Digital Filters Part 11 pdf

As shown in Table 5, the erms performances among the VdIIR VFD filters can be summarized as follows: The top performers for 0.95    0.9625 are the gradient-based designs with 35.. As

FdIIR VFD filters at  = 0.9,  = 0) At each iteration, the SOCP problems in (29), (37) and

(43) are solved using SeDuMi (Sturm, 1999) under MATLAB environment

6 Performance analysis

6.1 Error measurements and stability check

To evaluate the performances of each designed VFD filter, the maximum absolute error e max,

and the normalized root-mean-squared (RMS) error e rms of its (a) frequency responses, (b)

magnitude responses, and (c) fractional group delay responses are adopted and they are

defined, respectively, by

e maxmax ( , ) , e t [0,],t [ 0.5,0.5] (46)

1/2

0 0.5

0 0.5

( , ) ( , )

rms

d

e









 

e max,1maxe MAG( , ) , t [0,],t [ 0.5,0.5] (48)

1/2

0 0.5

0 0.5

( , ) ( , )

MAG rms

d

e









 

e max,2maxe FGD( , ) , t [0,],t [ 0.5,0.5] (50)

1/2

0 0.5

0 0.5

( , )

FGD rms

e

t dtd













 

where

e MAG( , )t  H e t( , )j H d( , )t (52)

e FGD( , ) t  ( , )t t (53)

In (53), τ(ω,t) denotes the actual fractional group delay of a designed VFD filter Since the design problem is formulated in the WLS sense (see (19)), so the e rms of the frequency responses is the most appropriate criterion for comparisons among different design

methods In case two designs have the same e rms, other error measurements shall be compared For each of the designed VdIIR VFD filters and AP VFD filters, a uniform grid

consisting of 1001 discrete fractional delay values t were used to ensure all these 1001 VFD

filters are stable By checking individual maximum pole radius to be within the unity circle, each of the designed VFD filters has been verified to be stable

6.2 IIR VFD filter performances

Based on the design specifications of Table 1, the error performances of the designed IIR VFD filters are summarized in Tables 3-4 The keywords adopted in Tables 3-4 are defined

as follows: The “Sequential design” refers to the minimization problem defined by (29) subject to (a) stability inequality constraints (35) for VdIIR VFD filter design; and (b) stability inequality constraints (34) for FdIIR VFD filter design The “Gradient-based design with (35)” refers to the minimization problem defined by (37) subject to stability inequality constraints (35) for an initial VdIIR VFD filter design, and followed by a local search The

“Gradient-based design with (34)” refers to the minimization problem defined by (37) subject to stability inequality constraints (34) for an initial FdIIR VFD filter design, and followed by a local search The “Gradient-based design with (43)” refers to the minimization problem defined by (43) for an initial VdIIR or FdIIR VFD filter design, and followed by a

local search Within each of the four sets of designs, the relative erms (in frequency responses) performances are ranked from top to bottom as shown in Tables 3-4 The top performer of each IIR VFD design method in Tables 3-4 is listed in Table 5

As shown in Table 5, the erms performances among the VdIIR VFD filters can be summarized

as follows: The top performers for 0.95    0.9625 are the gradient-based designs with (35) The top performers for 0.9    0.925 are the gradient-based designs with (43) The bottom performer is the two-stage design of (ZK) The performance of the sequential designs (29) ranks at the middle between the designs of (ZK) and the gradient-based designs with (35)

and with (43) As also shown in Table 5, the erms performances among the FdIIR VFD filters can be summarized as follows: The top performers for 0.925    0.9625 are the gradient-based designs with (43) but has an average performance for  = 0.9 The top performer for 

= 0.9 is the gradient-based design with (34) which has close but lower performances than those of the gradient-based designs with (43) for 0.925    0.95 The bottom performer for 0.925    0.9625 is (TCK) but it ranks second among all the FdIIR VFD designs for  = 0.9 Between (KJ) and the sequential design (29), the former ranks higher than those of the sequential designs (29) for 0.95    0.9625 but vice versa for 0.9    0.925 Comparing (KJ) and (TCK), the former yields better performances for 0.925    0.9625 but vice versa for  = 0.9.

α N D A R Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

1 49

25

(29) 9 -35.490 1.892e-3 -37.360 1.289e-3 1.763 2.754e-1

(35) 3 -50.347 3.683e-4 -50.402 2.923e-4 3.970e-1 6.042e-2

(43) 4 -46.317 4.790e-4 -46.373 3.607e-4 5.621e-1 7.708e-2

(ZK) 12 -11.622 2.766e-2 -12.295 2.402e-2 1.972 4.208e-1

28

(29) 8 -40.026 1.403e-3 -40.664 1.036e-3 1.160 1.823e-1

(35) 2 -50.808 3.444e-4 -51.710 2.318e-4 4.850e-1 7.108e-2

(43) 5 -45.817 4.981e-4 -48.255 3.327e-4 6.545e-1 9.443e-2

(ZK) 11 -12.042 2.623e-2 -13.067 2.268e-2 1.892 4.291e-1

31

(29) 7 -42.041 8.851e-4 -42.698 6.840e-4 9.504e-1 1.431e-1

(35) 1 -52.436 2.890e-4 -53.731 1.833e-4 4.442e-1 6.963e-2

(43) 6 -45.492 5.203e-4 -46.819 3.439e-4 6.152e-1 1.034e-1

(ZK) 10 -12.674 2.460e-2 -13.590 2.110e-2 1.797 4.203e-1

2 46

23

(29) 9 -43.309 8.175e-4 -46.118 5.256e-4 6.791e-1 1.095e-1

(35) 5 -57.964 1.563e-4 -57.970 1.230e-4 1.561e-1 2.346e-2

(43) 6 -55.398 2.194e-4 -56.439 1.629e-4 2.370e-1 3.347e-2

(ZK) 10 -17.857 1.511e-2 -18.471 1.328e-2 1.097 2.441e-1

26

(29) 8 -48.237 4.151e-4 -50.465 2.946e-4 3.830e-1 6.093e-2

(35) 3 -59.298 1.354e-4 -60.759 9.100e-5 1.680e-1 2.487e-2

(43) 4 -59.500 1.442e-4 -59.567 1.025e-4 1.855e-1 2.446e-2

(ZK) 11 -17.735 1.531e-2 -18.573 1.340e-2 1.021 2.346e-1

29

(29) 7 -48.984 3.667e-4 -49.148 2.845e-4 3.047e-1 4.843e-2

(35) 1 -60.500 1.171e-4 -63.434 7.782e-5 1.400e-1 2.453e-2

(43) 2 -59.982 1.310e-4 -60.924 9.276e-5 1.434e-1 2.400e-2

(ZK) 12 -11.036 2.871e-2 -12.351 2.526e-2 1.702 3.513e-1

3 41

21

(29) 9 -57.865 1.108e-4 -61.693 6.780e-5 1.306e-1 1.993e-2

(35) 5 -62.965 5.007e-5 -63.189 3.882e-5 5.270e-2 7.486e-3

(43) 6 -64.763 6.303e-5 -67.058 4.233e-5 7.008e-2 1.016e-2

(ZK) 10 -18.100 1.752e-2 -18.330 1.493e-2 4.667e-1 1.575e-1

24

(29) 7 -60.523 8.940e-5 -60.973 6.550e-5 9.716e-2 1.449e-2

(35) 4 -66.111 4.390e-5 -67.968 3.004e-5 5.477e-2 8.191e-3

(43) 3 -69.381 3.348e-5 -70.084 2.327e-5 4.344e-2 6.336e-3

(ZK) 11 -15.405 1.998e-2 -15.883 1.767e-2 6.691e-1 1.745e-1

27

(29) 8 -59.811 9.295e-5 -59.859 7.225e-5 7.450e-2 1.322e-2

(35) 2 -67.930 3.255e-5 -72.267 2.048e-5 4.415e-2 7.135e-3

(43) 1 -75.807 1.269e-5 -78.312 8.311e-6 2.229e-2 2.984e-3

(ZK) 12 -13.440 2.520e-2 -14.190 2.242e-2 1.020 2.197e-1

4 36

18

(29) 7 -70.872 3.336e-5 -74.955 2.250e-5 2.631e-2 4.264e-3 (35) 9 -71.177 3.592e-5 -71.466 2.760e-5 2.270e-2 3.510e-3 (43) 4 -71.255 2.661e-5 -73.122 1.942e-5 2.182e-2 3.217e-3 (ZK) 11 -20.667 1.381e-2 -20.070 1.113e-2 2.332e-1 1.109e-1

21

(29) 6 -71.817 3.311e-5 -73.389 2.411e-5 2.564e-2 3.895e-3 (35) 5 -72.620 2.730e-5 -73.472 1.881e-5 2.110e-2 3.541e-3 (43) 2 -79.979 7.880e-6 -83.184 5.360e-6 8.086e-3 1.170e-3 (ZK) 10 -21.880 1.139e-2 -22.079 9.317e-3 2.680e-1 1.033e-1

24

(29) 8 -71.882 3.488e-5 -72.448 2.545e-5 1.982e-2 3.541e-3 (35) 3 -75.763 2.294e-5 -77.805 1.494e-5 2.183e-2 3.434e-3 (43) 1 -83.278 6.257e-6 -85.250 4.068e-6 8.721e-3 1.314e-3 (ZK) 12 -14.311 2.847e-2 -14.477 2.483e-2 5.477e-1 1.958e-1 Table 3 Performances of VdIIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; A: Design method; (29): Sequential design; (35): Gradient-based design with (35); (43): Gradient-based design with (43); (ZK): (Zhao & Kwan, 2007); R: Rank; FGD: Fractional group delay)

α N D A R Freq Responses Mag Responses FGD Responses

e max e rms e max,1(dB) e rms,1 e max,2 e rms,2

1 54

27

(29) 12 -38.000 1.426e-3 -40.368 9.325e-4 1.556 2.398e-1 (34) 6 -51.464 2.796e-4 -52.628 2.229e-4 3.141e-1 4.812e-2 (43) 5 -49.821 2.791e-4 -49.826 2.345e-4 2.523e-1 4.390e-2 (KJ) 9 -39.632 5.615e-4 -39.696 4.623e-4 8.980e-1 1.365e-1 (TCK) 15 -30.303 2.429e-3 -31.218 1.974e-3 3.359 5.846e-1

30

(29) 11 -42.034 9.887e-4 -43.963 7.094e-4 1.014 1.559e-1 (34) 4 -50.852 2.683e-4 -53.605 1.810e-4 3.932e-1 6.088e-2 (43) 3 -49.940 2.663e-4 -51.336 1.906e-4 3.675e-1 5.526e-2 (KJ) 7 -40.645 5.044e-4 -41.407 3.952e-4 1.010 1.446e-1 (TCK) 14 -31.333 2.206e-3 -34.075 1.415e-3 3.364 6.026e-1

33

(29) 10 -43.634 6.475e-4 -45.398 4.989e-4 8.047e-1 1.196e-1 (34) 2 -50.271 2.647e-4 -54.681 1.649e-4 4.254e-1 6.933e-2 (43) 1 -58.117 1.360e-4 -59.459 1.055e-4 1.553e-1 2.391e-2 (KJ) 8 -40.973 5.101e-4 -42.615 3.681e-4 1.143 1.668e-1 (TCK) 13 -33.233 1.793e-3 -38.764 8.176e-4 2.853 5.160e-1

2 51 26

(29) 12 -46.106 4.757e-4 -49.348 3.021e-4 4.745e-1 7.514e-2 (34) 9 -56.847 1.423e-4 -59.984 1.015e-4 1.334e-1 2.122e-2 (43) 3 -60.282 1.172e-4 -62.605 9.084e-5 8.234e-2 1.344e-2 (KJ) 5 -55.680 1.241e-4 -58.979 8.890e-5 2.465e-1 3.491e-2 (TCK) 15 -38.816 8.603e-4 -38.917 7.661e-4 1.178 1.856e-1

α N D A R Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

1 49

25

(29) 9 -35.490 1.892e-3 -37.360 1.289e-3 1.763 2.754e-1

(35) 3 -50.347 3.683e-4 -50.402 2.923e-4 3.970e-1 6.042e-2

(43) 4 -46.317 4.790e-4 -46.373 3.607e-4 5.621e-1 7.708e-2

(ZK) 12 -11.622 2.766e-2 -12.295 2.402e-2 1.972 4.208e-1

28

(29) 8 -40.026 1.403e-3 -40.664 1.036e-3 1.160 1.823e-1

(35) 2 -50.808 3.444e-4 -51.710 2.318e-4 4.850e-1 7.108e-2

(43) 5 -45.817 4.981e-4 -48.255 3.327e-4 6.545e-1 9.443e-2

(ZK) 11 -12.042 2.623e-2 -13.067 2.268e-2 1.892 4.291e-1

31

(29) 7 -42.041 8.851e-4 -42.698 6.840e-4 9.504e-1 1.431e-1

(35) 1 -52.436 2.890e-4 -53.731 1.833e-4 4.442e-1 6.963e-2

(43) 6 -45.492 5.203e-4 -46.819 3.439e-4 6.152e-1 1.034e-1

(ZK) 10 -12.674 2.460e-2 -13.590 2.110e-2 1.797 4.203e-1

2 46

23

(29) 9 -43.309 8.175e-4 -46.118 5.256e-4 6.791e-1 1.095e-1

(35) 5 -57.964 1.563e-4 -57.970 1.230e-4 1.561e-1 2.346e-2

(43) 6 -55.398 2.194e-4 -56.439 1.629e-4 2.370e-1 3.347e-2

(ZK) 10 -17.857 1.511e-2 -18.471 1.328e-2 1.097 2.441e-1

26

(29) 8 -48.237 4.151e-4 -50.465 2.946e-4 3.830e-1 6.093e-2

(35) 3 -59.298 1.354e-4 -60.759 9.100e-5 1.680e-1 2.487e-2

(43) 4 -59.500 1.442e-4 -59.567 1.025e-4 1.855e-1 2.446e-2

(ZK) 11 -17.735 1.531e-2 -18.573 1.340e-2 1.021 2.346e-1

29

(29) 7 -48.984 3.667e-4 -49.148 2.845e-4 3.047e-1 4.843e-2

(35) 1 -60.500 1.171e-4 -63.434 7.782e-5 1.400e-1 2.453e-2

(43) 2 -59.982 1.310e-4 -60.924 9.276e-5 1.434e-1 2.400e-2

(ZK) 12 -11.036 2.871e-2 -12.351 2.526e-2 1.702 3.513e-1

3 41

21

(29) 9 -57.865 1.108e-4 -61.693 6.780e-5 1.306e-1 1.993e-2

(35) 5 -62.965 5.007e-5 -63.189 3.882e-5 5.270e-2 7.486e-3

(43) 6 -64.763 6.303e-5 -67.058 4.233e-5 7.008e-2 1.016e-2

(ZK) 10 -18.100 1.752e-2 -18.330 1.493e-2 4.667e-1 1.575e-1

24

(29) 7 -60.523 8.940e-5 -60.973 6.550e-5 9.716e-2 1.449e-2

(35) 4 -66.111 4.390e-5 -67.968 3.004e-5 5.477e-2 8.191e-3

(43) 3 -69.381 3.348e-5 -70.084 2.327e-5 4.344e-2 6.336e-3

(ZK) 11 -15.405 1.998e-2 -15.883 1.767e-2 6.691e-1 1.745e-1

27

(29) 8 -59.811 9.295e-5 -59.859 7.225e-5 7.450e-2 1.322e-2

(35) 2 -67.930 3.255e-5 -72.267 2.048e-5 4.415e-2 7.135e-3

(43) 1 -75.807 1.269e-5 -78.312 8.311e-6 2.229e-2 2.984e-3

(ZK) 12 -13.440 2.520e-2 -14.190 2.242e-2 1.020 2.197e-1

4 36

18

(29) 7 -70.872 3.336e-5 -74.955 2.250e-5 2.631e-2 4.264e-3 (35) 9 -71.177 3.592e-5 -71.466 2.760e-5 2.270e-2 3.510e-3 (43) 4 -71.255 2.661e-5 -73.122 1.942e-5 2.182e-2 3.217e-3 (ZK) 11 -20.667 1.381e-2 -20.070 1.113e-2 2.332e-1 1.109e-1

21

(29) 6 -71.817 3.311e-5 -73.389 2.411e-5 2.564e-2 3.895e-3 (35) 5 -72.620 2.730e-5 -73.472 1.881e-5 2.110e-2 3.541e-3 (43) 2 -79.979 7.880e-6 -83.184 5.360e-6 8.086e-3 1.170e-3 (ZK) 10 -21.880 1.139e-2 -22.079 9.317e-3 2.680e-1 1.033e-1

24

(29) 8 -71.882 3.488e-5 -72.448 2.545e-5 1.982e-2 3.541e-3 (35) 3 -75.763 2.294e-5 -77.805 1.494e-5 2.183e-2 3.434e-3 (43) 1 -83.278 6.257e-6 -85.250 4.068e-6 8.721e-3 1.314e-3 (ZK) 12 -14.311 2.847e-2 -14.477 2.483e-2 5.477e-1 1.958e-1 Table 3 Performances of VdIIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; A: Design method; (29): Sequential design; (35): Gradient-based design with (35); (43): Gradient-based design with (43); (ZK): (Zhao & Kwan, 2007); R: Rank; FGD: Fractional group delay)

α N D A R Freq Responses Mag Responses FGD Responses

e max e rms e max,1(dB) e rms,1 e max,2 e rms,2

1 54

27

(29) 12 -38.000 1.426e-3 -40.368 9.325e-4 1.556 2.398e-1 (34) 6 -51.464 2.796e-4 -52.628 2.229e-4 3.141e-1 4.812e-2 (43) 5 -49.821 2.791e-4 -49.826 2.345e-4 2.523e-1 4.390e-2 (KJ) 9 -39.632 5.615e-4 -39.696 4.623e-4 8.980e-1 1.365e-1 (TCK) 15 -30.303 2.429e-3 -31.218 1.974e-3 3.359 5.846e-1

30

(29) 11 -42.034 9.887e-4 -43.963 7.094e-4 1.014 1.559e-1 (34) 4 -50.852 2.683e-4 -53.605 1.810e-4 3.932e-1 6.088e-2 (43) 3 -49.940 2.663e-4 -51.336 1.906e-4 3.675e-1 5.526e-2 (KJ) 7 -40.645 5.044e-4 -41.407 3.952e-4 1.010 1.446e-1 (TCK) 14 -31.333 2.206e-3 -34.075 1.415e-3 3.364 6.026e-1

33

(29) 10 -43.634 6.475e-4 -45.398 4.989e-4 8.047e-1 1.196e-1 (34) 2 -50.271 2.647e-4 -54.681 1.649e-4 4.254e-1 6.933e-2 (43) 1 -58.117 1.360e-4 -59.459 1.055e-4 1.553e-1 2.391e-2 (KJ) 8 -40.973 5.101e-4 -42.615 3.681e-4 1.143 1.668e-1 (TCK) 13 -33.233 1.793e-3 -38.764 8.176e-4 2.853 5.160e-1

2 51 26

(29) 12 -46.106 4.757e-4 -49.348 3.021e-4 4.745e-1 7.514e-2 (34) 9 -56.847 1.423e-4 -59.984 1.015e-4 1.334e-1 2.122e-2 (43) 3 -60.282 1.172e-4 -62.605 9.084e-5 8.234e-2 1.344e-2 (KJ) 5 -55.680 1.241e-4 -58.979 8.890e-5 2.465e-1 3.491e-2 (TCK) 15 -38.816 8.603e-4 -38.917 7.661e-4 1.178 1.856e-1

29

(29) 11 -49.943 2.895e-4 -52.464 2.166e-4 2.821e-1 4.396e-2

(34) 8 -55.870 1.386e-4 -63.233 8.848e-5 1.524e-1 2.632e-2

(43) 2 -60.166 1.051e-4 -64.946 7.397e-5 8.715e-2 1.359e-2

(KJ) 4 -56.758 1.193e-4 -59.001 8.726e-5 1.691e-1 2.528e-2

(TCK) 14 -40.109 8.059e-4 -42.311 5.295e-4 1.314 2.294e-1

32

(29) 10 -51.166 2.425e-4 -52.046 1.934e-4 2.142e-1 3.369e-2

(34) 7 -55.703 1.382e-4 -61.363 9.540e-5 1.556e-1 2.623e-2

(43) 1 -58.723 1.018e-4 -65.813 7.060e-5 1.013e-1 1.683e-2

(KJ) 6 -55.965 1.287e-4 -55.998 9.835e-5 1.528e-1 2.498e-2

(TCK) 13 -41.867 6.935e-4 -48.144 3.326e-4 1.023 1.822e-1

3 46

23

(29) 12 -56.063 1.152e-4 -60.966 7.670e-5 1.237e-1 1.812e-2

(34) 3 -59.700 7.518e-5 -67.140 5.471e-5 4.434e-2 6.868e-3

(43) 4 -61.491 7.567e-5 -66.350 5.607e-5 3.709e-2 5.591e-3

(KJ) 10 -58.608 9.039e-5 -62.759 6.328e-5 8.504e-2 1.145e-2

(TCK) 13 -55.650 1.372e-4 -56.367 1.175e-4 1.242e-1 1.750e-2

26

(29) 7 -60.462 8.640e-5 -64.213 6.376e-5 6.447e-2 9.586e-3

(34) 6 -59.137 8.352e-5 -66.130 5.871e-5 6.708e-2 9.784e-3

(43) 2 -61.693 7.237e-5 -68.770 5.183e-5 3.782e-2 5.498e-3

(KJ) 9 -61.008 8.814e-5 -63.846 6.359e-5 5.162e-2 7.425e-3

(TCK) 14 -54.098 1.536e-4 -55.608 1.325e-4 2.001e-1 2.945e-2

29

(29) 5 -61.122 8.273e-5 -64.300 6.255e-5 5.129e-2 7.660e-3

(34) 11 -58.753 9.176e-5 -65.279 6.558e-5 7.955e-2 1.131e-2

(43) 1 -60.702 7.065e-5 -69.047 5.209e-5 3.796e-2 5.501e-3

(KJ) 8 -62.337 8.694e-5 -64.720 6.295e-5 4.210e-2 6.087e-3

(TCK) 15 -54.170 1.639e-4 -57.739 8.782e-5 2.696e-1 4.845e-2

4 41

21

(29) 8 -63.290 6.478e-5 -68.632 4.749e-5 2.587e-2 3.957e-3

(34) 1 -62.541 5.875e-5 -71.768 4.111e-5 2.003e-2 3.037e-3

(43) 5 -64.151 6.078e-5 -71.767 4.448e-5 1.876e-2 2.673e-3

(KJ) 11 -66.316 7.136e-5 -70.722 5.197e-5 7.839e-3 1.202e-3

(TCK) 2 -64.839 5.948e-5 -71.691 4.386e-5 2.400e-2 3.768e-3

24

(29) 6 -63.812 6.103e-5 -69.829 4.557e-5 1.439e-2 2.480e-3

(34) 3 -61.956 5.978e-5 -70.458 4.250e-5 2.073e-2 3.177e-3

(43) 4 -63.959 6.049e-5 -69.984 4.491e-5 1.615e-2 2.565e-3

(KJ) 12 -65.803 7.137e-5 -70.716 5.194e-5 1.140e-2 1.686e-3

(TCK) 14 -63.694 8.469e-5 -64.780 5.867e-5 6.538e-2 1.150e-2

27 (29) 7 -64.154 6.237e-5 -69.549 4.676e-5 1.283e-2 2.222e-3

(34) 9 -62.223 6.748e-5 -66.374 4.933e-5 1.815e-2 3.434e-3

(43) 10 -62.973 7.050e-5 -65.414 5.395e-5 1.670e-2 3.412e-3 (KJ) 13 -66.208 7.147e-5 -70.498 5.203e-5 1.101e-2 1.632e-3 (TCK) 15 -58.427 1.680e-4 -58.631 1.203e-4 7.196e-2 1.499e-2 Table 4 Performances of FdIIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; A: Design method; (29): Sequential design; (34): Gradient-based design with (34); (43): Gradient-based design with (43); (KJ): (Kwan & Jiang, 2009a); (TCK): (Tsui et al., 2007); R: Rank; FGD: Fractional group delay)

(29) (35) (43) (ZK) (29) (34) (43) (KJ) (TCK)

1 erms 8.851e-4 2.890e-4 4.790e-4 2.460e-2 6.475e-4 2.647e-4 1.360e-4 5.044e-4 1.793e-3

2 erms 3.667e-4 1.171e-4 1.310e-4 1.511e-2 2.425e-4 1.382e-4 1.018e-4 1.193e-4 6.935e-4

3 erms 8.940e-5 3.255e-5 1.269e-5 1.752e-2 8.273e-5 7.518e-5 7.065e-5 8.694e-5 1.372e-4

4 erms 3.311e-5 2.294e-5 6.257e-6 1.139e-2 6.103e-5 5.875e-5 6.049e-5 7.136e-5 5.948e-5

Table 5 Top-performed (erms) VFD filters from Tables 3-4 (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; (ZK): (Zhao & Kwan, 2007); (KJ): (Kwan & Jiang, 2009a); (TCK): (Tsui et al., 2007); R: Rank)

6.3 Allpass and FIR VFD filter performances

The error performances of the AP VFD filters designed by (KJ) and (LCR) and the FIR VFD filters designed by (KJ) and (LD) are summarized in Table 6 In general, the two AP VFD

filters achieve e rms improvements over the two FIR VFD filters (except for (LD) at  = 0.9625)

The top e rms performances of the AP VFD filters are (KJ) for 0.925    0.9625 and (LCR) for

 = 0.9

6.4 Optimal gradient-based designs with (43)

It can be observed in Tables 3-4 that the error performances of VdIIR and FdIIR VFD filters

at any specified cutoff frequency is a function of the mean group delay value D To

investigate this property further, consider the case of the gradient-based design with (43) in Table 5 in which it ranks top among VdIIR VFD filters for 0.9    0.925 and ranks top among FdIIR VFD filters for 0.925    0.9625 For each of the four cutoff frequencies, the error performances of the gradient-based designs with (43) for VdIIR and FdIIR VFD filters

versus mean group delay D (at a step size of 3) are, respectively, summarized in Tables 7-8 and their corresponding erms values versus D are plotted in Figs 1-8 From Tables 7-8, their mean group delay values D that yield minimum erms values are summarized in Table 9 For

comparisons, the erms performances of the AP and FIR VFD filters from Table 6 are also listed under Table 9 The magnitude responses and group delay responses of the widest

29

(29) 11 -49.943 2.895e-4 -52.464 2.166e-4 2.821e-1 4.396e-2

(34) 8 -55.870 1.386e-4 -63.233 8.848e-5 1.524e-1 2.632e-2

(43) 2 -60.166 1.051e-4 -64.946 7.397e-5 8.715e-2 1.359e-2

(KJ) 4 -56.758 1.193e-4 -59.001 8.726e-5 1.691e-1 2.528e-2

(TCK) 14 -40.109 8.059e-4 -42.311 5.295e-4 1.314 2.294e-1

32

(29) 10 -51.166 2.425e-4 -52.046 1.934e-4 2.142e-1 3.369e-2

(34) 7 -55.703 1.382e-4 -61.363 9.540e-5 1.556e-1 2.623e-2

(43) 1 -58.723 1.018e-4 -65.813 7.060e-5 1.013e-1 1.683e-2

(KJ) 6 -55.965 1.287e-4 -55.998 9.835e-5 1.528e-1 2.498e-2

(TCK) 13 -41.867 6.935e-4 -48.144 3.326e-4 1.023 1.822e-1

3 46

23

(29) 12 -56.063 1.152e-4 -60.966 7.670e-5 1.237e-1 1.812e-2

(34) 3 -59.700 7.518e-5 -67.140 5.471e-5 4.434e-2 6.868e-3

(43) 4 -61.491 7.567e-5 -66.350 5.607e-5 3.709e-2 5.591e-3

(KJ) 10 -58.608 9.039e-5 -62.759 6.328e-5 8.504e-2 1.145e-2

(TCK) 13 -55.650 1.372e-4 -56.367 1.175e-4 1.242e-1 1.750e-2

26

(29) 7 -60.462 8.640e-5 -64.213 6.376e-5 6.447e-2 9.586e-3

(34) 6 -59.137 8.352e-5 -66.130 5.871e-5 6.708e-2 9.784e-3

(43) 2 -61.693 7.237e-5 -68.770 5.183e-5 3.782e-2 5.498e-3

(KJ) 9 -61.008 8.814e-5 -63.846 6.359e-5 5.162e-2 7.425e-3

(TCK) 14 -54.098 1.536e-4 -55.608 1.325e-4 2.001e-1 2.945e-2

29

(29) 5 -61.122 8.273e-5 -64.300 6.255e-5 5.129e-2 7.660e-3

(34) 11 -58.753 9.176e-5 -65.279 6.558e-5 7.955e-2 1.131e-2

(43) 1 -60.702 7.065e-5 -69.047 5.209e-5 3.796e-2 5.501e-3

(KJ) 8 -62.337 8.694e-5 -64.720 6.295e-5 4.210e-2 6.087e-3

(TCK) 15 -54.170 1.639e-4 -57.739 8.782e-5 2.696e-1 4.845e-2

4 41

21

(29) 8 -63.290 6.478e-5 -68.632 4.749e-5 2.587e-2 3.957e-3

(34) 1 -62.541 5.875e-5 -71.768 4.111e-5 2.003e-2 3.037e-3

(43) 5 -64.151 6.078e-5 -71.767 4.448e-5 1.876e-2 2.673e-3

(KJ) 11 -66.316 7.136e-5 -70.722 5.197e-5 7.839e-3 1.202e-3

(TCK) 2 -64.839 5.948e-5 -71.691 4.386e-5 2.400e-2 3.768e-3

24

(29) 6 -63.812 6.103e-5 -69.829 4.557e-5 1.439e-2 2.480e-3

(34) 3 -61.956 5.978e-5 -70.458 4.250e-5 2.073e-2 3.177e-3

(43) 4 -63.959 6.049e-5 -69.984 4.491e-5 1.615e-2 2.565e-3

(KJ) 12 -65.803 7.137e-5 -70.716 5.194e-5 1.140e-2 1.686e-3

(TCK) 14 -63.694 8.469e-5 -64.780 5.867e-5 6.538e-2 1.150e-2

27 (29) 7 -64.154 6.237e-5 -69.549 4.676e-5 1.283e-2 2.222e-3

(34) 9 -62.223 6.748e-5 -66.374 4.933e-5 1.815e-2 3.434e-3

(43) 10 -62.973 7.050e-5 -65.414 5.395e-5 1.670e-2 3.412e-3 (KJ) 13 -66.208 7.147e-5 -70.498 5.203e-5 1.101e-2 1.632e-3 (TCK) 15 -58.427 1.680e-4 -58.631 1.203e-4 7.196e-2 1.499e-2 Table 4 Performances of FdIIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; A: Design method; (29): Sequential design; (34): Gradient-based design with (34); (43): Gradient-based design with (43); (KJ): (Kwan & Jiang, 2009a); (TCK): (Tsui et al., 2007); R: Rank; FGD: Fractional group delay)

(29) (35) (43) (ZK) (29) (34) (43) (KJ) (TCK)

1 erms 8.851e-4 2.890e-4 4.790e-4 2.460e-2 6.475e-4 2.647e-4 1.360e-4 5.044e-4 1.793e-3

2 erms 3.667e-4 1.171e-4 1.310e-4 1.511e-2 2.425e-4 1.382e-4 1.018e-4 1.193e-4 6.935e-4

3 erms 8.940e-5 3.255e-5 1.269e-5 1.752e-2 8.273e-5 7.518e-5 7.065e-5 8.694e-5 1.372e-4

4 erms 3.311e-5 2.294e-5 6.257e-6 1.139e-2 6.103e-5 5.875e-5 6.049e-5 7.136e-5 5.948e-5

Table 5 Top-performed (erms) VFD filters from Tables 3-4 (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; (ZK): (Zhao & Kwan, 2007); (KJ): (Kwan & Jiang, 2009a); (TCK): (Tsui et al., 2007); R: Rank)

6.3 Allpass and FIR VFD filter performances

The error performances of the AP VFD filters designed by (KJ) and (LCR) and the FIR VFD filters designed by (KJ) and (LD) are summarized in Table 6 In general, the two AP VFD

filters achieve e rms improvements over the two FIR VFD filters (except for (LD) at  = 0.9625)

The top e rms performances of the AP VFD filters are (KJ) for 0.925    0.9625 and (LCR) for

 = 0.9

6.4 Optimal gradient-based designs with (43)

It can be observed in Tables 3-4 that the error performances of VdIIR and FdIIR VFD filters

at any specified cutoff frequency is a function of the mean group delay value D To

investigate this property further, consider the case of the gradient-based design with (43) in Table 5 in which it ranks top among VdIIR VFD filters for 0.9    0.925 and ranks top among FdIIR VFD filters for 0.925    0.9625 For each of the four cutoff frequencies, the error performances of the gradient-based designs with (43) for VdIIR and FdIIR VFD filters

versus mean group delay D (at a step size of 3) are, respectively, summarized in Tables 7-8 and their corresponding erms values versus D are plotted in Figs 1-8 From Tables 7-8, their mean group delay values D that yield minimum erms values are summarized in Table 9 For

comparisons, the erms performances of the AP and FIR VFD filters from Table 6 are also listed under Table 9 The magnitude responses and group delay responses of the widest

band designs at α = 0.9625 obtained by the VdIIR and FdIIR VFD filters shown in Table 9 are

plotted in Figs 9-12

α OD A/F Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

α1

56,

56

A(KJ) -40.677 3.246e-4 N.A N.A 1.980 1.717e-1 A(LCR) -24.604 9.309e-3 N.A N.A 5.920e-1 1.374e-1

55,

28

F(KJ) 2.798 8.242e-1 -24.807 3.048e-3 2.117 1.761 F(LD) -31.994 3.573e-3 -31.997 2.933e-3 1.548 3.248e-1

α2

53,

53 A(LCR) -55.710 2.258e-4 A(KJ) -61.643 5.626e-5 N.A N.A N.A N.A 4.437e-1 3.779e-2 8.224e-2 2.181e-2

52,

26

F(KJ) -32.726 1.493e-3 -32.770 1.216e-3 8.027e-1 1.633e-1 F(LD) -38.421 1.552e-3 -38.432 1.229e-3 6.470e-1 1.459e-1

α3

48,

48 A(LCR) -73.920 1.265e-5 A(KJ) -70.691 1.264e-5 N.A N.A N.A N.A 2.011e-2 1.745e-3 2.991e-3 9.069e-4

47,

24

F(KJ) 2.474 7.957e-1 -42.609 3.731e-4 7.122e-1 1.732 F(LD) -50.268 3.654e-4 -50.411 2.917e-4 1.802e-1 3.536e-2

α4

43,

43

A(KJ) -80.513 4.987e-6 N.A N.A 5.892e-3 5.193e-4 A(LCR) -84.237 4.119e-6 N.A N.A 3.870e-4 1.044e-4

42,

21

F(KJ) -53.561 1.310e-4 -53.810 1.027e-4 7.986e-2 1.609e-2 F(LD) -59.247 1.354e-4 -59.572 1.015e-4 5.479e-2 1.223e-2 Table 6 Performances of allpass and FIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925,

4= 0.9; OD: Filter order and mean group delay (MAP, DAP) or (LFIR, DFIR); A: Allpass design,

F: FIR design; (KJ): (Kwan & Jiang, 2009a); (LCR): (Lee et al., 2008); (LD): (Lu & Deng, 1999);

FGD: Fractional group delay)

The relationship between numerator and denominator orders, and optimal mean group

delay of a VdIIR or FdIIR VFD filter is a subject of interest Table 10 summarizes such

relationships among those VdIIR and FdIIR VDF filters listed in Table 9 It can be observed

from Table 10 that as  changes from 0.9    0.9625, the ratio D/(N+M) changes from 0.64

to 0.67 for VdIIR VFD filters, and changes from 0.57 to 0.55 for FdIIR VFD filters Also, as

seen from Figs 1-8, for the higher wideband side with  = 0.9625 and 0.95, there is a mean

group delay value that yields a minimum erms value; but for the lower wideband side with 

= 0.925 and 0.9, each of the mean group delay curves shows that erms becomes lower much

earlier at smaller D before reaching its minimum erms value In other words, the mean group

delay requirement is lower for lower wideband cutoff frequencies From Table 10, in

general, the VdIIR VFD filters require slightly higher optimal mean group delay values D

than those of the corresponding FdIIR VFD filters

α N D R Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

α1 49

25 6 -46.317 4.790e-4 -46.373 3.607e-4 5.621e-1 7.708e-2

28 7 -45.817 4.981e-4 -48.255 3.327e-4 6.545e-1 9.443e-2

31 8 -45.492 5.203e-4 -46.819 3.439e-4 6.152e-1 1.034e-1

34 3 -55.689 1.709e-4 -56.650 1.203e-4 3.135e-1 4.301e-2

37 1 -56.746 1.157e-4 -56.792 8.227e-5 2.371e-1 3.090e-2

40 2 -54.753 1.333e-4 -55.272 8.621e-5 2.725e-1 3.913e-2

43 4 -52.061 1.811e-4 -54.511 1.181e-4 3.634e-1 5.468e-2

46 5 -48.664 2.877e-4 -48.979 2.016e-4 3.676e-1 6.420e-2

α2 46

23 7 -55.398 2.194e-4 -56.439 1.629e-4 2.370e-1 3.347e-2

26 6 -59.500 1.442e-4 -59.567 1.025e-4 1.855e-1 2.446e-2

29 5 -59.982 1.310e-4 -60.924 9.276e-5 1.434e-1 2.400e-2

32 2 -63.424 6.157e-5 -66.513 4.168e-5 1.025e-1 1.451e-2

35 1 -64.515 5.514e-5 -67.411 3.558e-5 1.019e-1 1.364e-2

38 3 -62.722 6.798e-5 -63.918 4.290e-5 1.184e-1 1.767e-2

41 4 -57.588 9.448e-5 -57.757 7.247e-5 1.200e-1 1.731e-2

44 8 -48.195 2.999e-4 -52.186 2.194e-4 5.620e-1 5.862e-2

α3 41

18 8 -49.959 3.716e-4 -50.563 2.537e-4 2.966e-1 4.916e-2

21 6 -64.763 6.303e-5 -67.058 4.233e-5 7.008e-2 1.016e-2

24 5 -69.381 3.348e-5 -70.084 2.327e-5 4.344e-2 6.336e-3

27 2 -75.807 1.269e-5 -78.312 8.311e-6 2.229e-2 2.984e-3

30 1 -75.789 1.082e-5 -80.087 6.474e-6 2.048e-2 3.090e-3

33 3 -71.425 1.823e-5 -71.675 1.433e-5 2.420e-2 3.420e-3

36 4 -67.853 2.618e-5 -69.170 1.809e-5 3.759e-2 5.315e-3

39 7 -59.463 7.159e-5 -61.018 5.770e-5 1.011e-1 1.101e-2

α4 36

12 8 -54.423 3.608e-4 -54.631 2.655e-4 2.113e-1 3.317e-2

15 7 -62.453 1.158e-4 -64.365 8.504e-5 7.312e-2 1.147e-2

18 6 -71.255 2.661e-5 -73.122 1.942e-5 2.182e-2 3.217e-3

21 3 -79.979 7.880e-6 -83.184 5.360e-6 8.086e-3 1.170e-3

24 2 -83.278 6.257e-6 -85.250 4.068e-6 8.721e-3 1.314e-3

27 1 -81.501 5.606e-6 -82.356 4.315e-6 6.449e-3 9.108e-4

30 4 -76.734 8.225e-6 -82.492 5.195e-6 1.332e-2 1.626e-3

33 5 -68.507 2.048e-5 -73.101 1.519e-5 2.204e-2 3.328e-3 Table 7 Performances of gradient-based design (43) of VdIIR VFD filters versus mean group delay (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; R: Rank; FGD: Fractional group delay)

band designs at α = 0.9625 obtained by the VdIIR and FdIIR VFD filters shown in Table 9 are

plotted in Figs 9-12

α OD A/F Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

α1

56,

56

A(KJ) -40.677 3.246e-4 N.A N.A 1.980 1.717e-1 A(LCR) -24.604 9.309e-3 N.A N.A 5.920e-1 1.374e-1

55,

28

F(KJ) 2.798 8.242e-1 -24.807 3.048e-3 2.117 1.761 F(LD) -31.994 3.573e-3 -31.997 2.933e-3 1.548 3.248e-1

α2

53,

53 A(LCR) -55.710 2.258e-4 A(KJ) -61.643 5.626e-5 N.A N.A N.A N.A 4.437e-1 3.779e-2 8.224e-2 2.181e-2

52,

26

F(KJ) -32.726 1.493e-3 -32.770 1.216e-3 8.027e-1 1.633e-1 F(LD) -38.421 1.552e-3 -38.432 1.229e-3 6.470e-1 1.459e-1

α3

48,

48 A(LCR) -73.920 1.265e-5 A(KJ) -70.691 1.264e-5 N.A N.A N.A N.A 2.011e-2 1.745e-3 2.991e-3 9.069e-4

47,

24

F(KJ) 2.474 7.957e-1 -42.609 3.731e-4 7.122e-1 1.732 F(LD) -50.268 3.654e-4 -50.411 2.917e-4 1.802e-1 3.536e-2

α4

43,

43

A(KJ) -80.513 4.987e-6 N.A N.A 5.892e-3 5.193e-4 A(LCR) -84.237 4.119e-6 N.A N.A 3.870e-4 1.044e-4

42,

21

F(KJ) -53.561 1.310e-4 -53.810 1.027e-4 7.986e-2 1.609e-2 F(LD) -59.247 1.354e-4 -59.572 1.015e-4 5.479e-2 1.223e-2

Table 6 Performances of allpass and FIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925,

4= 0.9; OD: Filter order and mean group delay (MAP, DAP) or (LFIR, DFIR); A: Allpass design,

F: FIR design; (KJ): (Kwan & Jiang, 2009a); (LCR): (Lee et al., 2008); (LD): (Lu & Deng, 1999);

FGD: Fractional group delay)

The relationship between numerator and denominator orders, and optimal mean group

delay of a VdIIR or FdIIR VFD filter is a subject of interest Table 10 summarizes such

relationships among those VdIIR and FdIIR VDF filters listed in Table 9 It can be observed

from Table 10 that as  changes from 0.9    0.9625, the ratio D/(N+M) changes from 0.64

to 0.67 for VdIIR VFD filters, and changes from 0.57 to 0.55 for FdIIR VFD filters Also, as

seen from Figs 1-8, for the higher wideband side with  = 0.9625 and 0.95, there is a mean

group delay value that yields a minimum erms value; but for the lower wideband side with 

= 0.925 and 0.9, each of the mean group delay curves shows that erms becomes lower much

earlier at smaller D before reaching its minimum erms value In other words, the mean group

delay requirement is lower for lower wideband cutoff frequencies From Table 10, in

general, the VdIIR VFD filters require slightly higher optimal mean group delay values D

than those of the corresponding FdIIR VFD filters

α N D R Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

α1 49

25 6 -46.317 4.790e-4 -46.373 3.607e-4 5.621e-1 7.708e-2

28 7 -45.817 4.981e-4 -48.255 3.327e-4 6.545e-1 9.443e-2

31 8 -45.492 5.203e-4 -46.819 3.439e-4 6.152e-1 1.034e-1

34 3 -55.689 1.709e-4 -56.650 1.203e-4 3.135e-1 4.301e-2

37 1 -56.746 1.157e-4 -56.792 8.227e-5 2.371e-1 3.090e-2

40 2 -54.753 1.333e-4 -55.272 8.621e-5 2.725e-1 3.913e-2

43 4 -52.061 1.811e-4 -54.511 1.181e-4 3.634e-1 5.468e-2

46 5 -48.664 2.877e-4 -48.979 2.016e-4 3.676e-1 6.420e-2

α2 46

23 7 -55.398 2.194e-4 -56.439 1.629e-4 2.370e-1 3.347e-2

26 6 -59.500 1.442e-4 -59.567 1.025e-4 1.855e-1 2.446e-2

29 5 -59.982 1.310e-4 -60.924 9.276e-5 1.434e-1 2.400e-2

32 2 -63.424 6.157e-5 -66.513 4.168e-5 1.025e-1 1.451e-2

35 1 -64.515 5.514e-5 -67.411 3.558e-5 1.019e-1 1.364e-2

38 3 -62.722 6.798e-5 -63.918 4.290e-5 1.184e-1 1.767e-2

41 4 -57.588 9.448e-5 -57.757 7.247e-5 1.200e-1 1.731e-2

44 8 -48.195 2.999e-4 -52.186 2.194e-4 5.620e-1 5.862e-2

α3 41

18 8 -49.959 3.716e-4 -50.563 2.537e-4 2.966e-1 4.916e-2

21 6 -64.763 6.303e-5 -67.058 4.233e-5 7.008e-2 1.016e-2

24 5 -69.381 3.348e-5 -70.084 2.327e-5 4.344e-2 6.336e-3

27 2 -75.807 1.269e-5 -78.312 8.311e-6 2.229e-2 2.984e-3

30 1 -75.789 1.082e-5 -80.087 6.474e-6 2.048e-2 3.090e-3

33 3 -71.425 1.823e-5 -71.675 1.433e-5 2.420e-2 3.420e-3

36 4 -67.853 2.618e-5 -69.170 1.809e-5 3.759e-2 5.315e-3

39 7 -59.463 7.159e-5 -61.018 5.770e-5 1.011e-1 1.101e-2

α4 36

12 8 -54.423 3.608e-4 -54.631 2.655e-4 2.113e-1 3.317e-2

15 7 -62.453 1.158e-4 -64.365 8.504e-5 7.312e-2 1.147e-2

18 6 -71.255 2.661e-5 -73.122 1.942e-5 2.182e-2 3.217e-3

21 3 -79.979 7.880e-6 -83.184 5.360e-6 8.086e-3 1.170e-3

24 2 -83.278 6.257e-6 -85.250 4.068e-6 8.721e-3 1.314e-3

27 1 -81.501 5.606e-6 -82.356 4.315e-6 6.449e-3 9.108e-4

30 4 -76.734 8.225e-6 -82.492 5.195e-6 1.332e-2 1.626e-3

33 5 -68.507 2.048e-5 -73.101 1.519e-5 2.204e-2 3.328e-3 Table 7 Performances of gradient-based design (43) of VdIIR VFD filters versus mean group delay (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; R: Rank; FGD: Fractional group delay)

α N D R Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

α1 54

24 9 -47.551 4.030e-4 -48.815 3.254e-4 3.946e-1 6.066e-2

27 8 -49.821 2.791e-4 -49.826 2.345e-4 2.523e-1 4.390e-2

30 7 -49.940 2.663e-4 -51.336 1.906e-4 3.675e-1 5.526e-2

33 1 -58.117 1.360e-4 -59.459 1.055e-4 1.553e-1 2.391e-2

36 2 -54.776 1.581e-4 -56.752 1.100e-4 2.200e-1 3.225e-2

39 3 -53.351 1.695e-4 -58.289 1.097e-4 3.108e-1 4.832e-2

42 4 -52.767 1.852e-4 -57.168 1.246e-4 3.521e-1 5.312e-2

45 5 -51.723 2.027e-4 -54.003 1.500e-4 3.394e-1 4.971e-2

48 6 -50.532 2.165e-4 -53.051 1.745e-4 3.007e-1 4.414e-2

α2 51

23 7 -57.352 1.585e-4 -57.948 1.258e-4 1.085e-1 1.823e-2

26 4 -60.282 1.172e-4 -62.605 9.084e-5 8.234e-2 1.344e-2

29 2 -60.166 1.051e-4 -64.946 7.397e-5 8.715e-2 1.359e-2

32 1 -58.723 1.018e-4 -65.813 7.060e-5 1.013e-1 1.683e-2

35 3 -56.737 1.073e-4 -63.980 7.180e-5 1.307e-1 1.956e-2

38 5 -56.078 1.210e-4 -60.347 8.811e-5 1.470e-1 2.142e-2

41 6 -57.176 1.354e-4 -58.376 1.015e-4 1.199e-1 1.825e-2

44 8 -54.520 1.590e-4 -57.346 1.155e-4 1.488e-1 2.299e-2

47 9 -51.036 2.173e-4 -58.471 1.441e-4 3.066e-1 5.044e-2

α3 46

17 9 -54.883 1.565e-4 -56.964 1.190e-4 1.131e-1 1.781e-2

20 8 -60.232 7.723e-5 -65.677 5.865e-5 3.142e-2 5.028e-3

23 5 -61.491 7.567e-5 -66.350 5.607e-5 3.709e-2 5.591e-3

26 2 -61.693 7.237e-5 -68.770 5.183e-5 3.782e-2 5.498e-3

29 1 -60.702 7.065e-5 -69.047 5.209e-5 3.796e-2 5.501e-3

32 3 -62.120 7.440e-5 -66.268 5.689e-5 2.962e-2 4.939e-3

35 4 -60.883 7.454e-5 -66.131 5.552e-5 4.267e-2 6.465e-3

38 7 -59.235 7.703e-5 -67.887 5.477e-5 6.825e-2 1.023e-2

41 6 -58.976 7.603e-5 -66.870 5.497e-5 6.936e-2 1.007e-2

α4 41

12 9 -55.792 1.883e-4 -58.359 1.342e-4 1.093e-1 1.991e-2

15 8 -62.408 7.731e-5 -65.923 5.838e-5 3.030e-2 5.618e-3

18 2 -63.307 5.875e-5 -71.407 4.177e-5 1.061e-2 1.921e-3

21 5 -64.151 6.078e-5 -71.767 4.448e-5 1.876e-2 2.673e-3

24 4 -63.959 6.049e-5 -69.984 4.491e-5 1.615e-2 2.565e-3

27 1 -63.586 5.820e-5 -70.713 4.244e-5 9.738e-3 1.712e-3

30 3 -61.756 5.975e-5 -70.908 4.170e-5 2.336e-2 3.916e-3

33 6 -62.236 6.151e-5 -70.075 4.376e-5 3.241e-2 4.699e-3

36 7 -61.444 6.189e-5 -68.939 4.454e-5 2.113e-2 3.729e-3

Table 8 Performances of gradient-based design (43) of FdIIR VFD filters versus mean group

delay (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; R: Rank; FGD: Fractional group delay)

(43) (43) (KJ) (LCR) (KJ) (LD)

erms 1.157e-4 1.360e-4 3.246e-4 9.309e-3 8.242e-1 3.573e-3

erms 5.514e-5 1.018e-4 5.626e-5 2.258e-4 1.493e-3 1.552e-3

erms 1.082e-5 7.065e-5 1.264e-5 1.265e-5 7.957e-1 3.654e-4

erms 5.606e-6 5.820e-5 4.987e-6 4.119e-6 1.310e-4 1.354e-4

Table 9 Performances (erms) of VFD filters selected from Tables 6-8 (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; (KJ): (Kwan & Jiang, 2009a); (LCR): (Lee et al., 2008); (LD): (Lu & Deng, 1999))

 D N M N+M D/(N+M)

VdIIR

1 37 49 6 55 0.6727

2 35 46 6 52 0.6731

3 30 41 6 47 0.6383

4 27 36 6 42 0.6429 FdIIR

1 33 54 6 60 0.5500

2 32 51 6 57 0.5614

3 29 46 6 52 0.5577

4 27 41 6 47 0.5745 Table 10 D/(N+M) for IIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9)

Fig 1 erms versus mean group delay D (VdIIR VFD filter, α = 0.9625, N = 49, M = 6)

α N D R Freq Responses Mag Responses FGD Responses

e max (dB) e rms e max,1(dB) e rms,1 e max,2 e rms,2

α1 54

24 9 -47.551 4.030e-4 -48.815 3.254e-4 3.946e-1 6.066e-2

27 8 -49.821 2.791e-4 -49.826 2.345e-4 2.523e-1 4.390e-2

30 7 -49.940 2.663e-4 -51.336 1.906e-4 3.675e-1 5.526e-2

33 1 -58.117 1.360e-4 -59.459 1.055e-4 1.553e-1 2.391e-2

36 2 -54.776 1.581e-4 -56.752 1.100e-4 2.200e-1 3.225e-2

39 3 -53.351 1.695e-4 -58.289 1.097e-4 3.108e-1 4.832e-2

42 4 -52.767 1.852e-4 -57.168 1.246e-4 3.521e-1 5.312e-2

45 5 -51.723 2.027e-4 -54.003 1.500e-4 3.394e-1 4.971e-2

48 6 -50.532 2.165e-4 -53.051 1.745e-4 3.007e-1 4.414e-2

α2 51

23 7 -57.352 1.585e-4 -57.948 1.258e-4 1.085e-1 1.823e-2

26 4 -60.282 1.172e-4 -62.605 9.084e-5 8.234e-2 1.344e-2

29 2 -60.166 1.051e-4 -64.946 7.397e-5 8.715e-2 1.359e-2

32 1 -58.723 1.018e-4 -65.813 7.060e-5 1.013e-1 1.683e-2

35 3 -56.737 1.073e-4 -63.980 7.180e-5 1.307e-1 1.956e-2

38 5 -56.078 1.210e-4 -60.347 8.811e-5 1.470e-1 2.142e-2

41 6 -57.176 1.354e-4 -58.376 1.015e-4 1.199e-1 1.825e-2

44 8 -54.520 1.590e-4 -57.346 1.155e-4 1.488e-1 2.299e-2

47 9 -51.036 2.173e-4 -58.471 1.441e-4 3.066e-1 5.044e-2

α3 46

17 9 -54.883 1.565e-4 -56.964 1.190e-4 1.131e-1 1.781e-2

20 8 -60.232 7.723e-5 -65.677 5.865e-5 3.142e-2 5.028e-3

23 5 -61.491 7.567e-5 -66.350 5.607e-5 3.709e-2 5.591e-3

26 2 -61.693 7.237e-5 -68.770 5.183e-5 3.782e-2 5.498e-3

29 1 -60.702 7.065e-5 -69.047 5.209e-5 3.796e-2 5.501e-3

32 3 -62.120 7.440e-5 -66.268 5.689e-5 2.962e-2 4.939e-3

35 4 -60.883 7.454e-5 -66.131 5.552e-5 4.267e-2 6.465e-3

38 7 -59.235 7.703e-5 -67.887 5.477e-5 6.825e-2 1.023e-2

41 6 -58.976 7.603e-5 -66.870 5.497e-5 6.936e-2 1.007e-2

α4 41

12 9 -55.792 1.883e-4 -58.359 1.342e-4 1.093e-1 1.991e-2

15 8 -62.408 7.731e-5 -65.923 5.838e-5 3.030e-2 5.618e-3

18 2 -63.307 5.875e-5 -71.407 4.177e-5 1.061e-2 1.921e-3

21 5 -64.151 6.078e-5 -71.767 4.448e-5 1.876e-2 2.673e-3

24 4 -63.959 6.049e-5 -69.984 4.491e-5 1.615e-2 2.565e-3

27 1 -63.586 5.820e-5 -70.713 4.244e-5 9.738e-3 1.712e-3

30 3 -61.756 5.975e-5 -70.908 4.170e-5 2.336e-2 3.916e-3

33 6 -62.236 6.151e-5 -70.075 4.376e-5 3.241e-2 4.699e-3

36 7 -61.444 6.189e-5 -68.939 4.454e-5 2.113e-2 3.729e-3

Table 8 Performances of gradient-based design (43) of FdIIR VFD filters versus mean group

delay (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; R: Rank; FGD: Fractional group delay)

(43) (43) (KJ) (LCR) (KJ) (LD)

erms 1.157e-4 1.360e-4 3.246e-4 9.309e-3 8.242e-1 3.573e-3

erms 5.514e-5 1.018e-4 5.626e-5 2.258e-4 1.493e-3 1.552e-3

erms 1.082e-5 7.065e-5 1.264e-5 1.265e-5 7.957e-1 3.654e-4

erms 5.606e-6 5.820e-5 4.987e-6 4.119e-6 1.310e-4 1.354e-4

Table 9 Performances (erms) of VFD filters selected from Tables 6-8 (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9; (KJ): (Kwan & Jiang, 2009a); (LCR): (Lee et al., 2008); (LD): (Lu & Deng, 1999))

 D N M N+M D/(N+M)

VdIIR

1 37 49 6 55 0.6727

2 35 46 6 52 0.6731

3 30 41 6 47 0.6383

4 27 36 6 42 0.6429 FdIIR

1 33 54 6 60 0.5500

2 32 51 6 57 0.5614

3 29 46 6 52 0.5577

4 27 41 6 47 0.5745 Table 10 D/(N+M) for IIR VFD filters (Keys: 1= 0.9625, 2= 0.95, 3= 0.925, 4= 0.9)

Fig 1 erms versus mean group delay D (VdIIR VFD filter, α = 0.9625, N = 49, M = 6)

Fig 2 erms versus mean group delay D (VdIIR VFD filter, α = 0.95, N = 46, M = 6)

Fig 3 erms versus mean group delay D (VdIIR VFD filter, α = 0.925, N = 41, M = 6)

Fig 4 erms versus mean group delay D (VdIIR VFD filter, α = 0.90, N = 36, M = 6)

Fig 5 erms versus mean group delay D (FdIIR VFD filter, α = 0.9625, N = 54, M = 6)

Fig 6 erms versus mean group delay D (FdIIR VFD filter, α = 0.95, N = 51, M = 6)

Fig 7 erms versus mean group delay D (FdIIR VFD filter, α = 0.925, N = 46, M = 6)