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.''