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Unformatted text preview: odic signal by adding the first n harmonics
and truncating a ll t he higher harmonics. These examples show t hat d ata t runcation
can occur in b oth time and frequency domain. O n t he surface, truncation appears
to be a simple problem of cutting off t he d ata a t a p oint where i t is deemed to be
sufficiently small. Unfortunately, this is n ot the case. Simple truncation can cause
some unsuspected problems.
Window Functions
T runcation operation may be regarded as multiplying a signal of a large width
by a window function of a smaller (finite) width. Simple truncation amounts to
using a r ectangular w indow WR(t) (Fig. 4.48a) in which we assign unit weight
to all the d ata within the window width (It I < ~\ and assign zero weight t o all
the d ata lying outside the window (It I >
I t is also possible t o use a window in
which the weight assigned to the d ata w ithin t he window may not be constant. In
a t riangular w indow WT(t), for example, the weight assigned t o d ata decreases
linearly over t he window width (Fig. 4.48b).
Consider a signal f (t) a nd a window function W (t). I f f (t) ~ F (w) and
wet) ~ W (w), a nd if t he windowed function fw(t) ~ Fw(w), t hen
1
f w(t) = f (t)w(t)
a nd
Fw(w) = 27r F(w) * W (w) f). According t o t he w idth property of convolution, i t follows t hat t he w idth of Fw(w)
equals t he s um of t he widths of F(w) a nd W (w). Thus, truncation of a Signal
increases its b andwidth by the amount of bandwidth of w (t). Clearly, t he t runcation
of a signal causes its spectrum to spread (or smear) by t he amount of the bandwidth
of wet). R ecall t hat t he signal bandwidth is inversely proportional t o t he signal
duration (width). Hence, the wider t he window, the smaller is its bandwidth, and
the smaller is t he s pectral s preading. T his result is predictable because a wider
window means we are accepting more d ata (closer approximation), which should
cause smaller distortion (smaller spectral spreading). Smaller window width (poorer
approximation) causes more spectral spreading (more distortion). There are also
other effects produced by the fact t hat W (w) is really not strictly bandlimited, and
its spectrum > 0 only asymptotically. This causes the spectrum of Fw(w)  + 0
a symptotically also a t t he same r ate as t hat of W (w), even though the F(w) may
be strictly bandlimited. Thus, windowing causes the spectrum of F(w) t o leak in
the b and w here i t is supposed t o b e zero. This effect is called l eakage. These twin
effects, t he s pectral spreading a nd t he leakage, will now be clarified by an example.
For an example, let us take f (t) = cos wot a nd a rectangular window WR(t) =
rect(~), i llustrated in Fig. 4.46b. T he reason for selecting a sinusoid for f (t) is
t hat its s pectrum consists of spectral lines of zero width (Fig. 4.46a). This choice
will make t he effect of spectral spreading and leakage clearly visible. The spectrum
of t he t runcated signal fw(t) is t he convolution of the two impulses of F(w) with
t he sin...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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