All Pass Filter
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## 1 Introduction to All Pass Filters

An all pass filter is a filter with a completely flat magnitude response across all frequencies. However, the phase is shifting by -180 °. This is accomplished by a pole zero pair where the real part of the pole equals -1 times the real part of the zero.

## 2 Basics of an All Pass Filter

For an all pass filter in the s-domain the transfer function is simple.

 $LaTeX: F_{AP}=\frac{s-a}{s+a}$

where

$LaTeX: \Re \left \{a \right \}>0$ for a stable filter.

In MATLAB an all pass filter with a -180 ° phase shift at 1 Hz would be formed with the following commands

 >> a = 1 * (2*pi);     % Convert 1 Hz into rad/sec
>> F_AP = tf([1 -a], [1 +a]);