IEEE Signal Processing - May 2018 - 130

can be performed in the pole-zero form.
The factored partial fraction expansion
method has a very low computational
complexity requirement, thus, it is ideal
for cases when the conversion must be
done on the fly. This comes at a price of
a low-order FIR filter in series with the
parallel sections.
The goal of this article is to raise
awareness about the numerical issues
related to the common way of converting
filters to the parallel form, with the hope
that the reader will find the alternative
methods appealing and useful in many
practical situations.

Acknowledgments
I am thankful to Prof. László Sujbert,
Prof. Julius Smith, and Prof. Vesa Välimäki
for their helpful comments. This work
was supported by the ÚNKP-16-4-III New
National Excellence Program of the Hungarian Ministry of Human Capacities.

Supplementary material
This article has supplementary downloadable material available at http://
ieeexplore.ieee.org, provided by the

sp educaTion

[1] A. V. Oppenheim, "One plus one could equal
three (and other favorite cliches)," IEEE Signal
Process. Mag., vol. 23, pp. 10-12, Jan. 2006.

130

Author
Balázs Bank (bank@mit.bme.hu) is
an associate professor in the Department of Measurement and Information
Systems, Budapest University of Technology and Economics, Hungary. From
2013 to 2016, he was an associate editor of IEEE Signal Processing Letters.
His research interests include physicsbased sound synthesis and filter design
for audio applications.

Acoustics Speech and Signal Processing, Florence,
Italy, May 2014, pp. 6692-6696.
[4] M. Price, S. Holden, and M. Sandler, "Accurate
parallel form filter synthesis," IEEE Electron. Lett.,
vol. 32, no. 22, pp. 2066-2067, Oct. 1996.
[5] A. Krukowski, I. Kale, and G. D. Cain, "Decomposition of IIR transfer functions into parallel
arbitrary-order IIR subfilters," in Proc. IEEE Nordic
Signal Processing Symp., Espoo, Finland, Sept.
1996, pp. 175-178.
[6] J. O. Smith. (2007, Sept.). Introduction to digital
filters with audio applications [Online]. Available:
http://ccrma.stanford.edu/˜jos/filters/
[7] B. Bank and J. O. Smith, "A delayed parallel filter structure with an FIR part having improved
numerical properties," in Proc. 136th Audio
Engineering Society Convention, preprint no. 9084,
Berlin, Germany, Apr. 2014.
[8] K. Steiglitz and L. E. McBride, "A technique for
the indentification of linear systems," IEEE Trans.
Automat. Contr., vol. AC-10, pp. 461-464, Oct.
1965.
[9] B. Bank, "Perceptually motivated audio equalization using fixed-pole parallel second-order filters,"
IEEE Signal Process. Lett., vol. 15, pp. 477-480,
May 2008.
[10] V. Välimäki and J. D. Reiss, "All about audio
equalization: Solutions and frontiers," Appl. Sci., vol.
6, no. 5, paper 129, May 2016.

References

[1] A. V. Oppenheim, R. W. Schafer, and J. R. Bruck,
Discrete-Time Signal Processing. Englewood Cliffs,
NJ: Prentice Hall, 1975.
[2] W. Chen, "Performance of cascade and parallel
IIR filters," J. Audio Eng. Soc., vol. 44, no. 3, pp.
148-158, 1996.
[3] J. A. Belloch, B. Bank, L. Savioja, A. Gonzalez,
and V. Välimäki, "Multi-channel IIR filtering of
audio signals using a GPU," in Proc. IEEE Int. Conf.

[11] G. Ramos and J. J. López, "Filter design method
for loudspeaker equalization based on IIR parametric
filters," J. Audio Eng. Soc., vol. 54, no. 12, pp. 1162-
1178, Dec. 2006.
[12] M. Tyril, J. A. Pedersen, and P. Rubak, "Digital
filters for low-frequency equalization," J. Audio Eng.
Soc., vol. 29, no. 1-2, pp. 36-43, Jan. 2001.

sp

(continued from page 116)

Danilo P. Mandic (d.mandic@
imperial.ac.uk) is a professor of signal
processing at Imperial College London,
United Kingdom. He has been working in
statistical, nonlinear, and biomedical signal processing and is a pioneer of hearables, in-ear sensing of vital signs, and
electroencephalograms from the ear canal.
He has a keen interest in signal processing
education, is a member of the IEEE Signal
Processing Society Education Technical
Committee, and his contributions were
recognized with the President's Award for
Excellence in Postgraduate Supervision
at Imperial College in 2014. He is a
Fellow of the IEEE.

References

author. The material includes MATLAB functions required for converting
IIR filters to parallel form and specific
scripts to generate the example figures of the article. The files can also
be downloaded at http://www.mit.bme
.hu/~bank/parconv.

[2] E. Richter and A. Nehorai, "Enriching the undergraduate program with research projects," IEEE
Signal Process. Mag., vol. 33, pp. 123-127, Jan.
2016.
[3] D. P. Mandic, S. Kanna, and A. G. Constantinides, "On the intrinsic relationship between the
least mean square and Kalman filters [Lecture
Notes]," IEEE Signal Process. Mag., vol. 32, pp.
117-122, Nov. 2015.
[4] R. Faragher, "Understanding the basis of the
Kalman filter via a simple and intuitive derivation,"
IEEE Signal Process. Mag., vol. 29, no. 5, pp. 128-
132, 2012.
[5] J. Humpherys and J. West, "Kalman filtering with
newton's method," IEEE Control Syst. Mag., vol. 30,
no. 6, pp. 49-51, 2010.
[6] W. von Rosenberg, T. Chanwimalueang, V.
Goverdovsky, N. S. Peters, C. Papavassiliou, and
D. P. Mandic, "Hearables: Feasibility of recording
cardiac rhythms from head and in-ear locations,"
Roy. Soc. Open Sci., vol. 4, no. 11, pp. 1-13,
2017.
[7] M. Akay, Biomedical Signal Processing. San
Diego, CA: Academic, 1994.
[8] A. V. Oppenheim, "A personal view of education," IEEE Signal Process. Mag., vol. 9, pp. 69-72,
Jan. 1992.

IEEE Signal Processing Magazine

|

May 2018

|

[9] M. Malik, J. T. Bigger, A. J. Camm, R. E.
K leiger, A. Ma l l ia n i, A. J. Moss, a nd P. J.
Schwartz, "Heart rate variability: Standards of measurement, physiological interpretation, and clinical
use," European Heart J., vol. 17, no. 3, pp. 354-381,
1996.
[10] J. F. Thayer, F. Ahs, M. Fredrikson, J. J. SollersIII, and T. D. Wager, "A meta-analysis of heart rate
variability and neuroimaging studies: Implications for
heart rate variability as a marker of stress and health,"
Neurosci. Biobehav. Rev., vol. 36, no. 2, pp. 747-
756, 2012.
[11] T. Chanwimalueang, W. von Rosenberg, and D.
P. Mandic, "Enabling R-peak detection in wearable
ECG: Combining matched filtering and Hilbert transform," in Proc. IEEE Int. Conf. Digital Signal
Processing, July 2015, pp. 134-138.
[12] S. M. Kay, Fundamentals of Statistical Signal
Processing: Estimation Theory. Englewood Cliffs,
NJ: Prentice Hall, 1993.
[13] V. Goverdovsky, W. von Rosenberg, T. Naka mu ra , D. L ooney, D. J. Sha r p, C. Papavassi l iou , M. J. Mor r el l , a nd D. P. Ma nd ic,
"Hearables: Multimodal physiological in-ear sensing," Nature Scientific Reports, vol. 7, no. 6948,
pp. 1-10, 2017.

sp
http://www.mit.bme.hu/~bank/parconv http://www.mit.bme.hu/~bank/parconv http://ccrma.stanford.edu/˜jos/filters/ http://ieeexplore.ieee.org http://ieeexplore.ieee.org
Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018
