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Unformatted text preview: EE 505 B, Autumn, 2011 Linear Systems 1 1 Linear Systems Definition 1. A linear system is a mapping of inputs to outputs that satisfy the principle of superposition. v ( x ) v [ m ] L w ( x ) w [ m ] Figure 1: Block diagram of a linear system. L must satisfy 1. L { v 1 + v 2 } = L { v 1 } + L { v 2 } , and 2. L { cv } = cL { v } . From these, one can easily show that L braceleftBigg N 1 k = c k v k bracerightBigg = N 1 k = c k L { v k } . For continuous (time, space, etc.) variables, we use integration instead of summation, and L braceleftbigg integraldisplay c ( ) v ( ) d bracerightbigg = integraldisplay c ( ) L { v ( ) } d . Example 1.1. Here is an example of a system that is nonlinear (although it may not seem so at first). L { v } = v + 1 To determine if the system is linear, we apply the rules to which all linear systems must obey: Evaluating the lefthand side and righthand side of rule 1 above, we have L { v 1 + v 2 } = v 1 + v 2 + 1 (LHS) L { v 1 } + L { v 2 } = v 1 + 1 + v 2 + 1 (RHS) = v 1 + v 2 + 2 therefore L { v 1 + v 2 } negationslash = L { v 1 } + L { v 2 } We could have also used rule 2 above. The easy way to apply this rule is to define a new variable = cv to evaluate the lefthand side. L { cv } = L { } = + 1 = cv + 1 (LHS) cL { v } = c ( v + 1 ) = cv + c (RHS) therefore L { cv } negationslash = cL { v } EE 505 B, Autumn, 2011 Linear Systems 2 There is actually a quick check for linearity. L { v } must equal 0 for v = 0, because L { } = L { } = L { } = 0 . 1.1 Operation of a linear system The operation of a linear system is defined by matrix multiplication for discrete systems, and integration for continuous systems. Discrete : vector w = L vector v where vector v R N , vector w R M , and L is an M N matrix. Here is another way of writing matrix multiplication that we didnt write down explicitly in the first class. w [ l ] = N 1 k = a lk v [ k ] for l = 0, . . . , m 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright l indicates the row of the matrix This equation describes the result of matrix multiplication row by row. Any finite dimensional linear system can be written as a matrix multiplication, and the properties of the matrix will define the properties of the linear system . For examples, symmetric matrices ( A T = A ) have orthogonal eigenvectors. Continuous : the infinitedimensional version of matrix multiplication for continu ous systems is w ( x ) = L { v ( x ) } = integraldisplay b a k ( x , y ) v ( y ) dy , where k ( x , y ) is called the kernel of integration. Again, special properties of the kernel correspond to special linear systems, e.g. k ( x , y ) = k ( y , x ) , or k ( x , y ) = k ( y , x ) if k is complex....
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 Spring '12
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