Handbook for Sound Engineers

Filters and Equalizers 799

23.5.1.2 FIR Length

The required number of taps (N) in an FIR filter at a
sample rate (fS) for a given transition band specified by a
width (ft ) and attenuation in dB(A) can be estimated as

(23-60)

As an example, we can calculate how may taps a

100 Hz, fourth order high-pass filter in a 48 kHz system

would use. Fourth order gives a roll-off of 24 dB per

octave, so the response will be 24 dB down by 50 Hz.

The transition band is for a 20 dB minimum attenuation

in the stop band and therefore is 50 × 20/24 or 42 Hz

wide and the desired attenuation is 24 dB so the equa-

tion gives us 48,000 × 20 × /(42 × 22) = 1049 taps. This

is a very long filter and introduces 520 samples or

10.8 ms of delay at the 48 kHz sample rate.

If we consider the same example, but for a 1000 Hz

cutoff frequency, everything scales by a factor of 10,

giving a much more acceptable filter length of 105

samples with a delay of 1.1 ms. This illustrates the limi-

tation of using FIR filters for low frequencies,

Fig. 23-24.

23.5.2 IIR Filters

There are many possible configurations of infinite

impulse response (IIR) filters, two of which are shown

in Figs. 23-25 and Fig. 23-26. They show the direct

form of a biquad filter in which the input and output

samples are passed into the delay line. The transpose

form has a sum between every delay and scaled copies

of the input and output samples are inserted into the

delay line. Direct form I is better suited to fixed point

implementation where it is important that the delayed

terms maintain as much precision as possible.

The biquad IIR filter is a second-order filter, and

forms the most common basis for higher-order IIR

filters. This form fits well with the transfer function

equations such as the Butterworth polynomials in Table

23-1. The feedback coefficients correspond to the poles

of the filter and the direct coefficients correspond to the

zeros. Each section can be represented as a short FIR

filter, but unlike the direct implementation of an FIR,

these sections can have complex coefficients, even if the

input and output are to be real only.

23.5.2.1 Calculation of Coefficients from Poles and

Zeros

IIR filters are designed in terms of the Z transform. In

this transform, the time domain representation of the

filter is used and the notation z^1 is used in place of the

common exponential terms in the discrete Fourier trans-

form:

Figure 23-23. Coefficients of a 100-tap low pass filter at 0.2

times the sample rate.

Figure 23-24. Increasing filter steepness with number of

FIR taps.

0.6

0.5

0.4

0.3

0.2

0.1

0

0.10 20 40 60 80 100

Tap number

Coefficient value

N

f 5 A

ft 22

---------=

Logarithmic frequency

Gain–dB

0



20



40



60



80

10

1

100

50

20

Figure 23-25. IIR Filter implementing a bi-quad section in

Direct Form I.

Figure 23-26. IIR filter implementing a bi-quad section in

Direct Form II.

a 0

a 2

a 2

b 1

b 2

Z^1

Z^1

Z^1

Z^1

3

a 0

a 2

a 2

b 2

b 2

Z^1

Z^1

3 3