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

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

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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...
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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.

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