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Unformatted text preview: rson) who can
give t he a nswers immediately, or go t o a wise m an a nd allow him a delay of one
year t o give us the answer! I f t he wise m an is t ruly wise, he may even be able t o
shrewdly guess t he future very closely with a delay of less t han a year by studying
trends. Such is t he case with noncausal s ystems-nothing more a nd n othing less.
e::. E xercise E Ll5 87 Lumped-Parameter and Distributed-Parameter Systems I n t he s tudy of electrical systems, we make use of voltage-current relationships
for various components ( Ohm's law, for example). I n doing so, we implicitly a ssume'
t hat t he c urrent in any system component (resistor, inductor, etc.) is t he s ame a t
every point throughout t hat c omponent. Thus, we assume t hat electrical signals are
propagated instantaneously throughout t he system. In reality, however, electrical
signals are electromagnetic space waves requiring some finite propagation time.
An electric current, for example, propagates through a component with a finite
velocity a nd therefore may exhibit different values a t different locations in t he same
component. Thus, a n electric current is a function not only of time b ut also of
space. However, if t he physical dimensions of a component are small compared t o
t he wavelength of t he signal propagated, we may assume t hat t he c urrent is c onstant
throughout t he component. T his is t he a ssumption made in l umped-parameter
s ystems, where each component is r egarded as being lumped a t one point in space.
Such a n a ssumption is justified a t lower frequencies (higher wavelength). Therefore,
in lumped-parameter models, signals can be assumed t o b e functions of time alone.
For such systems, t he s ystem equations require only one independent variable (time)
a nd therefore are ordinary differential equations.
In contrast, for d istributed-parameter s ystems such as transmission lines,
waveguides, antennas, a nd microwave tubes, t he s ystem dimensions cannot be assumed t o b e small compared to t he wavelengths of t he signals; thus the lumpedparameter a ssumption breaks down. T he signals here are functions of space as
well as of time, leading t o m athematical models consisting of partial differential
equations. 3 T he discussion in this book will be restricted to lumped-parameter systems only. 1 .7-6 Continuous-Time and Discrete-Time Systems D istinction between discrete-time a nd continuous-time signals is discussed in
Sec. 1.2-1. Systems whose inputs a nd o utputs a re continuous-time signals are
c ontinuous-time s ystems. O n t he o ther hand, systems whose inputs and outputs a re discrete-time signals are d iscrete-time s ystems. I f a c ontinuous-time
signal is sampled, t he r esulting signal is a discrete-time signal. We can process a
continuous-time signal by processing its samples with a discrete-time system. 1 .7-7 Analog and Digital Systems Analog a nd digital signals are discussed in Sec. 1.2-2. A system whose i nput
a nd o utput signals are analog is a n a nalog s ystem; a s ystem whose i nput a nd
o utput signals are digital is a d igital s ystem. A digital computer is a n example
of a digital (binary) system. Observe t hat a digital computer is a n e xample of a
system t hat is digital as well...
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