Handbook for Sound Engineers

792 Chapter 23

is 6 dB. This means that the 3 dB cutoff point has
moved somewhat lower.

We can analyze a filter in the Laplace domain in
terms of input signals of the form est with s defined as

(23-37)

where,

V is a value for exponential decay,

Z is 2Sf , f being the frequency.

This gives us a transfer function that can be
expressed as polynomial functions in s. A first order
low-pass filter is of the form

(23-38)

The value of p 0 defines the cutoff frequency. Adding
more sections in series progressively multiplies more
terms,

(23-39)

For a normalized version of Eq. 23-37, p 0 is set to be
one, and all other values in the sequence of pn can be
defined according to a formula. The exact formula used
depends on the most important characteristic of the
filter you are designing.

23.2.4.1 Butterworth

The Butterworth filter is maximally flat and has the
most linear phase response in the pass band but has the
slowest transition from pass band to stop band for a
given order. The polynomial transfer function in the
form of Eq. 23-37 can be constructed using a formula.

(23-40)

Eq. 23-40 gives the polynomials for an even order of
filter. To calculate the polynomial for an odd order, add
a term (s + 1), and then apply the formula with n = n–1.
Table 23-1 gives the calculated values for the Butter-
worth polynomials up to fifth order.

23.2.4.2 Linkwitz-Riley

The Linkwitz-Riley filter^2 is used in audio crossovers. It

is formed by cascading two Butterworth filters so that

the cutoff at the crossover frequency is 6dB. This

means that summing the low-pass and high-pass

responses will have a gain of 0 dB at crossover and all

other points.

23.2.4.3 Chebyshev I and II

Chebyshev filters have a steeper roll-off than the Butter-

worth filters but at the expense of a ripple in the

response. There are two forms of the Chebyshev filter.

Type I has a ripple in the pass band and maximum atten-

uation in the stop band. Type II is the reverse, with a flat

pass band and a ripple in the stop band that limits the

average attenuation.

The filter’s transfer function is defined in terms of a

ripple factor H

(23-41)

where,

Cn is the polynomial for the order n, as given in Table

23-2.

The magnitude of the ripple in decibels is

. (23-42)

s V+= jZ

h 1 s^1

sp+ 0

-------------------=

h 1 s^1

sp+ 0 sp+ 1

= --------------------------------------

Bn s s^222 kn1–+

2 n

------------------------S

©¹

++cos§·s 1

k 1=

n

2 ---

= –

Table 23-1. Butterworth Polynomials

Order Polynomial

1(s + 1)

2 (s^2 + 1.414s + 1)

(^3) (s^2 + 1)(s^2 + s + 1)
4 (s^2 + 0.765 + 1) (s^2 + 1.848s + 1)
5 (s + 1)(s^2 + 0.618s + 1)(s^2 + 1.618s + 1)
Table 23-2. Chebyshev Polynomials
Order Type I Type II
1 s 2 s
2 2 s^2 – 1 4 s^2 – 1s
3 4 s^3 – 3s 8 s^3 – 4s
(^48) s^4 – 8s^2 + 1 16 s^4 – 12s^2 + 1
5 16 s^5 – 20s^3 + 5s 32 s^5 – 32s^3 + 6s
H Z^1
1 H^2 Cn^2 Z
Z 0

• 
  

  ---

  
  

  =-----------------------------------
  rippledB 20 1
  1 +H^2

  
  

  ©¹
  ̈ ̧
  §·
  = log dB