Signal Processing and Linear Systems-B.P.Lathi copy

For this case using t he d ual of t he a pproach

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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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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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