What do Numerator and Denominator of a Filter mean?

 

The numerator of a filter modifies the signal in the frequency domain. On the other hand, the numerator denotes the zeros of a transfer function.

Poles of a transfer function tell us about the cut-off frequency of a filter.

Process of finding the cut-off frequency of a filter from z-transform

Let's assume the transfer function of a filter, H(z) = 1 / {1 - a*z^(-1)}

Where a is a constant. This represents a simple first-order recursive (IIR) low-pass filter

Then, make dominator, 1 - a*z^(-1) = 0

                      Or,  z = a

 Then calculate the magnitude of the pole: |a|

  If |a| is close to 1, then it is closest to the unit circle

The cut-off frequency in radians is determined by the angle θ between the real axis and the line connecting 0 to the pole a 

and θ = arctan{imag(a) / real(a)}