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

T he d irect link of t he a utocorrelation function t

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Unformatted text preview: above We, k nown a s t he u pper s ideband ( USB), a nd a p ortion t hat lies below We, known a s t he l ower s ideband ( LSB). S imilarly, t he s pectrum c entered a t - We h as u pper a nd lower sidebands. T his form o f m odulation is called d ouble s ideband ( DSB) m odulation for obvious reason. LSB -(mc-COm) - m c -(00"+0>,,,) F ig. 4 .32 M (w) + we) + M (w DSB spectrum t t ( Oc-Olm (b) USB O l, l i\+O)m I" An example of DSB-SC modulation. m_ In the time-domain approach, we work directly with signals in the time-domain. For the baseband signal m (t) = cos wmt, t he DSB-SC signal 'POSB.SC(t) is 'PDSB.SC(t) = m (t) cos wet = cos wmt cos wet 1 2'[cos (we + Wm)t + cos (we - Wm)t] (4.71) tPractical factors may impose additional restrictions on We. For instance, in broadcast applications, a radiating antenna can radiate only a narrowband without distortion. This restriction implies that to avoid distortion caused by the radiating antenna, wc/27l' B » 1. The broadcast band AM radio, for instance, with B = 5 kHz and the band of 550 to 1600 kHz for carrier frequency give a ratio of We/27l' B roughly in the range of 100 to 300. 280 4 C ontinuous-Time S ignal Analysis: T he F ourier T ransform T his result shows t hat when the baseband (message) signal is a single sinusoid of frequency t he modulated signal consists of two sinusoids: t he component of frequency W e + W m (the upper sideband), and the component offrequency Wc-W m ( the lower sideband). Figure 4.32b illustrates precisely the spectrum of 'PDsB-se (t). Thus, each component of frequency W m in the modulating signal results into two components of frequencies W e + W m and We - Wm in t he modulated signal. This being a DSB-SC (suppressed carrier) modulation, there is no component of the carrier frequency W e on the right-hand side of the above equation as expected.t • 4.7 A pplication t o C ommunications: A mplitude M odulation 281 Therefore, t he F ourier t ransform o f t he s ignal e (t) is Wm , e ( t) , t m(t) I , - . Lowpass filter 1 --_ _ _- ---1 (a) Demodulator (b) Spectrum of / 4\\\ ee,) 2ro c ~ Demodulation o f DSB-SC Signals T he D SB-SC m odulation t ranslates o r s hifts t he f requency s pectrum t o t he left a nd t he r ight b y We ( that is, a t +w e a nd - we), as seen from Eq. (4.70). To recover t he o riginal s ignal m et) f rom t he m odulated s ignal, we m ust r etranslate t he s pectrum t o i ts o riginal position. T he p rocess o f r ecovering t he s ignal from t he m odulated s ignal ( retranslating t he s pectrum t o i ts o riginal position) is referred t o as d emodulation, o r d etection. O bserve t hat i f t he m odulated s ignal s pectrum in Fig. 4.31c is s hifted t o t he left a nd t o t he r ight b y W e ( and h alved), we o btain t he s pectrum i llustrated in Fig. 4.33b, which c ontains t he d esired b aseband s pectrum i n a ddition t o a n u nwanted s pectrum a t ± 2we. T he l atter c an b e s uppressed b y a l owpass filter. T hus, d emodulation, w hich is a lmo...
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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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