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Unformatted text preview: obe w idth), we need t o i ncrease t he w indow width. 2 T o improve t he leakage behavior, we m ust s earch for t he c ause of t he slow decay
o f s idelobes. I n C hapter 3, we saw t hat t he F ourier s pectrum d ecays as 1 /w for
a signal w ith j ump d iscontinuity, a nd d ecays as 1/w2 for a continuous signal
whose first d erivative is discontinuous, a nd so o n.t S moothness o f a s ignal is
m easured b y t he n umber of continuous derivatives i t possesses. T he s moother
t he s ignal, t he f aster t he d ecay o f i ts s pectrum. T hus, we c an achieve a given
leakage b ehavior b y s electing a s uitably s mooth window.
3 F or a given window w idth, t he r emedies for t he t wo effects a re i ncompatible.
I f we t ry t o i mprove one, t he o ther d eteriorates. For instance, a mong a ll t he
windows o f a given w idth, t he r ectangular w indow has t he s mallest s pectral
s pread ( mainlobe w idth), b ut h as h igh level sidelobes, which decay slowly. A
t apered ( smooth) window of t he s ame w idth h as s maller a nd f aster decaying
sidelobes, b ut i t h as a w ider mainlobe.:j: B ut we can c ompensate for t he increased m ainlobe w idth by widening t he window. Thus, we c an r emedy b oth
t he side effects o f t runcation b y s electing a s uitably s mooth w indow o f sufficient
T here a re s everal well-known t apered-window f unctions, such as B artlett ( triangular), H anning ( von H ann), H amming, B lackman, a nd K aiser, which t runcate
t he d ata g radually. T hese windows offer different tradeoffs w ith r espect t o s pectral
s pread ( mainlobe w idth), t he p eak s idelobe m agnitude, a nd t he leakage rolloff r ate
as i ndicated i n T able 4.3. 8 ,9 O bserve t hat all windows are s ymmetrical a bout t he
o rigin (even f unctions of t). B ecause o f t his f eature, W(w) is a real function o f w;
t hat is, L W (w) is e ither 0 o r 7 r. Hence, t he p hase f unction o f t he t runcated s ignal
h as a m inimal a mount of distortion.
F igure 4 .47 s hows two well-known tapered-window functions, t he v on H ann
(or Hanning) w indow W HAN(X) a nd t he H amming window W HAM(X), W e have
intentionally u sed t he i ndependent v ariable x b ecause windowing c an b e p erformed
in t ime d omain as well as in frequency domain; so x could b e t o r w , d epending o n
t he a pplication.
tThis result was demonstrated for periodic sign:-ls. How~ver, it ~pplies t o aperiodic signals .als?
This is because w e showed in Chapter 4 that If fTo ( t) IS a periodic Signal formed by periodic
extension of an aperiodic signal f (t), tben the spectrum of fTo (t) is ( l/To times) the samples of
F(w). Thus, what is true of the decay rate of the spectrum of h o(t) is also true of the rate of
decay of F(w).
:j:A tapered window yields a higher mainlobe width because the effective width of a tapered :v~ndow
is smaller than t hat of the rectangular window (see Sec. 2.7-2 [Eq. (2.67)J for the defimtlOn of
effective width). Therefore, from the reciprocity of the signal width and its bandwidth, it follows
that the rectangular window mainlobe width is smaller t...
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