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Unformatted text preview: on u[k]. (8.3b) For example, e 3k = ( e 3 )k = ( 20.086)k. Similarly, 5k = e1.609k because 5 = e 1.609.
I n t he s tudy of discrete-time signals and systems, unlike t he continuous-time case,
t he form -yk proves more convenient t han t he form e Ak . Because of unfamiliarity
with exponentials with bases other t han e , exponentials of the form -yk may seem
inconvenient and confusing a t first. T he reader is urged to plot some exponentials
t o acquire a sense of these functions.
T he signal e Ak grows exponentially with k if Re A > 0 ('x in
RHP), and decays exponentially if Re ,x < 0 (,x in LHP). I t is c onstant or oscillates
with constant amplitude i fRe A = 0 (,x on t he imaginary axis). Clearly, the location
of ,x in the complex plane indicates whether the signal e>..k grows exponentially,
decays exponentially, or oscillates with constant frequency (Fig. 8.5a). A c onstant
signal (A = 0) is also an oscillation with zero frequency. We now find a similar
criterion for determining the n ature of -yk from t he location of -y in the complex
Figure 8.5a shows a complex plane (A-plane). Consider a signal e jOk • In this
case, ,x = j O lies on the imaginary axis (Fig. 8.5a), a nd therefore is a c onstantamplitude oscillating signal. This signal e jOk c an be expressed as -yk, where -y = e jo .
Nature o f u [ k] (8.3a) For example, e - O.3t = ( 0.7408)t because e - O.3 = 0.7408. Conversely, 4t = e1.386t
because In 4 = 1.386, t hat is, e1.386 = 4. I n t he s tudy of continuous-time signals
and systems we prefer the form e At r ather t han " (t. T he discrete-time exponential
can also be expressed in two forms as
h =e A or A =ln-y) T he d iscrete-time counterpart of t he u nit step function u(t) is u[k] (Fig. 8.4),
for k 2': 0
for k < 0 2 can be expressed in an alternate form as h =e A or A =ln-y) 2. D iscrete-Time U nit Step Function u[k] o e At (8.1) T his function, also called t he u nit impulse sequence, is shown in Fig. 8.3a. T he timeshifted impulse sequence 8 [k-m] is depicted in Fig. 8.3b. Unlike its continuous-time
counterpart c5(t), this is a very simple function without any mystery.
Later, we shall express an arbitrary input f [k] in terms of impulse components.
T he ( zero-state) system response t o i nput f [k] c an then be obtained as the sum of
system responses t o impulse components of f [k]. -2 (b) F ig. 8 .5 The A-plane, the -y-piane and their mapping. - yk: 544 8 Discrete-time Signals and Systems 8.2 Some Useful Discrete-time Signal models Because t he m agnitude of e is unity, hi = 1. Hence, when A lies on t he i maginary
axis, t he c orresponding "( lies on a circle of unit radius, centered a t t he origin (the
u nit c ircle i llustrated in Fig. 8.5b). Therefore, a signal " (k oscillates with constant
amplitude i f "( lies on t he u nit circle. Remember, also, t hat a c onstant signal
(A = 0, "( = 1 ) is a n oscillating signal with zero frequency. Thus, t he i maginary axis
in t he A-plane m aps into...
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