Category: CDS 101/110 Fall 2003

Identifiers: FN H0 H1 H2 H3 H4 H5 H6 H7 H8 L0.0 L1.1 L1.2 L10.1 L2.1 L2.2 L3.1 L3.2 L4.1 L4.2 L5.1 L5.2 L6.2 L8.1 L9.1 L9.2

Questions

• Please supply us with some numerical examples, for 1) convolution integral, 2) Jordan Form

• How does the impulse response set the initial condition to B since B is a matrix?

• Please supply us with some numerical examples, for 1) convolution integral, 2) Jordan Form
  Submitted by: mreiser
  Submitted on: October 22, 2003
  Identifier: L4.2

  1) The convolution integral tends to confuse most people who are seeing it for the first time, therefore there are many useful internet resources. One that is especially fun (and useful if you have not seen the 'graphical' version of convolution) is the "Joy of convolution" . There is also a "Joy of convolution - discrete version" , though presumably only the signals are discretized--the joy is not. Also let's not forget that Matlab easily performs convolutions for us, the command is 'conv'. Just make up any two functions and convolve them, e.g.:

  x = sin(-pi:0.1:pi); y = zeros(size(x)); y(40) = 1; plot(conv(x,y));

  2) As for the Jordan Form of a matrix, decent numerical examples show up in most Linear Algebra textbooks. Two that definitely have some are: Linear Algebra and Its Applications by Gilbert Strang, and Introduction to Linear Algebra and Differential Equations by John W. Dettman (Dover book). Most of the numerical examples can be very lengthy, spanning several pages of a book. The most detailed examples I've seen are in C. T. Chen's Linear System Theory and Design (older editions of the same book have even better examples. The older edition is called 'Introduction to Linear System Theory'). Without going to the library, the best way to see Jordan form examples is the use Matlab. Here is something to try:

  A = sym([12,32,66,116;-25,-76,-164,-294; 21,66,143,256;-6,-19,-41,-73]); [V,J] = jordan(A);

  this one should give a Jordan form with 2 jordan block of size 2 each. If you try this command on random matrices (e.g. A = rand(3)), you will find that you never see jordan blocks of size greater than 1. This is because the Jordan form is very sensitive to perturbations in the matrix entries (for fun, slightly perturb the A matrix above and see what happens).

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• How does the impulse response set the initial condition to B since B is a matrix?
  Submitted by: demetri
  Submitted on: October 22, 2003
  Identifier: L4.2

  There is some confusion in terminology and in notation here. First, you are presumably asking about the impulse function (or Dirac Delta Distribution, more technically). The impulse function and impulse response are two different things. The impulse is the infinite peak with unit area we discussed in class; the impulse response is the response of a system to an impulse input.

  Now, the impulse "function" has the effect of "setting the initial conditions" when used as an input. In the case where there is a single input, and hence the B matrix is a vector, it really does set the initial condition to B. When B is not just a vector, (i.e. a multi-input system) it is not clear which "impulse" you're referring to, since you need two inputs. If you put impulses in both input slots, you will get the initial condition corresponding to the sum across the columns of the B matrix.

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