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Unformatted text preview: T he cost of digital hardware continues t o halve every two or three years, while
performance o r c apacity doubles over t he s ame time period. And t here is no
end in sight yet t o t his breathtaking a nd relentless exponential progress in
digital technology. In recent years we have seen t he c ompact d isc-a digital
d evice-bury t he analog long-playing record; newspapers t ransmit p hotographs
in scanned digital form; a nd more recently t he shift in t he U nited S tates toward
a digital s tandard for high-definition television as opposed t o t he a nalog standard e mbraced by J apan a nd Europe. In contrast, analog technologies such as
paper, video, sound, a nd film do n ot decline rapidly in cost. I f a nything, t hey
become more expensive with time. For these a nd o ther reasons, it is only a
m atter of time before cost/performance curves cross, a nd d igital technologies
come t o d ominate in any given a rea of communication o r s torage technologies.
As mentioned earlier, digital signals come from a variety of sources. Some
sources such as computers are inherently digital. Some sources are analog, b ut a re
converted into digital form by a variety of techniques such as P CM m odulation.
/::, E xercise E S.2 American Standard Code for Information Interchange (ASCII) has 128 characters which are
binary coded. A certain computer generates 100,000 characters/so Show that
(a) 7 bits (binary digits) are required to encode each character.
(b) 700,000 bits/s are required to transmit the computer output. \ J A Historical N ote
G ottfried Wilhelm Leibnitz (1646-1716) was t he first mathematician t o work
o ut s ystematically the binary representation (using 1 's a nd D's) for any number.
tThe error in pulse detection can be made negligible. 5 Sampling 336 H e felt a spiritual significance in this discovery, reasoning t hat 1 r epresenting unity
was clearly a s ymbol for God, while 0 r epresented t he nothingness. Therefore, if all
numbers can be represented merely by t he use of 1 a nd 0, surely this is t he same
as saying t hat G od c reated t he universe o ut of nothing! 5 .1-4 5.1 T he Sampling T heorem where (assuming T 337 < To) D n = - 1 i To f(t)e-jnwot dt = - 1
~ ~ 0 iT f(t)e-jnwot dt 0 From Eq. (5.12) it follows t hat Dual o f t he Time-Sampling: The Spectral Sampling Theorem As in other cases, t he sampling theorem has its dual. In Sec. 5.1, we discussed
t he t ime-sampling theorem where we showed t hat a signal bandlimited t o B Hz can
be* reconstructed from the signal samples taken a t a r ate of :F. > 2B samples/so
Note t hat t he s ignal spectrum exists over t he frequency range - B t o B Hz. Therefore, 2B is t he s pectral w idth (not t he b andwidth, which is B ) of t he signal. This
fact means t hat a signal f (t) can be reconstructed from samples t aken a t a r ate:F.
g reater t han t he s pectral width (in Hz) of t he signal.
T he d ual o f t he time-sampling theorem is t he frequency-sampling theorem.
This theorem applies t o t imelimited signals,...
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