Signals and systems - Biomedical Engineering Desk Reference - page 66

2 Nu bits ¼ 10

V max 10

4 ; 096 ¼ 0 : 0024 volts

composed of several processes, but the two terms are
often used interchangeably, as they will be throughout
this text. To study and quantify complex processes, we
often impose rather severe simplifying constraints. The
most common assumption is that the process and its
components or subprocesses behave in a linear manner,
and that their basic characteristics do not change over
time. This assumption is referred to as the ''linear time-
invariant'' (LTI) model. Such an assumption allows us to
apply a powerful array of mathematical tools that are
known collectively as linear systems analysis. Of course,
most living systems change over time, are adaptive, and
are often nonlinear. Nonetheless, the power of linear
systems analysis is sufficiently seductive that assump-
tions or approximations are often made so that these
tools can be used. Linearity can be approximated by using
small-signal conditions where many systems behave
more or less linearly. Alternatively, piecewise linear ap-
proaches can be used where the analysis is confined to
operating ranges over which the system behaves linearly.
One approach to dealing with a process that changes over
time is to study that process within a short enough time
frame that it can be considered time-invariant.
The concept of linearity has a rigorous definition, but
the basic concept is one of proportionality of response. If
you double the stimulus into a linear system, you will get
twice the response. One way of stating this proportion-
ality property mathematically is the following: if the in-
dependent variables of linear function are multiplied by
a constant, k, the output of the function is simply mul-
tiplied by k :

V LSB ¼

2 12 ¼

Hence, the resolution would be 0.0024 V.
It is relatively easy, and common, to convert between
the analog and digital domains using electronic circuits
specially designed for this purpose. Many medical de-
vices acquire the physiological information as an analog
signal but convert it to digital format using an ADC so
that it can be processed using a computer. For example,
the electrical activity produced by the heart can be
detected using properly placed electrodes, and the
resulting signal, the ECG, is an analog-encoded signal.
This signal might undergo some preprocessing or condi-
tioning using analog electronics before being converted to
a digital signal using an ADC. The converted digital signal
would be sent to a computer for more complex pro-
cessing and storage. (In fact, conversion to digital format
is usually done even if the data are only to be stored for
later use.) Conversion from the digital to the analog
domain is possible using a digital-to-analog converter
(DAC). Most personal computers include both ADCs
and DACs as part of a sound card. This circuitry is spe-
cifically designed for the conversion of audio signals, but
can be used for other analog signals. Data transformation
cards designed as general-purpose ADCs and DACs are
readily available and offer greater flexibility in sampling
rates and conversion gains. These cards provide multi-
channel ADCs (usually eight to 16 channels) and several
channels of DAC.
Basic concepts that involve signals are often in-
troduced or discussed in terms of analog signals, but most
of these concepts apply equally well to the digital
domain. In this text, the equivalent digital domain
equation is often presented alongside the analog equation
to emphasize the equivalence. Many of the problems and
examples use a computer, so they obviously are being
implemented in the digital domain even if they are
presented as analog-domain problems.

y ¼ fðxÞ ;

where f is a linear function, then:

ky ¼ fðkxÞ [Eq. 2.3.3]

Note that:

ky ¼ ð fðkxÞdt

ky ¼ dfðkxÞ
dt

2.3.3 Linear signal analysis:
overview

and [Eq. 2.3.4]

Hence, differentiation and integration are linear op-
erations. The major transforms described in this text, the
Fourier transform and the Laplace transform, are also
linear processes.
Response proportionality, or linearity, is required for
the application of an important concept known as su-
perposition. Superposition states that if there are two
(or more) stimuli acting on the system, the system re-
sponds to each as if it were the only stimulus present.
The combined influence of the multiple stimuli is simply
the addition of each stimulus acting alone. This allows
complex stimuli to be broken down so that the problem

From a mechanistic point of view, all living systems are
composed of processes. These processes act, or interact,
through manipulation of molecular mechanisms, chem-
ical concentrations, ionic electrical current, and/or
mechanical forces and displacements. A physiological
process performs some operation(s) or manipulation(s)
in response to a specific input (or inputs), which gives
rise to a specific output (or outputs). In this regard,
a process is the same as a system and would be system-
atically represented as shown in Figure 2.3-1 . Sometimes
the term system is reserved for larger structures