# Design of discrete time second order all pass filter

I want to design a second order digital all pass filter with the transfer function given by $$H(z)=\frac{a_2 +a_1 z^{-1}+z^{-2}}{1+a_1 z^{-1}+a_2 z^{-2}}$$ The input to this filter has frequencies ranging from $$1$$ to $$8\ \rm kHz$$. The sampling frequency is $$20\ \rm kHz$$.

How to design the filter coefficients $$a_1, a_2$$ such that for a particular input frequency, I have to apply 180 degree phase shift? The frequency which is given a 180 degree phase shift will vary. So, how to vary the filter coefficients based on which frequency has to be phase shifted by 180 degree?

• That's a homework-style question, so you should show what you've tried and explain where you're stuck. Nov 25 '20 at 11:23
• I wanted to know where to start Nov 25 '20 at 12:16

1. What value does $$H(z_0)$$ have if there's a phase shift of $$180$$ degrees (for $$|z_0|=1$$)?
2. Equate $$H(z_0)$$ to that value and write down the resulting equation for $$a_1$$ and $$a_2$$ and $$z_0$$.
3. You want a stable all-pass filter, so choose $$a_2$$ such that a given desired pole radius $$r$$, $$0, is achieved if you assume two complex conjugate poles.
4. With that value of $$a_2$$, and with a given frequency $$\omega_0$$ for which the phase shift is $$180$$ degrees, solve the equation you got in 2. for $$a_1$$. The result will have the form $$a_1=f(a_2)\cos(\omega_0)$$ where $$f(\cdot)$$ is a simple affine function. Note that $$\omega_0$$ is normalized by the sampling frequency.