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Unformatted text preview: . This fact is true of numerical computation of direct
and inverse Fourier transforms, regardless of t he m ethod used. For example, if we
d etermine the Fourier transform by direct integration numerically, using Eq. (4.8a),
there will be a n error because the interval of integration A t c an never be made
zero. Similar r emarks apply t o numerical computation of the inverse transform.
Therefore, we s hould always keep in mind the n ature of this error in our results.
In our discussion (Fig. 5.14), we assumed f (t) t o b e a timelimited signal. I f f (t)
is not timelimited, we would need t o timelimit it because numerical computations
can work only w ith finite data. Further, this d ata t runcation causes error because
of spectral spreading (smearing) and leakage, as discussed in Sec. 4.9. T he leakage
also causes aliasing. Leakage can be reduced by using a tapered window for signal
truncation. B ut t his choice increases spectral spreading or smearing. The spectral
spreading can b e reduced by increasing the window width (Le. more data), which
increases To, a nd reduces :Fo (increases s pectral or f requency r esolution). Picket Fence Effect
T he n umerical computation method yields only the uniform sample values of
F (w) [or f (t)]. Using this method is like viewing a signal and its spectrum through a
"picket-fence." T he m ajor peaks of F (w) [or f (t)] could lie between two samples and
may remain hidden, a situation giving a false picture of the reality. Such misleading
results can be avoided by using a sufficiently large No, t he number of samples, which
increases resolution. We c an also use the spectral interpolation formula [Eq. (5.14»)
t o d etermine t he values of F (w) between samples. 5'.2 Numerical Computation of the Fourier Transform: The D FT N o-l Fr = i ke-Jrflok (5.18a) . N o-l "
f k - ~ L.., Fr e jrflok
No r =O 27r
flo = woT=No (5.18b) These equations define the direct and t he inverse d iscrete F ourier t ransforms
( DFT), w ith Fr t he direct discrete Fourier transform (DFT) of i k, a nd fk t he
inverse discrete Fourier transform (IDFT) of Fr. T he n otation fk ¢ =} Fr
is also used to indicate t hat fk a nd Fr are a D FT pair. Remember t hat fk is T o/No
times the k th sample of f (t) a nd Fr is t he r th sample of F (w). Knowing the sample
values of f (t), we c an compute the sample values of F (w)-and vice v ersa-using
t he D FT. Note, however, t hat fk is a function of k (k = 0 ,1,2, . .. , No - 1) r ather
t han o f t a nd t hat Fr is a function of r (r = 0 ,1,2, . .. , No - 1) r ather t han of
w. Moreover, both fk a nd Fr are periodic sequences of period No (Figs. 5.14e and
5.14f). Such sequences are called N o-periodic s equences. T he proof of the D FT
relationships in Eqs. (5.18) follows directly from the results of the sampling theorem.
The sampled signal f (t) (Fig. 5.14c) can be expressed as
N o-l L 7 (t) = f (kT)b (t - kT) (5.19) k=O
Since b (t - kT) ¢ =} e - jkwT , t he Fourier transform of Eq. (5.19) yields
N o-l L F (w) = f (kT)e- jkwT (5.20) k=O
B ut from Fig. 5.1e [or Eq. (5.4)], i t...
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