by Craig Lindley
There has been a lot written about the design and implementation of digital
filters over the years and I periodically survey the literature to try and keep
up with it. It seemed the right time to look around again when I decided to
build a digital color organ  which would require digital filtering. During
this latest survey, I came upon Lattice Wave Digital Filters (LWDFs) which
I was not familiar with. These filters piqued my interest as they had
intriguing properties directly applicable to my application.
Typically, when I find a technology that interests me, I go overboard in my research and spend
lots of hours playing with it in order to
understand it. LWDFs were no exception. This article, and the accompanying design tool, were a result of my
investigation into LWDFs which were
just too cool to pass up. We will get to
LWDFs shortly, but first a little context.
There exists two major classes of
digital filters referred to as Infinite
Impulse Response (IIR) — sometimes
called a recursive filter — and Finite
Impulse Response (FIR) or convolution filters. Both of these filter types
have advantages and disadvantages
which make them each suitable for a
variety of applications.
IIR filters might be used, for
example, in applications where cost is
a concern because they can be
implemented using a minimum of
hardware/software resources. FIR
filters would be used in applications
where linear phase response is
required such as in high end audio
applications, the processing of sensor
data, or possibly radio telescope
data. FIR filters use more resources to
implement to the same order as an
equivalent IIR filter.
68 November 2007
Typically, IIR filters are not as
stable as their FIR counterparts. This
fact is reflected in their names. Infinite
Impulse Response filters will typically
ring for a period of time after being
subjected to an impulse waveform.
Finite Impulse Response filters are
better damped and any ringing of the
filter will be of much shorter duration.
IIR filters also have sensitivity to
word length limitations and coefficient
round off errors which makes their
implementation tricky. All this is to say
that testing is very important for any
digital filter you design (regardless of
topology) to make sure it is performing per your requirements.
IIR filters were appropriate for the
application I had in mind because of
their inherent efficiencies, but only
if their stability issues could be
controlled or eliminated. This is where
LWDFs came into play.
All of the information I found on
these filters indicated they were
extremely stable, had a large dynamic
range, and weren’t sensitive to either
limited word lengths or round off
errors. LWDFs don’t correct the
erratic phase response of classic IIR
filters but phase response is not
important in my application.
LWDFs were first proposed by
Professor Alfred Fettweis in 1971 and
are backed with an extensive amount
of complex network theory which one
does not need to understand to apply.
A lot of information can be found on
the Internet describing these filters.
The Resources sidebar details some of
the most useful papers I found.
Specifically, the TI application note
and the paper by Gazsi were the most
helpful to me and my understanding
of this technology.
LWDFs are a cascade of first and
second order all pass filter sections.
Each all pass section is made up of a
two port adapter (picture a little box
with two inputs in1 and in2 and two
outputs out1 and out2) along with a
one sample delay element. Every
adapter has a gamma value (gamma
values range in value between -1 and
+1) which controls the response of the
all pass section. Adapters are
implemented in software using a set
of network formulas which require
a single multiplication and three
The LWDFs we are discussing use
four types of adapters as filter building
blocks. The number and the variety of
adapters used along with the gamma