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Unformatted text preview: l2 o !1=1 !1=O.5 F ig. 4.35 Tone-modulated AM (a) J1 = 0.5. (b) J1 = 1. Figures 4.35a and b show the modulated signals corresponding to J1 = 0.5 and J1 = 1,
respectively. • 286 4 Continuous-Time Signal Analysis: T he Fourier Transform 4.7 Application t o Communications: Amplitude Modulation In this ma.nner, during each positive cycle, t he capacitor charges u p t o t he p eak
voltage of the i nput signal a nd t hen decays slowly until t he n ext positive cycle.
Thus, the o utput voltage v c(t) follows t he envelope of t he i nput. T he c apacitor
discharge between positive peaks, however, causes a ripple signal of frequency W e
i n t he o utput. T his ripple can be reduced by increasing t he t ime constant R G so
t hat t he c apacitor discharges very little between the positive peaks ( RG :s; l /w e ).
Making R G t oo large, however, would make it impossible for t he c apacitor voltage
t o follow t he envelope (see Fig. 4.36b). Thus, R G should be large compared t o
l /w e b ut s hould b e small compared t o 1/27r B , where B is t he highest frequency in
met). Incidentally, these two conditions also require t hat We » 27r B , a condition
necessary for a well-defined envelope.
T he e nvelope-detector o utput v e(t) is A + m et) plus a ripple of frequency We'
T he dc t erm A c an be blocked o ut by a capacitor or a simple R G highpass filter.
T he ripple m ay b e reduced further by another (lowpass) R G filter. I n t he case
of audio signa.ls, t he speakers cannot respond to the high frequency ripple, and
therefore, t hey a ct as lowpass filters themselves. 4 .7-3 (a) Baseband Spectrum 0 -21tH 21tB <0 - (b) DSB Spectrum LT Ii (d) LSB Spectrum Figures 4 .37a a nd 4.37b show t he b aseband spectrum M (w), a nd t he s pectrum
of t he D SB-SC m odulated signal m et) cos wet. T he DSB s pectrum in Fig. 4.37b
has two sidebands: the upper sideband (USB) a nd t he lower sideband (LSB), b oth
c ontaining complete information of M(w) [see Eq. (4.10)]. Clearly, i t is r edundant
t o t ransmit b oth sidebands, a process which requires twice t he b andwidth of the
baseband signal. A scheme where only one sideband is t ransmitted is known as s ingle s ideband ( SSB) t ransmission, which requires only one-half t he b andwidth
of t he DSB s ignal. Thus, we t ransmit only t he u pper sidebands (Figures 4.37c) or
only t he lower sidebands (Fig. 4.37d).
A n SSB s ignal can be coherently (synchronously) demodulated. For example,
multiplication o f a USB signal (Fig. 4.37c) by cos wet shifts its s pectrum t o t he left
a nd t o t he r ight by W e, yielding t he s pectrum in Fig. 4.37e. Lowpass filtering of
this signal yields t he desired baseband signal. T he case is similar with LSB signal.
Hence, d emodulation of SSB signals is identical to t hat of DSB-SC signals, a nd t he
synchronous d emodulator in Fig. 4.33a can demodulate SSB signals. Note t hat we
a re talking of S SB signals without a n a dditional carrier. Hence, they are suppressed
carrier signals (SSB-SC) .
• E xampl...
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