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Appendix I

Frequency response shapes for loudspeakers [1]

Abstract

Means for obtaining high-pass filter functions using (1) Bessel polynomials, (2) Butterworth polynomials, and (3) Chebyshev polynomials are covered in this appendix.

Keywords: Bessel polynomials; Butterworth polynomials; Chebyshev polynomials; Frequency; High-pass filter; Ripple factor.

In the following high-pass filter functions G(s), the order of the function is denoted by N and the frequency at which the magnitude of the response is

1 / \sqrt{2}

$1 / \sqrt{2}$

(that is, 3 dB below the pass band level) is denoted by ω _3dB.

Synchronous

G (s) = {(\frac{s}{s + ω_{0}})}^{N}

$G (s) = {(\frac{s}{s + ω_{0}})}^{N}$

where

ω_{0} = ω_{3 dB} \sqrt{2^{1 / N} - 1}

$ω_{0} = ω_{3 dB} \sqrt{2^{1 / N} - 1}$

Bessel

The Bessel polynomials are generated from the following power series in s:

B_{n} (s) = \sum_{k = 0}^{n} a_{k} s^{k}

$B_{n} (s) = \sum_{k = 0}^{n} a_{k} s^{k}$

where s = jω and

a_{k} = \frac{(2 n - k)!}{2^{n - k} k! (n - k)!}

$a_{k} = \frac{(2 n - k)!}{2^{n - k} k! (n - k)!}$

Also, we define a frequency scaling factor γ such that

| B_{n} (j γ) | = a_{0} \sqrt{2} = \frac{2 n!}{2^{n - 1 / 2} n!}

$| B_{n} (j γ) | = a_{0} \sqrt{2} = \frac{2 n!}{2^{n - 1 / 2} n!}$

Bessel polynomials

First order: B ₁(s) = s + 1
Second order: B ₂(s) = s ² + 3s + 3
Third order: B ₃(s) = s ³ + 6s ² + 15s + 15
Fourth order: B ₄(s) = s ⁴ + 10s ³ + 45s ² + 105s + 105
Fifth order: B ₅(s) = s ⁵ + 15s ⁴ + 105s ³ + 420s ² + 945s + 945
Sixth order: B ₆(s) = s ⁶ + 21s ⁵ + 210s ⁴ + 1260s ³ + 4725s ² 10,395s + 10,395

If the real parts of the roots or poles are α ₁, α ₂,…α _N and the imaginary parts of the roots or poles are β ₁, β ₂,…β _N, then

ω_{n} = \frac{γ}{\sqrt{α_{n}^{2} + β_{n}^{2}}}

$ω_{n} = \frac{γ}{\sqrt{α_{n}^{2} + β_{n}^{2}}}$

Q_{n} = \frac{\sqrt{α_{n}^{2} + β_{n}^{2}}}{2 α_{n}}

$Q_{n} = \frac{\sqrt{α_{n}^{2} + β_{n}^{2}}}{2 α_{n}}$

Odd order

G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{(N - 1) / 2}}{Q_{(N - 1) / 2}} + s + ω_{(N - 1) / 2}^{2}) (s + ω_{(N + 1) / 2})}

$G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{(N - 1) / 2}}{Q_{(N - 1) / 2}} + s + ω_{(N - 1) / 2}^{2}) (s + ω_{(N + 1) / 2})}$

Even order

G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{N / 2}}{Q_{N / 2}} s + ω_{N / 2}^{2})}

$G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{N / 2}}{Q_{N / 2}} s + ω_{N / 2}^{2})}$

Butterworth

Odd order

G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{3 dB}}{Q_{1}} s + ω_{3 dB}^{2}) (s^{2} + \frac{ω_{3 dB}}{Q_{2}} s + ω_{3 dB}^{2}) \dots (s^{2} + \frac{ω_{3 dB}}{Q_{(N - 1) / 2}} s + ω_{3 dB}^{2}) (s + ω_{3 dB})}

$G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{3 dB}}{Q_{1}} s + ω_{3 dB}^{2}) (s^{2} + \frac{ω_{3 dB}}{Q_{2}} s + ω_{3 dB}^{2}) \dots (s^{2} + \frac{ω_{3 dB}}{Q_{(N - 1) / 2}} s + ω_{3 dB}^{2}) (s + ω_{3 dB})}$

The poles lie on a circle of radius ω _3dB, each at an angle of θ _n to the real axis, where

Q_{n} = \frac{1}{2 \cos θ_{n}}

$Q_{n} = \frac{1}{2 \cos θ_{n}}$

and

θ_{n} = \pm \frac{n}{N} π, n = 1,2, \dots (N - 1) / 2

$θ_{n} = \pm \frac{n}{N} π, n = 1,2, \dots (N - 1) / 2$

Even order

G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{3 d B}}{Q_{1}} s + ω_{3 d B}^{2}) (s^{2} + \frac{ω_{3 d B}}{Q_{2}} s + ω_{3 d B}^{2}) \dots (s^{2} + \frac{ω_{3 d B}}{Q_{N / 2}} s + ω_{3 d B}^{2})}

$G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{3 d B}}{Q_{1}} s + ω_{3 d B}^{2}) (s^{2} + \frac{ω_{3 d B}}{Q_{2}} s + ω_{3 d B}^{2}) \dots (s^{2} + \frac{ω_{3 d B}}{Q_{N / 2}} s + ω_{3 d B}^{2})}$

where

θ_{n} = \pm \frac{n - 1}{2 N} π, n = 1,2, \dots N / 2

$θ_{n} = \pm \frac{n - 1}{2 N} π, n = 1,2, \dots N / 2$

Chebyshev

Chebyshev polynomials

First order: C ₁(Ω) = Ω
Second order: C ₂(Ω) = 2Ω² − 1
Third order: C ₃(Ω) = 4Ω³ − 3Ω
Fourth order: C ₄(Ω) = 8Ω⁴ − 8Ω² + 1
Fifth order: C ₅(Ω) = 16Ω⁵ − 20Ω³ + 5Ω
Sixth order: C ₆(Ω) = 32Ω⁶ − 48Ω⁴ + 18Ω² − 1

where

Ω = \frac{ω}{ω_{P}}

$Ω = \frac{ω}{ω_{P}}$

and ω _P is the pass-band limit, which is defined as the frequency at which the final 0 dB crossing occurs before roll-off. Let R be the maximum permitted magnitude of the pass-band ripples in dB. Thus we define a ripple factor by

ε = \sqrt{10^{0.1 R} - 1}

$ε = \sqrt{10^{0.1 R} - 1}$

Also, we define a frequency scaling factor γ by

γ = \frac{ω_{3 dB}}{ω_{p}} = \cosh (\frac{1}{N} arccosh (\frac{1}{ε}))

$γ = \frac{ω_{3 dB}}{ω_{p}} = \cosh (\frac{1}{N} arccosh (\frac{1}{ε}))$

and then find the roots of the polynomial

1 + ε^{2} C_{N}^{2}

$1 + ε^{2} C_{N}^{2}$

so that the imaginary parts of the roots α ₁, α ₂,…α _N give the real parts of the poles and the real parts of the roots β ₁, β ₂,…β _N give the imaginary parts of the poles. Then

ω_{n} = \frac{γ}{\sqrt{α_{n}^{2} + β_{n}^{2}}}

$ω_{n} = \frac{γ}{\sqrt{α_{n}^{2} + β_{n}^{2}}}$

Q_{n} = \frac{\sqrt{α_{n}^{2} + β_{n}^{2}}}{2 α_{n}}

$Q_{n} = \frac{\sqrt{α_{n}^{2} + β_{n}^{2}}}{2 α_{n}}$

Odd order

G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{(N - 1) / 2}}{Q_{(N - 1) / 2}} s + ω_{(N - 1) / 2}^{2}) (s + ω_{(N - 1) / 2})}

$G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{(N - 1) / 2}}{Q_{(N - 1) / 2}} s + ω_{(N - 1) / 2}^{2}) (s + ω_{(N - 1) / 2})}$

Even order

G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{N / 2}}{Q_{N / 2}} s + ω_{N / 2}^{2})}

$G (s) = \frac{s^{N}}{(s^{2} + \frac{ω_{1}}{Q_{1}} s + ω_{1}^{2}) (s^{2} + \frac{ω_{2}}{Q_{2}} s + ω_{2}^{2}) \dots (s^{2} + \frac{ω_{N / 2}}{Q_{N / 2}} s + ω_{N / 2}^{2})}$

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Table of Contents for Appendix I. Frequency response shapes for loudspeakers [1]

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Synchronous

Bessel

Odd order

Even order

Butterworth

Odd order

Even order

Chebyshev

Odd order

Even order

Table of Contents for
Appendix I. Frequency response shapes for loudspeakers [1]