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Unformatted text preview: sary condition for BIBO s tability
in such systems. In other words, show t hat if Eq. (2.65) is n ot satisfied, t hen t here
exists a b ounded i nput t hat produces a n u nbounded output.
Hint: Assume t hat a system exists for which h it) violates Eq. (2.65) and yet produces
an o utput t hat is b ounded for every bounded input. Establish contradiction in this
s tatement by considering an input f it) defined by f (tl - T) = 1 when h iT) 2': 0 a nd
f (tl - T) = - 1 when hiT) < 0, where t l is some fixed instant. ,u, I / f- Signal R epresentation By
F ourier Series
T his c hapter i s i mportant f or b asic u nderstanding o f s ignal r epresentation a nd
s ignal c omparison. I n C hapter 2 we e xpressed a n a rbitrary i nput f (t) a s a s um o f
i ts i mpulse c omponents. T he L TI s ystem ( zero-state) r esponse t o i nput f (t) w as
o btained b y s umming t he s ystem's r esponses t o a ll t hese c omponents i n t he f orm
o f t he c onvolution i ntegral. T here a re i nfinite p ossible w ays o f e xpressing a n i nput
f (t) i n t erms o f o ther s ignals. F or t his r eason, t he p roblem o f s ignal r epresentation
in t erms o f a s et o f s ignals is v ery i mportant i n t he s tudy o f s ignals a nd s ystems.
T his c hapter a ddresses t he i ssue o f r epresenting a s ignal a s a s um o f i ts c omponents.
T he p roblem is s imilar t o t hat o f r epresenting a v ector i n t erms o f i ts c omponents. Fig. P 2.7-1
~.7-1 1.7-2 l .7-3 D ata a t a r ate of 1 million pulses per second are t o b e transmitted over a certain
communications channel. T he u nit step response g(t) for t his channel is shown in
( a) E xplain i f t his channel can t ransmit d ata a t t he required rate.
( b) C an a n a udio signal consisting of components with frequencies up t o 15kHz be
t ransmitted o ver this channel with reasonable fidelity?
A c ertain communication channel has a bandwidth of 10 kHz. A pulse of 0.5 ms
d uration is t ransmitted over this channel.
( a) D etermine t he w idth (duration) of t he received p ulse..
( b) F ind t he m aximum r ate a t which these pulses can be transmItted over thIS channel
without interference between the successive pulses.
4 A first-order LTI C system has a characteristic root >. = _ 10 .
( a) D etermine T r , t he rise time of its u nit s tep i nput response.
( b) D etermine t he b andwidth of this system.
(c) D etermine t he r ate a t which the information pulses can be transmItted through
this system. Signals and Vectors
T here is a p erfect a nalogy b etween s ignals a nd v ectors; t he a nalogy is so s trong
t hat t he t erm 'analogy' u nderstates t he r eality. S ignals a re n ot j ust like v ectors.
S ignals a re v ectors! A v ector c an b e r epresented a s a s um o f i ts c omponents i n a
v ariety o f w ays, d epending u pon t he c hoice o f c oordinate s ystem. A s ignal c an a lso
b e r epresented a s a s um o f i ts c omponents i n a v ariety o f w ays. L et u s b egin w ith
s ome b asic v ector c oncepts a nd t hen a pply t hese c oncepts t o s ignals. 3.1-1 Compon...
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