IIR - Bilinear transform - simplifying H(z)

[Scoridd]

Homework Statement

Hello,

I'm an audio engineer with an interest in creating/implementing audio processing techniques and I'm trying to learn IIR filter design via the Bilinear Transform method. I've looked through various books and online resources but I keep on getting stuck when it comes to simplifying H(z) in the final stages of the process. I'm sure the process is relatively straight forward but I cannot figure it out. I can usually 'get' most mathematical concepts but I've had no formal training, so I assume there is something that I'm missing here...

Homework Equations

In the most recent example, I have H(z) as...

$\frac{0.3752^{2}} {\frac{1-z^{-1}}{1+z^{-1}}^{2} + \sqrt{2}(0.3752)\frac{1-z^{-1}}{1+z^{-1}} + (0.3752)^{2}}$

...and I need to simplify and collect terms of like powers of z in order to get to here...

$\frac{0.0842(1+z^{-1})^2} {1 - 1.0281z^-1 + 0.3651z^-2}$

The Attempt at a Solution

Unfortunately, I do not know how. I began by calculating the squared and sqroot tems then mutliplying all terms by (1 + z-1). But then I get stuck. Mostly, I'm miffed by the z-2 in the answer, where does this come from? Also, I don't see how I arrive at the coefficients.

I'd be very greatful if someone could explain how I might go about doing this. This is for my own understanding, so I need to know how I'd go about simplifying this - I already have the answer after all! I just need pointing in the right direction.

 

[Scoridd]

If it helps, I am trying to follow this...

http://www.youtube.com/watch?v=5RoPKr94_-w&t=44m0s