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Unformatted text preview: 86 Lecture 21 Eigen Values and Eigen Vectors: To study linear systems we will develop the notion of eigen functions. This notion was used in first part of the course to introduce Fourier transforms. Let us look at some examples of linear systems. 1. Let S → space of all DT signals. Consider a linear transformation given by L : S → S . Let the output be given by y [ n ] = ∞ summationdisplay k =-∞ h [ k ] x [ n − k ] , where h [ n ] is some fixed DT signal. Observation1: There exist a special class of signals satisfying the following property: output is equal to an amplitude scaled version of the input. They are called eigen functions (or eigen vectors). If x [ n ] = e jω n , then y [ n ] = ∞ summationdisplay k =-∞ h [ n ] e jω ( n- k ) = e jω n ∞ summationdisplay k =-∞ h [ n ] e- jω k = e jω n H ( e jω ) Observation2: Every DT signal can be expressed as a linear combination of complex exponentials x [ n ] = 1 2 π integraldisplay π- π X ( e jω ) e jωn dω, where X ( e jω ) denotes the amplitude scaling associated with the signal e jωn . Observation3: These eigen functions are orthogonal ( e jω n , e jω 1 n ) = 0 , for ω negationslash = ω 1 Since these eigen functions are orthogonal, we can compute X ( e jω ) from x [ n ] as: ( x [ n ] , e jω n ) = 1 2 π integraldisplay π- π X ( e jω ) ( e jωn , e jω n ) dω = X ( e jω ) or X ( e jω ) = ∞ summationdisplay n =-∞ x [ n ] e- jω n 87 2. Let S → set of all DT signals. L : S → S , where y [ n ] = x [2 n ] ....
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- Spring '11
- Linear Algebra, Eigenvalue, eigenvector and eigenspace, Eigenfunction, Eigen vectors, Eigen functions