Unformatted text preview: t (Fig. 2.7c). Therefore,
f (t) a nd f ( r) will have t he s ame graphical representations. S imilar r emarks a pply
t o get) a nd g (r) (Fig. 2.7d).
T he f unction g (t  r) is n ot as easy t o c omprehend. To u nderstand w hat t his
function looks like, let us s tart w ith t he f unction g (r) (Fig. 2.7d). Timeinversion
of t his f unction (refiection a bout t he v ertical axis r = 0) yields g(  r) (Fig. 2.7e).
L et u s denote this function by <p(r) V' <p(r) = g (r) E xercise E 2.B Now <p( r ) s hifted by t s econds is <p( r  t), given by Using the convolution table, determine <p(r  t)
Answer: (~ e' + ~e2') u (t) V' 6 E xercise E 2.9
For an LTIC system with the unit impulse response h(t) = e  2 'u(t), determine the zerostate
response y et) if the input f (t) = sin 3 tu(t). Hint: Use the convolution table Pair 12 with suitable values for a
Answer: {3 e, and >.. h [ 3e2'+v'i3cos(3t146.32°)]u(t) 127 Multiple Inputs
M ultiple i nputs t o L TI systems c an b e t reated b y a pplying t he s uperposition
principle. E ach i nput is considered separately, w ith all o ther i nputs a ssumed t o b e
zero. T he s um o f all t hese i ndividual system responses c onstitutes t he t otal s ystem
o utput w hen all t he i nputs a re applied simultaneously. 2 .42 T he input is f (t) = l Oe 3'u(t), and the response yet) is = 1 0e 'u(t) S ystem R esponse t o E xternal I nput: T he Z eroState Response D f(t) T he impulse response h (t) for this system, as obtained in Example 2.3, is yet) 2.4 Or h [ 3e2tv'i3cos(3t+33.68°)]u(t) V' = g [(r  t)l = get  r) T herefore, we first timeinvert g (r) t o o btain g (r) a nd t hen t imeshift g (r) b y
t t o o btain get  r). F or positive t, t he s hift is t o t he r ight (Fig. 2.7f); for negative
t, t he s hift is t o t he left (Fig. 2.7g).
T he p receding discussion gives us a graphical i nterpretation o f t he f unctions
f ( r) a nd 9 (t  r). T he convolution c( t) is t he a rea u nder t he p roduct o f t hese two
functions. Thus, t o c ompute c(t) a t some positive i nstant t = t1, we first o btain
g(  r) by inverting g (r) a bout t he v ertical axis. Next, we rightshift or delay g(  r)
by t1 t o o btain 9(t1  r) (Fig. 2.7f), a nd t hen we multiply t his f unction by f er),
giving us t he p roduct f (r)g(t1  r) (Fig. 2.7f). T he a rea A l u nder t his p roduct 128 2 TimeDomain Analysis of ContinuousTime Systems 1 1 1 f"'I'
2 0 2.4 System Response t o E xternal Input: T he ZeroState Response 129 is C (tl), t he value of crt) a t t = t l' We c an therefore plot C (tl) = A l on a curve
describing crt), as shown in Fig. 2.7i. Observe t hat t he a rea under the product
f (r)g(  r) in Fig. 2.7e is c(O), t he value of t he convolution for t = 0 ( at t he origin).
A similar procedure is followed in computing t he value of crt) a t t = t2, where
t2 is negative (Fig. 2.7g). In this case, t he function g(  r) is shifted by a negative
amount ( that is, leftshifted) t o o btain 9(t2  r). Multiplication of this function with
f (r) yields the product f (r)g(t2 ...
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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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