Hi,
I am working on the following problem.
I have a function A(t) which is the "envelope" of a signal that satisfies the following conditions:
1) it is positive on an interval [a,b]
2) has a single maximum at the midpt of the interval
3) is symmetric about the maximum
I have some defined frequency w such that the period for this frequency, tau, is much smaller than b-a. Now I sample N points from the interval [a,b], t1,...tN. Although the points may not be uniformly sampled, we will assume that N is large (in the sense that the number of points per period is large, or in signal processing language, the signal has a high "oversampling factor"). I define the sum:
S1 = Sum from i=1 to N ( A(ti) )
and wish to consider the sum:
S2 = Sum from i=1 to N ( A(ti) cos(2*w*ti) )
I want to claim that S2 > tau, it follows that S2 will be very small.
QUESTIONS:
1) Is this argument correct if A was a Gaussian envelope?
2) How do I generalize this argument to handle more general envelopes that satisfy the conditions given above? I've tried considering Taylor series expansions of A about its maximum, but I can't seem to get anywhere.
Thank you very much,
Juno