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Unformatted text preview: 0 (lowpass spectrum), and will pass through the lowpass filter a t t he
output, yielding the o utput 1 m(t) cos 8.
I f 8 is c onstant, the phase asynchronism merely yields an attenuated o utput (by a
factor cos 8). Unfortunately, in practice, 8 is often the phase difference between the carriers
generated by two distant generators, and varies randomly with time. This variation would
result in an o utput whose gain varies randomly with time.
(b) In the case of frequency error, the demodulator carrier is cos (we + il.w)t. T his
situation is very similar to the phase error case in (a) with 8 replaced by (il.w)t. Following
the analysis in p art (a), we c an express the demodulator product e(t) as e(t) = met) cos wet cos (we
1 + il.w)t = 2m(t)[cos (il.w)t + cos (2we + il.w)tJ 282 4 C ontinuous-Time Signal Analysis: T he F ourier Transform The spectrum o f the component ~m(t) cos (2wc + tl.w)t is centered at ±(2wc + tl.w). Consequently, this component will be filtered out by the lowpass filter a t the output. The
component ~7n(t)cos(tl.w)t is the signal m (t) multiplied by a low frequency carrier of
frequency tl.w. T he spectrum of this component is centered a t ±tl.w. In practice, the
frequency error (tl.w) is usually very small. Hence, the signal ~m(t) cos (tl.w)t (whose
spectrum is centered a t ±tl.w) is a lowpass signal and passes through the lowpass filter at
t he o utput, resulting in the output ~m(t) cos (tl.w)t. T he output is t he desired signal m (t)
multiplied by a very low frequency sinusoid cos (tl.w)t. Clearly, the output in this case is
not merely an attenuated replica of the desired signal m (t), b ut represents m (t) multiplied
by a time-varying gain cos (tl.w)t. If, for instance, the transmitter and the receiver carrier
frequencies differ just by 1 Hz, the output will be the desired signal m (t) multiplied by
a time-varying signal whose gain goes from the maximum to 0 every half second. This
is like some restless child fiddling with the volume control knob of a receiver, going from
maximum volume t o zero volume every half second. This kind of distortion (called the
b eat e ffect) is beyond repair. • 4.7 Application t o C ommunications: A mplitude M odulation m(t) / '\
(a) A + m(t»O A + m (t) f orallt t t ( b) Amplitude Modulation (AM) For t he s uppressed c arrier scheme j ust discussed, a receiver m ust g enerate a
c arrier i n frequency a nd p hase synchronism w ith t he c a;rier a t t he t ransmitter t hat
m ay b e l ocated h undreds o r t housands o f miles away. T his s ituation calls for a
s ophisticated receiver, which could b e q uite costly. T he o ther a lternative is for
t he t ransmitter t o t ransmit a c arrier A cos wet [along w ith t he m odulated signal
m (t) cos wet] s o t hat t here is no need t o g enerate a carrier a t t he receiver. I n t his
case t he t ransmitter n eeds t o t ransmit m uch larger power, a r ather e xpensive procedure. I n p oint-to-point c ommunications, where t here is one t ransmitter fo...
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