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Unformatted text preview: soid cos
Here we make one basic observation. Because cos ( - x) = cos (x), (n- cos ( -Ok + 0) = cos (Ok - 0) (8.6) T his shows t hat b oth cos (Ok + 0) a nd cos ( -Ok + 0) have t he same frequency ( 0).
Therefore, t he f requency o f cos (Ok + 0) i s 101. A continuous-time sinusoid cos w t s ampled every T seconds yields a discretetime sequence whose k th element ( at t = k T) is cos w kT. T hus, the sampled signal
I[k] is given b y I[k] = cos w kT
= cos Ok where 0 = wT (8.7) Clearly, a continuous-time sinusoid cos w t s ampled every T seconds yields a discretetime sinusoid cos Ok, where 0 = wT. Superficially, it may appear t hat a discretetime sinusoid is a continuous-time sinusoid's cousin in a striped suit. As we shall
see, however, some of t he p roperties of discrete-time sinusoids are very different
from those of continuous-time sinusoids. In t he c ontinuous-time case, t he period
of a sinusoid can take on any value; integral, fractional, or even irrational. T he
discrete-time signal, in contrast, is specified only a t i ntegral values of k. Therefore,
t he period must be a n integer (in terms of k ) or a n integral multiple of T (in terms
of variable t). S ome Peculiarities o f D iscrete-Time Sinusoids
T here are two unexpected properties of discrete-time sinusoids which distinguish t hem from their continuous-time relatives.
1. A c ontinuous-time sinusoid is always periodic regardless of the value of its
frequency w. B ut a discrete-time sinusoid cos Ok is periodic only if 0 is 211"
t imes some rational number (
is a r ational number).
2. A continuous-time sinusoid cos w t has a unique waveform for each value of w.
I n contrast, a sinusoid cos Ok does n ot have a unique waveform for each value
of O. In fact, discrete-time sinusoids with frequencies separated by multiples of
211" are identical. Thus, a sinusoid cos Ok = cos (0+211")k = cos (0+411")k = ....
We now examine each of these peculiarities. £r 1 N ot All D iscrete-Time S inusoids Are Periodic
A discrete-time signal I[k] is said t o be N o-periodic if I[k] = I[k + No] (8.8) 548 8 Discrete-time Signals a nd Systems for some positive integer No. T he smallest value of No t hat satisfies Eq. (8.8) is
t he p eriod of f[k]. F igure 8.9 shows a n example of a periodic signal of period 6.
Observe t hat e ach period contains 6 samples (or values). I f we consider t he first
cycle to s tart a t k = 0, t he l ast sample (or value) in this cycle is a t k = No - 1 = 5
( not a t k = N o = 6). Note also t hat, by definition, a periodic signal must begin a t
k = - 00 ( everlasting signal) for t he reasons discussed in Sec. 1.2-4. 8.2 549 Some Useful Discrete-time Signal models ,I!I'III,III '1lI,Ill ,... ,IiI. III.!! L~ ;:': III
- 16 -8 8. 1 6. '. ~a) k __ J[k] ·.l!llllllIIlII [l) 1111 !11111111 L
- 12 -6 F ig. 8 .9 D iscrete-time periodic signal. I f a signal cos n k is No-periodic, t hen
cos n k = cos n (k
= cos (Ok + No) (c) + n No) T his result is possible only if n No is a n integral multiple of 271'; t hat is, n No = 271'm m integer F ig. 8 .10 or P...
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