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Differential and Difference Equation Representations of LTI Systems M dk dk ak k y (t ) bk k x(t ) dt dt k 0 k 0 N N M k 0 k 0 ak y[n k ] bk x[n k ] ( N , M ) order of the equation = # of energy storing devices in the system. Often N M and the order is referred to as N . To solve equations of this kind it is required to have initial values coming from the past (memory) in general for an order N system, N values are required. It is very often used to select t 0 or n0 as the time instants to evaluate the initial conditions. y[ N ], y[ N 1],..... y[1], y (t ) t 0 , d d2 dn y (t ) , 2 y (t ) ,..... n y (t ) dt dt t 0 dt t 0 t 0 Recursive Formulas M N k 0 k 1 y[n] bk x[n k ] ak y[n k ] An operator L is said to be linear if it satisfies the two linearity conditions: 1) 2) L(cy ) cL( y) for any function y and any constant c . L( y1 y2 ) L( y1 ) L( y 2 ) for any two functions y1 and y 2 . Otherwise it is said to be nonlinear. L is an operator which can be expressed in the form L L( y) F ( x, y' , y' ' ,...., y (n ) ) where n 1 is an integer and F is a function of (n+1) variables. A differential operator An operator L is called a linear differential operator if It turns out that any linear differential operator L L is a differential operator and if L is linear. can in fact be expressed in the form L( y) P0 ( x) y P1 ( x) y ' ......Pn ( x) y n where P0 , operator. P1 ,..... Pn are functions of x , and conversely any operator of this form is a linear differential A system y (t ) H {x(t )} described by a linear operator is a linear system. If the system is time invariant A system P0 , P1 ,..... Pn are not functions of t, y(t ) H {x(t )} described by a linear differential operator is linear system with memory. Example 2.15 n 1 Find the first two values of y[n] when x[n] u[n] , and y[1] 1, y[2] 2 2 1 y[n] x[n] 2 x[n 1] y[n 1] y[n 2] 4 Solving Differential and Difference Equations The homogeneous Equation dk ak k y (t ) 0 dt k 0 N N y (t ) ci e ri t (h) i 1 ri are the N solutions of the characteristic equation N a r k 0 k k 0 Discrete time LTI systems N a k 0 k y[n k ] 0 N y [n] ci ri (h) n i 1 ri are the N solutions of the characteristic equation N a r k 0 Example 2.17 Determine the Homogeneous solution k N k 0 Example 2.18 First Order Recursive System Find the homogeneous solution. y[n] y[n 1] x[n] Problem 2.16 d2 d d y(t ) 5 y(t ) 6 y(t ) 2 x(t ) x(t ) 2 dt dt dt The Particular Solution Example 2.20 RC Circuit Continued x(t ) cos(0t ) Problem 2.16 (continued) x(t ) e t The Complete Solution Example 2.18 First Order Recursive System Find the Complete solution of: y[n] y[n 1] x[n] n when 1 x[n] u[n] and y[1] 8 2 The Natural Response Zero Input Response Example 2.17 Determine the NaturalResponse C 1, R 1 , and i(0) 2V Example 2.25 First Order Recursive System Find the natural response for y[1] 8 y[n] 1 y[n 1] x[n] 4 The forced Response Zero State Response Example 2.26 First Order Recursive System n 1 Find the forced response for x[n] u[n] 2 1 y[n] y[n 1] x[n] 4 Example 2.17 Determine the Forced Response C 1, R 1 , and x(t ) costu (t ) V The Resonance Phenomenon