You can start Matlab (version 7.0) on your workstation by typing the command

matlab

in a command window. After starting up, you will get a Matlab window. To get help on any specific command, such as “plot”, you can type the following

help plot

in the “Command Window” portion of the Matlab window. You can do a keyword search for commands related to a topic by using the following.

lookfor topic

You can get an interactive help window using the function

helpdesk

or by following the Help menu near the top of the window.

Every element in Matlab is a matrix. So, for example, the Matlab command

creates a matrix named “a” with dimensions of 1×3. The variable “a” is stored in what is called the Matlab workspace. The operation

b = a .'

stores the transpose of “a” into the vector “b”. In this case, “b” is a 3×1 vector.

Since each element in Matlab is a matrix, the operation

c = a * b

computes the matrix product of “a” and “b” to generate a scalar value for “c” of

Often, you may want to apply an operation to each element of a vector. For example, you many want to square each value of “a”. In this case, you may use the following command.

c = a .* a

The dot before the * tells Matlab that the multiplication should be applied to each corresponding element of “a”. Therefore the .* operation is not a matrix operation. The dot convention works with many other Matlab commands such as divide ./ , and power .^. An error results if you try to perform element-wise operations on matrices that aren't the same size.

Note also that while the operation a .' performs a transpose on the matrix "a", the operation a ' performs a conjugate transpose on "a" (transposes the matrix and conjugates each number in the matrix).

Matlab has two methods for saving sequences of commands as standard files. These two methods are called scripts and functions. Scripts execute a sequence of Matlab commands just as if you typed them directly into the Matlab command window. Functions differ from scripts in that they accept inputs and return outputs, and variables defined within a function are generally local to that function.

A script-file is a text file with the filename extension ".m". The file should contain a sequence of Matlab commands. The script-file can be run by typing its name at the Matlab prompt without the .m extension. This is equivalent to typing in the commands at the prompt. Within the script-file, you can access variables you defined earlier in Matlab. All variables in the script-file are global, i.e. after the execution of the script-file, you can access its variables at the Matlab prompt. For more help on scripts click here.

To create a function called func, you first create a text file called func.m. The first line of the file must be

function output = func(input)

where input designates the set of input variables, and output are your output variables. The rest of the function file then contains the desired operations. All variables within the function are local; that means the function cannot access Matlab workspace variables that you don't pass as inputs. After the execution of the function, you cannot access internal variables of the function. For more help on functions click here.

Compute these two integrals. Do the computations manually.

For help on the following topics, visit the corresponding link: Plot Function, Stem Command, and Subplot Command.

It is common to graph a discrete-time signal as dots in a Cartesian coordinate system. This can be done in the Matlab environment by using the stem command. We will also use the subplot command to put multiple plots on a single figure.

Start Matlab on your workstation and type the following sequence of commands.

n = 0:2:60;
y = sin(n/6);
subplot(3,1,1)
stem(n,y)

This plot shows the discrete-time signal formed by computing the values of the function sin ( t / 6 ) at points which are uniformly spaced at intervals of size 2. Notice that while sin ( t / 6 ) is a continuous-time function, the sampled version of the signal, sin(n/6), is a discrete-time function.

A digital computer cannot store all points of a continuous-time signal since this would require an infinite amount of memory. It is, however, possible to plot a signal which looks like a continuous-time signal, by computing the value of the signal at closely spaced points in time, and then connecting the plotted points with lines. The Matlab plot function may be used to generate such plots.

Use the following sequence of commands to generate two continuous-time plots of the signal sin ( t / 6 ) .

n1 = 0:2:60;
z = sin(n1/6);
subplot(3,1,2)
plot(n1,z)
n2 = 0:10:60;
w = sin(n2/6);
subplot(3,1,3)
plot(n2,w)

As you can see, it is important to have many points to make the signal appear smooth. But how many points are enough for numerical calculations? In the following sections we will examine the effect of the sampling interval on the accuracy of computations.

For help on the following topics, click the corresponding link: MatLab Scripts, MatLab Functions, and the Subplot Command.

One common calculation on continuous-time signals is integration. Figure 2.1 illustrates a method used for computing the widely used Riemann integral. The Riemann integral approximates the area under a curve by breaking the region into many rectangles and summing their areas. Each rectangle is chosen to have the same width Δt, and the height of each rectangle is the value of the function at the start of the rectangle's interval.

Figure 2.1. 

Illustration of the Riemann integral

To see the effects of using a different number of points to represent a continuous-time signal, write a Matlab function for numerically computing the integral of the function s i n² ( 5 t ) over the interval [ 0 , 2 π ] . The syntax of the function should be I=integ1(N) where I is the result and N is the number of rectangles used to approximate the integral. This function should use the sum command and it should not contain any for loops!

Next write an m-file script that evaluates I(N) for 1≤N≤100, stores the result in a vector and plots the resulting vector as a function of N. This m-file script may contain for loops.

Repeat this procedure for a second function J=integ2(N) which numerically computes the integral of exp(t) on the interval [0,1].