Electrical Engineering - Mathematical Question from Signal Processing

Author

Mathematical Question from Signal Processing

junoexpress

2006-01-12, 2:21 am

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 << S1.

ARGUMENT:
My argument would be to say that S2 is like an approximation of the DFT
for A. If I consider A to be a Gaussian like signal with s dev sigma
for example, I can see that S2, its DFT will have the form
exp(-w^2/sigma^2)
and since we said [b-a] >> 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

Salmon Egg

2006-01-12, 2:21 am

On 1/11/06 9:20 PM, in article
1137043259.822120.168370@g47g2000cwa.googlegroups.com, "junoexpress"
<mathimagical@netscape.net> wrote:

> 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

<snip>

This is two complicated for me to work on without a motivation. It reads to
much like a side lemma in a math book.

Bill

-- Ferme le Bush

Salmon Egg

2006-01-12, 7:21 pm

On 1/11/06 9:48 PM, in article BFEB2A78.147AC%salmonegg@sbcglobal.net,
"Salmon Egg" <salmonegg@sbcglobal.net> wrote:

> This is two complicated for me to work on without a motivation. It reads to
> much like a side lemma in a math book.

I cannot believe how badly I spelled too and too.
-- Ferme le Bush

2006-01-12, 10:21 pm

"junoexpress" <mathimagical@netscape.net> wrote in
news:1137043259.822120.168370@g47g2000cwa.googlegroups.com:

>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
>

snip

good, keep working on your homework.

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Don Kelly

2006-01-13, 12:21 am

Why make things more complex. Think.
What is the maximum value of the cos(...) factor in each term.

--

Don Kelly @shawcross.ca
remove the X to answer
----------------------------
"junoexpress" <mathimagical@netscape.net> wrote in message
news:1137043259.822120.168370@g47g2000cwa.googlegroups.com...
> 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 << S1.
>
> ARGUMENT:
> My argument would be to say that S2 is like an approximation of the DFT
> for A. If I consider A to be a Gaussian like signal with s dev sigma
> for example, I can see that S2, its DFT will have the form
> exp(-w^2/sigma^2)
> and since we said [b-a] >> 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
>

daestrom

2006-01-14, 12:21 pm

"Don Kelly" <dhky@shaw.ca> wrote in message
news:MJFxf.323476$ki.72961@pd7tw2no...
> Why make things more complex. Think.
> What is the maximum value of the cos(...) factor in each term.
>

That was my first impression also. If the second sum is the same samples,
but some of them multiplied by a value less than one, and some others even
negative, it would seem 'obvious' that the second sumation must be less than
the first.

Exactly how much less would be a function of how often cos(2*w*ti) is not
==1. If the 'number of points per period is large', then the number of
times that 'cos(2*w*ti) <<1' is also 'large'.

daestrom

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