linear_systems

# So what is the point of all of this where is this

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So, what is the point of all of this? Where is this leading? The matrix describes the linear system, so the properties of the matrix are the prop- erties of the linear system. 3 The natural basis for LTI systems 3.1 Eigenvectors of a discrete LTI system The eigenvectors of the cyclic convolution matrix are complex exponentials, w k N = e j 2 π N k (you saw these complex exponentials in the discrete Fourier transform matrix). Note that, w N here is not the output of the linear system. w is a symbol used in digital signal processing textbooks. For N = 4: w 0 4 = 1 w 1 4 = j w 2 4 = 1 w 3 4 = j vector w 4 = 1 j 1 j We can check that this vector is an eigenvector of a cyclic convolution matrix. We want to check that L vector w 4 = λ vector w 4 .

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EE 505 B, Autumn, 2011 Linear Systems 8 If the statement that the eigenvectors of a cyclic convolution matrix are complex is true, then we should be able to select any cyclic convolution matrix, e.g, L = 1 4 3 2 2 1 4 3 3 2 1 4 4 3 2 1 . (We could have selected any numeric values to form the matrix.) Now we need to show that vector w 4 is an eigenvector of this matrix. 1 4 3 2 2 1 4 3 3 2 1 4 4 3 2 1 1 j 1 j = 1 + 4 j 3 2 j 2 + j 4 3 j 3 + 2 j 1 4 j 4 + 3 j 2 j = 2 + 2 j 2 2 j 2 2 j 2 + 2 j =( 2 + 2 j ) 1 j 1 j therefore vector w 4 is an eigenvector of the matrix, and has an eigenvalue of 2 + 2 j . An aside: division by a complex number (in case you have forgotten): 2 2 j 2 + 2 j = ( 2 2 j )( 2 2 j ) ( 2 + 2 j )( 2 2 j ) = 4 + 4 j + 4 j 4 4 + 4 = j (The trick: multiply top and bottom by the complex conjugate of the bottom value.) There are four eigenvectors and eigenvalues for the cyclic convolution matrix with N = 4 (you can verify these if you want).
EE 505 B, Autumn, 2011 Linear Systems 9 Eigenvectors of L Eigenvalues of L 1 1 1 1 10 1 j 1 j 2 + 2 j 1 1 1 1 2 1 j 1 j 2 2 j Complex exponentials of the form w kl N = e j 2 π N kl are eigenvectors of a discrete linear system. 3.2 Eigenfunctions of a continuous LTI system For continuous LTI systems, instead of eigenvectors, we have eigenfunctions, and we make the claim (which we substantiate below) that the eigenfunctions of linear systems are also complex exponentials. Recall that the convolution integral is w ( x )=( h v )( x ) = integraldisplay h ( x y ) v ( y ) dy . We want to write this differently, because it can make calculations easier. To do this, let z = x y then dz = dy and y = x z . The limits on the integral become, y → − then z y then z → − ,

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EE 505 B, Autumn, 2011 Linear Systems 10 so, w ( x ) = integraldisplay h ( z ) v ( x y ) dz = integraldisplay h ( z ) v ( x z ) dz therefore, ( h v )( x ) =( v h )( x ) .
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