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

T he s pectrum centered a t we would have a bandwidth

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

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