Robust Algorithms for Chebyshev Polynomials and Related - - PowerPoint PPT Presentation

▶
robust algorithms for chebyshev polynomials and related
SMART_READER_LITE
LIVE PREVIEW

Robust Algorithms for Chebyshev Polynomials and Related - - PowerPoint PPT Presentation

Apr 21, 2023 101 likes •488 views

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Robust Algorithms for Chebyshev Polynomials and Related Approximations Miroslav Vl cek vlcek@fd.cvut.cz Department of

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

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

slide-4
SLIDE 4

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Chebyshev polynomials - approximation view

(1 − x2) dy dx 2 = n2 (1 − y2)

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0.5 1 1.5

x → Tn(x)

  • Miroslav Vlˇ

cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-5
SLIDE 5

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Parametric solutions of differential equation

(1 − x2) dy dx 2 = n2 (1 − y2) ⇓ dy

  • 1 − y2 = n

dx √ 1 − x2 x = cos Φ De Moivre’s formula y = cos n Φ ⇒ y(x) = Tn(cos Φ)

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-6
SLIDE 6

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

The second order differential equation

(1 − x2) dy dx 2 = n2 (1 − y2) ⇓ (1 − x2) d2y dx2 − x dy dx + n2 y = 0 y(x) ≡ Tn(x) =

n

  • k=0

t(k)xk .. is a polynomial of variable x

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-7
SLIDE 7

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Recursive evalution of the coefficients t(k)

given n initialisation t(n) = 2n−1 t(n − 1) = 0 recursive body (for k = n − 2 to 0 t(k) = −(k + 2)(k + 1) n2 − k2 t(k + 2) end)

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-8
SLIDE 8

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

  • Algorithm produces the coefficients t(k) for Chebyshev

polynomials Tn(x) as expected.

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 T0 1 T1 1 T2

  • 1

2 T3

  • 3

4 T4 1

  • 8

8 T5 5

  • 20

16 T6

  • 1

18

  • 48

32 T7

  • 7

56

  • 112

64 T8 1

  • 32

160

  • 256

128 T9 9

  • 120

432

  • 576

256

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-9
SLIDE 9

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

Tn(x) = n 2

n

  • m=0

(−1)m(n − m − 1)! m! (n − 2m)! (2 x)n−2m the algorithm computes t(k) for quite high order polynomials (n ≈ 100′s)

  • The maximum of the coefficients appears at ≈
  • (2)

2 × n

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-10
SLIDE 10

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Absolute values of the coefficients n = 100

20 40 60 80 100 2 4 6 8 10 12 x 10

36 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-11
SLIDE 11

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Symmetrical Zolotarev polynomials-approximation

(1 − x2)(x2 − κ′2) dy dx 2 = 4 m2x2 (1 − y2)

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 1 2 3 4 5 6

x = dn(u|κ) → Tm(2 cn2(u|κ) − 1)

  • Miroslav Vlˇ

cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-12
SLIDE 12

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Parametric solutions of differential equation

(1 − x2)(x2 − κ′2) dy dx 2 = 4 m2x2 (1 − y2) ⇓ dy

  • 1 − y2 = 2m

xdx

  • (1 − x2)(x2 − κ′2)

x = dn(u|κ) Jacobi elliptic functions y = Tm(2 cn2(u|κ) − 1) ⇒ y(x) = Tm 2x2 − 1 − κ′2 1 − κ′2

  • Miroslav Vlˇ

cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-13
SLIDE 13

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

The second order differential equation

(1 − x2)(x2 − κ′2) dy dx 2 = 4 m2x2 (1 − y2) ⇓ x(x2−κ′2)

  • (1 − x2)d2y

dx2 − x dy dx

  • +κ′2(1−x2)dy

dx +4 m2x3 y = 0 y(x) ≡ Tm 2x2 − 1 − κ′2 1 − κ′2

  • =

m

  • k=0

b(2k)x2k .. is a polynomial of variable x

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-14
SLIDE 14

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Recursive evalution of the coefficients b(2k)

given m initialisation b(2m) = 22m−1 (1 − κ′2)m , b(2m + 2) = 0 recursive body (for k = m − 1 to 0 b(2k) = −(1 + κ′2)(2k + 2)(2k + 1) 4m2 − 4k2 b(2k + 2) +κ′2 (2k + 4)(2k + 2) 4m2 − 4k2 b(2k + 4) end)

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-15
SLIDE 15

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

An alternative representation

y(x) ≡ Tm 2x2 − 1 − κ′2 1 − κ′2

  • =

m

  • k=0

a(2k)T2k(x) .. is developed in terms of Chebyshev polynomials Inserting y(x) =

m

  • k=0

a(2k)T2k(x) in differential equation x(x2−κ′2)

  • (1 − x2)d2y

dx2 − x dy dx

  • +κ′2(1−x2)dy

dx +4 m2x3 y = 0 we obtain...

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-16
SLIDE 16

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Recursive algorithm for the coefficients a(2k)

given m initialisation a(2m) = 1 (1 − κ′2)m a(2m + 2) = a(2m + 4) = 0 recursive body (for k = m − 1 to 0 −

  • m2 − k2

a(2k) = +

  • 3(m2 − (k + 1)2) + (2k + 2)(2k + 1)κ′2

a(2k + 2) +

  • 3(m2 − (k + 2)2) + (2k + 4)(2k + 5)κ′2

a(2k + 4) +

  • m2 − (k + 3)2

a(2k + 6) end )

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-17
SLIDE 17

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Comments

  • Properties of Jacobi elliptic functions as

κ′2 + κ2cn2(u|κ) = dn2(u|κ) convert parametric representation to the explicit form

  • The second order differential equation produces recursive

algorithms for coefficients b(2k) and a(2k)

  • The dynamic range of coefficients a(2k) is far better than

b(2k) 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

slide-18
SLIDE 18

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Dynamic range of both representations m = 50

20 40 60 80 100 −1 −0.5 0.5 1 x 10

5

a(2k) 20 40 60 80 100 −6 −4 −2 2 4 6 x 10

37

b(2k)

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-19
SLIDE 19

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Equiripple polynomials

−1 −0.5 0.5 1 −2 2 −1 −0.5 0.5 1 −2 2 −1 −0.5 0.5 1 −2 2 −1 −0.5 0.5 1 −2 2

x → x → x → Tn(x) x = dn(u|κ) → T2m(cn(u|κ)) Zm,2,m(x|κ) Zp,q(x|κ)

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-20
SLIDE 20

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Differential equations - approximation view

f(x) dy dx 2 = n2g(x)(1 − y2

i )

(1) ya : f(x) = 1 − x2 g(x) = 1 yb : f(x) = (1 − x2)(x2 − κ′2) g(x) = x2 yc : f(x) = (1 − x2)(x2 − x2

p)(x2 − x2 s )

g(x) = (x2 − x2

m)2

yd : f(x) = (1 − x2)(x − xp)(x − xs) g(w) = (x − xm)2

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-21
SLIDE 21

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Linear differential equation

By derivation of eq. (1) we obtain d2y dx2 + 1 2 f ′(x) f(x) − g′(x) g(x) dy dx + n2 g(x) f(x) y = 0 ,

  • r in canonical form

d2y dx2 − d dx ln

  • g(x)

f(x) dy dy + n2 g(x) f(x) y = 0 .

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-22
SLIDE 22

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Wronskian and general solutions

Two linearly independent solutions of eq. (2) y1(x) a y2(x), form Wronskian determinant W(y1(x), y2(x)) ≡ y1(x)y′

2(x) − y′ 1(x)y2(x) =

= exp

  • d

dx ln

  • g(x)

f(x) dx =

  • g(x)

f(x) .

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-23
SLIDE 23

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Solutions

In order to avoid relatively complicated parametric representation...

Zp,q(A p

n (u|κ)) = (−1)p 1

2          ϑ1

  • v − π

2 p p + q

  • ϑ1
  • v + π

2 p p + q

  

n

+     ϑ1

  • v + π

2 p p + q

  • ϑ1
  • v − π

2 p p + q

  

n

    A p

n (u|κ) = 1

2     ϑ1

  • v − π

2 p p + q

  • ϑ1
  • v + π

2 p p + q + ϑ1

  • v + π

2 p p + q

  • ϑ1
  • v − π

2 p p + q

  

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-24
SLIDE 24

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

... the recursive algorithms

for the coefficients b(m) and a(m) have been developed. Zp,q(x|κ) =

n

  • m=0

b(m)xm Zp,q(x|κ) =

n

  • m=0

a(m)Tm(x)

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-25
SLIDE 25

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Zolotarev polynomial

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −3 −2 −1 1 2 3

x → Z7,9(x|0.6) xp xs

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-26
SLIDE 26

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Application in Filter Design

  • Design of digital FIR filters: DC-Notch FIR filters and Comb

FIR filters

  • Based on differential equation developed for Chebyshev

polynomials as Tn(λ w + λ − 1) and Tn [λ Tr(w)]

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-27
SLIDE 27

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Example of DC-Notch: ωpT = 0.15π, a = −1.2446 dB

0.2 0.4 0.6 0.8 1 −60 −50 −40 −30 −20 −10 ω T/ π 20log|H(ej ω T)| [dB] a [dB] ωp T/ π

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-28
SLIDE 28

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Example of Comb: ∆ωpT = 0.066π, a = −0.65 dB

20 log |H(eω T )|

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −60 −50 −40 −30 −20 −10 a [dB] ∆ ω T

ω T/π →

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-29
SLIDE 29

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Differential Equation for Tn(λ w + λ − 1)

  • 1 − w2 + 2 1 − λ

λ (1 − w) d2F(w) dw2 −

  • w − 1 − λ

λ dF(w) dw + n2F(w) = 0 ... by substitution of F(w) =

n

  • m=0

α(m) Tm we obtain

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-30
SLIDE 30

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Recursive algorithm for coefficients

given n (integer value), λ (real value) initialization α(n) = λn , α(n + 1) = α(n + 2) = α(n + 3) = 0 body (for k = 2 ... n + 1) α(n + 1 − k) = { 2 [(k − 1)(2n + 1 − k) − ((1 − λ)/λ)(n + 1 − k)(2n + 1 − 2k)] α(n + 2 − k) + 4 ((1 − λ)/λ)(n + 2 − k) α(n + 3 − k) − 2 [(k − 3)(2n + 3 − k) − ((1 − λ)/λ)(n + 3 − k)(2n + 7 − 2k)] α(n + 4 − k) + (k − 4)(2n + 4 − k) α(n + 5 − k) } / k(2n − k) (end loop on k) α(0) = α(0) 2 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-31
SLIDE 31

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Differential Equation for Tn [λ Tr(w)]

Ur−1(w)

  • κ2 − T 2

r (w)

(1 − w2)d2Fnr(w) dw2 − w dFnr(w) dw

  • − r(1 − κ2)Tr(w)dFnr(w)

dw + n2r2Ur−1(w)

  • 1 − T 2

r (w)

  • Fnr(w) = 0

... by substitution of Fnr(w) =

n×r

  • m=0

˜ a(m) Tm(w) we obtain

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-32
SLIDE 32

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Basic recursive algorithm for ˜ a(nr − 2µr) = α(n − 2µ)

given n (even integer), r (integer), λ > 1 (real) initialization κ = 1 λ α(n) = λn α(n + 2) = α(n + 4) = α(n + 6) = 0 body (for µ = 1 ... n 2 ) α(n − 2µ) = { α(n − 2(µ − 1)) × h (1 − κ2)(n − (2µ − 1))(n − (2µ − 2)) + 3(µ − 1)(n − (µ − 1)) i −α(n − 2(µ − 2)) × h (1 − κ2)(n − (2µ − 4))(n − (2µ − 5)) + 3(µ − 2)(n − (µ − 2)) i +α(n − 2(µ − 3))(µ − 3)(n − (µ − 3)) } / µ(n − µ) (end loop on µ) α(0) = α(0) 2 Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-33
SLIDE 33

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Robustness of Algorithms

Strange filter specification ωpT = 0.00001π, a = −0.01 dB requires to evaluate expression 1 − T259524 (1.00000000024674w + 0.00000000024674) + 1 T259524(1.00000000049348) + 1

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-34
SLIDE 34

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Robustness of Algorithms

0.2 0.4 0.6 0.8 1 −300 −250 −200 −150 −100 −50 ω T/ π 20log|H(ej ω T)| [dB]

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-35
SLIDE 35

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

1 Chebyshev polynomials and their relatives were presented

in unified approach

2 recursive algorithms were developed 3 the role of the linear differential equations was emphasized 4 numerical robustness was shown

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-36
SLIDE 36

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

References

1 P

. Zahradn´ ık, M. Vlˇ cek: Note on the design of an equiripple DC-notch FIR filter, IEEE Trans. on Circuits and Systems II, vol. CAS - 54, no. 2, February 2007, pp.196-199.

2 P

. Zahradn´ ık, M. Vlˇ cek: Analytical Design Method for Optimal Equiripple Comb FIR Filters, IEEE Trans. on Circuits and System II., vol. CAS - 52, no. 2, February 2005, pp.112-115.

3 P

. Zahradn´ ık, M. Vlˇ cek: Fast Analytical Design Algorithms for FIR Notch Filters, IEEE Trans. on Circuits and Systems, vol. CAS - 51, no. 3, March 2004, pp. 608- 623.

4 M. Vlˇ

cek, R. Unbehauen : Zolotarev Polynomials and Optimal FIR Filters, IEEE Trans. on Signal Processing, vol. SP - 47, no. 3, March 1999, pp. 717 - 730.

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007

slide-37
SLIDE 37

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion

Thank you for your attention !

Miroslav Vlˇ cek Robust Algorithms, Harrachov, August 19-25, 2007