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 circleThe cut-off frequency in radians is determined by the angle

**Î¸**between the real axis and the line connecting**0**to the pole**a**

and

**Î¸****= arctan{imag(a) / real(a)}**