Raised cosine
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In mathematics, the raised cosine is a function commonly used in wireless communications:
![\mathbf{X}(\omega) = \left \{ \begin{matrix} 1/f_0, & \left|\omega\right| \le \omega_1 \\ \\ \frac{0.5}{f_0} \left[1 + \cos\left(\frac{\left|\omega\right| - \omega_1}{2 \alpha f_0}\right)\right], & \omega_1 < \left|\omega\right| < \omega_2 \\ \\ 0, & \left|\omega\right| > \omega_2 \end{matrix} \right.](../../../math/f/4/0/f40fead66600a373e0266d8788cfb36f.png)
with the following inverse Fourier transform:
![\mathbf{x}(t) = \mathrm{sinc}(f_0 t)\cdot \frac{\cos\left(\pi \alpha f_0 t\right)}{\left[1 - \left(2 \alpha f_0 t\right)^2\right]}](../../../math/a/b/6/ab6ca80a04a2b8cdc20323c2414bff43.png)
where:
- sinc(x) is the normalized sinc function
![\alpha \in \left[0, 1\right]](../../../math/f/2/0/f20dccd16e4673987dfa98eebfc291c0.png)
being the pulse length
radians
For this function, α acts as a spreading parameter in the frequency spectrum of 
. If α = 0 then 
becomes the sinc function.
[edit] Sample plot
| Color | f0 | α |
| Black | 2 | 1 |
| Blue | 2 | 0.75 |
| Red | 2 | 0.5 |
| Green | 2 | 0.25 |
| Purple | 2 | 0 |
The above table maps the plots shown to the parameters used to generate the plot. Things to notice:
- For α = 0 the function is the sinc function
- As α decreases the more the plot looks like a sinc function
- All plots go through y = 1 at x = 0
- All plots have the same roots, which is a function of f0
- All of the roots are multiples of
