# What is the Roll-Off factor of a Root Raised Cosine Filter?

When designing a root raised cosine filter on Matlab, there is an option to set its "roll-off factor". What does this mean? I have been looking around for a while but I can't seem to find a clear explanation. Any help will be appreciated.

## 1 Answer

Think of the root-raised cosine as a family of functions. The roll-off factor is a parameter that chooses one of the possible functions.

All root-raised cosine pulses look similar to the sinc pulse. The interesting thing about the roll-off factor is that it controls two features of the function:

• The rate at which the function's lobes (or ripples) decrease. At one extreme, with roll-off set to 0, they decrease slowly (as $1/t$, with $t$ equal to time). At the other end, when the roll-off factor is 1, they decrease as $1/t^3$.

• As the roll-off factor increases, so does the function's bandwith. Basically, if the function's bandwidth is $B$ when the roll-off is 0, then it becomes $2B$ when the roll-off is 1.

You can choose any value for the roll-off factor between 0 and 1, so you can select the function with the ripple and bandwidth that are best suited to your needs.

• So if the roll-off changes the bandwidth, then it also changes the cut-off frequency because the 3dB point would have moved right? I would have thought the bandwidth of the raised cosine filter would be determined solely by the value you define for it's cutoff frequency and the roll-off would determine the transition bandwidth because if you decide to set the rolloff factor then you do not set the transition bandwidth. – KillaKem Oct 24 '14 at 9:01
• @KillaKem, in my answer I've used the absolute definition of bandwidth; that is, the function's spectrum becomes $0$ for $|f|>B$. If you define bandwidth as the cut-off frequency, then you're right: the roll-off factor will affect the transition bandwidth. If the roll-off is zero, then the transition bandwidth is essentially zero. If it's one, then the transition bandwidth is roughly equal to the pass band. – MBaz Oct 24 '14 at 13:48