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Unformatted text preview: n c ontrast, if t he i nput is very different from t he n atural mode,
(  ), is l arge, a nd t he s ystem responds poorly. T his is precisely w hat we s et o ut t o
prove.
We h ave p roved t he above assertion for a singlemode (firstorder) system. I t
c an b e generalized t o a n n thorder s ystem, which has n c haracteristic modes. T he
i mpulse r esponse h (t) of such a system is a l inear combination of its n modes.
Therefore, i f f (t) is similar t o a nyone of t he modes, t he c orresponding response
will b e high; if i t is similar t o n one of t he modes, t he response will b e small. Clearly,
t he c haracteristic m odes are very influential in determining system response t o a
given input.
I t w ould b e t empting t o c onclude on t he basis of Eq. (2.66) t hat if t he i nput
is i dentical t o t he c haracteristic mode, so t hat ( = )" t hen t he r esponse goes t o
infinity. R emember, however, t hat if ( = )" t he n umerator o n t he r ighthand side
of Eq. (2.66) also goes t o zero. We shall s tudy t his complex behavior (resonance
p henomenon) l ater in t his section.
We s hall now show t hat m ere i nspection o f t he impulse response h (t) (which is
composed o f c haracteristic m odes), reveals a g reat deal a bout t he s ystem behavior. 2 .72 t Fig. 2 .18 Effective duration of an impulse response. T here is no single satisfactory definition of effective signal d uration (or width)
applicable t o every s ituation. For t he s ituation d epicted in Fig. 2.18, a reasonable
definition of t he d uration h (t) w ould be Th, t he w idth of t he r ectangular pulse
h(t). T his r ectangular pulse h (t) h as a n a rea i dentical t o t hat of h (t) a nd a h eight
identical t o t hat of h (t) a t s ome s uitable i nstant t = to. I n Fig. 2.18, to is chosen
as t he i nstant a t which h (t) is maximum. According t o t his d efinition,t or J~oo h (t) dt
h(to) Th = (2.67) Now if a system has a single m ode Response Time o f a System: The System Time Constant Like h uman beings, systems have a certain response time. I n o ther words, when
a n i nput ( stimulus) is applied t o a s ystem, a certain a mount of t ime e lapses before
t he s ystem fully responds t o t hat i nput. T his t ime lag or response t ime is called
t he s ystem t ime c onstant. As we shall see, a system's t ime c onstant is equal t o
t he w idth o f i ts i mpulse response h(t).
A n i nput 8(t) t o a s ystem is i nstantaneous (zero d uration), b ut i ts r esponse
h (t) h as a d uration Th. T herefore, t he s ystem requires a t ime Th t o r espond fully t o
t his i nput, a nd we are justified in viewing Th as t he s ystem's response t ime o r time w ith), n egative a nd r eal, t hen h (t) is m aximum a t t
T herefore, according t o E q. (2.67) 1 o w ith value h(O) A. 00 Th = 1
A 0 Ae At dt 1
=  ,
) (2.68) t T his definition is satisfactory when h (t) is a single, mostly positive (or mostly negative) pulse
Such systems a re lowpass systems. T his d efinition should n ot b e a pplied indiscriminat...
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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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