Signal Processing and Linear Systems-B.P.Lathi copy

T he response of a n l tic system t o t he e

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 (D2 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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online