Unformatted text preview: 18.03 Muddy Card responses, April 21, 2006 1. LTI = Linear, Time Invariant. This is a property of an operator or a system. An operator (which is a rule L that converts one function of time to another one) is linear if L ( f + g ) = L ( f )+ L ( g ) and L ( cf ) = cL ( f ) for functions f and g and constants c . Differention D is linear, adding 1 (i.e. sending f ( t ) to f ( t ) + 1) is not. An operator is time invariant if ( Lf )( t − a ) = L ( f ( t − a )) for any constant a and any function f . This is more or less the same as commuting with D : LD = DL . Differentiation is time independent, while multiplication by t , sending f ( t ) to tf ( t ), is not: ( Lf )( t − a ) = ( t − a ) f ( t − a ), while L ( f ( t − a )) = tf ( t − a ). The system represented by a time independent operator does not change through time. A differential operator is LTI when it has the form p ( D ) = a n + a 1 D + a I for constants D n + · · · a , . . . , a n . This has been our main object of study. It has a characteristic polynomial, gotten ....
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