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

However if m n i t is impossible for h t t o have any

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: y~k) (0) is t he value of t he k th d erivative o f Yn(t) a t t = O. We c an e xpress t his c ondition f or various values of n ( the s ystem order) as follows: n n n n = 1: = 2: = 3: = 4: =1 Yn(O) = 0 a nd Yn(O) = 1 Yn(O) = Yn(O) = 0 a nd iin(O) = 1 Yn(O) = Yn(O) = iin(O) = 0 a nd iin(O) = Yn(O) 1 (2.21) a nd so on. I f t he o rder o f P (D) is less t han t he o rder of Q (D), bn = 0, a nd t he i mpulse t erm bn 8(t) i n h ( t ) is zero. • I n t he above discussion, we have assumed m ::::: n , a s specified by Eq. (2.17b). A ppendix 2.1 shows t hat t he e xpression for h (t) a pplicable t o all possible values of m a nd n is given b y h (t) = P (D)[Yn(t)u(t)] where Y n(t) is a linear combination of t he c haracteristic modes of t he s ystem s ubject t o i nitial conditions (2.20). T his e xpression reduces t o Eq. (2.19) when m ::::: n . D etermination o f t he i mpulse response h (t) u sing t he p rocedure in t his s ection is relatively simple. However, in C hapter 6 we shall discuss a nother, even simpler m ethod using t he L aplace transform. f::,. Determine the unit impulse response of LTIC systems described by the equations: (a) ( b) ( e) E xample 2 .3 Determine t he unit impulse response h( t} for a system specified by the equation (D2 + 3D + 2) yet} This is a second-order system (n = 2) = D f(t} 2C2 e- 2t (2.23a) (2.23b) T he initial conditions are [see Eq. (2.21) for n = 2] and Yn(O} = 0 Setting t = 0 in E qs. (2.23a) and (2.23b), and substituting the above initial conditions, we obtain 2C2 Solution of these two simultaneous equations yields - Cl - This is a second-order system with bn = b2 = O. First we find the zero-input component for initial conditions y(O-} = 0, and y(O-} = 1 Y zi=dsoive('D2y+3*Dy+2*y=0', 'y(O} = 0', ' Dy(O} = 1', ' t') Yzi • - exp(-2*t)+exp(-t) Since P (D) = D , we differentiate the zero-input response: P Yzi=symdiff(Yzi) PYzi • 2 *exp(-2*t)-exp(-t) T herefore h(t} 0=C!+C2 1= C omputer E xample C 2.3 Determine the impulse response h(t} for an LTIC system specified by the differential equation (D2 + 3D + 2)y(t} = D f(t} Differentiation of t his equation yields = - cle- t - Answers: (a) 38(t) - e- 2t u(t) ( b) (2 - e- 2t )u(t) (e) (1 - t)e-tu(t) \ l 8 (A2 + 3A + 2) = (A + 1}(A + 2) Yn(t} (D + 2)y(t) = (3D + 5 )f(t) D(D + 2)y(t) = (D + 4 )f(t) (D2 + 2D + l)y(t) = D f(t) (2.22) having the characteristic polynomial T he characteristic roots of this system are A = - 1 and A = - 2. Therefore Yn(t} = c le- t + c2e- 2t E xercise E 2.4 (2.24) = b28(t) + [Dyo(t)]u(t} = (2e- 2t - e-t)u(t} 0 118 2 Time-Domain Analysis of Continuous-Time Systems S ystem Response t o Delayed Impulse I f h(t) is t he r esponse of a n LTIC system t o t he i nput 8(t), t hen h(t - T) is t he response of this s ame s ystem t o t he i nput 8(t - T). T his conclusion follows from t he t ime-invariance p roperty of LTIC systems. Thus, by knowing t he u nit impulse response h(t), we c an d etermine t he system response t o a delayed impulse 8(t - T). 2.4 System Response t o E xternal Input: T he Z ero-State Response 119 t [ ( I) [ (nil,;) 2 .4 System Response t o External Input: Z ero-state Response 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