Frequency Transformation with Pascal Matrix Equations

Frequency transformation with Pascal matrix
equations is a method for transforming an electronic filter (analogue
or digital) into another filter. The technique is based on frequency
transformation in the s-domain, bilinear z-transform with pre-warping
frequency, inverse bilinear transformation and a very useful
application of the Pascal’s triangle that simplifies computing and
enables calculation by hand when transforming from one filter to
another. This paper will introduce two methods to transform a filter
into a digital filter: frequency transformation from the s-domain into
the z-domain; and frequency transformation in the z-domain. Further,
two Pascal matrix equations are derived: an analogue to digital filter
Pascal matrix equation and a digital to digital filter Pascal matrix
equation. These are used to design a desired digital filter from a given
filter.

Authors:

• Phuoc Si Nguyen

Keywords:

• Frequency transformation
• Bilinear z-transformation
• Pre-warping frequency
• Digital filters
• Analog filters
• Pascal’s triangle.

References:

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(Addison-Wesley, 1993)
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12, (1), pp. 5–8
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in the design of IIR Filters’, IEEE, 2011, 9, (3), pp. 355–366
[4] Chivapreecha, S.: ‘Bilinear s-z frequency transformation using matrix
Pascal operation’, IEEE Communications and Information Technology,
2005, 1, pp. 764–767
[5] Phuoc S. N.: ‘Bilinear z-transformation using Pascal’s triangle’,
International Journal of Scientific and Engineering Research, 2013, 4,
pp. 1974–1978.