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

# Fourier Transform Question

Started by ●August 2, 2004

Reply by ●August 2, 20042004-08-02

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 AdamsIf 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

Reply by ●August 2, 20042004-08-02

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 AdamsYou 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

Reply by ●August 2, 20042004-08-02

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

Reply by ●August 3, 20042004-08-03

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 1Z2Thanks! I followed OK until;>I'd like to write this as > > int_{-infinity}^infinity G(w) X(w) dwHow did you make this step? Regards Bob Adams

Reply by ●August 3, 20042004-08-03

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 1Z2Interesting ... 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