Unformatted text preview: e,
o r realizable with infinite time delay. Therefore, in practice, there is always a n e rror
in reconstructing a signal from its samples. Moreover, practical signals are n ot bandlimited, which causes a n a dditional error (aliasing error) in signal reconstruction
from its samples. Aliasing error can be eliminated by bandlimiting a signal t o its
effective bandwidth.
T he s ampling theorem is very i mportant i n signal analysis, processing, a nd
t ransmission because it allows us t o replace a continuoustime signal with a discrete
sequence o f numbers. Processing a continuoustime signal is therefore equivalent to
processing a discrete sequence of numbers. This leads us directly into t he a rea of
digital filtering (discretetime systems). In t he field of communication, t he t ransmission o f a c ontinuoustime message reduces t o t he t ransmission of a sequence of
numbers. This opens doors t o m any new techniques of communicating continuoustime signals by pulse trains.
T he d ual of t he sampling theorem s tates t hat for a signal timelimited t o T
seconds, its s pectrum F(w) c an be reconstructed from t he samples of F(w) t aken
a t uniform intervals n ot g reater t han I /T Hz. I n o ther words, t he s pectrum should
be sampled a t a r ate n ot less t han T s amples/Hz.
To compute t he d irect or inverse Fourier transform numerically, we need a relationship between t he samples of j (t) a nd F(w). T he sampling theorem a nd i ts dual
provide such a quantitative relationship in t he form o f a discrete Fourier transform
(DFT). T he D FT c omputations are greatly facilitated by a fast Fourier transform
( FFT) a lgorithm, which reduces t he n umber of computations from something o f t he
o rder o f N J t o No log No. W e show t hat 2::: N ol
e imflok = m = 0, ±No, ±2No, . .. { No
0 o therwise k=O Recall t hat n oNa = 2rr a nd N ol 2::: k =O e imflok References
(5.43) = 1 for m = 0, ± No, ± 2No , . .. , so t hat N ol
e imflok = 2::: 1 = No for m = 0, ±No, ±2No, . .. k=O To c ompute t he s um for o ther values of m, we note t hat t he s um o n t he l efthand side
3mflo
of Eq. (5.43) is a geometric progression w ith common ratio a = e
. T herefore,
(see Sec. B.74)
N ol e imfloNo _ " " ei mflok = e3mflO  l
.
~ 1 1. Linden, D.A., "A Discussion of S ampling Theorem," Proc. I RE, vol. 47, pp
12191226, J uly 1959. 2. Bracewell, R.N., The Fourier Transform and Its Applications, 2nd revised ed.,
McGrawHill, New York, 1986.
3. B ennett, W.R., Introduction to Signal T ransmission, McGrawHill, New York,
1970.
4. L athi, B.P., L inear S ystems and Signals, BerkeleyCambridge Press, C armichael, CA, 1992.
5. Cooley, J .W., a nd J .W., Tukey, "An A lgorithm for t he Machine Calculation of
Complex Fourier Series," M athematics o f C omputation, Vol. 19, pp. 297301,
April 1965. =0 Problems k =O t Actually,
where W 5 .11
~ is a conservative figure because some multiplications corresponding t o t he cases No = I ,j, e tc., a...
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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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