Application Of Fir Filters To Loudspeakers Crossovers; Fir Overview; What Are Iir Filters And Fir Filters; Pros And Cons Of Iir And Fir Filters - Crown I-Tech 4x3500HD DriveCore Series Operation Manual

16 Application of FIR Filters to Loudspeaker Crossovers

16.1 FIR Overview

Digital filters may be further classified into two general types: Infinite
Impulse Response (IIR) and Finite Impulse Response (FIR). Fundamen-
The powerful DSP processing in the latest Crown I-Tech HD Series
tally, IIR filters include feedback (recursion) in their implementation
amplifiers allow sophisticated Finite Impulse Response filters as well.
while FIR filters do not. The feedback in an IIR filter means that the
FIR filters offer a number of very strong advantages when used to
impulse response of the filter theoretically goes on forever, thus the
implement DSP electronic loudspeaker crossovers.
"Infinite" term in the name. The impulse response of a FIR filter, on the
other hand, is finite because there is no feedback. Simplified structures
Although high-order IIR filters, which are based on conventional
for both these types of filters are shown in Fig. 16.1.
analog filters, can be designed to provide high stop-band roll-off rates,
the resultant filter phase response is highly nonlinear. This
significantly complicates crossover design and implementation.
As compared to IIR filters, FIR filters can be straightforwardly designed
to allow extremely narrow crossover overlap between adjacent drivers
with high stop-band attenuation and very-high rolloff rates. In-phase
and linear-phase crossovers can easily be designed. Minimizing
overlap dramatically reduces polar lobing and vastly improves off-axis
response through the crossover region.
This white paper briefly describes the characteristics and pros/cons of
FIR and IIR filters, discusses the desirable attributes of FIR filters, and
SUM
IN OUT
closes with a set of example measurements on a two-way loudspeaker
system to illustrate the practical application of IIR and FIR crossover
filters. In addition, audio demonstration files are available on Crown's DELAY
IN DELAY
web site that demonstrates the effects described in this white paper
(see comments at the end of this report).
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16.2 What are IIR Filters and FIR Filters?

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A filter modifies certain characteristics of a signal such as amplitude/
phase frequency response and wave-shape in a desired manner. This
can be done either in a purely analog manner with a real piece of
hardware such a non-computer based equalizer or filter set, or with a
Fig. 16.1 Simplified filter structures of an IIR (top) and FIR
computer-based instrument running mathematical algorithms using
(bottom) filter
digital signal processing techniques. These algorithms can be either
The IIR structure on the left feeds back weighted the delayed versions of
implemented in hardware and/or software. The term "digital filter"
the output signal back to the input and thus sets up a form of
refers to the specific hardware or software routine that performs the
recirculation which effectively goes on for forever once a signal is
filtering algorithm.
applied to the input. The FIR structure on the right feeds weighted and
delayed signals to the output ounly and thus can't recirculate the
signals. Effectively there is only a single path through the FIR filter.
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I-Tech HD DriveCore Series Power Amplifiers

16.3 Pros and Cons of IIR and FIR Filters

The following table lists several characteristics of the two types of
filters and their pros and cons.
Characteristic
Linear Phase
Response
High Rolloff Rates
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Implement
Complex Filters
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IN
DELAY DELAY DELAY DELAY DELAY DELAY
Stability
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Computational
SUM
Complexity
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OUT
Implementation
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16.4 Desirable Attributes of FIR Filters

Linear Phase
FIR filters can be designed to have exact linear phase. Linear-phase
SUM
filters provide minimal modification to the wave shape of a signal and
also greatly simplify crossover design and implementation because the
OUT
filter does not change the phase of the drivers being crossed over.
A linear phase crossover does not mean that the overall response of the
crossover including driver response is linear phase. This can be true
only if the individual drivers themselves are linear phase or can be
equalized to be linear phase.
Electronic crossovers using conventional non-linear phase filters such
as IIR filters can artificially increase the crest factor of a signal and thus
decrease the headroom in the transmission channel.
I-Tech HD DriveCore Series Power Amplifiers
16 Application of FIR Filters to Loudspeaker Crossovers

16.5 High Rolloff and Steep Slopes

FIR filters can be designed to have extremely high stop-band rolloffs
and exceptionally steep slopes which greatly minimizes crossover
driver overlap. In a conventional analog or analog-based IIR
crossover, driver overlap can extend over two or three octaves. FIR
IIR Filter FIR Filter
crossovers dramatically restrict the operating overlap bandwidth of the
crossover which considerable reducers the range over which both
Not Possible Possible
upper and lower range drivers are radiating in the same frequency
range. Very narrow overlaps of one-third octave or less can be
Yes, but with Yes, without
implemented with FIR filters.
phase distortion phase distortion
In addition, the extremely steep slopes of FIR filters offer greater driver
No Yes
protection and reduced distortion. Beyond the driver's linear frequency
range, energy is attenuated so rapidly that most non-linearity's cease to
be a problem. The driver does not need to be as well behaved outsied
Conditionally Naturally Stable
its frequency range. Power handling capability of HF drivers is much
improved. The narrower crossover region also lessens the need for
Massive CPU
precise driver time alignment since the overlap region is so small.
Few CPU cycles
cycles
Difficult to
Common

16.6 Stop-band Attenuation

achieve with
Practice
limited DSP Associated with the very high stop-band rolloff of an FIR filter, is the
associated extremely high stop-band attenuation. This minimizes
interaction between adjacent drivers such as a low-frequency woofer
signal bleeding into a tweeter and thus causing intermodulation
distortion. In a home theater setup, high stop-band attenuation of the
subwoofer minimizes subjective localization of the woofers due to
hight-frequency bleed through.
Operation Manual Operation Manual
16.7 Polar Lobing Error

16.9 FFT Convolution

Polar lobing is a potential problem when spatially separated The resource requirements of a high-performance FIR filter in terms of
non-coincident drivers are crossed over. Throug the crossover region cpu cycles per sample, datapath bandwidth and memory footprint can
both drivers are radiating simultaneously. This may cause a narrowing exceed those of an IIR solution by several orders of magnitude. Key to
of the coverage pattern and the creation of a directional lobe. It is very an efficient FIR implementation is the use of Fast Fourier Transform
desirable that this lobe face straight ahead, and not directionally (FFT) techniques to accelerate the FIR convolution process, which is
wander with frequency. If it does, lobing error occurs. usually thought of as a time domain operation. Time domain
convolution is something to be avoided, because it is extremely
Polar lobing error is minimized when the low- and high-pass sections expensive in the computational sense. Fortunately, signal processing
of the crossover are in-phase with each other throughout the crossover theory tells us that multiplication in the frequency domain is equivalent
are in-phase with each other throughout the crossover region. This is of convolution in the time domain. This is important because
an attribute of the "Linquitz-Riley" (LR) type of crossover rewponses. multiplication is very efficient in comparison to convolution. Of course,
Polar lobing can also be minimized by reducing the crossover overlap an efficient and speedy means of moving between the time and
with zero-phase sharp-cutoff filters such as provided by FIR filters. frequency domains is also required. This is where the FFT comes in.
Using the FFT to transform back and forth between the time and
frequency domains so as to replace convolution with multiplication is

16.8 Crown's Implementation of FIR Filters

referred to as FFT Convolution. Figure 16.2 shows a block diagram of
Crown's FIR filter implementation uses state-of-the-art digital signal
the FFT convolution process.
processing techniques which are highly optimized for the DSP engine
in the I-Tech HD series of amplifiers.
X
input
FFT
FFT
lter impulse response
Fig. 16.2 FFT conbolution block diagram. here the input and filter
impulse responses are both individually FFT'd and multiplied and then
inverse FFT'd to genterate the output.
output
inv FFT
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