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Unformatted text preview: odic s ignal I[k] b y c onsidering J[k] a s a l imiting c ase o f
a p eriodic signal w ith t he p eriod a pproaching infinity. 10.1 Periodic Signal Representation by Discrete-Time
Fourier Series (DTFS) A p eriodic s ignal o f p eriod No is called a n N o-periodic signal. F igure 8 .9 shows
a n e xample o f a p eriodic s ignal o f p eriod 6. A c ontinuous-time p eriodic s ignaf o f
p eriod To c an b e r epresented a s a t rigonometric F ourier series c onsisting o f a sinusoid o f t he f undamental f requency Wo = ~, a nd a ll i ts h armonics ( sinusoids o f
f requencies t hat a re i ntegral m ultiples o f wo). T he e xponential f orm o f t he F ourier
series consists o f e xponentials e jOt , e ±jwot, e ±j2wot, e ±j3w0 t, .... F or a p arallel d evelopment o f t he d iscrete t ime c ase, recall t hat t he f requency o f a s inusoid o f p eriod
No is n o = 21r/No. H ence, a n N o-periodic d iscrete-time s ignal I[k] c an b e represented b y a d iscrete-time F ourier series w ith f undamental f requency n o = 'flo
a nd i ts h armonics. A s i n t he c ontinuous-time case, we m ay u se a t rigonometric o r
a n e xponential f orm o f t he F ourier series. B ecause o f i ts c ompactness a nd e ase o f
m athematical m anipulations, t he e xponential f orm is preferable t o t he t rigonometric. F or t his r eason we s hall b ypass t he t rigonometric form a nd go d irectly t o t he
e xponential f orm o f t he d iscrete-time F ourier series.
T he e xponential F ourier series consists o f t he e xponentials e jOk e ±jflok e ±j2flo k
e ±jnflok, . .. , a nd s o o n. T here w ould b e a n i nfinite numb~r o f h~rmonics:
617 10 Fourier Analysis o f D iscrete-Time Signals 10.1 P eriodic Signal R epresentation b y Discrete-Time Fourier Series ( DTFS) 619 e xcept for t he p roperty proved in Sec. 8.2: t hat d iscrete-time exponentials whose
frequencies are s eparated b y 21r ( or integral multiples o f 21r) a re identical because We now have a discrete-time Fourier series ( DTFS) r epresentation of a n N o-periodic
signal f[k] a s 6 18 N o-I J[k] = (10.1) L v rejrrlok (10.8) r =O T he consequence of this result is t hat t he r th h armonic is identical t o t he ( r+No)th
n armonic. To d emonstrate t his, let 9n d enote t he n th h armonic ejnrlok. T hen w here
1 Vr No (10.2)
9r = 9r+No = 9r+2No = . .. = 9r+mNo m , i nteger (10.3) T hus, t he first harmonic is identical t o t he (No + 1) st h armonic, t he s econd harmonic
i s i dentical t o t he ( No+2)nd h armonic, a nd so on. I n o ther words, t here a re o nly No
i ndependent h armonics, a nd t hey r ange over a n i nterval 21r ( because t he h armonics
a re s eparated by n o =
We m ay choose t hese No i ndependent h armonics as
e jrrlok over 0 ::; r ::; No - 1, or over - 1 ::; r ::; No - 2, or over 1 ::; r ::; No, o r over
a ny o ther s uitable choice for t hat m atter. E veryone o f t hese s ets will have t he s ame
narmonics, a lthough i n different order. L et us t ake t he first choice (0 ::; r ::; No - 1).
T his choice corresponds t o e xponential...
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