MIT18_03S10_c15 - 18.03 Class 15 March 8 2010 Operators...

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Unformatted text preview: 18.03 Class 15 , March 8, 2010 Operators, polynomial signals, resonance [1] Operators [2] Resonance [3] Polynomial signals Several different topics today, and a respite from the gain game. [1] Operators function Just as number ------------> number operator function ------------> function The *differentiation operator D takes x to x' : Dx = x' . For example, D sin(t) = cos(t) , D x^n = n x^{n-1} , D8 = 0 . We can iterate: D^2 = x" . There's also the "identity operator": Ix = x And we can take linear combinations of operators: (D^2 + 2D + 2I) x = x" + 2x' + 2x . The characteristic polynomial here is p(s) = s^2 + 2s + 2 , and it's irresistible to write D^2 + 2D + 2I = p(D) so x" + 2x' + 2x = p(D) x This formalism lets us discuss linear equations of higher order with no extra work. Such an equation has the form an x^{(n)} + ... + a1 x' + a0 x = q(t) (*) It has a characteristic polynomial p(s) = a_n s^n + ... + a_1 s + a_0 and so it can be written p(D) x = q(t) Now we can say that the *operator* p(D) = an D^n + ... + a_1 D + a_0 I represents the system. This is a "linear, time-invariant differential operator." In the systems and signals yoga, an input signal determines a function q(t) , and the system response x satisfies Lx = q . We could write x = L^{-1} q and indeed most of this course is about finding ways to "invert" these operators. The Exponential Response Formula lets us find a solution of p(D) x = B e^{rt} very efficiently, as long as (1) r is not a root of p(s) and (2) B is constant Today we'll see how to deal with the first limitation....
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This note was uploaded on 01/18/2012 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c15 - 18.03 Class 15 March 8 2010 Operators...

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