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

Clearly t he t runcation of a signal causes its

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 width. 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...
View Full Document

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

Ask a homework question - tutors are online