how to design LPH and HPF from APF I came across a concept that says we can design LPH and HPF from APF when the transfer function of the two APFs given as

$$\begin{align}A_0(z)&amp;=1\\
A_1(z)&amp;=\frac{-a+z^{-1}}{1-az^{-1}}\\\\
H_{LP}(z)&amp;=A_0(z)+A_1(z)\\
H_{HP}(z)&amp;=A_0(z)-A_1(z)\end{align}$$

i tried to implement this in matlab but i did not get the desired frequency response..kindly help me with the matlab code for the same..
Since you don't say what response you're actually getting from Matlab, this is all guesswork. 

So let's do the basics: A little formula juggling.

$$\begin{align}A_0(z)&amp;=1\\
A_1(z)&amp;=\frac{-a+z^{-1}}{1-az^{-1}}\\[4em]
H_{LP}(z)&amp;=A_0(z)+A_1(z)\\
   &amp;= 1 + \frac{-a+z^{-1}}{1-az^{-1}}\\
   &amp;= \frac{{1-az^{-1}}-a+z^{-1}}{1-az^{-1}}\\
   &amp;= \frac{(1-a)\quad +\quad (1-a)z^{-1}}{1-az^{-1}}\\[4em]
H_{HP}(z)&amp;=A_0(z)-A_1(z)\\
   &amp;= 1 - \frac{-a+z^{-1}}{1-az^{-1}}\\
   &amp;= \frac{{1-az^{-1}}+a-z^{-1}}{1-az^{-1}}\\
   &amp;= \frac{(1+a)\quad +\quad (-1-a)z^{-1}}{1-az^{-1}}
\end{align}$$

So, taken from these formulas, the filter coefficients for $H_{LP}$ are $b=[1-a,1-a]$, $a=[1,-a]$, and  for $H_{HP}$ it's $b=[1+a,-1-a]$, $a=[1,-a]$ for the canonical form of recursive filters

$$H= \frac{\sum\limits_{i=0}^N b_i z^{-i}}{\sum\limits_{i=0}^M a_i z^{-i}}\text.$$

Plugging that into scipy's scipy.signal.freqz for different $a$ yielded (abscissa: normalized frequency $\in[0,\pi[$, ordinate: magnitude):

$H_{LP,a}$ for $a\in\{0,0.1,\dots,0.9,1\}$:



$H_{HP,a}$ for $ a\in\{0,0.1,\dots,0.9,1\}$:



which both very nicely match the terms low and high pass filters, respectively.