02 Analog Filters

١١ Chebyshev Filters Spring 2008 © Ammar Abu-Hudrouss -Islamic University Gaza Slide ٢ Digital Signal Processing C N (x) is N th order Chebyshev polynomial defined as Chebychev I filter ( ) O + = O 2 2 2 1 1 ) ( N C H  It has ripples in the passband and no ripple in the stop band It has the following transfer function ( ) ¦ ¹ ¦ ´ ¦ > s = ÷ ÷ 1 ) cosh cosh( 1 ) cos cos( 1 1 x x N x x N x C N The Chebyshev polynomial can be generated by recursive technique ( ) ( ) ( ) O ÷ O O = O ÷ + 1 1 2 k k k C C C ( ) ( ) O = O = O 1 0 and 1 C C ٢ Slide ٣ Digital Signal Processing Chebyshev polynomial characteristics Chebyshev polynomial ( ) 1 all for 1 s O s O N C ( ) N C N all for 1 1 = ( ) 1 1 interval the in occurs of roots the All s s ÷ O x C N ( ) ¦ ¹ ¦ ´ ¦ + = even 1 1 odd 1 0 2 2 N N H  ( ) 2 2 1 1 1  + = H The filter parameter c is related to the ripple as shown in the figures Slide ٤ Digital Signal Processing Filter roots The poles of LP prototype filter H (p ) are given by ( ) ( ) ( ) ( ) N k j p k k k , , 3 , 2 , 1 cos cosh sin sinh 2 2  = + ÷ =     | . | \ | = ÷   1 sinh 1 1 2 N ( ) N k k 2 1 2   ÷ = And the order of the filter is given by ( ) ( ) s A A p s N O ÷ ÷ > ÷ ÷ 1 10 / 10 / 1 cosh 1 10 / 1 10 cosh ٣ Slide ٥ Digital Signal Processing  p = 2500 rad/s ,  s = 12.5 Krad/s, For the LP prototype filter A s = 30 dB, A p = 0.5 dB O p =  p /  p = 1 rad/s O s =  s /  p = 5 rad/s Chebyshev filter type I Example: Design lowpass Chebyshev filter with maximum gain of 5 dB. A p = 0.5-dB ripple in the passband, a bandwidth of 2500 rad/sec. A stopband frequency of 12.5 Krad/sec, and A s = 30 dB or more for O > O s ( ) ( ) ( 3 2676 . 2 ) 5 ( cosh 1 10 / 1 10 cosh 1 05 . 0 3 1 = = ÷ ÷ = ÷ ÷ N To determine the order of the filter 34931 . 0 5 . 0 ) 1 log( 10 2 = = +   To calculate the ripple factor Slide ٦ Digital Signal Processing Chebyshev filter type I The transfer function 626456 . 0 and 02192 . 1 313228 . 0 ÷ ± ÷ = j p k To determine H 0 we know that H (p = 0) = 1 (N is odd) 591378 . 0 34931 . 0 1 sinh 2 1 1 2 = | . | \ | = ÷  ( ) ( )( ) 62645 . 0 142447 . 1 626456 . 0 2 0 + + + = p p p H p H ( )( ) 62645 . 0 142447 . 1 1 0 H = 715693 . 0 0 = H And to have 5 dB gain we multiply H 0 by 1.7783 ( ) ( )( ) ( )( ) 62645 . 0 142447 . 1 626456 . 0 7783 . 1 715693 . 0 2 + + + = p p p p H ٤ Slide ٧ Digital Signal Processing Chebyshev filter type I When we substitute by p =s/2500 ( ) ( )( ) 1566 10 714 1566 10 886 . 19 6 2 12 + × + + × = s s s s H Slide ٨ Digital Signal Processing Chebyshev filter type II This class of filters is called inverse Chebyshev filters. Its transfer function is derived by applying the following 1) Frequency transformation of O’ = 1/O in H (j O) of lowpass normalized prototype filter to achieve the following highpass filter 2) When it subtracted from one we get transfer function of Chebyshev II filter ( ) ' / 1 1 1 ' 1 2 2 2 O + = | | . | \ | O n C j H  ( ) ( ) ' / 1 1 1 1 ' / 1 1 1 1 2 2 2 2 O + = O + ÷ n n C C   ٥ Slide ٩ Digital Signal Processing Chebyshev filter type II It has ripples in the passband and ripple in the stop band It has the following transfer function Slide ١٠ Digital Signal Processing 1. The frequencies are normalised by  s instead of  p . 2. The ripple factor  i is calculated by 2 . The order of the filter is given by 3. The poles of LP prototype filter H (p) are given by Chebyshev filter type II Design ( ) ( ) ( ) ( ) | | N k j p k k k , , 3 , 2 , 1 cos cosh sin sinh / 1 2 2  = + ÷ =     | . | \ | = ÷   1 sinh 1 1 2 N ( ) N k k 2 1 2   ÷ = ( ) ( ) ( ) p A A p s N O ÷ ÷ > ÷ ÷ / 1 cosh 1 10 / 1 10 cosh 1 10 / 10 / 1 ( ) 1 10 / 1 1 . 0 ÷ = s A i  ٦ Slide ١١ Digital Signal Processing 5. The zeros of H ( p ) is given by The filter transfer function is calculated by 6. Restore the magnitude scale by calculating H 0 . 7. Restore the frequency scale to  by frequency transformation Chebyshev filter type II Design ( ) ( ) ( ) [ [ = = ÷ O + = n k k m k k p p p H p H 1 1 2 0 2 0 ¸ ¸ 2 / , , 3 , 2 , 1 sec 0 N m k j z k k k = = ± = O ± =   Slide ١٢ Digital Signal Processing  p = 1000 rad/s ,  s = 2000 krad/s, For the LP prototype filter O p =  p /  s = 0.5 rad/s O s =  s /  s = 1 rad/s Chebyshev filter type II Example: Design lowpass Chebyshev II filter that has 0-dB ripple in the passband,  p = 1000 rad/sec, A p = 0.5dB, A stopband frequency of 2000 rad/sec and A s = 40 dB. ( ) ( ) ( 5 82 . 4 ) 2 ( cosh 1 10 / 1 10 cosh 1 05 . 0 4 1 = = ÷ ÷ = ÷ ÷ N To determine the order of the filter 995 . 99 / 1 40 ) 1 log( 10 2 = = +   ٧ Slide ١٣ Digital Signal Processing Chebyshev filter type II The transfer function ( ) ( ) ( ) 66 0.78777026 - and , 11 j0.4853890 5 0.52479948 - , 75 0.61087031 6 0.15595592 ± ± ÷ = j p k ( ) . 1.05965847 995 . 99 sinh 5 1 1 2 = = ÷  ( ) ( )( ) ( ) ( )( ) 6 0.78770266 47 0.51101698 1.04959897 76 0.39747221 52 0.31183118 1.7013 1.0515 2 2 2 2 2 2 0 + + + + + + + = p p p p p p p H p H The zeros is calculated by ¸ ¸ 2 / , , 2 , 1 sec 0 n m k j j z k k k = = = O ± =   1.7013 1.0515 2 1 j z j z ± = ± = Slide ١٤ Digital Signal Processing Chebyshev filter type II H (s) can be achieved by substituting by p =s/2000 To determine H 0 we know that H (p = 0) = 1 (N is odd) 15999426 . 0 2002 . 3 1 0 H = 0.049995 0 = H ( ) ( ) 0.15999426 65725515 0 54997 1 30818905 2 1491328 2 2002 . 3 04 . 4 2 3 4 5 2 4 0 + + + + + + + = p . p . p . p . p p p H p H ٨ Slide ١٥ Digital Signal Processing Elliptic filter is more efficient than Butterworth and Chebychev filters as it has the smallest order for a given set of specifications The drawback it suffers from high nonlinearity in the phase, especially near the band edges. Elliptic filter The filter order requires to achieve a given set of specifications in passband ripple  1 and  2 Chebyshev polynomial Chebyshev polynomial characteristics C N    1 for all   1 CN 1  1 for all N All the roots of C N   occurs in the interval  1  x  1 The filter parameter  is related to the ripple as shown in the figures 1 2  H 0    1 1   2  H 1  2 N odd N even 1 1  2 Digital Signal Processing Slide ٣ Filter roots The poles of LP prototype filter H (p ) are given by pk   sinh 2 sin  k   j cosh 2  cos k  2  k  1 1 sinh 1  N   k  1. N 2k  1 2N And the order of the filter is given by N cosh 1 10 As / 10  1 / 10 cosh 1  s   Ap / 10 1  Digital Signal Processing Slide ٤ ٢ .2.3. . 7783 H  p  p 2 0.62645 Slide ٦ Digital Signal Processing ٣ .5 Krad/s.7156931. Ap = 0. Ap = 0.5-dB ripple in the passband.626456 The transfer function H  p  p 2 H0  0.715693 And to have 5 dB gain we multiply H0 by 1.142447  p  0. a bandwidth of 2500 rad/sec. and As = 30 dB or more for  > s p = 2500 rad/s .7783  0.02192 and  0.591378 2  0.626456 p  1.5 Digital Signal Processing Slide ٥   0.5 Krad/sec. For the LP prototype filter As = 30 dB.1424470.05  1  2.Chebyshev filter type I Example: Design lowpass Chebyshev filter with maximum gain of 5 dB.62645 H 0  0.313228  j1.2676  3 cosh 1 (5) 3   To calculate the ripple factor 10 log(1   2 )  0. A stopband frequency of 12.62645  To determine H0 we know that H (p = 0) = 1 (N is odd) 1 H0 1.142447 p  0.5 dB p = p / p = 1 rad/s s = s / p = 5 rad/s To determine the order of the filter N cosh 1 10  1 / 10 0.34931 Chebyshev filter type I 1 1    2  sinh 1    0.626456 p  1.34931  p k  0. s = 12. Chebyshev filter type I When we substitute by p =s/2500 H s    19.886 1012 s 2  1566s  714 10 6 s  1566  Digital Signal Processing Slide ٧ Chebyshev filter type II This class of filters is called inverse Chebyshev filters. Its transfer function is derived by applying the following 1) Frequency transformation of ’ = 1/ in H (j ) of lowpass normalized prototype filter to achieve the following highpass filter  1  1   H   1   2C 2 1 / '  j '  n 2 2) When it subtracted from one we get transfer function of Chebyshev II filter 1 1  1   Cn2 1 / ' 1  2 1 1  C 1 / ' 2 2 n Digital Signal Processing Slide ٨ ٤ .  . The order of the filter is given by N cosh 1 10 As / 10  1 / 10 cosh 1 1 /  p    Ap / 10 1  k  1. The frequencies are normalised by s instead of p. N 3.Chebyshev filter type II It has ripples in the passband and ripple in the stop band It has the following transfer function Digital Signal Processing Slide ٩ Chebyshev filter type II Design 1. The ripple factor i is calculated by  i  1 / 10 0.2.1As  1 2 . 2. The poles of LP prototype filter H (p) are given by pk  1 / sinh  2 sin  k   j cosh  2  cos  k  2  1 1 sinh 1   N   k  2k  1 2N Digital Signal Processing Slide ١٠ ٥ .3. s = 2000 krad/s. Restore the magnitude scale by calculating H0. Ap = 0. The zeros of H ( p ) is given by zk   0k   j sec k k  1.5dB.2. Restore the frequency scale to  by frequency transformation Digital Signal Processing Slide ١١ Chebyshev filter type II Example: Design lowpass Chebyshev II filter that has 0-dB ripple in the passband. A stopband frequency of 2000 rad/sec and As = 40 dB.Chebyshev filter type II Design 5.995 To determine the order of the filter N cosh 1 10  1 / 100. 7.05  1  4. For the LP prototype filter p = p / s = 0.5 rad/s s = s / s = 1 rad/s 10 log(1   2 )  40   1 / 99.3. p = 1000 rad/sec.82  5 cosh 1 (2) 4   Digital Signal Processing Slide ١٢ ٦ . . m  N / 2 The filter transfer function is calculated by H  p  2 H 0 k 1 p 2   0 k m    p  p  k 1 k n 6. p = 1000 rad/s . 7877702666  The zeros is calculated by z k   j 0k  j sec  k k  1.1491328 p 4  2.04959897 p  0.3974722176 2 2      1.0515 z2   j1. .05965847.6108703175.04 p 2  3.3118311852 p  0.995  1.70132 p  0.0.30818905 p 3  1. m  n / 2  z1   j1.5110169847  p  0. and .485389011 . 5 p k   0.54997 p 2  0.Chebyshev filter type II 1  2  sinh 1 99. .15999426 To determine H0 we know that H (p = 0) = 1 (N is odd)   1 3.65725515 p  0.2002 p 5  2.15999426 H 0  0.2002H 0 0.787702666   Digital Signal Processing Slide ١٣ Chebyshev filter type II H  p  H 0 p 4  4.049995 H (s) can be achieved by substituting by p =s/2000 Digital Signal Processing Slide ١٤ ٧ .7013 The transfer function H  p   p H 0 p 2  1.155955926  j 0.524799485  j0.0.05152 p 2  1.2. Elliptic filter Elliptic filter is more efficient than Butterworth and Chebychev filters as it has the smallest order for a given set of specifications The drawback it suffers from high nonlinearity in the phase. The filter order requires to achieve a given set of specifications in passband ripple 1 and 2 Digital Signal Processing Slide ١٥ ٨ . especially near the band edges.