 Complex pole-zero pairs in all-pass filters help control the frequency where phase shifts occur.To create an all-pass implementation with real coefficients, the complex all-pass filter can be cascaded with an all-pass that substitutes formula_16 for formula_11, leading to the Z-transform implementation
which is equivalent to the difference equation
where formula_20 is the output and formula_21 is the input at discrete time step formula_22.Filters such as the above can be cascaded with unstable or mixed-phase filters to create a stable or minimum-phase filter without changing the magnitude response of the system.