w.adams@verizon.net> wrote:
> >Suppose I have a signal x(t) with Fourier Transform X(jw). It is known
> >that if one scales the time axis by a factor K, then the frequency
> >axis is scaled by 1/K. I am curious about non-linear scalings of the
> >time axis. Is there a way to derive the Fourier Transform of
> >x(exp(t)), given that one knows the Fourier Transform of x(t)??
> 
> With the change of variables exp(t)=s, and assuming convergence 
> 
> int_{-infinity}^infinity exp(-ikt) x(exp(t)) dt
> = int_0^infinity s^(-ik-1) x(s) ds
> 
> I'd like to write this as 
> 
> int_{-infinity}^infinity G(w) X(w) dw
> 
> where G(w) is the Fourier transform of Heaviside(s) s^(-ik-1).
> That doesn't quite work when k is real because it's too singular at s=0, 
> but you can either replace k by k+i epsilon or Heaviside(s) by
> Heaviside(s-epsilon) where epsilon>0, and take the limit as epsilon->0.
> 
> Robert Israel                                israel@math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel 
> University of British Columbia            
> Vancouver, BC, Canada V6T 1Z2

Interesting ... your first formula looks a lot like the Mellin
Transform(Z), with the complex variable Z evaluated for imaginary
values only. I wonder then if the Mellin transform of x(t) evaluated
on the imaginary axis gives the same result as the Fourier Transform
of x(exp(t)) ??

Regards

Bob Adams