Digital Filters Part 11 pdf

Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 191 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   max ( , ) , [0, ], [ 0.5,0.5] max e e t t        (46) 1/2 0.5 2 0 0.5 0.5 2 0 0.5 ( , ) ( , ) rms d e t dtd e H t dtd                        (47)   ,1 max ( , ) , [0, ], [ 0.5,0.5] max MAG e e t t        (48) 1/2 0.5 2 0 0.5 ,1 0.5 2 0 0.5 ( , ) ( , ) MAG rms d e t dtd e H t dtd                        (49)   ,2 max ( , ) , [0, ], [ 0.5,0.5] max FGD e e t t        (50) 1/2 0.5 2 0 0.5 ,2 0.5 2 0 0.5 ( , ) FGD rms e t dtd e t dtd                       (51) where ( , ) ( , ) ( , ) j MAG d e t H e t H t      (52) ( , ) ( , ) FGD e 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 e rms (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 e rms 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 e rms 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. Digital Filters192 α N D A R Fre q . Res p onses Ma g . Res p onses FGD Res p onses 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 Fre q . Res p onses Ma g . Res p onses FGD Res p onses 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 Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 193 α N D A R Fre q . Res p onses Ma g . Res p onses FGD Res p onses 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 Fre q . Res p onses Ma g . Res p onses FGD Res p onses 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 Digital Filters194 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)  VdII R FdII R ( 29 ) ( 35 ) ( 43 ) ( ZK ) ( 29 ) ( 34 ) ( 43 ) ( KJ ) ( TCK )  1 e rms 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 R 3 1 2 4 4 2 1 3 5  2 e rms 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 R 3 1 2 4 4 3 1 2 5  3 e rms 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 R 3 2 1 4 3 2 1 4 5  4 e rms 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 R 3 2 1 4 4 1 3 5 2 Table 5. Top-performed (e rms ) 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 e rms values versus D are plotted in Figs. 1-8. From Tables 7-8, their mean group delay values D that yield minimum e rms values are summarized in Table 9. For comparisons, the e rms 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 Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 195 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)  VdII R FdII R ( 29 ) ( 35 ) ( 43 ) ( ZK ) ( 29 ) ( 34 ) ( 43 ) ( KJ ) ( TCK )  1 e rms 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 R 3 1 2 4 4 2 1 3 5  2 e rms 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 R 3 1 2 4 4 3 1 2 5  3 e rms 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 R 3 2 1 4 3 2 1 4 5  4 e rms 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 R 3 2 1 4 4 1 3 5 2 Table 5. Top-performed (e rms ) 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 e rms values versus D are plotted in Figs. 1-8. From Tables 7-8, their mean group delay values D that yield minimum e rms values are summarized in Table 9. For comparisons, the e rms 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 Digital Filters196 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 Fre q . Res p onses Ma g . Res p onses FGD Res p onses e max ( dB ) e rms e max,1 ( dB ) e rms,1 e max,2 e rms,2 α 1 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 α 2 53, 53 A ( KJ ) -61.643 5.626e-5 N.A. N.A. 4.437e-1 3.779e-2 A ( LCR ) -55.710 2.258e-4 N.A. N.A. 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 α 3 48, 48 A ( KJ ) -70.691 1.264e-5 N.A. N.A. 2.011e-2 1.745e-3 A ( LCR ) -73.920 1.265e-5 N.A. N.A. 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 α 4 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 (M AP , D AP ) or (L FIR , D FIR ); 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 e rms value; but for the lower wideband side with  = 0.925 and 0.9, each of the mean group delay curves shows that e rms becomes lower much earlier at smaller D before reaching its minimum e rms 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 Fre q . Res p onses Ma g . Res p onses FGD Res p onses e max ( dB ) e rms e max,1 ( dB ) e rms,1 e max,2 e rms,2 α 1 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 α 2 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 α 3 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 α 4 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) Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 197 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 Fre q . Res p onses Ma g . Res p onses FGD Res p onses e max ( dB ) e rms e max,1 ( dB ) e rms,1 e max,2 e rms,2 α 1 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 α 2 53, 53 A ( KJ ) -61.643 5.626e-5 N.A. N.A. 4.437e-1 3.779e-2 A ( LCR ) -55.710 2.258e-4 N.A. N.A. 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 α 3 48, 48 A ( KJ ) -70.691 1.264e-5 N.A. N.A. 2.011e-2 1.745e-3 A ( LCR ) -73.920 1.265e-5 N.A. N.A. 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 α 4 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 (M AP , D AP ) or (L FIR , D FIR ); 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 e rms value; but for the lower wideband side with  = 0.925 and 0.9, each of the mean group delay curves shows that e rms becomes lower much earlier at smaller D before reaching its minimum e rms 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 Fre q . Res p onses Ma g . Res p onses FGD Res p onses e max ( dB ) e rms e max,1 ( dB ) e rms,1 e max,2 e rms,2 α 1 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 α 2 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 α 3 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 α 4 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) Digital Filters198 α N D R Fre q . Res p onses Ma g . Res p onses FGD Res p onses e max ( dB ) e rms e max ,1 ( dB ) e rms ,1 e max ,2 e rms ,2 α 1 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 α 2 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 α 3 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 α 4 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)  VdII R FdII R AP FIR ( 43 ) ( 43 ) ( K J) ( LCR ) ( K J) ( LD )  1 D 37 33 56 56 28 28 e rms 1.157e-4 1.360e-4 3.246e-4 9.309e-3 8.242e-1 3.573e-3  2 D 35 32 53 53 26 26 e rms 5.514e-5 1.018e-4 5.626e-5 2.258e-4 1.493e-3 1.552e-3  3 D 30 29 48 48 24 24 e rms 1.082e-5 7.065e-5 1.264e-5 1.265e-5 7.957e-1 3.654e-4  4 D 27 27 43 43 21 21 e rms 5.606e-6 5.820e-5 4.987e-6 4.119e-6 1.310e-4 1.354e-4 Table 9. Performances (e rms ) 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. e rms versus mean group delay D (VdIIR VFD filter, α = 0.9625, N = 49, M = 6) Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 199 α N D R Fre q . Res p onses Ma g . Res p onses FGD Res p onses e max ( dB ) e rms e max ,1 ( dB ) e rms ,1 e max ,2 e rms ,2 α 1 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 α 2 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 α 3 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 α 4 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)  VdII R FdII R AP FIR ( 43 ) ( 43 ) ( K J) ( LCR ) ( K J) ( LD )  1 D 37 33 56 56 28 28 e rms 1.157e-4 1.360e-4 3.246e-4 9.309e-3 8.242e-1 3.573e-3  2 D 35 32 53 53 26 26 e rms 5.514e-5 1.018e-4 5.626e-5 2.258e-4 1.493e-3 1.552e-3  3 D 30 29 48 48 24 24 e rms 1.082e-5 7.065e-5 1.264e-5 1.265e-5 7.957e-1 3.654e-4  4 D 27 27 43 43 21 21 e rms 5.606e-6 5.820e-5 4.987e-6 4.119e-6 1.310e-4 1.354e-4 Table 9. Performances (e rms ) 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. e rms versus mean group delay D (VdIIR VFD filter, α = 0.9625, N = 49, M = 6) Digital Filters200 Fig. 2. e rms versus mean group delay D (VdIIR VFD filter, α = 0.95, N = 46, M = 6) Fig. 3. e rms versus mean group delay D (VdIIR VFD filter, α = 0.925, N = 41, M = 6) Fig. 4. e rms versus mean group delay D (VdIIR VFD filter, α = 0.90, N = 36, M = 6) Fig. 5. e rms versus mean group delay D (FdIIR VFD filter, α = 0.9625, N = 54, M = 6) Fig. 6. e rms versus mean group delay D (FdIIR VFD filter, α = 0.95, N = 51, M = 6) Fig. 7. e rms versus mean group delay D (FdIIR VFD filter, α = 0.925, N = 46, M = 6) [...]... stable variable fractional delay IIR filters IEEE Transactions on Circuits and Systems II, Vol 54, No 1, (January 2007), pp 86–90, ISSN 1057-7130 Complex Coefficient IIR Digital Filters 209 9 X Complex Coefficient IIR Digital Filters Zlatka Nikolova, Georgi Stoyanov, Georgi Iliev and Vladimir Poulkov Technical University of Sofia Bulgaria 1 Complex Coefficient IIR Digital Filters – Basic Theory 1.1 Introduction... least-squares design for variable fractional delay FIR filters IEEE Transactions on Circuits and Systems II, Vol 46, No 8, (August 1999), pp 1035–1040, ISSN 1057-7130 208 Digital Filters Lu, W.-S (1999) Design of stable IIR digital filters with equiripple passbands and peakconstrained least-squares stopbands IEEE Transactions on Circuits and Systems II, Vol 46, No 11, (November 1999), pp 1421-1426, ISSN 1057-7130... reductions in mean group delay values of (a) VdIIR VFD filters versus AP VFD filters range approximately from 1.5 to 1.6 times; and (b) FdIIR VFD filters versus AP VFD filters are higher and range approximately from 1.7 to 1.6 times The maximum pole radius versus fractional delay t of the four VdIIR VFD filters as listed in Table 9 and the four AP VFD filters designed by (KJ) and (LCR) are plotted with... complex digital filtering have been successfully solved but scientific and technological advances challenge researchers with new problems or require new and better solutions to existing problems 210 Digital Filters In this chapter we examine IIR (Infinite Impulse Response) digital filters only They are more difficult to synthesize but are more efficient and selective than FIR (Finite Impulse Response) filters. .. delay (VFD) digital filters with variable and fixed denominators Both sequential and gradient-based design approaches in the weighted least-squares (WLS) sense are adopted The results obtained are compared to other design methods for IIR, allpass, and FIR VFD filters In the sequential design method, the Levy’s method is adopted along with an iterative reweighting technique 206 Digital Filters to transform... AP, and FIR) of VFD filters, but also depend on the effectiveness of its design method 9 References Brandenstein, H & Unbehauen, R (1998) Least-squares approximation of FIR by IIR digital filters IEEE Transactions on Signal Processing, Vol 46, No 1, (January 1998), pp 2130, ISSN 1053-587X Brandenstein, H & Unbehauen, R (2001) Weighted least-squares approximation of FIR by IIR digital filters IEEE Transactions... Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 207 Deng, T.-B (2001) Discretization-free design of variable fractional-delay FIR filters IEEE Transactions on Circuits and Systems II, Vol 48, No 6, (June 2001), pp 637–644, ISSN 1057-7130 Deng, T.-B (2006) Noniterative WLS design of allpass variable fractional-delay digital filters IEEE Transactions on Circuits... for 0.925    0.9625 (see Table 9) (c) When compared to the corresponding AP VFD filters (KJ; LCR) shown in Table 6, the following FdIIR VFD filters could achieve improved erms performances: (i) (29) over (LCR) for  = 0.9625 (see Table 5); (ii) (34) over (KJ; LCR) for  = 0.9625 and over (LCR) for  = 0.95 204 Digital Filters (see Table 5); (iii) (43) over (KJ; LCR) for  = 0.9625 and over (LCR) for...Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 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) 201 202 Digital Filters Fig 8 erms versus mean group delay... variable fractionaldelay digital filters IEEE Transactions on Circuits and Systems I, Vol 55, No 5, (June 2008), pp 1248–1256, ISSN 1549-8328 Levy, E C (1959) Complex curve fitting IRE Transactions on Automatic Control, Vol AC-4, (May 1959), pp 37-43, ISSN 0096-199X Lu, W.-S., Pei, S.-C., & Tseng, C.-C (1998) A weighted least-squares method for the design of stable 1-D and 2-D IIR digital filters IEEE Transactions . FIR filters. IEEE Transactions on Circuits and Systems II, Vol. 46, No. 8, (August 1999), pp. 1035–1040, ISSN 1057-7130. Digital Filters2 08 Lu, W S. (1999). Design of stable IIR digital filters. 1.942e-5 2.182e-2 3.217e-3 ( ZK ) 11 -20.667 1.381e-2 -20.070 1 .113 e-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 . mean group delay values of (a) VdIIR VFD filters versus AP VFD filters range approximately from 1.5 to 1.6 times; and (b) FdIIR VFD filters versus AP VFD filters are higher and range approximately