Theo Verelst Filter Page
Last Update: april 5, 2005
Introduction
Filters in electronics and electrical engineering are usually circuits
with allow certain frequencies in a circuit to be emphasized or others
suppressed.
In general, linear filters in Network Theory are circuits with at least
on input and one output with overall stable behaviour, influenced by at
least one significant frequency dependent element a: a reactance,
usually a capacitor or a coil, which in the continuous realm are the
only such elements except for delay lines or transmission lines.
In the digital domain much confusion seems to exist about what a filter
is exactly, and what can be said about such circuits, though in EE the
fine discipline of Network Theory and Circuits and Systems, and
certainly Electronics can be consulted to answer a lot of important
questions, but it is essential in the case of sampled, Time Discrete
systems that we get our theoretics right before assuming all kinds of
wild things about Digital Filters.
In the strong and theoretical discipline of Network Theory (wherein the
author holds a Master of Science Degree from Delft University), we can
for continuous circuits, that is 'real' electronics circuits with
actual parts, compute the behaviour of any
filter with infinite accuracy, given we know the part characteristics,
even following computerizable computation steps. This is, as EEs will
know, a major part of the first and second year foundation laid in the
study for any such engineer, and of profound theoretical value, and a
fairly attractive theory, at least I found it pleasing enough.
Basically, by filling all the circuit elements in a complex matrix
according to given rules (which can be proven mathematically) and
inverting this matrix some way (for isntance gaussian elimination with
row exchange) we can find all the complex voltages and currents in any
physically realizable network (with no bad short circuits) and from
this compute the transfer function and the frequency (or time)
behaviour of the network or filter. For any network, no matter how big
or complex. On this page I'll see if I can pay some attention to this,
and automated ways to do this and also to compute the transfer function
symbolically, which can be advantageous, and is interesting, and maybe
I'll make a bwise/maxima combination program to do this automatically.
To begin with , my first motivation is to be able to say theoretically
founded things about audio filters, which then can be realized also in
the digital (sampled) domain, with certain precautions taken, and which
will lead to good and decent filters for Digital Signal Processing
application in a sound synthesizer which I have made, and which
already works.
Because of the popularity of the subject, starting already years ago
when I was in highschool and programmed such things on a TRS80 computer
(...) it is important to understand a lot of appoximations and
theoretical errors which are made in the field of digital signal
processing and filtering therein, so we start with a section about the
most inportant theoretical considerations in the digital/sampled domain
of signal processing, even though it might seem more natural to start
with an introduction to analog circuits and filtering an computation in
that field.
Digital Signals and their theory
It can hardly be overstated or overstressed: digital signals and their
processing is not as almighty as some computer oriented people may want
to think, there are stringent, binding, and essential theoretically
founded rules which must be obeyed to take part in this branch of sport
to be ultimately succesfuill and to be theoretically sound. Note that
the essence of analog signal processing and computations is well
founded already for about a century at least, from the time of the
early radios and the advent of quantum physics (both areas I'm
certainly not new to), and that the relative newcomer digital signal
processing since the foundation time of computers of the modern kind
(lets say after Turing and Neumann) has received a lot of attention and
in electrical engineering has a solid basis (to begin with) in the
first or second year course of Information Theory.
Lets phrase the most important 'laws' from that field.
First, a continuous signal can be sampled at a certain rate, the
'sampling rate' preferably fixed (without getting into even more
advanced problems) by 'sampling' the voltage or magnitude of the analog
signal at a precise moment in time, the sample moment, and then stored
in some memory for later use.
Second, in most, though certainly not all, cases, that signal is not
just sampled, or made discrete, in time sense, but also in amplitude
sense, which is called quantization of the signal amplitude, which
usually takes place in the Analog to Digital converter unit. So we have
a continuous signal which goes into this unit, and at the output we
have a time and amplitude discrete signal, in the form of a string of
zeros and ones, like in computer words, which (hopefully) represent the
same signal.
Now there appear two problems: the amplitude quantisation cannot be
undone, and the chopping of the signal into samples can possibly not be
undone. Once the voltage of a sample has been measured and stored in a
computer with some accuracy, lets say 16 bits, it is impossible to know
from a sample more accurately than a 2^-16 th part what the
original signal voltage was, and that is unrecoverable, so hopefully
that error is small or unnoticable in practice (which it rarely is).
The sampling accuracy is another matter. Shannon theory says that when
the original analog, unsamped (and unquantized) signal contains no
frequency components (according to Fourier analysis) higher than half
the sampling frequency, and the sampling is perfect, without
quantisation errors, the original analog (!) sugnal can be regenerated
from the samples by a perfect reconstruction filter, in error free
form, as in mathematically correct. That would of course require the
use of infinite accuracy samples, but still, that is a powerfull thesis
to start from.
The problem is how we can make sure that we comply with this frequency
range condition, called the Nyquist rate sampling frequency, because
most natural, analog signals will have all kinds of frequencies in
them. Some signals may have a gairly certain upper frequency in its
frequency analysis picture, so we can take the sampling rate a bit
higher than twice this highest frequency. Most signals will not by
nature be hard-bounded to a certain frequency range, so would possibly
have to be filtered to cut off all frequencies higher than the Nyquist
required upper frequency.
This filtering takes us immedeately to one of the big issues of the
field: assuring the Shannon Sampling theorem is not not violated,
filtering a signal to contain no frequencies above half the sampling
frequency is NEVER perfectly possible in practice, except for very
special cases, and then usually only theoretically.
So if we start with a signal which is not frequency limited to half the
sampling rate, we always have a problem digitising and sampling it, per
definition. There are solutions which for certain practical cases and
for a certain required accuracy may work, and even work well, but the
whole idea of perfect filtering is not quite possible. At the very
least all filters of practical kind will have phase distortion
problems, and also no filter perferctly filters all frequencies above a
certain frequency, there is at best talk of a certain diminishing of
higher frequencies, which hopefully is sufficient for the purpose. For
instance when higher frequencie components start to become smaller in
the signal amplitude then the quanitsation error, it might be
considered sufficient filtering.
The next source of error is when we go from the discrete or computer
signal back to the analog domain, for instance the DA converter of a
sound card.and is called the signal reconstruction problem. The
reconstruction filter is that wonderfull idea of taking the samples of
the time discrete signal., and putting them back into the analog
domain, which in theory can be done without error at all, assuming
correct sampling has taken place. In practice the length and especially
the complexity of the reconstruction filter is pretty amazing, to get
close to perfection. CD players and all other digital audio
equipment will not get close to doing this reconstruciton perfectly,
really, believe me. It depens on what you call perfect enough, but the
higher frequencies require a lot of work to get reconstructed well, on
the basis of what is called the sinc function for each sample.
See for instance Sampled Signal Reconstruction in Tcl driven Maxima .
The basic 1st order electronic filter analyzed
r:1000;
c:1/1000000;
h(w):=(1/(%i*w*c))/(r+1/(%i*w*c));
plot2d([abs(h(2*%pi*w)),carg(h(2*%pi*w))],[w,0,10000]);
float(abs(h(2*%pi/(2*%pi*r*c))));
