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

T he p eak s idelobe m agnitude is 0217 times the

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Unformatted text preview: han that of a tapered window. o F ig.4.47 Hanning and Hamming windows. T here a re h undreds o f windows, each w ith differing characteristics. B ut t he choice d epends o n a p articular a pplication. T he r ectangular w indow h as t he n arrowest mainlobe. T he B artlett ( triangle) window (also called t he F ejer o r C esaro) is inferior i n a ll respects t o t he H anning window. For t his r eason i t is r arely used in practice. H anning is preferred over H amming i n s pectral a nalysis because i t h as f aster sidelobe decay. For filtering applications, on t he o ther h and, t he H amming window is t he choice because i t h as t he s mallest sidelobe m agnitude for a given mainlobe width. T he H amming window is t he m ost w idely used, general p urpose window. T he K aiser window, which uses 10(0'), t he Bessel function o f t he o rder 0, is more versatile a nd a djustable. S electing a p roper v alue o f a ( 0::; a ::; 10) allows t he d esigner t o t ailor t he w indow t o s uit a p articular a pplication. T he p arameter a c ontrols t he m ainlobe a nd s ide lobe trade-off. W hen a = 0, t he K aiser window is t he r ectangular window. For a = 5.4414, i t is t he H amming window, a nd w hen a = 8.885, i t is t he B lackman window. As a increases, t he m ainlobe w idth i ncreases a nd t he s idelobe level decreases. T able 4 .3 S ome W indow F unctions a nd T heir C haracteristics Mainlobe Width Windoww(t) Rolloff Rate dB/oct Peak Sidelobe Level in dB 4" -6 - 13.3 8" Rectangular: rect(,f.) - 12 - 26.5 8" - 18 - 31.5 8" -6 - 42.7 1 2.. r- - 18 - 58.1 11.2'71' -6 - 59.9 ( a = 8.168) T 2 Bartlett: Do( ft.) 3 Hanning: 0.5 [1 4 Hamming: 0.54 + 0.46 cos ( 2;t) 5 Blackman: 0.42 + 0.5 cos ( 2 ;') + 0.08 cos (~ ) 6 Kaiser: T + cos ( 2;')1 T T Io 1 ::; a ::; 10 -r 306 4.9-1 4 Continuous-Time Signal Analysis: T he Fourier Transform 4.10 Summary Filter Design Using Windows 3 We shall design a n i deallowpass filter of bandwidth W r ad/s. For this filter, t he impulse response h(t) = ~sinc(Wt) (Fig. 4.48c) is noncausal and, therefore, unrealizable. T runcation of h(t) by a suitable window (Fig. 4.48a) makes i t realizable, although the reSUlting filter is now an approximation t o t he desired ideal filter.t We shall use a rectangular window WR(t) a nd a triangular (Bartlett) window WT(t) t o t runcate h(t), a nd then examine t he resulting filters. T he t runcated impulse responses hR(t) and hT(t) for t he two cases are depicted in Fig. (4.48d). 1 ~t-----, 1 1 - - - -.... ~ o o ~t-----' Hence, the windowed filter transfer function is the convolution of H (w) with the Fourier transform of the window, as illustrated in Fig. 4.48e and f. We make the following observations. ~t-----I e: and e: * 1. T he windowed filter spectra show s pectral s preading a t t he edges, and instead of a s udden switch there is a g radual transition from the passband to the s topband of the filter. T he t ransition b and is smaller (211" / T r ad/s) for the rectangular case...
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