Raised Cosines

Date: 01/28/2001 at 19:26:55
From: Deborah
Subject: Raised cosines

Hi Dr. Math,

Can you explain to me in the simplest way possible what raised cosines 
are and how they work? I am a biology student and I know basic 
single-variable calculus and elementary linear algebra.

Thank you.

Date: 01/29/2001 at 10:56:30
From: Doctor Douglas
Subject: Re: Raised cosines

Hi Deborah, 

I've come across "raised cosines" in the field of signal processing, 
where it is simply used to mean a sinusoidal function (a sine or 
cosine) that also has a constant, or an "offset" term:

   f(t) = C + A cos(wt)

where C is a constant, and Acos(wt) is the usual sinusoidal part with
which you're probably familiar (w is a kind of frequency, and is also 
a constant with respect to t). I think you won't have too much trouble 
using functions such as f(t), above - they're fairly easy to integrate 
and differentiate, for example:

  df/dt = -wA sin(wt)

In a biological context, a set of such functions added together may 
give a person's blood pressure as a function of time. If we tried to 
model the pressure with just one f, as above, then the maximum 
pressure would be C+A (the systolic pressure) and the minimum would be 
C-A (diastolic pressure).  Of course the actual pressure behavior is 
quite a bit more complicated that that given by a single raised 
cosine.

It's also possible that "raised cosine" means something else entirely.
If you find this to be the case, let us know!

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/