9 Bandstop filter transfer function

The transfer function of the band-stop filter can be obtained by mapping the low-pass prototype transfer function, using the following mapping function

The entire procedure for obtaining the band-stop transfer function may include the following steps:

  • Step 1:

    Generate band-pass filter specification

  • Step 2:

    Convert the band-pass filter specification into the symmetrical one

    Parameters of the symmetrical specification must satisfy the following conditions:

  • Step 3:

    Convert the symmetrical band-stop specification into the equivalent low-pass prototype using the following expressions

  • Step 4:

    Generate the low-pass prototype transfer function

  • Step 5:

    Map the low-pass prototype transfer function into the desired band-pass transfer function


Sample

The transfer function of the low-pass prototype is obtained 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-stop transfer function of the filter for which (9.5) is an equivalent low-pass prototype, can be obtained by z-domain frequency mapping. Replacing variable with mapping function

the band-stop filter transfer function can be expressed as follows:

Note, that are complex conjugate zeros and poles of the low-pass prototype transfer function. Therefore, the factors and are real numbers. The pairs and are complex conjugate numbers as well. Consequently, by analogy with (8.6), expression (9.7) can be presented in the form of a rational polynomial with real coefficients, and it satisfies the requirements to the filter transfer function.

Matheonics Technology Inc, 2009

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