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Ch 5.17 - Equation 5.56 (p.133-134)
November 21, 2016
12:12 am
Sidora83
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It mention that the equation : 1 + α1z-1 + α2z-2
can be factored to: (1-Z1z-1)(1-Z2z-1)
with some algebra.

Would anyone have the algebra steps to arrive at that conclusion?
Im a bit confuse about how this was deduced.

Thanks!
Yan

November 21, 2016
5:55 pm
Sidora83
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Never mind. I was looking too deep into it :P.
Just realized the conjugate pair was added in to replace the "pre-FOIL" coefficients, with the knowledge the imaginary part would cancel out, leaving real numbers.

July 25, 2017
3:50 pm
P
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I did have exactly the same question. I wanted to know the algebra steps involved. Digging a little bit, I found this link with the explanation behind the algebra. Very useful and interesting.

https://www.dsprelated.com/freebooks/filters/Pole_Zero_Analysis_I.html

Here is a little summary

"we can write the general transfer function for the recursive LTI digital filter as

(...)

In the same way that

can be factored into

we can factor the numerator and denominator to obtain.

Hope this helps!

Thanks.

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