Delay and Sum Beamforming Equation Derivation

[mastrepolo]

Homework Statement

I have to simplify this beam form (equation 1) which simplifies to equation 2 and then finally to equation 3.

Homework Equations

equation 1: e^-ix((1-e^y)/(1-e^z)) where x = Beta*M_(1/2), y = beta*M, z= Beta

equation 2: sin(M*Beta/2)/(sin(Beta/2))

equation 3: M((sinc(M*Beta/2pi))/(sinc(Beta/2pi))) where sinc(x) = sin(pi*x)/pi*x

Eulors: e^jx = cos(x) +j*sin(x)

The Attempt at a Solution

I have tries to use eulors equation to change it to sine sand cosine but I can't seem to get the proper cancellations i am looking for.

 

[haruspex]

equation 1: e^-ix((1-e^y)/(1-e^z)) where x = Beta*M_(1/2), y = beta*M, z= Beta

What is M_(1/2)?
Do you mean $e^{-i\frac{\beta}2(M-1)}\frac{1-e^{i\beta M}}{1-e^{i\beta}}$?

 

[mastrepolo]

Sorry about the late response. It is M_1/2 as a subscript, I attached a picture. I am going to the math department for some guidance since I can't seem to figure this one out. Any help would be appreciated,

 

[haruspex]

It is M_1/2 as a subscript,

So what is the relationship between that and M? To get your eqn 2 I need it to equal (M-1)/2.

 

[mastrepolo]

So what is the relationship between that and M? To get your eqn 2 I need it to equal (M-1)/2.

To get it in this form we had to assume M = 2*M_1/2+1. so . There were some other step to get it into the equation form but this is where it originated from.

 

[haruspex]

To get it in this form

Are you saying to get eqn 1 you already assumed that?
I assume you know how to write sin x in terms of e^ix.
Can you see how to write 1-e^-ix as some function f(x) multiplied by a sine function (not necessarily sin(x), exactly).