## 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  $\hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).$

### z=FFTshift(signal,100,1); %positive delay

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