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Intro to ECE Design
Drs. Butera/Williams
Module:
Digital Signals/Representations
2008, Douglas B. Williams and Robert J. Butera, Jr.
Page 1
Objectives
Upon completion of this module, you should be able to:
•
understand uniform quantizers, including dynamic range and sources of error,
•
represent numbers in two’s complement binary form,
•
assign binary symbols to quantized signal values, and
•
scale a signal to fit within a specified range.
What are signals?
As we saw in the
“Volts
and Amps”
module,
sensors typically convert
observed data into time
varying voltages. In other
words the sensors act as
transducers
that convert
environmental data into a
corresponding current
flow that can then be
observed and measured
with electronics as a
changing voltage. The resulting data are functions of time with units in volts and are
known as ‘signals.’
What does such a voltage signal look like? Signals are commonly observed and graphed
as functions of time:
Such a function is known as a
continuoustime signal
. These signals are defined for all
values of
t
and take on a continuous range of voltage values.
Problem: Digital computers cannot store this type of data.
+

Voltage
sensor
environment
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View Full DocumentIntro to ECE Design
Drs. Butera/Williams
Module:
Digital Signals/Representations
2008, Douglas B. Williams and Robert J. Butera, Jr.
Page 2
Numerical quantities represented by computers have finite precision, as they must be
represented by a small number of bits. We have to
quantize
the values in
v
(
t
) into the
finite precision values that can be stored in however many bits are allowed per number.
Quantization
Assume that the observed voltage is known to range from a minimum of
v
min
to a
maximum of
v
max
. A
quantizer
is a system that divides this continuous range of values
into a finite number of discrete values. The most common quantizers are uniform
quantizers that divide the range of values into equalsized intervals. Here are two typical
uniform quantizers for a system with 3 bits of precision.
For
N
bits of precision a quantizer has 2
N
output levels.
Thus, these 3bit quantizers have
8 levels. Midstep quantizers are generally preferred over midrise quantizers as they are
less sensitive to noise at low signal levels. Let
Δ
denote the size of the quantization levels
in these uniform quantizers. Note that for any input value of
v
(
t
) between
−Δ
/2 and
Δ
/2
the output of the midstep quantizer is the same, while the midrise quantizer will change
output levels whenever the input crosses zero. For a small signal that is approximately
zero but is varying slightly because of noise or similar observational variations, the
midrise quantizer’s toggling between
−Δ
/2 and
Δ
/2 can actually magnify the effect of
these small distortions. The midstep quantizer is slightly asymmetric with more negative
output levels than positive, but that difference becomes insignificant as the number of bits
increases.
Question: The LEGO Mindstorms NXT sensors use 10 bits to encode data. How
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