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Unformatted text preview: which are duals of bandlimited signals.
We now prove t hat t he s pectrum F (w) of a signal timelimited t o T seconds can be
reconstructed from t he samples of F (w) t aken a t a r ate R > T ( the signal width)
samples per H ertz.
F igure 5 .13a shows a signal f (t) t hat is t ime limited to T seconds along with its
Fourier transform F (w). A lthough F (w) is complex in general, it is adequate for
o ur line of reasoning t o show F (w) as a real function. 1
Dn =  F(nwo)
To
~his r esult indicates t hat t he coefficients o f t he Fourier series for f ro(t) are ( liTo)
times t he sample values of t he s pectrum F (w) t aken a t intervals of w00 T his means
t hat t he s pectrum o f t he periodic signal fro (t) is t he s ampled spectrum F (w ), as
i llustrated in Fig. 5.13b. Now as long as T < To, t he successive cycles of f (t)
a ppearing in fro{t) d o n ot overlap, so t hat f (t) c an be recovered from f ru(t). Such
recovery implies indirectly t hat F (w) c an be reconstructed from its samples. These
samples are separated by t he f undamental frequency :Fo = l iTo Hz of t he periodic
signal fTo(t). T he condition for recovery is t hat To ;::: T; t hat i st
1
:Fo ~  Hz
T
Therefore, t o b e able t o r econstruct t he s pectrum F (w) from t he samples of F (w),
the samples should b e t aken a t frequency intervals not greater t han :Fo = l iT Hz.
I f R is t he s ampling r ate ( samples/Hz), t hen
1
(5.13)
R = :Fo ;::: T s amples/Hz
Spectral Interpolation T he s pectrum F (w) of a signal f (t) t imelimited t o T seconds can be reconstructed from t he samples of F (w). For this case, using t he d ual of t he a pproach
employed t o derive t he signal interpolation formula in Eq. (5.10), we o btain the
spectral interpolation formula 1 <a) <a) F (w) = L F(nwo) sinc (W2T  n1T) 21T
W o=  • :o To
r.=..!..  ( b) (e) F ig. 5 .13 ~ F (w) = JOO f (t)ejwtdt =  00 r io f (t)ejwtdt 00 ""'" Dnejnwot
~
n =oo The spectrum F(w) of a unit duration signal f (t) is sampled at the intervals of 1 Hz
or 21T rOO/s (the Nyquist rate). The samples are: (5.12) and F(±21Tn) = 0 (n 21T
w o= To = 1, 2, 3, . .. ) We use the interpolation formula (5.14) to construct F(w) from its samples. Since all
but one of the Nyquist samples are zero, only one term (corresponding to n = 0) in the
summation on the righthand side of Eq. (5.14) survives. Thus, with F(O) = 1 and T = 1,
we obtain r We now c onstruct fro (t), a periodic signal formed by r epeating f (t) every To
seconds ( To> T), as depicted in Fig. 5.13b. This periodic signal can be expressed
by t he e xponential Fourier series
f To (t) = E xample 5 .4 F(O) = 1 Periodic repetition of a signal amounts to sampling its spectrum. (5.14) T n F(w) = sinc (i) (5.15) For a signal of unit duration this is the only spectrum with the sample values F(O) = 1
and F(21Tn) = 0 (n ' " 0). No other spectrum satisfies these conditions. •
tThls r esult assumes t hat I (t) does n ot have impulses a t t = 0 o r Fo < l /r. T. I f i mpulses d o o ccur, t hen 338 5 .2 5 Sampling 5.2 Numerica...
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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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