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Unformatted text preview: c s pectrum of the window function. Because the convolution of any function
with an impulse is t he function itself (shifted a t t he location of t he impulse), the
resulting s pectrum of the truncated signal is (1/2rr times) the two sinc pulses at
±wo, as depicted in Fig. 4.46c. Comparison of spectra F(w) a nd Fw(w) reveals the
effects of t runcation. These are:
T he s pectral lines of F(w) have zero width. B ut the truncated signal is spread
out by 47r / T a bout each spectral line. T he amount of spread is equal t o t he 303 t - t nOl f it) - £0 0 £0 0 £ 0_ ( aj -T 2" 1 W R(t ) I L RolloffRa'e t_ 2 £0 _ ( b) o l~j
.T• Mainlobe 1"-;. (£0)1
in dB Z!l>t
T £0 - F ig. 4 .46 Windowing and its effects.
w idth of the mainlobe of t he window spectrum. One effect of this s pectral
s preading (or smearing) is t hat if f (t) h as two spectral components of frequencies differing by less t han 47r / T r ad/s ( 2/T Hz), they will be indistinguishable
in t he t runcated signal. T he result is loss of spectral resolution. We would like
t he s pectral spreading (mainlobe width) t o b e as small as possible.
2 In addition to the main lobe spreading, the t runcated signal also has sidelobes
which decay slowly with frequency. The spectrum of f (t) is zero everywher~
except a t ±wo· On t he other hand, t he t runcated signal spectrum Fw(w) is zero
nowhere because of sidelobes. These sidelobes decay asymptotically as 1/w.
T hus, the truncation causes spectral l eakage in the band where the spectrum
of the signal f (t) is zero. T he p eak s idelobe m agnitude is 0.217 times the
~ainlobe m agnitude (13.3 dB below the peak mainlobe magnitude). Also, the
~Idelobes decay a t a r ate 1/w, which is - 6 d B/octave (or - 20 d B/decade). This
IS t he r olloff r ate of side lobes. We w ant smaller sidelobes with a faster rate
of decay (high rolloff rate). Figure 4.46d shows IW R (w ) I (in dB) as a function
of w. T his plot clearly shows the mainlobe and side lobe features, with the first
sidelobe amplitude - 13.3 dB below the mainlobe amplitude, and the sidelobes
decaying a t a r ate of - 6 d B/octave (or - 20 dB p er decade). 3 04 4 C ontinuous-Time S ignal Analysis: T he F ourier Transform 4.9 D ata T runcation: W indow F unctions 305 So far, we h ave d iscussed t he effect of signal t runcation ( truncation i n t ime
d omain) o n t he s ignal s pectrum. B ecause o f t ime-frequency duality, t he effect o f
s pectral t runcation ( truncation in frequency d omain) o n t he s ignal s hape is s imilar. Remedies for S ide Effects o f Truncation
F or b etter r €sults, we m ust t ry t o m inimize t he t runcation's t win s ide effects,
t he s pectral s preading ( mainlobe width) a nd leakage (sidelobe). L et us consider
each o f t hese ills.
T he s pectral s pread ( mainlobe w idth) o f t he t runcated s ignal is equal t o t he
b andwidth o f t he window function w(t). W e know t hat t he s ignal b andwidth
is inversely p roportional t o t he s ignal w idth ( duration). H ence, t o r educe t he
s pectral s pread ( mainl...
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