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 tradeoff. 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
4.91 4 ContinuousTime 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. ~tI 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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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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