# Chebyshev poles into transfer function

I have obtained a set of three poles for a third order Chebyshev filter as shown below:

$$p_k=\rm -0.2471+0.9660j,\quad -0.2471-0.9660j, \quad -0.4942.$$

However I am unsure of how to actually implement these into the Chebyshev prototype filter equation?

Any help would be greatly appreciated!

The transfer function is

$$H(s)=\frac{g}{\prod_k(s-p_k)}\tag{1}$$

where $g$ is a scaling constant which is usually chosen such that the maximum value of the magnitude of $H(j\omega)$ equals $1$.

For a third order (or any odd-order) Chebyshev filter, a maximum occurs at $\omega=0$, so the gain constant can be derived from the requirement $H(0)=1$:

$$g=\prod_k(-p_k)=-\prod_kp_k$$

For your example the gain constant is $g=0.49134$.

• Thank you - May I ask when calculating these poles in the solutions given above my imaginary part does not match up. Using the standard equations with v=0.47 and n =3 I get the imaginary part to equal - +0.965, 0 and -0.96 for n = 1, 2 and 3. Where am I making a mistake? – Tricks Aug 15 '16 at 23:07
• @Tricks: Complex poles must come in complex conjugate pairs. I'm not sure what the problem is, but you could add your calculations to your questions to show how you arrive at those values. – Matt L. Aug 16 '16 at 7:40
• @MattL. Could you please provide some good reference link for details on scaling constant. Three of my books are passing it by giving some "magic formula" or are constructed different way - and it is not working. – Jan Filip Feb 16 '18 at 20:33
• @JanFilip: It's probably best to ask a new question describing your problem, but "Digital Filter Design" by Parks and Burrus is generally a good reference. – Matt L. Feb 16 '18 at 21:08