Frequency Transformation with Pascal Matrix Equations

Authors: Phuoc Si Nguyen

Abstract:

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.

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

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1338618

References:

[1] Emmanuel, C., Ifeacor, B. & Jervis, W.: ‘Digital signal processing’ (Addison-Wesley, 1993)
[2] Konopacki J.: ‘The frequency transformation by matrix operation and its application in IIR filters design’, IEEE Signal Processing Letters, 2005, 12, (1), pp. 5–8
[3] Garcia-Ugalde, F. J.: ‘Z-transform by Pascal matrix and its applications 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.