Need Help Solving This Really Difficult Quadratic Equation!

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March 24th 2010, 10:57 AM
TomMUFC

Need Help Solving This Really Difficult Quadratic Equation!

Hi all this relates to a Digital Signal Processing question, but it's the maths I can't understand when attempting to solve the quadratic equation.

This was originally part of a transfer function (denominator) and basically reads:

$1-2rCos(\omega_cT)z^-1 + r^-2$

Multiplying this line to positive values of z, to give the quadratic the NOW equation. I'm attempting to solve for r:

$z^2-2rCos(\omega_cT)z^1 +r^2$

a=1
b= $2rcos(\omega_cT)$
c= $r^2$

But this where I fall on, and where I need help, because I cannot for the life of me solve with another unknown in it.

$(2rcos(\omega_cT) +/- [sqrt](-(2rcos(\omega_cT))^2-4r^2)$

all divided by 2

I can change the $b^2$ in the brackets to become = $4r^2cos^2(\omega_cT)$ but then I've become snookered. Can anyone help? I can't go any further unfortunatley.

Apologies my Mathtags are terrible! (Can't do the quadratic properly).
March 24th 2010, 11:49 AM
Grandad

Hello TomMUFC

Quote:

Originally Posted by TomMUFC

Hi all this relates to a Digital Signal Processing question, but it's the maths I can't understand when attempting to solve the quadratic equation.

This was originally part of a transfer function (denominator) and basically reads:

$1-2rCos(\omega_cT)z^-1 + r^-2$ $\color{blue}\text{ Grandad says: should this be: }1-2r\cos(\omega_cT)z^-1 + r^2z^{-2}=0\text{ ?}$

Multiplying this line to positive values of z, to give the quadratic the NOW equation. I'm attempting to solve for r:

$z^2-2rCos(\omega_cT)z^1 +r^2$

a=1
b= $2rcos(\omega_cT)$
c= $r^2$

But this where I fall on, and where I need help, because I cannot for the life of me solve with another unknown in it.

$(2rcos(\omega_cT) +/- [sqrt](-(2rcos(\omega_cT))^2-4r^2)$

all divided by 2

I can change the $b^2$ in the brackets to become = $4r^2cos^2(\omega_cT)$ but then I've become snookered. Can anyone help? I can't go any further unfortunatley.

Apologies my Mathtags are terrible! (Can't do the quadratic properly).

Assuming the correction I've suggested, then you need to solve:
$z^2-2r\cos(\omega_cT)z +r^2=0$

In which case, you're on the right track:
$z = \frac{2r\cos(\omega_cT)\pm\sqrt{[-2r\cos(\omega_cT)]^2-4r^2}}{2}$
$= \Big(\cos(\omega_cT)\pm\sqrt{\cos^2(\omega_cT)-1}\Big)r$

Now $\cos^2(\omega_cT)-1 = -\sin^2(\omega_cT)$

$\Rightarrow \sqrt{\cos^2(\omega_cT)-1} = i\sin(\omega_cT)$

So we get:
$z = \Big(\cos(\omega_cT)\pm i\sin(\omega_cT)\Big)r$

If you know that you can take the positive sign, then this will reduce to:
$z = re^{i\omega_cT}$

Does that make sense?

Grandad
March 24th 2010, 12:31 PM
TomMUFC

Yes you are correct. That is fantastic Grandad.

Now see this, I don't know what you know about transfer functions (regarding DSP) but the original question was to "find the range of (real) values of r for which the filter is stable?"

Now a filter is stable if the poles all lie within the "unit circle" of the plane (i.e. <1).

To obtain pole positions you would need to obtain the roots of the demominator (which I gave). Normally answers come in a complex format (real/imaginary) I think so for the filter to be stable the complex numbers need to be less than 1. But now I'm confused. You don't happen to know anything regarding this do you?
March 24th 2010, 01:07 PM
Grandad

Hello TomMUFC

Quote:

Originally Posted by TomMUFC

Yes you are correct. That is fantastic Grandad.

Now see this, I don't know what you know about transfer functions (regarding DSP) but the original question was to "find the range of (real) values of r for which the filter is stable?"

Now a filter is stable if the poles all lie within the "unit circle" of the plane (i.e. <1).

To obtain pole positions you would need to obtain the roots of the demominator (which I gave). Normally answers come in a complex format (real/imaginary) I think so for the filter to be stable the complex numbers need to be less than 1. But now I'm confused. You don't happen to know anything regarding this do you?

I'm afraid you're on your own as far as the electronics is concerned. (I took A level Physics in 1961!) But $z = re^{i\omega_cT}\Rightarrow |z| = r$ , so if you are sure that $|z| < 1$ , then $-1 < r < 1$ . That's the best I can do.

Grandad
March 24th 2010, 01:49 PM
TomMUFC

That's great Grandad, if you're not right I'm sure you are close to it. I can always see a phD student with regards to the problem, the main thing was solve r from the quadratic. I was backed by my lecturer who suggested that the correct way was the solve the denominator using the quadratic formula. So if you're happy with your answer then that's good. It was just that maths I couldn't understand there, he said that the cosine function can solved/broken down with the use of a trig identity. I just wasn't sure what it was. Thanks anyway.