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...Overview Algebraic Homework #1 (2008) Q1: Eigenvectors of some linear operators in matrix form. In each case, find the eigenvalues, the eigenvectors, the dimensions of the eigenspaces, and whether a basis can be chosen from the eigenvectors. 0 1 A. A = . 0 0 a b c B. B = c a b (assume a > b > c > 0 ). b c a 0 1 Observe that B commutes with T = 0 0 1 0 eigenvectors of T. cos sin C. C = . sin ...
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Overview Algebraic Homework #1 (2008) Q1: Eigenvectors of some linear operators in matrix form. In each case, find the eigenvalues, the eigenvectors, the dimensions of the eigenspaces, and whether a basis can be chosen from the eigenvectors. 0 1 A. A = . 0 0 a b c B. B = c a b (assume a > b > c > 0 ). b c a 0 1 Observe that B commutes with T = 0 0 1 0 eigenvectors of T. cos sin C. C = . sin cos Do the eigenvectors form a basis? Hint: 0 1 , and find the eigenvalues and 0 3 0 D. D = 0 0 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 . 1 0 Q2: Adjoints, etc. A. in Work the vector space of finite dimension N over the complex numbers. Use the standard inner product x, y = xk yk Given an operator A in matrix form (specified by k =1 N x1 z1 N , z = and z = Ax , then z = an array akl , so that if x = Akl xl ), find the k l =1 x z N N matrix form of its adjoint A* . B. Work in the vector space of complex-valued functions of time, and using the inner product f , g = f (t ) g (t )dt . Find the adjoint of the time-translation operator ( DT f ) (t ) = f (t + T ) . C. Set up as in B. Find the adjoint of the linear operator A, where Af is defined by ( Af )(t ) = A(t, ) f ( )d .
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Cornell >> WWW-USERS >> 08 (Fall, 2008)
Algebraic Overview Homework #1 (2008) Answers Q1: Eigenvectors of some linear operators in matrix form. In each case, find the eigenvalues, the eigenvectors, the dimensions of the eigenspaces, and whether a basis can be chosen from the eigenvectors. ...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
Algebraic Overview Homework #1 (2008) Answers Q1: Eigenvectors of some linear operators in matrix form. In each case, find the eigenvalues, the eigenvectors, the dimensions of the eigenspaces, and whether a basis can be chosen from the eigenvectors. ...
Cornell >> MED >> 0809 (Fall, 2008)
Algebraic Overview Homework #1 (2008) Answers Q1: Eigenvectors of some linear operators in matrix form. In each case, find the eigenvalues, the eigenvectors, the dimensions of the eigenspaces, and whether a basis can be chosen from the eigenvectors. ...
Cornell >> LST >> 01 (Fall, 2008)
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Cornell >> LST >> 0809 (Fall, 2008)
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Cornell >> MED >> 0809 (Fall, 2008)
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Cornell >> LST >> 08 (Fall, 2008)
Linear Systems Theory Homework #1 (2008) Q1. Some basic properties of Fourier transforms pairs, f ( ) = f (t )e it dt (1) and 1 f (t ) = 2 f ( )eit d . (2) A. Let g (t ) = eiat f (t ) . Find g ( ) in terms of f ( ) . B. Let g (t ) ...
Cornell >> LST >> 0809 (Fall, 2008)
Linear Systems Theory Homework #1 (2008) Q1. Some basic properties of Fourier transforms pairs, f ( ) = f (t )e it dt (1) and 1 f (t ) = 2 f ( )eit d . (2) A. Let g (t ) = eiat f (t ) . Find g ( ) in terms of f ( ) . B. Let g (t ) ...
Cornell >> MED >> 0809 (Fall, 2008)
Linear Systems Theory Homework #1 (2008) Q1. Some basic properties of Fourier transforms pairs, f ( ) = f (t )e it dt (1) and 1 f (t ) = 2 f ( )eit d . (2) A. Let g (t ) = eiat f (t ) . Find g ( ) in terms of f ( ) . B. Let g (t ) ...
Cornell >> LST >> 08 (Fall, 2008)
Linear Systems Theory Homework #1 (2008) Answers Q1. Some basic properties of Fourier transforms pairs, f ( ) = f (t )e it dt (1) and 1 f (t ) = 2 f ( )eit d . (2) A. Let g (t ) = eiat f (t ) . Find g ( ) in terms of f ( ) . Beginni...
Cornell >> LST >> 0809 (Fall, 2008)
Linear Systems Theory Homework #1 (2008) Answers Q1. Some basic properties of Fourier transforms pairs, f ( ) = f (t )e it dt (1) and 1 f (t ) = 2 f ( )eit d . (2) A. Let g (t ) = eiat f (t ) . Find g ( ) in terms of f ( ) . Beginni...
Cornell >> MED >> 0809 (Fall, 2008)
Linear Systems Theory Homework #1 (2008) Answers Q1. Some basic properties of Fourier transforms pairs, f ( ) = f (t )e it dt (1) and 1 f (t ) = 2 f ( )eit d . (2) A. Let g (t ) = eiat f (t ) . Find g ( ) in terms of f ( ) . Beginni...
Cornell >> WWW-USERS >> 01 (Fall, 2008)
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Cornell >> WWW-USERS >> 0809 (Fall, 2008)
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Cornell >> MED >> 0809 (Fall, 2008)
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Cornell >> WWW-USERS >> 0809 (Fall, 2008)
Slepian functions, K=5 0.08 14 0.06 12 0.04 10 0.02 8 0 6 -0.02 4 -0.04 FT of Slepian functions, K=5 -0.06 2 -200 -100 0 100 200 -200 -100 0 100 200 ...
Cornell >> MED >> 0809 (Fall, 2008)
Slepian functions, K=5 0.08 14 0.06 12 0.04 10 0.02 8 0 6 -0.02 4 -0.04 FT of Slepian functions, K=5 -0.06 2 -200 -100 0 100 200 -200 -100 0 100 200 ...
Cornell >> WWW-USERS >> 08 (Fall, 2008)
Noise and Variability Homework #1 (2008) Q1: Power spectra of some simple noises A. Poisson noise. A Poisson noise n(t ) is a sequence of delta-function pulses, each occurring independently, at some rate r. (More formally, it is a sum of pulses of w...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
Noise and Variability Homework #1 (2008) Q1: Power spectra of some simple noises A. Poisson noise. A Poisson noise n(t ) is a sequence of delta-function pulses, each occurring independently, at some rate r. (More formally, it is a sum of pulses of w...
Cornell >> MED >> 0809 (Fall, 2008)
Noise and Variability Homework #1 (2008) Q1: Power spectra of some simple noises A. Poisson noise. A Poisson noise n(t ) is a sequence of delta-function pulses, each occurring independently, at some rate r. (More formally, it is a sum of pulses of w...
Cornell >> WWW-USERS >> 08 (Fall, 2008)
Noise and Variability Homework #1 (2008), answers Q1: Power spectra of some simple noises A. Poisson noise. A Poisson noise n(t ) is a sequence of delta-function pulses, each occurring independently, at some rate r. (More formally, it is a sum of pul...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
Noise and Variability Homework #1 (2008), answers Q1: Power spectra of some simple noises A. Poisson noise. A Poisson noise n(t ) is a sequence of delta-function pulses, each occurring independently, at some rate r. (More formally, it is a sum of pul...
Cornell >> MED >> 0809 (Fall, 2008)
Noise and Variability Homework #1 (2008), answers Q1: Power spectra of some simple noises A. Poisson noise. A Poisson noise n(t ) is a sequence of delta-function pulses, each occurring independently, at some rate r. (More formally, it is a sum of pul...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
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Cornell >> WWW-USERS >> 26 (Fall, 2008)
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Cornell >> MED >> 0809 (Fall, 2008)
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Cornell >> WWW-USERS >> 0809 (Fall, 2008)
simple channel noise 1 0.5 0 0 4 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 power spectrum by multitaper method 10 10 2 10 0 10 -2 10 -4 10 -6 10 -8 NW=1 k=1 NW=3 k=5 NW=7 k=13 analytic -5 10 10 -4 10 -3 10 -2 ...
Cornell >> MED >> 0809 (Fall, 2008)
simple channel noise 1 0.5 0 0 4 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 power spectrum by multitaper method 10 10 2 10 0 10 -2 10 -4 10 -6 10 -8 NW=1 k=1 NW=3 k=5 NW=7 k=13 analytic -5 10 10 -4 10 -3 10 -2 ...
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Cornell >> WWW-USERS >> 08 (Fall, 2008)
Nonlinear Systems Theory Homework #1 (2008) Laguerre Polynomials. These are classically defined as the orthonormal polynomials with respect to the weight exp( x) for x 0 . Here we calculate orthogonal (not necessarily orthonormal) polynomials with r...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
Nonlinear Systems Theory Homework #1 (2008) Laguerre Polynomials. These are classically defined as the orthonormal polynomials with respect to the weight exp( x) for x 0 . Here we calculate orthogonal (not necessarily orthonormal) polynomials with r...
Cornell >> MED >> 0809 (Fall, 2008)
Nonlinear Systems Theory Homework #1 (2008) Laguerre Polynomials. These are classically defined as the orthonormal polynomials with respect to the weight exp( x) for x 0 . Here we calculate orthogonal (not necessarily orthonormal) polynomials with r...
Cornell >> WWW-USERS >> 08 (Fall, 2008)
Nonlinear Systems Theory Homework #1 (2008) Answers Laguerre Polynomials. These are classically defined as the orthonormal polynomials with respect to the weight exp( x) for x 0 . Here we calculate orthogonal (not necessarily orthonormal) polynomial...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
Nonlinear Systems Theory Homework #1 (2008) Answers Laguerre Polynomials. These are classically defined as the orthonormal polynomials with respect to the weight exp( x) for x 0 . Here we calculate orthogonal (not necessarily orthonormal) polynomial...
Cornell >> MED >> 0809 (Fall, 2008)
Nonlinear Systems Theory Homework #1 (2008) Answers Laguerre Polynomials. These are classically defined as the orthonormal polynomials with respect to the weight exp( x) for x 0 . Here we calculate orthogonal (not necessarily orthonormal) polynomial...
Cornell >> WWW-USERS >> 0809 (Fall, 2008)
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Cornell >> MED >> 0809 (Fall, 2008)
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