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Unformatted text preview: lative shape of t he envelope remains t he s ame [proportional t o F(!1) in
Eq. (10.22)J. In t he limit, as No ~ 0 0, t he fundamental frequency !1o ~ 0, a nd
'Or  + O. T he s eparation between successive harmonics, which is !1o, is a pproaching
zero (infinitesimal), a nd t he s pectrum becomes so dense t hat i t appears continuous.
B ut as t he n umber of harmonics increases indefinitely, t he harmonic amplitudes
'Or become vanishingly small (infinitesimal). We have a strange s ituation of having n othing o f e verything. T his phenomenon is already discussed in C hapter 4,
where we showed t hat these are t he classic characteristics of a familiar phenomenon
(the density function).
Let us see what happens mathematically as t he p eriod No ~ 0 0. According t o
Eq. (10.22)
00 L F(r!1 o) = f[kJejrflok (10.24) k=oo 10.2 J[kJ = L F(r!1o)ejrflok (~: )  + 0 0, !1o  + 0 a nd fNo[kJ f[kJ = lim flo~O L
r=<No> + 00 e jrflok (10.26) (10.27) E quation (10.26) can be expressed as (10.28) T he range r = < No> implies t he interval o f No n umber of harmonics, which is
N oL'l.!1 = 271" according t o Eq. (10.27). In t he limit, t he r ighthand side of Eq.
(10.29) becomes the integral f [kJe jflk (10.31) T he integral on t he r ighthand side of Eq. (10.30) is called t he F ourier i ntegral. We have now succeeded in representing a n a periodic signal J[kJ by a Fourier
integral ( rather t han a Fourier series). This integral is basically a Fourier series (in
t he limit) with fundamental frequency fl.!1 ~ 0, as seen in Eq. (10.28). T he a mount
of t he e xponential e jr6flk is F(rL'l.!1)L'l.!1/271". T hus, the function F(!1) given by Eq.
(10.31) acts as a spectral function, which indicates t he relative amounts of various
exponential components of f[kJ.
We call F(!1) t he (direct) discretetime Fourier transform ( DTFT) o f f[k], a nd
J[kJ t he inverse discretetime Fourier transform ( IDTFT) o f F(!1). T his can be
represented as
J[kJ = F  1 {F(!1n a nd T he s ame information is conveyed by t he s tatement t hat J[kJ a nd F(!1) a re a
(discretetime) Fourier transform pair. Symbolically, this is expressed as
{ =} F(!1) T he Fourier transform F(!1) is t he frequencydomain description of f[kJ. N ature o f Fourier S pectra We now discuss several i mportant features o f t he d iscretetime Fourier transform a nd t he s pectra associated with it.
T he Fourier S pectra a re C ontinuous F unctions o f !1.
I t is helpful t o keep in mind t hat t he Fourier integral in Eq. (10.30) is basically
a Fourier series with fundamental frequency fl.!1 a pproaching zero [Eq. (1O.28)J.
Therefore, most o f t he discussion a nd p roperties of Fourier series apply t o t he
Fourier transform as well. T he successive harmonics are separated by t he fundamental frequency L'l.!1, which approaches zero. T his fact makes t he s pectra continuous
functions of !1.
T he Fourier S pectra a re Periodic Functions o f !1 with Period 271"
According t o Eq. (10.31) i t follows t hat L
00 (10.29) (10.3...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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