# American Institute of Mathematical Sciences

September  2019, 9(3): 319-326. doi: 10.3934/naco.2019021

## Frequency interval model reduction of complex fir digital filters

 1 Department of Mechatronics Engineering, Kulliyah of Engineering, International Islamic University Malaysia, 53100 Jalan Gombak, Malaysia 2 School of Electrical and Electronics Engineering, University of Western Australia, 35 Stirling Highway, WA 6009, Australia 3 Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad, 211004, India

Received  May 2018 Revised  April 2019 Published  May 2019

In this paper, a model reduction method for FIR filters with complex coefficients based on frequency interval impulse response Gramians is developed. The advantage of the proposed method is that only one Lyapunov equation needs to be solved in order to obtain the information regarding the frequency interval controllability and observability of the system. In addition this method overcomes the limitations of using cross Gramians which are not applicable for filters with complex coefficients. The effectiveness of the proposed method is demonstrated by a numerical example.

Citation: Ahmad Jazlan, Umair Zulfiqar, Victor Sreeram, Deepak Kumar, Roberto Togneri, Hasan Firdaus Mohd Zaki. Frequency interval model reduction of complex fir digital filters. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 319-326. doi: 10.3934/naco.2019021
##### References:
 [1] P. Benner, P. Kürschner and J. Saak, Frequency-limited balanced truncation with low-rank approximations, SIAM Journal on Scientific Computing, 38 (2016), A471–A499. doi: 10.1137/15M1030911. [2] X. Chen and T. Parks, Design of FIR filters in the complex domain, IEEE Transactions on Acoustics, Speech, and Signal Processing, 35 (1987), 144-153. [3] D. W. Ding, X. Du and X. Li, Finite-frequency model reduction of two-dimensional digital filters, IEEE Trans. Autom. Control, 60 (2015), 1624-1629. doi: 10.1109/TAC.2014.2359305. [4] X. Du, F. Fan, D. W. Ding and F. Liu, Finite-frequency model order reduction of discrete-time linear time-delayed systems, Nonlinear Dynamics, X (2016), 1-12. doi: 10.1007/s11071-015-2496-0. [5] X. Du, A. Jazlan, V. Sreeram, R. Togneri, A. Ghafoor and S. Sahlan, A frequency limited model reduction technique for linear discrete systems, Proceedings of the 2013 Australian Control Conference, 421–426. [6] W. Gawronski and J. Juang, Model reduction in limited time and frequency intervals, International Journal of Systems Science, 21, 349–376. doi: 10.1080/00207729008910366. [7] J. Gryka, I. Kale and G. D. Cain, Complex IIR filter design through balance model reduction of FIR prototypes, Electronics Letters, 31 (1995), 1332-1334. [8] M. Imran and A. Ghafoor, Frequency limited model reduction techniques With error bounds, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 65 (2018), 86–90. [9] M. Imran and A. Ghafoor, Model reduction of descriptor systems using frequency limited Gramians, J. Franklin Inst., 352 (2015), 33-51. doi: 10.1016/j.jfranklin.2014.10.013. [10] A. Jazlan, V. Sreeram, H. R. Shaker, R. Togneri and H. B. Minh, Frequency interval cross Gramians for linear and bilinear systems, Asian Journal of Control, 19 (2017), 22-34. doi: 10.1002/asjc.1330. [11] D. Kumar, V. Sreeram and X. Du, Model reduction using parameterized limited frequency interval Gramians for 1-D and 2-D separable denominator discrete-time systems, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 65 (2018), 2571-2580. [12] X. Li, C. Yu and H. Gao, Frequency limited $H_{\infty}$ model reduction for positive systems, IEEE Trans. Autom. Control, 60 (2015), 1093-1098. doi: 10.1109/TAC.2014.2352751. [13] M. A. Masnadi-Shirazi, A. Zollanvari and M. A. Amin, Complex digital Laguerre filter design with weighted least square error subject to magnitude and phase constraints, Signal Processing, 88 (1987), 796. [14] W. A. Mousa, Frequency-space wavefield extrapolation using infinite impulse response digital filters: is it feasible?, Geophysical Prospecting, 61 (2013), 504-515. [15] M. Okuda, M. Kiyose, M. Ikehara and S. Takahashi, Equiripple design in complex domain for FIR digital filters by transforming desired response, Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 84 (2001), 30. [16] H. R. Shaker and M. Tahavori, Frequency-interval model reduction of bilinear systems, IEEE Transactions on Automatic Control, 59 (2014), 1948-1953. doi: 10.1109/TAC.2013.2295661. [17] C. Tseng and S. Lee, Designs of fractional derivative constrained 1-D and 2-D FIR filters in the complex domain, Signal Processing, 95 (2014), 111. [18] D. L. Wang and A. Zilouchian, Model reduction of discrete linear systems via frequency-domain balanced structure, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 47 (2000), 830-837. doi: 10.1109/81.852936. [19] K. Xu and Y. Jiang, An approach to H2 $\omega$ model reduction on finite interval for bilinear systems, Journal of the Franklin Institute, 354 (2017), 7429-7443. doi: 10.1016/j.jfranklin.2017.08.037.

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##### References:
 [1] P. Benner, P. Kürschner and J. Saak, Frequency-limited balanced truncation with low-rank approximations, SIAM Journal on Scientific Computing, 38 (2016), A471–A499. doi: 10.1137/15M1030911. [2] X. Chen and T. Parks, Design of FIR filters in the complex domain, IEEE Transactions on Acoustics, Speech, and Signal Processing, 35 (1987), 144-153. [3] D. W. Ding, X. Du and X. Li, Finite-frequency model reduction of two-dimensional digital filters, IEEE Trans. Autom. Control, 60 (2015), 1624-1629. doi: 10.1109/TAC.2014.2359305. [4] X. Du, F. Fan, D. W. Ding and F. Liu, Finite-frequency model order reduction of discrete-time linear time-delayed systems, Nonlinear Dynamics, X (2016), 1-12. doi: 10.1007/s11071-015-2496-0. [5] X. Du, A. Jazlan, V. Sreeram, R. Togneri, A. Ghafoor and S. Sahlan, A frequency limited model reduction technique for linear discrete systems, Proceedings of the 2013 Australian Control Conference, 421–426. [6] W. Gawronski and J. Juang, Model reduction in limited time and frequency intervals, International Journal of Systems Science, 21, 349–376. doi: 10.1080/00207729008910366. [7] J. Gryka, I. Kale and G. D. Cain, Complex IIR filter design through balance model reduction of FIR prototypes, Electronics Letters, 31 (1995), 1332-1334. [8] M. Imran and A. Ghafoor, Frequency limited model reduction techniques With error bounds, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 65 (2018), 86–90. [9] M. Imran and A. Ghafoor, Model reduction of descriptor systems using frequency limited Gramians, J. Franklin Inst., 352 (2015), 33-51. doi: 10.1016/j.jfranklin.2014.10.013. [10] A. Jazlan, V. Sreeram, H. R. Shaker, R. Togneri and H. B. Minh, Frequency interval cross Gramians for linear and bilinear systems, Asian Journal of Control, 19 (2017), 22-34. doi: 10.1002/asjc.1330. [11] D. Kumar, V. Sreeram and X. Du, Model reduction using parameterized limited frequency interval Gramians for 1-D and 2-D separable denominator discrete-time systems, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 65 (2018), 2571-2580. [12] X. Li, C. Yu and H. Gao, Frequency limited $H_{\infty}$ model reduction for positive systems, IEEE Trans. Autom. Control, 60 (2015), 1093-1098. doi: 10.1109/TAC.2014.2352751. [13] M. A. Masnadi-Shirazi, A. Zollanvari and M. A. Amin, Complex digital Laguerre filter design with weighted least square error subject to magnitude and phase constraints, Signal Processing, 88 (1987), 796. [14] W. A. Mousa, Frequency-space wavefield extrapolation using infinite impulse response digital filters: is it feasible?, Geophysical Prospecting, 61 (2013), 504-515. [15] M. Okuda, M. Kiyose, M. Ikehara and S. Takahashi, Equiripple design in complex domain for FIR digital filters by transforming desired response, Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 84 (2001), 30. [16] H. R. Shaker and M. Tahavori, Frequency-interval model reduction of bilinear systems, IEEE Transactions on Automatic Control, 59 (2014), 1948-1953. doi: 10.1109/TAC.2013.2295661. [17] C. Tseng and S. Lee, Designs of fractional derivative constrained 1-D and 2-D FIR filters in the complex domain, Signal Processing, 95 (2014), 111. [18] D. L. Wang and A. Zilouchian, Model reduction of discrete linear systems via frequency-domain balanced structure, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 47 (2000), 830-837. doi: 10.1109/81.852936. [19] K. Xu and Y. Jiang, An approach to H2 $\omega$ model reduction on finite interval for bilinear systems, Journal of the Franklin Institute, 354 (2017), 7429-7443. doi: 10.1016/j.jfranklin.2017.08.037.
Magnitude Response of 35th order IIR Filter
Magnitude Response of 30th order IIR Filter
Magnitude Response of 23rd order IIR Filter
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