Signal Processing and Linear Systems-B.P.Lathi copy

Signal Processing and Linear Systems-B.P.Lathi copy

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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 k) ( 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'; f t=cos(3"'pi*t/7); 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 s ta...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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