Science Fair Project Encyclopedia
Raised cosine
Raised cosine is a function commonly used in wireless communications.
![\mathbf{x}(t) = \frac{\sin\left(\pi f_0 t\right)\cos\left(\pi \alpha f_0 t\right)}{\pi f_0 t\left[1 - \left(2 \alpha f_0 t\right)^2\right]}](a/../upload/math/92a7de5ff6ed6b1736e91fe1e999cd5d.png)
![\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.](a/../upload/math/f40fead66600a373e0266d8788cfb36f.png)
where
- α ∈
![\left[0, 1\right]](a/../upload/math/078c04b251da1da2aee75daa308d0945.png)
is the continuous Fourier transform of 
(note that the normalization of the transform here is slightly inconsistent with the other Wikipedia pages, however)
- ω ∈
radians
For this function, α acts as a spreading parameter in the frequency spectrum of .
If α = 0 then becomes the sinc function.
Sample plot
| Color | f0 | α |
| Black | 2 | 2 |
| 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
Last updated: 08-25-2005 09:08:56
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details