Filter transformation

The filter transformation is used in the context of the filter design to implement electronic filters between different filter types such as low-pass filters , high-pass filters or band-pass filters .

General

Electronic filters are designed using a desired transfer function H (s). The Butterworth filters , Tschebyscheff filters , Bessel filters or Cauer filters are examples of the usual transfer functions, which are used in particular for analog, time-continuous filters in different filter orders .

For dimensioning, a general prototype filter is usually assumed in the form of a standardized low-pass filter. During normalization, which is used to design the filter design independently of specific frequency values, all frequency-dependent parameters are related to the cut-off frequency of the prototype filter . In this simplified form, the filter parameters of the transfer function, which in analog filters are described by the values ​​of electrical components such as resistors , capacitors and coils , can be specified in general in the form of tables or stored in databases.

For the implementation of a concrete filter in the form of an electrical circuit , the dimensioned prototype filter is denormalized and the filter is transformed into the final filter type. Since low-pass filters are mostly used as prototype filters in the filter tables, the transformation from the low-pass to the high-pass and the band-pass or bandstop is particularly important. However, like any transformation, the transformation can also take place in the opposite direction. The filter transformations are also used in an adapted form in the context of time-discrete digital filters. In particular when analog filter circuits are simulated in the context of digital signal processing, such as in the case of wave digital filters .

Low-pass-high-pass transformation

Properties of the TP-HP transformation

The low-pass-high-pass transformation (TP-HP) is used to convert any low-pass into a high-pass. The transfer function H _HP (s ′) is formed from the transfer function of the low-pass filter H _TP (s ) by the following substitution:

${\displaystyle s'={\frac {1}{s}}}$

with s = jω or s ′ = jω ′ as parameters of the angular frequency . In the electrical circuit, this implementation means that the standardized components such as capacitors C and coils L , which characterize the pole or zero positions , are interchanged:

${\displaystyle R=R',\qquad L={\frac {1}{C'}},\qquad C={\frac {1}{L'}}}$

Low-pass-band-pass transformation

Properties of the TP-BP transformation

The low-pass-band-pass transformation (TP-BP) is used to convert any low-pass into a band-pass. The absolute frequency response of the low-pass filter is mirrored on a logarithmic frequency scale at the center frequency of the band-pass filter. The order is thus doubled by the TP-BP transformation and only symmetrical bandpass filters with an even filter order are possible.

The transfer function H _BP (s ′) is formed from the transfer function of the low-pass filter H _TP (s) with the following transformation :

${\displaystyle s'={\frac {1}{2\pi B}}\left(s+{\frac {1}{s}}\right)}$

with the parameter B which describes the bandwidth of the bandpass. In the electrical circuit, this implementation means, as shown in the adjacent figure, that the standardized capacitors C of the low-pass filter are implemented in series resonant circuits and coils L in parallel resonant circuits in the bandpass filter.

Special transformations such as the Zdunek filter transformation exist for asymmetrical bandpass filters .

Low pass bandstop transformation

The low-pass bandstop transformation (TP-BS) can be viewed as a combination of the TP-HP and TP-BP transformation: The application of the TP-BP transformation to a high-pass filter results in a bandstop.

Summarized from the transfer function of the low-pass filter H _TP (s), the following transformation is used to form the transfer function H _BS (s ′) of the bandstop filter, the parameter B of the bandwidth is identical to that of the TP-BP transformation:

${\displaystyle s'={\frac {2\pi B}{\left(s+{\frac {1}{s}}\right)}}}$

literature

• Ulrich Tietze, Christoph Schenk: Semiconductor circuit technology . 12th edition. Springer, Berlin 2002, ISBN 3-540-42849-6 .
• Fred H. Irons: Active Filters for Integrated Circuit Applications . Artech House, 2005, ISBN 1-58053-896-7 .

Individual evidence

1. ^ Rudolf Saal, Walter Entenmann: Handbook for the filter design . 2nd Edition. Hüthig, 1988, ISBN 3-7785-1558-6 (filter tables).
2. ↑ W. Winkelnkemper: Unsymmetrical bandpass and bandstop digital filters . IEEE Electronics Letters, 5, 23rd Edition, 1969, pp. 585 to 586 ( ieee.org [PDF]).