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Unformatted text preview: c haracteristics c an b e e asily a ltered
s imply b y c hanging t he p rogram.
3. A g reater v ariety o f filters c an b e r ealized b y d igital s ystems.
4. Very low frequency filters, i f r ealized b y c ontinuous-time s ystems, r equire p rohibitively b ulky c omponents. S uch is n ot t he c ase w ith d igital f ilters.
5. D igital s ignals c an b e s tored e asily o n m agnetic t apes o r d isks w ithout d eterioration o f s ignal q uality.
6. M ore s ophisticated s ignal processing a lgorithms c an b e u sed t o p rocess d igital
7. D igital f ilters c an b e t ime s hared, a nd t herefore c an s erve a n umber o f i nputs
s imultaneously. P ractical Realization o f D iscrete-Time S ystems
T hese e xamples show t hat t he b asic e lements r equired i n t he r ealization of
d iscrete-time s ystems a re t ime d elays, s calar m ultipliers, a nd a dders ( summers). tThe terms d iscrete-time and c ontinuous-time qualify the nature of a signal along the time axis
(horizontal axis). The terms analog and digital, in contrast, qualify the nature of the signal
amplitUde (vertical axis). 5 68 8 D iscrete-time Signals a nd Systems 8. Using integrated circuit technology, they can be fabricated in small packages
requiring low power consumption.
Some more advantages of using digital signals are listed in Sec. 5.1-3. 8 .6 Summary Signals specified only a t discrete instants such as t = 0, T, 2T, 3T, . .. , kT are
discrete-time signals. Basically, it is a sequence of numbers. Such a signal may be
viewed as a function o f t ime t , where the signal is defined o r specified only a t t = kT
w ith k any positive o r negative integer. The signal therefore may be denoted as
f (kT). Alternately, such a signal may be viewed as a function of k, where k is any
positive or negative integer. T he l atter approach results in a more compact notation
s uch as f [k], which is convenient a nd easier t o m anipulate. A system whose inputs
a nd o utputs a re discrete-time signals is a discrete-time system.
In t he s tudy o f continuous-time systems, exponentials with t he n atural base;
t hat is, exponentials of t he form e At, where A is comple~ in general, are more natural
a nd convenient. In contrast, in t he s tudy of discrete-time systems, exponentials with
a general base; t hat is, exponentials o f t he form '"Yk, where '"Y is complex in general,
a re m ore convenient. One form of exponential can be readily converted t o t he o ther
form by noting t hat eAk = '"Yk, where'"Y = e \ or A = In'"Y, a nd A as well as '"Y
a re complex in general. T he e xponential '"Yk grows exponentially with k i f bl > 1
h outside t he unit circle), a nd decays exponentially if I'"YI < 1 h within t he unit
circle). I f I'"YI = 1; t hat is, if'"Y lies on t he u nit circle, t he e xponential is e ither a
c onstant or oscillates with a constant amplitude.
Discrete-time sinusoids have two properties n ot s hared by their continuoustime cousins. First, a discrete-time sinu...
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