Fig 1 describes the proposed system. Keywords IIR, FIR, inverse Chebyshev, Elliptic, LMS, RLS.

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1 Design of approximately linear phase sharp cut-off discrete-time IIR filters using adaptive linear techniques of channel equalization. IIT-Madras R.Sharadh, Dual Degree--Communication Systems Abstract: Sharp cut-off IIR digital filters have a relatively much lower order than the design for the same cut-off frequency but the filters can have an exactly linear phase. This paper looks at a compromise in which the IIR filter is followed by a filter, which adaptively tries to linearize the overall phase response but leaves the magnitude response unchanged. Keywords IIR,, inverse Chebyshev, Elliptic,,. In the design of brick wall digital filters, IIR designs can achieve very sharp characteristics with a relatively low order. But the phase response is highly non-linear. Fir designs can be designed to have exactly linear phase but in order to achieve identical fall-off rates of IIR filters they need to be of relatively larger order. In this paper, a system is proposed where the IIR filter (designed for a particular cutoff frequency and decay rate)is followed by a filter, which tries to adaptively compensate for the non-linear phase introduced by the former. The filter however leaves the magnitude response unchanged. Fig 1 describes the proposed system Now, the learning signal is the output (suitably delayed) of another filter, which has the same magnitude characteristics as the IIR but has linear phase. This linear phase system can be designed a trial and error process using scientific software but quite predictably, the order of the system will be, relative to the IIR system, quite large.

2 Now, white noise or a pseudo random binary sequence is driven through the filter. The output is then passed through the IIR filter, followed by the adaptive phase equalizer. The filter output is also driven through a delay line (length N) and the delay line output is the learning signal. Thus, N is the designed group delay of the cascaded system. The updating of equalizer weights was done by the Least Mean Squared () or Recursive Least Squares () algorithms. White Noise or PRBS source Linear Phase IIR Phase Equalizer (Length M) Delay Line Length N Fig 1 Note 1) The Linear Phase and IIR" have the same magnitude response characteristics. 2) The updating of the tap-weights of the phase equalizer has been done recursively, through the or algorithms

3 Inverse Chebyshev Filter Elliptic Filter N M Stdev Stdev % 15.91% 7.77% 18.3% % 22.3% 8.51% 22% % 6.5% 3.87% 9.42% % 9.43% 4.67% 12.4% % 2.73% 2.23% 5.05% % 3.9% 1.33% 5.2% N M Stdev Stdev % 17.3% 8% 22% % 19.3% 8.22% 24% % 8.57% 4.08% 12.27% % 11.22% 4.38% 13.88% % 4.6% 2.4% 6.92% % 2.9% 0.93% 2% Table 1 Table 2 Note 1) The tables list the observed ripples and standard deviation in the group delay as a percentage of N, the designed group delay. 2) (in the above tables) is defined as (max(group delay)- min(group delay))/2 3) For the algorithm, µ was chosen as ) For the algorithm, δ was chosen as 0.01 and forget factor λ as ) The low pass Elliptic and Inverse Chebyshev filters, were designed for a cutoff frequency of 0.21 Hz with maximum stop-band and pass band ripples of 0.01 and 0.1 respectively. The Elliptic filter was a 6th order IIR filter and the equivalent linear phase was of length 111. The Inverse Chebyshev filter was a 9th order IIR filter and the equivalent linear phase was of length ) In the following diagrams, for both cases, the phase equalizer used was of length 25.

4 Lowpass Inverse Chebyshev ω c =0.21 Hz Lowpass Elliptic ω c =0.21 Hz Magnitude comparison Magnitude comparison Group delays ( with M=N=25) Group delays ( with M=N=25) Group delays ( with M=N=25) Group delays ( with M=N=25)

5 Results The phase response attained an approximately constant group delay in the pass band. The ripples were found to decrease as the equalizer order was made higher and the delay-line length was also made larger. The ripples were observed to have a maximum at the pass band edge and were a minimum when the equalizer order was the same as the length of the delay line. They were not completely eliminated, even as the order was made very large. In most cases, the algorithm gave lower ripples in the group delay than the algorithm. s observed in the pass-band magnitude response of the cascaded system were observed to reduce as the equalizer order was increased. Also, the stop band response of the cascaded system had smaller ripples. System type No. of multipliers No. of adders Inverse 9x2=18 16 Chebyshev Linear (103+1)/2= phase Proposed sys 9x2+25=43 40 Table 4 Relative complexity References 1) Alan V. Oppenheim, Ronald W. Schafer, John R. Buck, Discrete Time Signal Processing Second Edition, Prentice-Hall, ) Sanjit K. Mitra, Digital signal processing -- A computer based approach, Tata McGraw-Hill Edition ) Simon Haykin, Adaptive Filter Theory- Third Edition, Prentice Hall, 1996 System type No. of multipliers No. of adders Elliptic 6x2=12 10 Linear phase (111+1)/2= Proposed sys 6x2+25=37 34 Table 3 Relative complexity

MULTIRATE IIR LINEAR DIGITAL FILTER DESIGN FOR POWER SYSTEM SUBSTATION

MULTIRATE IIR LINEAR DIGITAL FILTER DESIGN FOR POWER SYSTEM SUBSTATION Riyaz Khan 1, Mohammed Zakir Hussain 2 1 Department of Electronics and Communication Engineering, AHTCE, Hyderabad (India) 2 Department