An alternative method

The alternative method we propose first Fourier transforms the time series p(t). Then it applies the traditional log-Fourier transformation (described above) to the spectrum of the original time trace. We call this new domain the Fourier-log-Fourier domain(${\bf FLF}$). We denote $\tau_{1}$ as the variable of the new domain after the original $\alpha$ stretch.

In this domain the intended trace stretch is carried out by a simple scaling of the series' coefficients:  

$${P_{1}(\tau_{1})} = {P(\tau_{1}) e^{i \tau_{1} \log{1\over{\alpha}}}}$$ (4)

Comparing (2) and (4), we find that the stretching in the ${\bf FL}$ and in the ${\bf FLF}$ domain differ only in the exponantial scaling factor.

The number of samples $n_{\omega}$ for a well sampled representation of our input data in the ${\bf FLF}$ domain is:

 

$${n_{\omega}} = {2 f_{\max} t_{\max} \log({f_{\max}\over{f_{\min}}})}.$$ (5)

This expression derives directly from equation (3) by interchanging the role of the frequency and time axis. Especially, note the symmetry of the non logarithmic factor of (3) and (5).

3|c|
3|c|Logarithmic trace stretch - An overview
3|c|
Interpolation	Log-Fourier transformation FL	Fourier-Log-Fourier transformation FLF
		${\bf F}ourier: p(t) \rightarrow p(\omega)$
	${\bf L}og:~~~~~~p(t) \rightarrow p(s)$	${\bf L}og:~~~~~p(\omega) \rightarrow p(\nu)$
	${\bf F}ourier: p(s) \rightarrow p(\sigma)$	${\bf F}ourier: p(\nu) \rightarrow p(\tau)$
Interpolation according to (1):	$\alpha$ stretch by multiplication:	$\alpha$ stretch by multiplication:
$p(t) \rightarrow p(t_{1})$	$p(\sigma) \rightarrow p(\sigma_{1})$	$p(\tau) \rightarrow p(\tau_{1})$
3|c|Disadvantages
Each stretch requires a new interpolation.	Overhead calculations: FL. Long traces after logarithmic stretch.	Comparatively,expensive overhead calculations: FLF.
3|c|Advantages
Easy to implement. No overhead calculations.	Each $\alpha$ stretch is a simple multiplication.	Each $\alpha$ stretch is a simple multiplication. Shorter traces in the transform domain.