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Unformatted text preview: ob. 2.2-1 if
(D2 + 4 D + 4) y (t) = D f(t)
2 .2-3 and ya(O) = 3, ya(O) = - 4.
Repeat Prob. 2.2·1 if
D (D 2 .2-4 + 1)y(t) = (D + 2 )f(t) = (3D + 2 )f(t) and ya(O) = ya(O) = 1.
Repeat Prob. 2.2-1 if
(D2 + 9) y(t) and ya(O) = 0, ya(O) = 6.
2 .2-5 Repeat Prob. 2.2-1 if
(D2 + 4 D + 13) y(t) = 4(D + 2 )f(t) 2. An LTIC s ystem is unstable if, a nd o nly if, either one or b oth of t he following
conditions e xist: (i) a t l east one r oot is in t he R HP; (ii) there are r epeated
r oots on t he i maginary axis. with ya(O) = 5, ya(O)
2 .2-6 Repeat Prob. 2.2·1 if = 15.98. 3. A n L TIC s ystem is marginally stable if, a nd only if, there are no roots in t he
R HP, a nd t here a re some u nrepeated r oots on t he i maginary axis. with ya(O) = 4, ya(O)
2 .2-7 Repeat Prob. 2.2-1 if =3 According t o a n a lternative definition of s tability- bounded-input b oundedoutput ( BIBO) s tability-a s ystem is s table if every bounded i nput p roduces a
b ounded o utput. O therwise t he s ystem is ( BIBO) u nstable. Asymptotically s table
s ystem is always BIBO-stable. T he converse is n ot necessarily true, however.
C haracteristic b ehavior of a system is extremely i mportant b ecause i t d etermines n ot only t he s ystem response t o i nternal conditions (zero-input behavior),
b ut also t he s ystem response t o e xternal i nputs ( zero-state behavior) a nd t he system stability. T he s ystem response t o e xternal i nputs is determined by t he i mpulse
response, which itself is m ade u p o f characteristic modes. T he w idth o f t he i mpulse
response is c alled t he t ime c onstant o f t he s ystem, which indicates how fast t he
s ystem c an r espond t o a n i nput. T he t ime c onstant plays a n i mportant role in determining such d iverse s ystem behavior as t he r esponse time a nd filtering properties
of t he s ystem, dispersion of pulses, a nd t he r ate of pulse transmission t hrough t he
s ystem. D2(D + 1)y(t) = and ya(O) (D (D2 + 2 )f(t) = - 1. + 1) (D2 + 5D + 6) y(t) = D f(t) with ya(O) = 2, ya(O) = - 1 and ya(O) = 5.
2 .3-1 Find the unit impulse response of a system specified by t he equation
2 .3-2 + 4 D + 3) y(t) = (D + 5 )f(t) Repeat Prob. 2.3-1 if
(D 2 +5D+6)y(t)= ( D2+7D+ll)f(t) 2 .3-3 Repeat Prob. 2.3-1 for the first· order allpass filter specified by t he equation
(D 2 .3-4 + 1)y(t) = - (D - l)f(t) Find the unit impulse response of an LTIC system specified by the equation
(D2 + 6D + 9) y(t) = (2D + 9) f (t) 166 2 T ime-Domain A nalysis o f C ontinuous-Time S ystems . 4-1 I f c(t) = f (t) * get), t hen show t hat Ac = A fAg, where A f, Ag, a nd Ac are t he a reas
under f (t), get), a nd c(t), respectively. Verify this a rea p roperty o f convolution in
Examples 2.6 a nd 2.8 . . 4 -2 I f f (t) * get) = c(t), t hen show t hat f eat) * g(at) = I~ic(at). T his t ime-scaling
p roperty o f convolution s tates t hat if b oth f (t) a nd get) are time-scaled by a, t heir
convolution is also time-scaled by a ( and multiplied by I l/al). . 4-3 Show t hat t he convolution of an odd a nd a n even function is a n odd function a nd t...
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