Robust Algorithms for Chebyshev Polynomials and Related - PowerPoint PPT Presentation

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Robust Algorithms for Chebyshev Polynomials and Related Approximations Miroslav Vlˇ cek vlcek@fd.cvut.cz Department of Applied Mathematics, Czech Technical University in Prague Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Contents 1 Introduction 2 Chebyshev polynomials 3 Symmetrical Zolotarev polynomials 4 Overview 5 Application in Filter Design 6 Conclusion Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Introduction Chebyshev Polynomials and their Relatives approximation view numerical view nonlinear differential eq. ⇒ linear differential eq. ⇓ ⇓ parametric solution recursive algorithms Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Chebyshev polynomials - approximation view � dy � 2 = n 2 ( 1 − y 2 ) ( 1 − x 2 ) dx 1.5 • • • • 1 0.5 T n ( x ) 0 −0.5 • • • −1 −1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x → Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Parametric solutions of differential equation � dy � 2 = n 2 ( 1 − y 2 ) ( 1 − x 2 ) dx ⇓ dy dx 1 − y 2 = n √ 1 − x 2 � x = cos Φ De Moivre’s formula y = cos n Φ y ( x ) = T n ( cos Φ) ⇒ Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion The second order differential equation � dy � 2 = n 2 ( 1 − y 2 ) ( 1 − x 2 ) dx ⇓ ( 1 − x 2 ) d 2 y dx 2 − x dy dx + n 2 y = 0 n t ( k ) x k y ( x ) ≡ T n ( x ) = � k = 0 .. is a polynomial of variable x Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Recursive evalution of the coefficients t ( k ) given n t ( n ) = 2 n − 1 initialisation t ( n − 1 ) = 0 recursive body ( for k = n − 2 to 0 t ( k ) = − ( k + 2 )( k + 1 ) t ( k + 2 ) n 2 − k 2 end ) Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion • Algorithm produces the coefficients t ( k ) for Chebyshev polynomials T n ( x ) as expected. x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 T 0 1 0 0 0 0 0 0 0 0 0 T 1 0 1 0 0 0 0 0 0 0 0 T 2 -1 0 2 0 0 0 0 0 0 0 T 3 0 -3 0 4 0 0 0 0 0 0 T 4 1 0 -8 0 8 0 0 0 0 0 T 5 0 5 0 -20 0 16 0 0 0 0 T 6 -1 0 18 0 -48 0 32 0 0 0 T 7 0 - 7 0 56 0 -112 0 64 0 0 T 8 1 0 -32 0 160 0 -256 0 128 0 T 9 0 9 0 -120 0 432 0 -576 0 256 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Comments • The second order differential equation converts parametric representation to the explicit form • In contrast to the explicit formula n ( − 1 ) m ( n − m − 1 )! T n ( x ) = n ( 2 x ) n − 2 m � m ! ( n − 2 m )! 2 m = 0 the algorithm computes t ( k ) for quite high order polynomials ( n ≈ 100 ′ s ) � ( 2 ) × n • The maximum of the coefficients appears at ≈ 2 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Absolute values of the coefficients n = 100 36 x 10 12 10 8 6 4 2 0 0 20 40 60 80 100 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Symmetrical Zolotarev polynomials -approximation � dy � 2 ( 1 − x 2 )( x 2 − κ ′ 2 ) = 4 m 2 x 2 ( 1 − y 2 ) dx 6 • 5 T m (2 cn 2 ( u | κ ) − 1) 4 3 2 • • • • 1 0 • • −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x = dn( u | κ ) → Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Parametric solutions of differential equation � dy � 2 ( 1 − x 2 )( x 2 − κ ′ 2 ) = 4 m 2 x 2 ( 1 − y 2 ) dx ⇓ dy xdx 1 − y 2 = 2 m ( 1 − x 2 )( x 2 − κ ′ 2 ) � � x = dn ( u | κ ) Jacobi elliptic functions � 2 x 2 − 1 − κ ′ 2 � y = T m ( 2 cn 2 ( u | κ ) − 1 ) ⇒ y ( x ) = T m 1 − κ ′ 2 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion The second order differential equation � dy � 2 ( 1 − x 2 )( x 2 − κ ′ 2 ) = 4 m 2 x 2 ( 1 − y 2 ) dx ⇓ ( 1 − x 2 ) d 2 y dx 2 − x dy + κ ′ 2 ( 1 − x 2 ) dy � � dx + 4 m 2 x 3 y = 0 x ( x 2 − κ ′ 2 ) dx m � 2 x 2 − 1 − κ ′ 2 � b ( 2 k ) x 2 k y ( x ) ≡ T m � = 1 − κ ′ 2 k = 0 .. is a polynomial of variable x Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Recursive evalution of the coefficients b ( 2 k ) given m 2 2 m − 1 initialisation b ( 2 m ) = ( 1 − κ ′ 2 ) m , b ( 2 m + 2 ) = 0 recursive body ( for k = m − 1 to 0 b ( 2 k ) = − ( 1 + κ ′ 2 )( 2 k + 2 )( 2 k + 1 ) b ( 2 k + 2 ) 4 m 2 − 4 k 2 + κ ′ 2 ( 2 k + 4 )( 2 k + 2 ) b ( 2 k + 4 ) 4 m 2 − 4 k 2 end ) Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion An alternative representation m � 2 x 2 − 1 − κ ′ 2 � y ( x ) ≡ T m a ( 2 k ) T 2 k ( x ) � = 1 − κ ′ 2 k = 0 .. is developed in terms of Chebyshev polynomials m Inserting y ( x ) = a ( 2 k ) T 2 k ( x ) in differential equation � k = 0 ( 1 − x 2 ) d 2 y dx 2 − x dy + κ ′ 2 ( 1 − x 2 ) dy dx + 4 m 2 x 3 y = 0 � � x ( x 2 − κ ′ 2 ) dx we obtain... Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Recursive algorithm for the coefficients a ( 2 k ) given m 1 initialisation a ( 2 m ) = ( 1 − κ ′ 2 ) m a ( 2 m + 2 ) = a ( 2 m + 4 ) = 0 recursive body ( for k = m − 1 to 0 m 2 − k 2 � a ( 2 k ) = � − 3 ( m 2 − ( k + 1 ) 2 ) + ( 2 k + 2 )( 2 k + 1 ) κ ′ 2 � a ( 2 k + 2 ) � + 3 ( m 2 − ( k + 2 ) 2 ) + ( 2 k + 4 )( 2 k + 5 ) κ ′ 2 � a ( 2 k + 4 ) � + m 2 − ( k + 3 ) 2 � a ( 2 k + 6 ) � + end ) Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Comments • Properties of Jacobi elliptic functions as κ ′ 2 + κ 2 cn 2 ( u | κ ) = dn 2 ( u | κ ) convert parametric representation to the explicit form • The second order differential equation produces recursive algorithms for coefficients b ( 2 k ) and a ( 2 k ) • The dynamic range of coefficients a ( 2 k ) is far better than b ( 2 k ) and it enables to evaluate safely the symmetrical Zolotarev polynomial of order ( n ≈ 1000 ′ s ) Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Dynamic range of both representations m = 50 5 37 x 10 x 10 6 1 4 0.5 a(2k) b(2k) 2 0 0 −2 −0.5 −4 −1 −6 20 40 60 80 100 20 40 60 80 100 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Equiripple polynomials T 2 m (cn( u | κ )) 2 2 T n ( x ) 0 0 −2 −2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x → x = dn( u | κ ) → Z m, 2 ,m ( x | κ ) 2 2 Z p,q ( x | κ ) 0 0 −2 −2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x → x → Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19 -25, 2007