ECE 421  Sum 2010
Notes Set 7:
Filter Design by PoleZero Placement
1
INTRODUCTION
In this set of Notes we consider a digital filter design method which is done entirely within
the digital frequency
z
domain. As such, it does not depend on an underlying analog form
like an ODE or transfer function
H s
. This method is called digital filter design using
polezero placement, and it extensively uses
Z
transforms. You can review the Supplement
Notes on
Z
Transforms (accessible from the course homepage) if you need to refresh your
knowledge of
Z
transforms.
Here is a simple example we can use to motivate our digital filter design. Suppose we had
an narrowband analog information signal which was being acquired in a wideband noisy
analog environment. Suppose further that we want to store only the digital information
signal (and not the noise) on a digital medium, like memory or disc. We basically have two
approaches for removing the noise and then digitizing the information signal:
We could try to build an analog filter which would pass the analog information signal
and filter out the analog noise. Then we could sample the output of this analog filter,
digitize this output, and then store it as a sequence of digital bytes or words.
We could instead filter the noisy signal with a bandlimiting antialiasing filter, and
then sample (digitize) the noisy bandlimited analog signal. Then we could pass these
samples through a digital filter designed to separate the digital information compo
nent from the noise.
The output of this digital filter would then be stored as a sequence of digital words
representing the filtered, “less noisy” information signal.
This set of Notes examines aspects of the digital filtering approach above, in which we
construct filters which operate directly on digital samples. In this introduction we will only
examine a very basic approach for designing such digital filters. Digital Filtering is a very
broad topic and we will see only an introductory approach. However, many concepts used
in advanced filter designs will be demonstrated in this introduction.
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ECE 421  Sum 2010
Notes Set 7:
Filter Design by PoleZero Placement
2
ADDITIVE NOISE
To gain more understanding of digital filtering, let’s consider in more detail the noisy ac
quisition process described on the previous page. Let information signal be
s t
and let the
additive noise be
v t
. This noise may be from physical processes unrelated to the signal,
in which case we would like to “filter out” the noise before we store the information. An
analog block diagram for the noise addition process is shown below:
x(t) = s(t) + v(t)
+
s(t)
Information
Noise
v(t)
Mathematically we have the noisy acquired signal is given by the sum of the information
and noise:
x t
s t
v t
1
At the acquistion sensor we only have access to
x t
; that is, we can not tell which part
of
x t
is the desired
s t
and which part is the noisy
v t
. In applying digital filtering
to this noiseremoval problem, an AnalogtoDigital Converter will be sampling the noisy
analog
x t
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 Summer '08
 HALLEN
 Digital Signal Processing, Frequency, Signal Processing, θ, polezero placement

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