The transfer function of the band-pass filter can be obtained by mapping the low-pass prototype transfer function, using the following function
The entire procedure of obtaining the band-pass transfer function may include the following steps:
Generate band-pass filter specification
Convert the band-pass filter specification into the symmetrical one
Parameters of the symmetrical specification must satisfy the following conditions:
Convert the symmetrical band-pass specification into the equivalent low-pass prototype using the following expressions
Generate the low-pass prototype transfer function
Map the low-pass prototype transfer function into the desired band-pass transfer function
Another way of obtaining the transfer function is by cascading the low-pass filter and high-pass filter. Usually, this approach is used for wide band band-pass filters.
The transfer function of the low-pass 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 band-pass transfer function of the filter for which (8.5) is an equivalent low-pass prototype, can be obtained by z-domain frequency mapping. Replacing variable with mapping function,
the band-pass transfer function can be expressed as follows:
Note, that are complex zeros and poles of the low-pass 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 band-pass 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 band-pass transfer function are complex conjugate pairs as well. Therefore, the transfer function of the band-pass filter for which (8.5) is an equivalent low-pass prototype, can be given in the form of a rational polynomial with real coefficients.