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Unformatted text preview: (0.8k) is n ot periodic. 8 Discrete-time Signals a nd Systems 550
6, E xercise E 8.3
S tate w ith reasons if t he following sinusoids are periodic. I f p eriodic, find t he p eriod.
( i) cos
( ii) cos (~k) (iii) cos (y'1Tk)
Ans: ( i) P eriodic: period No = 14. ( ii) a nd (iii) Aperiodic: fl.j27T i rrational.
'V (¥- <::) C omputer E xample C S.3
S ketch a nd verify if cos (¥-k) is periodic.
According t o E q. ( 8.9b), t he s mallest value of m t hat will m ake No = m (~)
m ( ¥) a n i nteger is 3. T herefore, No = 14. T his r esult means cos (¥-k) is periodic a nd
i ts p eriod is 14 s amples in t hree cycles of i ts envelop. T his a ssertion c an b e verified by t he
following M ATLAB c ommands:
t =-5*pi:pi/lOO:5*pi; t =t';
p lot(t,ft,':'), h old o n
k =-15:15; k =k';
f k=cos(k * 3*pi/7);
s tem(k,fk), h old o ff
<::) 2 8.2 Some Useful Discrete-time Signal models 551 T his result shows t hat a sinusoid cos ( nk + 8) can always be expressed a s
cos ( nfk + 8), where -11' ::s: nf < 11' ( the fundamental frequency range). T he readel;.
s hould get used t o t he fact t hat t he r ange o f discrete-time frequencies is only 211'.
We m ay select this range t o b e from -11' t o 11' o r from 0 t o 211', o r a ny other interval of
width 211'. I t is most convenient t o use t he r ange from -11' t o 11'. A t times, however,
we shall find it convenient t o use t he r ange from 0 t o 211'. T hus, in t he discrete-time
world, frequencies can be considered t o lie only in t he f undamental frequency range
(from -11' t o 11', for instance). Sinusoids of frequencies outside t he f undamental
frequencies do exist technically. B ut physically, t hey c annot be distinguished from
t he sinusoids of frequencies within t he f undamental range. Thus, a discrete-time
sinusoid of any frequency, no m atter how high, is identical t o a sinusoid of some
frequency within t he f undamental range (-11' t o 11').
T he above results, derived for discrete-time sinusoids, are also applicable t o
d iscrete-time exponentials of t he form e jrlk . For example
m, integer N onuniqueness o f D iscrete-Time Sinusoid Waveforms A c ontinuous-time sinusoid cos w t has a unique waveform for every value of w
in t he r ange 0 to 0 0. Increasing w results in a sinusoid of ever increasing frequency.
Such is n ot t he case for t he discrete-time sinusoid cos Ok because (8.12) Here we have used t he fact t hat e±j27Tn = 1 for all integral values of n . T his result
means t hat discrete-time exponentials of frequencies separated by integral multiples
of 211' a re identical.
F urther R eduction in t he F requency Range o f D istinguishable D iscrete-Time
S inusoids cos ( 0 ± 211'm)k = cos ( nk ± 211'mk)
Now, if m is a n integer, m k is also a n integer, a nd t he above equation reduces to
cos ( 0 ± 211'm)k = cos n k m integer (8.10) T his r esult shows t hat a discrete-time sinusoid of frequency 0 is indistinguishable
from a sinusoid of frequency n plus or minus a n integral multiple of 211'. This
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