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

Energy spectral density from autocorrelation function

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Unformatted text preview: odulation I n a mplitude m odulation, t he a mplitude A o f t he c arrier A cos (wet + l1e) is varied in some m anner w ith t he b aseband ( message)t signal met) (known as t he tThe term baseband is used to designate the band of frequencies of the signal delivered by the source or the input transducer. 4 Continuous- Time S ignal Analysis: T he F ourier Transform 4.7 m et) 278 T he r elationship o f B t o We is of interest. F igure 4.31c shows t hat We ?: 27l' B in o rder t o avoid t he o verlap o f t he s pectra c entered a t ± we. I f We < 27l' B , t he s pectra o verlap a nd t he i nformation o f m (t) is l ost in t he p rocess o f m odulation, a loss w hich, m akes i t i mpossible t o g et b ack m (t) from t he m odulated s ignal m (t) cos w et.t (Modulated Signal) (Modulating Signal) (a) Modulator Application t o C ommunications: A mplitude M odulation 279 • E xample 4 .17 For a baseband signal m (t) = cos wmt, find the DSB signal, and sketch its spectrum. Identify the upper and lower sidebands. M (w) m et) We shall work this problem in the frequency-domain as well as the time-domain in order to clarify the basic concepts of DSB-SC modulation. In the frequency-domain approach, we work with the signal spectra. The spectrum of the baseband signal m (t) = cos wmt is given by - M(w) = 7l'[6(w - wm ) + 6(w (b) + wm)] The spectrum consists of two impulses located a t ± wm , as depicted in Fig. 4.32a. The DSB-SC (modulated) spectrum, as indicated by Eq. (4.70), is t he baseband spectrum in Fig. 4.32a shifted to the right and the left by We (times one-half), as depicted in Fig. 4.32b. This spectrum consists of impulses a t ±(we - Wm) and ±(we + Wm). T he spectrum beyond We is the upper sideband (USB), a nd t he one below We is t he lower sideband (LSB). Observe t hat t he DSB-SC spectrum does not have the component of the carrier frequency We. This is why it is called d ouble s ideband-suppressed c arrier ( DSB-SC) . M (m) (c) F ig. 4 .31 (a) DSB-SC modulation. m odulating s ignal). T he f requency We a nd t he p hase (Je a re c onstant. W e c an assume (Je = 0 w ithout a loss of generality. I f t he c arrier a mplitude A is m ade d irectly p roportional t o t he m odulating s ignal m (t), t he m odulated s ignal is m (t) cos wet (Fig. 4.31a). As was i ndicated e arlier [Eq. (4.41)], this t ype o f m odulation s imply shifts t he s pectrum o f m (t) t o t he c arrier frequency (Fig. 4.31c). T hus, i f m (t) t hen m (t) cos wet < ===} < ===} 1 Z [M(w rrJ2 USB t t LSB - we)] (4.70) R ecall t hat M (w - we) is M (w) s hifted t o t he r ight by We a nd M (w + we) is M (w) s hifted t o t he l eft b y We' T hus, t he p rocess o f m odulation s hifts t he s pectrum o f t he m odulating s ignal t o t he left a nd t he r ight b y We' N ote also t hat i f t he b andwidth o f m (t) is B Hz, t hen, as i ndicated i n Fig. 4.31c, t he b andwidth of t he m odulated s ignal is 2 B Hz. We also observe t hat t he m odulated s ignal s pectrum c entered a t We is c omposed o f two p arts: a p ortion t hat lies...
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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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