Forums

Summation problem

Started by Jerry Wolf December 27, 2006
In the course of understanding a cross-correlation problem (this is not
homework!), I need to evaluate:

     sum (n = 0 to N) exp (j * n * theta)

or equivalently:

     sum (n = 0 to N) cos(n*  theta) + j * sum(n = 0 to N) sin(n *
theta)

where N is 150 or so and theta is something rather small, on the order
of 0.001 * pi.

I'd guess these summations are somewhat common and I could find them in
some listing or other, but my lone math handbook lacks them, and so far
I haven't found any such listing on the web.  Can anyone give me a
pointer to them?  Many thanks.

cheers,
  jerry wolf
  spaceflight systems corp.

Jerry Wolf wrote:
> In the course of understanding a cross-correlation problem (this is not > homework!), I need to evaluate: > > sum (n = 0 to N) exp (j * n * theta) > > or equivalently: > > sum (n = 0 to N) cos(n* theta) + j * sum(n = 0 to N) sin(n * > theta) > > where N is 150 or so and theta is something rather small, on the order > of 0.001 * pi. > > I'd guess these summations are somewhat common and I could find them in > some listing or other, but my lone math handbook lacks them, and so far > I haven't found any such listing on the web. Can anyone give me a > pointer to them? Many thanks. > > cheers, > jerry wolf > spaceflight systems corp.
Maybe you could approximate the sum with a definite integral and get a formula that way, since you'd be integrating sines and cosines which are easy to solve. John
I appealed for help a little too soon.  Further creative googling and
link-following led me to:

Weisstein, Eric W. "Exponential Sum Formulas." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/ExponentialSumFormulas.html

and also
  http://mathworld.wolfram.com/Cosine.html
  http://mathworld.wolfram.com/Sine.html
where the needed formulas were found.

cheers,
  jerry

"Jerry Wolf" <jjwolf22@verizon.net> writes:

> In the course of understanding a cross-correlation problem (this is not > homework!), I need to evaluate: > > sum (n = 0 to N) exp (j * n * theta) > > or equivalently: > > sum (n = 0 to N) cos(n* theta) + j * sum(n = 0 to N) sin(n * > theta) > > where N is 150 or so and theta is something rather small, on the order > of 0.001 * pi. > > I'd guess these summations are somewhat common and I could find them in > some listing or other, but my lone math handbook lacks them, and so far > I haven't found any such listing on the web. Can anyone give me a > pointer to them? Many thanks.
Are you asking about the relationship sum (n = 0 to N) z^n = (1 - z^(N+1)) / (1 - z) ??? -- % Randy Yates % "I met someone who looks alot like you, %% Fuquay-Varina, NC % she does the things you do, %%% 919-577-9882 % but she is an IBM." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr