# Significance of poles in a Transfer Function

Sorry for asking this basic question, but I am new to signal processing and have this doubt for a long time.

I have been studying signal modelling and have $$H(z) = B_q(z)/A_p(z)$$ where $$A_p(z)$$ represents the poles of the Filter's Transfer function.
When we increase poles, $$A_p(z)$$ increases in terms and the signal approximation becomes more accurate to the desired signal (I've understood this while doing Pade's approximation on a discrete signal in MATLAB, I increased the poles and the approximation error reduced).

However, when they teach us the basic definitions of poles and zeroes of a Transfer function, I've been told that poles are those values where the system is unstable.

Now, how can adding poles to the filter system make its approximation more accurate, while at the same time increasing its unstable region? Is this a tradeoff that design engineers have to take note of?

Please forgive if this question sounds too much basic, it's just a doubt I have and my professor doesn't intend to explain because it's too rudimentary for his calibre.

• Oh! I've had luck, I never met a prof that said something like "this is too rudimentary for my calibre"; I've met professors who said, "that is basic stuff. I can't really explain this in any time frame that I have to spare, I'm afraid you'll have to rework these basics on your own"; I really hope your professor is nice to you, and a helpful person overall. Commented Oct 27, 2022 at 8:55