EECS 551 PROBLEM SET #4 Fall 1997 ASSIGNED: Sept. 25, 1997 DUE DATE: Oct. 2, 1997 Read Section 4.1-Section 4.3 of V&K. Note similarities between this material and Section 3.1-Section 3.3. This week's theme: Applications of basis function representations. 1. Solve the differential equation d^2x/dt^2+2=0 on the interval (0,pi) with boundary conditions x(0)=x(pi)=0 using basis functions as follows: a. Show that sin(nt),n an integer, is a complete orthogonal basis set over the interval (0,pi) of L^2 functions x(t) such that x(0)=x(pi)=0. b. Substitute x(t)=sum_n A_nsin(nt) in the diff equation and compute A_n. c. Find the solution directly and confirm its Fourier expansion is (b). NOTE: Make the function odd in t to expand in sines. 2. Basis functions in electromagnetics (from EECS 331): A long conducting cylinder of radius a is split lengthwise. The two halves are maintained at potentials v_1 and v_2. Show that the potential in the cylinder at radius r from the axis is v(r,theta)=(v_1+v_2)/2+(v_1-v_2)/(pi/2) times sum_n[(-1)^(n-1)]/[2n-1](r/a)^(2n-1)cos((2n-1)theta HINT: Remember (?) that the solution to Laplace's equation is v(r,theta)=sum_n(r/a)^n[A_ncos(ntheta)+B_nsin(ntheta)] and find the Fourier series expansion of x(t)=v_1,|t|le pi/2;v_2,pi/2 ge|t|. 3. Legendre polynomials are a complete orthogonal basis for L^2[-1,1]. a. Compute them by performing a Gram-Schmidt orthonormalization of {1,t,t^2,t^3...} using the L^2[-1,1] inner product (x_1(t),x_2(t))=int_-1^1x_1(t)x_2^*(t)dt. b. Expand x(t)=t^3,|t|le1 in Legendre polynomials. c. Expand x(t)=e^-|t|,|t|le1 in Legendre polynomials. Find the first three coefficients of the expansion. 4. V&K Problem #3.6 Note this is D3 Daubechies filter and B(z) is R(z). 5. V&K Problem #3.11a,b For (a): Note P(z=e^{j2pi/3})=0. For (b): Multiply given P(z) by z^{-3} and use G=2H^{-T} (p.17 of my notes) Note it is no longer true that H_1(z)=H_0(-z)! See p.120-1 and 137 The following exchange was noted between President ``Silent Cal'' Coolidge (during whose administration ``America lost its audio'') and a heckler: "Pres. Coolidge, I bet I can get you to say 3 words.'' ``You lose.''