Abstract
This article describes the practical way to build digital IIR filters for
audio applications. Those filters can be used
for equalization and crossover band splitting. See also digital filter design
source code in the "Algorithms and Sources" section of this
site.
The filter structure
Probably the best and the most common way to implement a filter would be in the form of cascaded sections. First, we consider the
building blocks: the first and the second order sections.
Each first order section has 3 coefficients: B0, B1 for
non-recursive part and A1 for recursive part. Also, the section has a memory
array Z of 4 elements.
At the each sample, the first order section performs the
following operations:
Acc = X*B0 + Z[0]*B1 - Z[1]*A1;
Z[1] = Acc;
Z[0] = X;
X is an input of filter, Acc is an output.
Each second order biquad section has 5 coefficients: B0,
B1, B2 for non-recursive part, and A1, A2 for the recursive part. Also, the
section has a memory array Z of 4 elements.
At the each sample, the second order section performs the
following operations:
Acc
= X*B0 + Z[0]*B1 + Z[1]*B2 - Z[2]*A1 - Z[3]*A2;
Z[3] = Z[2];
Z[2] = Acc;
Z[1] = Z[0];
Z[0] = X
X is an input of filter, Acc is an output.
The first order section can be viewed as the second order
section with B2 = A2 = 0. That may be useful if the filter is implemented in
hardware.
We are using bilinear transform (BLT) with frequency warping to compute
the coefficients for different first and second order filter types.
Input parameters:
PI
= 3.14159265...
Fc = cutoff frequency (Hz)
Fs = sample rate (Hz)
Q = quality factor (for second order sections)
G = gain (for parametric EQ filter also known as peaking EQ)
Lowpass 1-st order:
H(S) = 1/(
S + 1)
W = tan (PI*Fc/Fs)
N = 1/(1+W)
B0 = W*N
B1 = B0
A1 = N*(W-1)
Highpass 1-st order:
H(S) = S/(S + 1)
W = tan (PI*Fc/Fs)
N = 1/(1+W)
B0 = N
B1 = -B0
A1 = N*(W-1)
Lowpass 2-nd order:
H(S) = 1/(S^2 + S/Q + 1)
W = tan (PI*Fc/Fs)
N = 1/(W^2 + W/Q + 1)
B0 = N*W^2
B1 = 2*B0
B2 = B0
A1 =
2*N*(W^2 - 1)
A2 =
N*(W^2 - W/Q + 1)
Highpass 2-nd order:
H(S) = S^2/(S^2 + S/Q + 1)
W = tan (PI*Fc/Fs)
N = 1/(W^2 + W/Q + 1)
B0 = N
B1 = -2*N
B2 = B0
A1
= 2*N*(W^2 - 1)
A2
= N*(W^2 - W/Q + 1)
Bandpass 2-nd order:
H(S) = (S/Q)/(S^2 + S/Q + 1)
W = tan (PI*Fc/Fs)
N = 1/(W^2 + W/Q + 1)
B0 = N*W/Q
B1 = 0.0
B2 = -B0
A1 =
2*N*(W^2 - 1)
A2 =
N*(W^2 - W/Q + 1)
Parametric Equalizer 2-nd order:
H(S) = (S^2 + S*G/Q + 1)/(S^2+ S/Q + 1)
Bandwidth = Fc/Q
This is required for constant bandwidth:
If G < 1 then Q = Q*G
A = G/Q
W = tan (PI*Fc/Fs)
N = 1/(W^2 + W/Q + 1)
B0 = N*(W^2 + W*A + 1)
B1 =
2* N*(W^2 - 1)
B2 =
N*(W^2 - W*A + 1)
A1 = B1
A2 =
N*(W^2 - W/Q + 1)
Multiple cascaded filters
The first order lowpass or highpass section has a slope of
6dB/Octave, and the second order section has a slope of 12dB/Octave. If the
better slope is required, you have to cascade several sections.
The design of cascaded filters is
a pretty wide topic. Here we represent only the main results for Butterworth,
Linkwitz-Riley and Bessel types.
Butterworth filter is known as a filter with maximum flat frequency
response. It has a good frequency response with no ripple; however the phase
response may be quite nonlinear especially for high order filters. Due to that
fact, the Butterworth filters with the order of 4 and above have large
oscillations in the step response.
Linkwitz-Riley (LR) is common type of filter for crossovers. The
main feature of LR is that the sum of high passed and low passed signals has a
flat frequency response. LR
consists of two identical Butterworth filters in series. Because of that fact,
the step and the frequency responses of LR are similar to that of the
Butterworth filter.
Bessel filter has a good step response with very little overshoot;
however the frequency response is worse then that of Butterworth filter. Bessel
filters have near linear phase response. The applications of Bessel are where
the phase relations are critical.
Butterworth
All stages of Butterworth filter have the same cutoff
frequency Fc. The parameter which is different for the stages is the filter Q.
Here is the table for Q for Butterworth up to 6-th order:
1: ---
2: 0.71
3: --- 1.0
4: 0.54 1.31
5: --- 0.62 1.62
6: 0.52 0.71 1.93
If the value of Q is skipped, it means this is a first
order section. Otherwise it is a second order section.
Linkwitz-Riley
LR consists of two identical Butterworth filters in series.
Use the same table that is used for Butterworth filter design.
Bessel
Bessel design is a little bit more complicated, since not
only Q but the Fc also differs from the stage to stage.
You should multiply the Fc for each stage by the following
coefficients:
1: 1.00
2: 1.27
3: 1.32 1.45
4: 1.60 1.43
5: 1.50 1.76 1.56
6: 1.90 1.69
1.60
The corresponding Q values are:
1: ---
2: 0.58
3: --- 0.69
4: 0.81 0.52
5: ---- 0.92 0.56
6: 1.02 0.61 0.51
If the value of Q is skipped, it means this is a first order section.
Otherwise it is a second order section.
Note that the digital Bessel filter is designed using BLT. That introduces
a phase response error due to the frequency warping. In the other words,
the linear phase will be considerably distorted if Fc is higher then Fs/4
approximately. You should use impulse invariant approach or other phase
preserving technique to design Bessel filter for high frequencies.
Practical Implementation
of IIR filters
A good way to implement audio IIR filters on the PC would be with the
use of the floating point. You need the single precision floating point
for filter coefficients and double precision for Acc and filter feedback
path. Integer calculations are actually slower then the floating point
on the PC platform.
If you are using a general
purpose integer DSP, you are recommended to have MAC operations with 32x32 bit
or better precision. The very minimal accuracy for the satisfactory results in
the audio processing is 24x24 bit. On the 16-bit DSPs, you have to implement
32-bit processing in software. Otherwise the sound will suffer quantization
noise artifacts especially with the low Fc filters.
The audio processing consumes
considerable computing power due to the high sampling rates. For example, the
10-band stereo equalizer with the Fs of 48 kHz takes about 10MIPS from the DSP.
You can also use specialized ICs
with the hardware implemented filters. The examples of those ICs are Analog
Devices AD1954 or Texas Instruments TAS3004. The ICs have the biquad sections
built in. You only have to compute and upload the coefficients for the filters.
The artifacts of IIR filters
The main artifacts of IIR filtering are the quantization
noise and the limit cycle oscillations.
The truncation or rounding of
the IIR accumulator at the output of filter creates quantization noise. This
noise is fed into the filter recursive path. The noise is multiplied by the IIR
recursive path transfer function. The impact of this noise source is very
significant in the low cutoff frequency filters of the second order, since the
recursive path gain is proportional to the second power of Fc/Fs ratio. The
filter stages with high Q can also suffer from this effect because the gain is
proportional to Q.
For the audio applications, the
quantization noise can be a problem for crossover, EQ and bass boost filters
with the cutoff frequency below 1kHz.
The best way to reduce the quantization noise is improve
the arithmetic accuracy. For the practical audio applications, I would recommend
to have at least 32 bit data precision in IIR feedback path. That will keep the
IIR quantization noise level well below -100dB in most cases.
As for the filter coefficients,
the accuracy of 20 bits is good enough for the recursive path. For the
non-recursive path, 16 bit coefficients with scaling are sufficient. If the
filter is built using the floating point, single precision float accuracy
coefficients are adequate for audio applications.
The MAC operation should have at
least 52 bits integer or double floating point precision.
If the filter has many stages,
the noise contributions of the stages can add, which makes the situation worse.
In the multi cascaded filters, it is useful to sort the stages by decreasing Q.
Place the stage with the highest Q at the input, then the stage with
lower Q, and so on. The idea is to filter out the higher noise of the high Q
stage by the next stages with the lower noise. That is an efficient method for
highpass, lowpass, bandpass and bandstop type of response. The approach works
very well for Butterworth, Bessel and especially Chebyshev filters. For the
elliptic filters, you also have to optimize the locations of poles and zeroes
for each stage.
The disadvantage of placing high
Q stages in front is a possibility of an overflow due to the high gain for
integer filter implementations. This is not an issue if the filters are
implemented in the floating point arithmetic.
The other way to reduce the
quantization noise is the noise shaping. Noise shaping is feeding the
accumulator truncation error back into the filter. That allows for better SNR at
low frequencies for the cost of an increased noise at the high frequencies. The
noise shaping with higher order error feedback can significantly improve the SNR,
however the added complexity and limited performance makes it less attractive,
then the increased precision arithmetic.
The limit cycles are the low
amplitude oscillations which may occur in IIR filters. The reason for those
oscillations is a finite precision of the arithmetic operations. The limit cycle
existence depends on the particular filter coefficients and the input data. The
filters with high Q, low Fc/Fs ratio and the rounding of the accumulator at the
output rather then with truncation have a higher probability of a limit cycle
behavior.
Usually the amplitude of limit
cycle oscillations does not exceed several LSBs. The methods to avoid the limit
cycles are the following:
- Improve the precision of filter arithmetic
- Implement a center clipping
nonlinear function
- Dithering (adding a random
noise with the amplitude of several LSBs)
- Gating: blocking the filter if
the energy of the signal is below a certain limit
Recommended Reading
"Digital Filters: Basics and Design"
Dietrich Schlichtharle
Springer Verlag
ISBN 3-540-66841-1
