8 Bandpass filter transfer function
The transfer function of the bandpass filter can be obtained by mapping the lowpass prototype transfer function, using the following function
The entire procedure of obtaining the bandpass transfer function may include the following steps:

Step 1
Generate bandpass filter specification

Step 2
Convert the bandpass filter specification into the symmetrical one
Parameters of the symmetrical specification must satisfy the following conditions:

Step 3
Convert the symmetrical bandpass specification into the equivalent lowpass prototype using the following expressions

Step 4
Generate the lowpass prototype transfer function

Step 5
Map the lowpass prototype transfer function into the desired bandpass transfer function
Another way of obtaining the transfer function is by cascading the lowpass filter and highpass filter. Usually, this approach is used for wide band bandpass filters.
Sample
The transfer function of the lowpass prototype is given as follows
Such transfer functions occur for the elliptic or inverse Chebyshev approximations. Zeros are complex conjugate; poles are complex conjugate as well.
The bandpass transfer function of the filter for which (8.5) is an equivalent lowpass prototype, can be obtained by zdomain frequency mapping. Replacing variable with mapping function,
the bandpass transfer function can be expressed as follows:
Note, that are complex zeros and poles of the lowpass prototype transfer function.
This expression can be rearranged to the form where all coefficients in the nominator and denominator would be real numbers. To compute the poles of (8.6), 2 quadratic equations may be solved:
The roots of these equations are:
Replacing conjugate poles in (8.7) with the expressions (8.7) can be rearranged as follows
Introducing new variable,
the roots (8.8) can be written as
Using the trigonometric form of complex numbers, all 4 poles of bandpass filter can be written as follows
It is easy to see that pairs of roots and are complex conjugate. Consequently, the denominator of transfer function (8.6) can be presented in the form of a polynomial with real coefficients.
In the same way, it can be shown that zeros of the bandpass transfer function are complex conjugate pairs as well. Therefore, the transfer function of the bandpass filter for which (8.5) is an equivalent lowpass prototype, can be given in the form of a rational polynomial with real coefficients.