Raised cosine

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In mathematics, the raised cosine is a function commonly used in wireless communications:

with the following inverse Fourier transform:

[\mathbf(t) = \mathrm(f_0 t)\cdot \frac]

where:

sinc(x) is the normalized sinc function
[\alpha \in \left[0, 1right]]
[f_0=\frac, T ]being the pulse length
[\omega \in \left(-\infty, \infty \right)] radians
[\omega_1 = \left(1 - \alpha\right) \pi f_0]
[\omega_2 = \left(1 + \alpha\right) \pi f_0]

For this function, [\alpha] acts as a spreading parameter in the frequency spectrum of [\mathbf(\omega)]. If [\alpha = 0] then [\mathbf(t)] becomes the sinc function.

Sample plot

Color	[f_0]	[\alpha]
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 [\alpha = 0] the function is the sinc function
As [\alpha] 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 [f_0]
All of the roots are multiples of [\frac]

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