Unformatted text preview: s e jrrlok for r = 0, 1, 2, . .. , No  1. T he
F ourier series for a n Noperiodic signal f[k] c onsists o f o nly these No h armonics,
a nd c an b e expressed as i:a. N oI f[k] = L n _ 21r
"0 v rejrrlok No r=O (lOA) T o c ompute coefficients V r i n t he F ourier series ( lOA), we multiply b oth sides of
(1004) by e jmrlok a nd s um over k from k = 0 t o (No  1).
N oI L N oI N oI f[k]ejmrlok = k =O LL k=O V rej(rm)rlok (10.5) r =O T he r ighthand sum, after interchanging t he o rder o f s ummation, r esults in
N oI ~ Vr [ NOI ~ e j(rm)rlok (10.6) T he i nner s um, according t o E q. (5043), is zero for all values of r t m . I t is nonzero
w ith a value No only when r = m . T his fact m eans t he o utside s um h as only one
t erm V mN o ( corresponding t o r = m ). T herefore, t he r ighthand side o f E q. (10.5)
i s e qual t o VmNO, a nd
N oI L = 21r N oI L f [k]ejrrlok 1 0.11 Fourier Spectra o f a Periodic Signal f[k] T he F ourier series consists of No c omponents T he frequencies o f t hese c omponents a re 0, n o, 2 n o, . .. , (No  l)no w here n o =
21r/No. T he a mount o f t he r th h armonic is V r. We c an p lot t his a mount V r ( the
F ourier coefficient) as a function o f n . S uch a plot, called t he F ourier s pectrum o f
f[k], gives us, a t a glance, t he g raphical p icture o fthe a mounts o f v arious harmonics
of f[k].
I n general, t he F ourier coefficients V r a re complex, a nd t hey c an b e r epresented
in t he p olar form as
(10.10)
T he p lot o f IVrl vs. n is called t he a mplitude s pectrum a nd t hat o f L Vr vs. n is
called t he a ngle (or phase) s pectrum. T hese two plots t ogether a re t he frequency
s pectra o f J[k]. K nowing t hese s pectra, we c an r econstruct o r s ynthesize f[k] according t o E q. (10.8). Therefore, t he F ourier (or frequency) s pectra, which are a n
a lternative way of describing a signal f[k], a re i n every way equivalent (in t erms
o f t he i nformation) t o t he p lot o f f[k] a s a function o f k. T he F ourier s pectra o f
a signal c onstitute t he f requencydomain d escription of f[k]' i n c ontrast t o t he
t imedomain description, where f[k] is specified as a function o f t ime ( k).
T he r esults are very similar t o t he r epresentation o f a continuoustime periodic
signal by a n e xponential Fourier series except t hat, generally, t he c ontinuoustime
signal s pectrum b andwidth is infinite, a nd c onsists of a n infinite n umber o f exponential components (harmonics). T he s pectrum o f t he d iscretetime periodic signal,
in c ontrast, is b andlimited a nd h as a t m ost No c omponents.
Periodic Extension o f Fourier Spectrum
N oI L a nd r =O No (10.9) No k =O N ote t hat i f 4>[r] is a n N operiodic function of r , t hen f[k]ejmrlok = V mNo ~ = O bserve t hat D TFS e quations (10.8) a nd (10.9) are identical (within a scaling constant) t o t he D FT e quations (5.18b) a nd ( 5.18a).t Therefore, we c an c ompute t he
D TFS coefficients using t he efficient F FT a lgo...
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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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