Simple transform, wicked algebra.

[brendan_foo]

Hey folks,

I'm just searching for a generalized result here. Due to lack of note taking I am in a bind for this Fourier transform. The transform is...

$$\int_{-a/2}^{a/2} \cos (\frac{\pi x}{a}) e^{-j\beta s x} dx$$.

Of course, this is a general Fourier transform, although variables such as f and t are omitted as this is related to antennas.

Anyways, I figure that this is a product between a cosine function
$$cos(\pi x/a)$$
and a unit rectangular pulse function between -a/2 and a/2. Of course, the Fourier modulation theorem states that we should have a transform of the form:

$$F(s) \sim sinc(0.5 a \beta s - \pi /a) + sinc(0.5 a \beta s + \pi /a)$$.
with scalar coefficients omitted for the sake of simplicity.

Anyways, I believe there to be an analytical expression for the sum of these Sinc functions, and it is allegedly a cosine function of certain parameters.

Anyways, the transform itself is complete from the general expression above, but the algebra is long winded to form it into another trig function.. Can anyone see a nice, glaring solution to this?

Cheers guys...
Brendan

 

[mathman]

cos(u)=(e^iu+e^-iu)/2

Substitute into your integral and evaluation is trivial.

 

[brendan_foo]

Absolutely, and this general "Fourier property" indeed comes from the expansion into what I have listed above...the sum of sinc functions.

This is not the problem I am having. I am aiming to some how combine that sum of Sinc functions into a single trigonometric function. I am lead to believe that it is possible. When I leave the integral in its canonical exponential form, there is a lot of algebraic work to be done. I was just wondering if someone could see off the top of their head, or with a quick scribble on paper what this could be..

-Brendan

 

[cristo]

Perhaps this may be of use: $$\sin(x\pm y)=\sin x\cos y \pm \cos x\sin y$$

 

[brendan_foo]

I am almost sure that it is... Is this one of these occasions that I'm going to have to slug through it and feel good at the end if/when I get to the desired result?

This is just a possible question that may be posed, to determine a far E/H field using Huygen's principle and the argument passed to the F.T describes the field pattern of a horn-antenna aperture.

Back to the grind...
-Brendan