Biomedical Engineering Reference
In-Depth Information
tal signal processors operates on binary numbers that are sets of bits. A bit is
quantal; it contains the smallest possible chunk of information: a single ON or OFF
signal. More useful binary numbers are created by aggregating between 8 and 80
bits. The accuracy or resolution (
q
) of binary numbers is determined by the number
of bits they contain: An 8-bit binary number can represent 2
8
or 1 of 256 possible
states at any given time; a 16-bit number, 2
16
or 65,536 possible states. If one were
using an 8-bit number to represent an analog signal, the binary number would have,
at best, a resolution of approximately 0.4% (1/256) over its range of measurement.
Assuming, for example, that a converter was designed to measure voltages ranging
from
1.0V, the step size of an 8-bit converter would be about 7.8 mV and a
16-bit converter about 30
−
1.0 to
+
V. Commercial EEG monitoring systems use 12 to 16
bits of resolution. More bits also create a wider dynamic range with the possibility of
recovery from more artifact; however, more bits increase the expense dramatically.
Digital signals are also quantized in time, unlike analog signals, which vary
smoothly from moment to moment. When translation from analog to digital occurs,
it occurs at specific points in time, whereas the value of the resultant digital signal at
all other instants in time is indeterminate. Translation from the analog to digital
world is known as sampling, or digitizing, and in most applications is set to occur at
regular intervals. The reciprocal of the sampling interval is known as the sampling
rate (
fs
) and is expressed in hertz (Hz or samples per second). A signal that has been
digitized is commonly written as a function of a sample number,
i
, instead of analog
time,
t
. An analog voltage signal written as
v
(
t
), would be referred to, after conver-
sion, as
v
(
i
). Taken together, the set of sequential digitized samples representing a
finite block of time is referred to as an epoch.
When sampling is performed too infrequently, the fastest sine waves in the
epoch will not be identified correctly. When this situation occurs, aliasing distorts
the resulting digital data. Aliasing results from failing to meet the requirement of
having a minimum of two points within a single sinusoid. If sampling is not fast
enough to place at least two sample points within a single cycle, the sampled wave
will appear to be slower (longer cycle time) than the original.
Aliasing is familiar to observers of the visual sampled-data system known as cin-
ema. In a movie, where frames of a scene are captured at rate of approximate 24 Hz,
rapidly moving objects such as wagon wheel spokes often appear to rotate slowly or
even backwards. Therefore, it is essential to always sample at a rate more than twice
the highest expected frequency in the incoming signal (Shannon's sampling theorem
[25]). Conservative design actually calls for sampling at a rate 4 to 10 times higher
than the highest expected signal, and to also use a lowpass filter prior to sampling to
eliminate signals that have frequency components that are higher than expected.
Lowpass filtering reduces high-frequency content in a signal, just like turning down
the treble control on a stereo system. In monitoring work, EEG signals have long
been considered to have a maximal frequency of 30 or 40 Hz, although 70 Hz is a
more realistic limit. In addition, other signals present on the scalp include power-line
interference at 60 Hz and the EMG, which, if present, may extend above 100 Hz. To
prevent aliasing distortion in the EEG from these other signals, many digital EEG
systems sample at a rate above 250 Hz (i.e., a digital sample every 4 ms).
μ
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