Future Scope

The proposed Differential Evolution algorithm is able to reduce hardware complexity when the filter constraints are relaxed and many feasible solutions exist. The new escape ap-proach can be developed whereby the search agent is able to escape a deep valley. One such scheme could be to develop Particle Swarm optimization inspired local and global best learning approach.

The current objective in the filter design procedure has been to reduce the adder cost of implementing the finite word length FIR filter. However, it is observed that the problem is multi modal. Thus, a new problem at a lower level of abstraction can be formulated taking into account the full adder cost of implementation. Also, the actual synthesis of the filter can carried out using tools such as Synopsys Design Compiler to make precise latency and power analysis.

The proposed Differential Evolution algorithm can also be used for other discrete prob-lems. It can be used to design cascade form FIR filters. It can also be modified for solving

mixed problems by modifying the population members to include floating point parameters and utilizing the original mutation scheme for these parameters.

References

[1] H. K. Kwan, Digital Filters and Systems. ISP Lab, University of Windsor, 2015. 8 [2] L. R. Rabiner and B. Gold, “Theory and application of digital signal processing,”

Englewood Cliffs, NJ, Prentice-Hall, Inc., 1975. 777 p., vol. 1, 1975. 8

[3] T. Parks and J. McClellan, “A program for the design of linear phase finite impulse response digital filters,” Audio and Electroacoustics, IEEE Transactions on, vol. 20, no. 3, pp. 195–199, Aug 1972. 11

[4] T. Baran, D. Wei, and A. Oppenheim, “Linear Programming Algorithms for Sparse Filter Design,” Signal Processing, IEEE Transactions on, vol. 58, no. 3, pp. 1605–1617, March 2010. 13

[5] A. Jiang and H. K. Kwan, “WLS Design of Sparse FIR Digital Filters,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 60, no. 1, pp. 125–135, Jan 2013. 13

[6] Y. C. Lim, “Frequency-response masking approach for the synthesis of sharp linear phase digital filters,” Circuits and Systems, IEEE Transactions on, vol. 33, no. 4, pp.

357–364, Apr 1986. 13

[7] Y. Lim and Y. Lian, “Frequency-response masking approach for digital filter design:

complexity reduction via masking filter factorization,” Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, vol. 41, no. 8, pp. 518–525, Aug 1994. 13

[8] D. Kodek and K. Steiglitz, “Comparison of optimal and local search methods for de-signing finite wordlength FIR digital filters,” Circuits and Systems, IEEE Transactions on, vol. 28, no. 1, pp. 28–32, Jan 1981. 14

[9] D. Kodek, “Design of optimal finite wordlength FIR digital filters using integer pro-gramming techniques,” Acoustics, Speech and Signal Processing, IEEE Transactions on, vol. 28, no. 3, pp. 304–308, Jun 1980. 14

[10] Y. Lim and S. Parker, “FIR filter design over a discrete powers-of-two coefficient space,” Acoustics, Speech and Signal Processing, IEEE Transactions on, vol. 31, no. 3, pp. 583–591, Jun 1983. 14

[11] Y. Lim, “Design of discrete-coefficient-value linear phase FIR filters with optimum normalized peak ripple magnitude,” Circuits and Systems, IEEE Transactions on, vol. 37, no. 12, pp. 1480–1486, Dec 1990. 15

[12] H. Samueli, “An improved search algorithm for the design of multiplierless FIR filters with powers-of-two coefficients,” Circuits and Systems, IEEE Transactions on, vol. 36, no. 7, pp. 1044–1047, Jul 1989. 15

[13] Y. C. Lim, R. Yang, D. Li, and J. Song, “Signed power-of-two term allocation scheme for the design of digital filters,” Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, vol. 46, no. 5, pp. 577–584, May 1999. 16

[14] D. Bull and D. Horrocks, “Primitive operator digital filters,” Circuits, Devices and Systems, IEE Proceedings G, vol. 138, no. 3, pp. 401–412, June 1991. 16

[15] P. Cappello and K. Steiglitz, “Some complexity issues in digital signal processing,”

Acoustics, Speech and Signal Processing, IEEE Transactions on, vol. 32, no. 5, pp.

1037–1041, Oct 1984. 17

[16] A. Yurdakul and G. Dundar, “Fast and efficient algorithm for the multiplierless real-isation of linear DSP transforms,” Circuits, Devices and Systems, IEE Proceedings -, vol. 149, no. 4, pp. 205–211, Aug 2002. 17, 23

[17] M. Potkonjak, M. Srivastava, and A. Chandrakasan, “Multiple constant multiplica-tions: efficient and versatile framework and algorithms for exploring common subex-pression elimination,” Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, vol. 15, no. 2, pp. 151–165, Feb 1996. 17

[18] A. Dempster and M. Macleod, “Use of minimum-adder multiplier blocks in FIR digital filters,” Circuits and Systems II: Analog and Digital Signal Processing, IEEE Trans-actions on, vol. 42, no. 9, pp. 569–577, Sep 1995. 17, 18, 24, 52,54

[19] ——, “Multiplication by an integer using minimum adders,” in Mathematical Aspects of Digital Signal Processing, IEE Colloquium on, Feb 1994, pp. 11/1–11/4. 18,19,52 [20] Y. J. Yu and Y. C. Lim, “Design of Linear Phase FIR Filters in Subexpression Space Using Mixed Integer Linear Programming,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 54, no. 10, pp. 2330–2338, Oct 2007. 20,21, 89

[21] Y. Yu and Y. Lim, “Optimization of Linear Phase FIR Filters in Dynamically Ex-panding Subexpression Space,” Circuits, Systems and Signal Processing, vol. 29, no. 1, pp. 65–80, 2010. 23

[22] D. Shi and Y. J. Yu, “Design of Linear Phase FIR Filters With High Probability of Achieving Minimum Number of Adders,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 58, no. 1, pp. 126–136, Jan 2011. 23,89

[23] C.-Y. Yao, W.-C. Hsia, and Y.-H. Ho, “Designing Hardware-Efficient Fixed-Point FIR Filters in an Expanding Subexpression Space,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 61, no. 1, pp. 202–212, Jan 2014. 23,89

[24] W. B. Ye and Y. J. Yu, “Two-Step Optimization Approach for the Design of Mul-tiplierless Linear-Phase FIR Filters,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 62, no. 5, pp. 1279–1287, May 2015. 23, 89

[25] ——, “Single-Stage and Cascade Design of High Order Multiplierless Linear Phase FIR Filters Using Genetic Algorithm,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 60, no. 11, pp. 2987–2997, Nov 2013. 24, 86

[26] A. Antoniou and W.-S. Lu, Practical optimization: algorithms and engineering appli-cations. Springer Science & Business Media, 2007. 27

[27] “OPTI Toolbox,” accessed: 2016-01-02. [Online]. Available: http://www.i2c2.aut.ac.

nz/Wiki/OPTI/index.php/ 31, 65

[28] B. Borchers, “CSDP, A C library for semidefinite programming,” Optimization Methods and Software, vol. 11, no. 1-4, pp. 613–623, 1999. [Online]. Available:

http://dx.doi.org/10.1080/10556789908805765 31

[29] K. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: A Practical Ap-proach to Global Optimization (Natural Computing Series). Secaucus, NJ, USA:

Springer-Verlag New York, Inc., 2005. 34

[30] J. Brest, S. Greiner, B. Boskovic, M. Mernik, and V. Zumer, “Self-Adapting Control Parameters in Differential Evolution: A Comparative Study on Numerical Benchmark Problems,” Evolutionary Computation, IEEE Transactions on, vol. 10, no. 6, pp. 646–

657, Dec 2006. 35

[31] Y. Liu, W. neng Chen, Z. hui Zhan, Y. Lin, Y.-J. Gong, and J. Zhang, “A Set-Based Discrete Differential Evolution Algorithm,” in Systems, Man, and Cybernetics (SMC), 2013 IEEE International Conference on, Oct 2013, pp. 1347–1352. 35

[32] W.-N. Chen, J. Zhang, H. Chung, W.-L. Zhong, W. gang Wu, and Y. hui Shi, “A Novel Set-Based Particle Swarm Optimization Method for Discrete Optimization Problems,”

Evolutionary Computation, IEEE Transactions on, vol. 14, no. 2, pp. 278–300, April 2010. 35

[33] A. Shahein, Q. Zhang, N. Lotze, and Y. Manoli, “A Novel Hybrid Monotonic Local Search Algorithm for FIR Filter Coefficients Optimization,” Circuits and Systems I:

Regular Papers, IEEE Transactions on, vol. 59, no. 3, pp. 616–627, March 2012. 40, 86

[34] W. B. Ye and Y. J. Yu, “A polynomial-time algorithm for the design of multiplierless linear-phase FIR filters with low hardware cost,” in Circuits and Systems (ISCAS), 2014 IEEE International Symposium on, June 2014, pp. 970–973. 48

[35] H. Samueli, “On the design of FIR digital data transmission filters with arbitrary mag-nitude specifications,” Circuits and Systems, IEEE Transactions on, vol. 38, no. 12, pp. 1563–1567, Dec 1991. 55

[36] M. Aktan, A. Yurdakul, and G. Dundar, “An Algorithm for the Design of Low-Power Hardware-Efficient FIR Filters,” Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 55, no. 6, pp. 1536–1545, July 2008. 48,64, 89

[37] L. Aksoy, P. Flores, and J. Monteiro, “Exact and Approximate Algorithms for the Fil-ter Design Optimization Problem,” Signal Processing, IEEE Transactions on, vol. 63, no. 1, pp. 142–154, Jan 2015. 86,89

[38] K. Johansson, O. Gustafsson, and L. Wanhammar, “Bit-Level Optimization of Shift-and-Add Based FIR Filters,” in Electronics, Circuits and Systems, 2007. ICECS 2007.

14th IEEE International Conference on, Dec 2007, pp. 713–716. 81