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().
We denote as the variable of the new domain after the
original stretch.

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

(4) |

Comparing (2) and (4), we find that the stretching in the and in the domain differ only in the exponantial scaling factor.

The number of samples for a well sampled representation of our input data in the domain is:

(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 |

Interpolation according to (1): | stretch by multiplication: | stretch by multiplication: |

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 stretch is a simple multiplication. | Each stretch is a simple multiplication.
Shorter traces in the transform domain. |

11/18/1997