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Unformatted text preview: l C omputation of t he Fourier Transform: T he D FT 339 N umerical Computation o f t he Fourier Transform:
T he Discrete Fourier Transform ( OFT) Numerical computation of t he Fourier transform of I (t) requires sample values
of I (t) b ecause a digital computer can work only with discrete d ata (sequence of
numbers). Moreover, a computer can compute F(w) only a t some discrete values
of w [samples o f F(w )]. We therefore need t o r elate t he samples of F(w) t o samples
of j (t). T his t ask c an be accomplished by using t he results of the two sampling
theorems developed in Sec. 5 .l.
We begin w ith a t imelimited signal I (t) (Fig. 5.14a) and its spectrum F(w) (Fig.
5.14b). Since j (t) is timelimited, F(w) is nonbandlimited. For convenience, we shall
show all s pectra as functions of t he frequency variable F ( in Hertz) r ather t han w.
According t o t he sampling theorem, t he s pectrum F(w) of t he s ampled signal f (t)
consists of F(c.v) r epeating every Fs Hz where Fs = l iT. T his is d epicted in Figs.
5.14c a nd 5 .14d.t I n t he n ext step, t he s ampled signal in Fig. 5.14c is repeated
periodically every To seconds, as illustrated in Fig. 5.14e. According to t he s pectral
sampling theorem, such a n o peration results in sampling t he s pectrum a t a r ate
o f To samples p er Hz. This sampling r ate means t hat t he samples are spaced a t
Fo = l iTo Hz, as depicted in Fig. 5.14f.
T he a bove discussion shows t hat, when a signal j (t) is s ampled a nd t hen periodically repeated, t he corresponding spectrum is also sampled a nd periodically
repeated. O ur goal is t o r elate t he samples of j (t) t o the samples of F(w). -..,
~ 0"'1 j One i nteresting observation in Figs. 5.14e a nd 5.14f is t hat No, t he n umber of
t he number
samples of t he signal in Fig. 5.14e in one period To is identical t o
of samples of t he s pectrum in Fig. 5.14f in one period Fs. T he reason is No, a nd ' - Fs
N0 - Fo (5.1Ba) 1
F s= T and 1
F o=- (5.1Bb) B ut, because
N o= - = - =No T (5.1Bc) Fo Aliasing and Leakage in Numerical Computation F igure 5.14f shows t he presence of aliasing in t he samples o f t he s pectrum
F(w). This aliasing error can be reduced as much as desired by increasing the
sampling frequency Fs (decreasing t he sampling interval T =
T he aliasing
can never be eliminated for timelimited I (t), however, because its spectrum F(w) is
n onbandlimited. Had we s tarted o ut w ith a signal having a bandlimited spectrum
F (w), t here would be no aliasing in t he s pectrum in Fig. 5.14f. Unfortunately such i.). t There is a m ultiplying c onstant l iT for t he s pectrum i n Fig. 5 .l4d [see Eq. (5.4)], b ut t his is
i rrelevant t o o ur discussion here. s: e- " 'I t N umber o f Samples To
N o=y ? -:::- 1 3 t 340 5 Sampling a signal is n ontimelimited and its repetition (in Fig. 5.14e) would result in signal
overlapping (aliasing in the time domain). I n t his case we shall have t o contend
with errors in signal samples. In other words, in computing the direct or inverse
Fourier t ransform numerically, we c an reduce the error as much as we wish, b ut t he
error can never be eliminated...
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