*Introduction
to Computation in Matlab*

*Prof. Brian L. Evans, Dept. of ECE, The University of Texas, Austin,
Texas USA*

Matlab's forte is numeric calculations with matrices and vectors. A vector can be defined as

**vec = [1 2 3 4];**

The first element of a vector is at index 1. Hence, vec(1) would return 1. A way to generate a vector with all of its 10 elements equal to 0 is

**zerovec = zeros(1,10);**

Two vectors, **a** and **b**, can be used in
Matlab to represent the left hand side and right hand side, respectively, of a
linear constant-coefficient difference equation:

a(3) y[n-2] + a(2) y[n-1] + a(1) y[n] = b(3) x[n-2] + b(2) x[n-1] + b(1) x[n]

The representation extends to
higher-order difference equations.
Assuming zero initial conditions, we can derive the transfer function.
The transfer function can also be represented using the two vectors
**a** (negated feedback coefficients) and
**b** (feedforward coefficients).
For the second-order case, the transfer function becomes

We can factor a polynomial by using the roots command.

Here is an example of values for vectors **a** and **b**:

**a = [ 1 6/8 1/8];**

**b = [ 1 2 3];**

For an asymptotically stable transfer function, i.e. one for which the region of convergence includes the unit circle, the frequency response can be obtained from the transfer function by substituting z = exp(j w). The Matlab command freqz implements this substitution:

**[h, w] = freqz(b, a, 1000);**

The third argument for freqz
indicates how many points to use in uniformly sampling the points on the unit
circle. In this example, freqz returns two arguments: the vector of frequency
response values **h** at samples of the
frequency domain given by **w**.
One can plot the magnitude response on a linear scale or a decibel scale:

**plot(w, abs(h));**

**plot(w, 20*log10(abs(h)));**

The phase response can be computed using a smooth phase plot or a discontinous phase plot:

**plot(w, unwrap(angle(h)));**

**plot(w, angle(h));**

One can obtain help on any function by using the help command, e.g.

**help freqz**

As an example of defining and computing with
matrices, the following lines would define a 2 x 3 matrix **A**, then define
a 3 x 2 matrix **B**, and finally compute the matrix **C** that is the
inverse of the transpose of the product of the two matrices **A** and **B**:

**A = [1 2 3; 4 5 6];
B = [7 8; 9 10; 11 12];
C = inv((A*B)');**

**Matlab Tutorials, Help and Training**

Here are excellent Matlab tutorials:

https://stat.utexas.edu/training/software-tutorials#matlab

http://www.mathworks.com/academia/student_center/tutorials/mltutorial_launchpad.html

The following Matlab tutorial book is a useful reference:

Duane C. Hanselman and Bruce Littlefield, *Mastering MATLAB*, ISBN 9780136013303,
Prentice Hall, 2011.

A full version of Matlab is available for your personal computers via a university site license:

https://users.ece.utexas.edu/~bevans/courses/realtime/homework/matlab.html

The first few homework assignments will help step you through Matlab.Technical support is provided through free consulting services from the Department of Statistics and Data Sciences. Simple queries can be e-mailed to stat.consulting@austin.utexas.edu. You can book an appointment via the free consulting services page.

**Running Matlab in Unix**

On the Unix machines in the ECE Learning Resource Center, you can run Matlab by typing

module load matlab

matlab

When Matlab begins running, it will automatically
execute the commands in your Matlab initialization file, if you have one. On
Unix systems, the initialization file must be ~/matlab/startup.m where ~ means
your home directory.