## lunedì 1 novembre 2010

### Time shift of signal using FFT in matlab

Time shift of signal using FFT in matlab:

An integrable function is a function ƒ on the real line that is Lebesgue-measurable and satisfies
$\int_{-\infty}^\infty |f(x)| \, dx < \infty.$

### Basic properties

Given integrable functions f(x), g(x), and h(x) denote their Fourier transforms by $\hat{f}(\xi)$, $\hat{g}(\xi)$, and $\hat{h}(\xi)$ respectively. The Fourier transform has the following basic properties (Pinsky 2002).
Linearity
For any complex numbers a and b, if h(x) = (x) + bg(x), then  $\hat{h}(\xi)=a\cdot \hat{f}(\xi) + b\cdot\hat{g}(\xi).$
Translation
For any real number x0, if h(x) = ƒ(x − x0), then  $\hat{h}(\xi)= e^{-2\pi i x_0\xi }\hat{f}(\xi).$
Modulation
For any real number ξ0, if h(x) = e2πixξ0ƒ(x), then  $\hat{h}(\xi) = \hat{f}(\xi-\xi_{0})$.
Scaling
For a non-zero real number a, if h(x) = ƒ(ax), then  $\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)$.     The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then  $\hat{h}(\xi)=\hat{f}(-\xi)$.
Conjugation
If $h(x)=\overline{f(x)}$, then  $\hat{h}(\xi) = \overline{\hat{f}(-\xi)}.$
In particular, if ƒ is real, then one has the reality condition$\hat{f}(-\xi)=\overline{\hat{f}(\xi)}.$
And if ƒ is purely imaginary, then  $\hat{f}(-\xi)=-\overline{\hat{f}(\xi)}.$
Convolution
If $h(x)=\left(f*g\right)(x)$, then  $\hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).$

### Python for Seismology

The pytho tools for seismology

OBSPY (http://www.obspy.org/) is an open-source project dedicated to provide a Python framework for processing seismological data. It provides parsers for common file formats and seismological signal processing routines which allow the manipulation of seismological time series (see Beyreuther et. al. 2010).
The goal of the ObsPy project is to facilitate rapid application development for seismology.

A great tutorial page, help us to use this tools for our seismological analysis.

Why we have to switch to this Python based tools? The graphical power is really strong in Python, really better than matlab, and also is a free open source tools, that include the instruments for various kind of analysis