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MATLAB ReviewJ. M. Sebeson, DeVry University, 2009This course
makes use of MATLAB for calculations and the plotting of functions.
MATLAB has extensive capabilities, but we will focus on just a few
of them. These notes will review the following topics on the use of
MATLAB:1. Variables and operations: scalars, vectors, and matrices;
uniformly spaced vector sequences; element-by-element
calculations.2. Representation of complex numbers, magnitude and
angle.3. Common built-in functions; trig and exponential
functions.4. Entering and plotting functions and signals.5.
Polynomials and their roots.6. MATLAB help. Tabbing for command
names; lookfor command.7. Symbolic Math toolbox; transforms and
inverse transforms; ordinary differential equations
Topic 1 Variables and operationsNumbers and variables in MATLAB
can be represented as scalars, vectors (either row or column
strings) or matrices (row and column arrays). For example:
>> a=[2] % A scalara =
2
>> b=[1,2,3,4] % A row vectorb =
1 2 3 4
>> c=[1,2,3;4,5,6;7,8,9] % A matrixc =
1 2 3
4 5 6
7 8 9A screen shot of the MATLAB window for the above example is
given below (MATLAB version R2011a). Commands are always entered in
the Command Window at the >> prompt and the results are
always displayed in the Command Window. The other windows provide
other information, but this is not important for our discussion in
these notes.
Continuing with our example, row vectors can be turned into
column vectors and vice versa with the transpose character ().
>> b' % Changes the row vector b to a column vectorans
=
1
2
3
4
In the construction of signals, it is often necessary to create
a vector of values over a certain range in evenly spaced intervals.
For example, suppose you wish to generate the function
over the interval of t = 0 to t = 1 second in intervals of 0.01
seconds. The commands would be:
>> t=0:.01:1;
>> s=sin(2*pi*5*t); % The number pi = 3.1416.. is just
written pi in MATLAB> length(s)
ans =
101
Notice the use of the colon (:) in the construction of the t
vector. Both the t and the s vectors would contain 101
elements.
Other useful commands for constructing vectors with evenly
spaced values are linspace and logspace. (See their help
descriptions.)When carrying out certain operations with vectors,
such as addition, subtraction, and element-by-element
multiplication or division, the length of the operand vectors must
be the same (that is, they must have the same number of elements).
If you wish to carry out element-by-element operations, you must
use the dot-operation syntax. For example:>> b=[1,2,3,4];
>> d=[5,6,7,8];>> b.*d % b and d have the same
length. The .* defines the operation of element-by-element %
multiplication of b and dans =
5 12 21 32>> b.^3 % note the use of .^ to define the
operation of cubing each element of vector b
ans =
1 8 27 64
Topic 2 Complex numbers
In MATLAB the unity imaginary number (that is, ) is represented
either by i or j. (Therefore, never use these letters as variable
names.) To see this, take the squares of i and j:>> i*i
ans =
-1
>> j*j
ans =
-1
MATLAB always represents complex numbers in their rectangular
form:
>> x=exp(-j*pi/4) % A complex number in polar form
x =
0.7071 - 0.7071i % Rectangular form
>> real(x) % The real part of xans =
0.7071 >> imag(x) % The imaginary part of xans =
-0.7071
Given a complex number, you can find the components of its polar
form as follows:
>> x=5*exp(j*2.5) % Polar formx =
-4.0057 + 2.9924i % Rectangular form>> abs(x) % Finds the
magnitude of xans =
5
>> angle(x) % Finds the angle of x in radiansans =
2.5000
By default, MATLAB represents angles in radians. It is a good
idea to always use radians for angular measure in mathematics.Topic
3 Common functions in MATLAB
The functions in MATLAB can be found in the built-in help
documentation. Basic mathematical functions can be listed with the
command help elfun. Some of the more specialized mathematical
functions can be listed with the command help specfun. (Try some of
the functions that are returned by these commands.) Especially
useful functions in signal processing are the trigonometric
functions ( sin, cos, etc) and the exponential function ex.
>> sin(pi/2) % note that 90 degrees is pi/2 radiansans
=
1
>> exp(1) % This calculates e^1, the base of natural
logarithms.ans =
2.7183
> format long % This commands extends the number of decimal
places displayed
>> exp(1)
ans =
2.71828182845905
Topic 4 Plotting graphs - continuous and discrete signalsMATLAB
has very sophisticated graphics capabilities that can be explored
in the help documentation. However, the most basic commands for
plotting are plot (connects the data points with continuous lines)
or stem (creates stem plots of discrete data). Example 1: Plot the
sum of two equal amplitude sinusoids of frequencies 50 Hz and 100
Hz from 0 to 50 milliseconds. Use an increment of .01 ms in the
plot. Note that the signal to be plotted is
In the above equation, f1 = 50 Hz and f2 =100 Hz and t is in
units of seconds. This can also be written as:
In this equation, T1 is the period of f1 and T2 is the period of
f2. T and t can be in any units of time as long as they are the
same units (milliseconds in this example).
>> t=0:.01:50; % A time vector from 0 to 50 in increments
of .01. This vector is in ms.>> f1=50; % Frequencies in
Hz>> f2=100;
>> T1=1000/f1; % Periods in milliseconds (there are 1000
milliseconds per second)>> T2=1000/f2;
>> s=sin(2*pi*t/T1)+sin(2*pi*t/T2); % Note that t and T
would have the same units (ms)>> plot(t,s)
Example 2: Using the same signal as above, plot the
discrete-time signal (digital signal) that would result from
sampling s(t) at a frequency of 500 Hz (that is, a sample every 2
milliseconds). For this example, we will generate a stem plot,
because that is the standard way to represent discrete-time signals
and functions. We can use the same code as above except that we
will make the time vector run from 0 to 50 milliseconds with an
increment of 2 ms and use stem in place of plot.>> t=0:2:50;
% Now the t vector has an increment of 2 instead of .01
>> f1=50; % Frequencies in Hz
>> f2=100;
>> T1=1000/f1; % Periods in milliseconds
>> T2=1000/f2;
>> s=sin(2*pi*t/T1)+sin(2*pi*t/T2);
>> stem(t,s)
To see that the above plot is in fact a sample of the continuous
function, we can re-plot the original graph on the stem plot by
using the hold function.
>> f1=50;
>> f2=100;
>> T1=1000/f1;
>> T2=1000/f2;
>> t=0:2:50; % Time vector with a 2 ms interval
>> s=sin(2*pi*t/T1)+sin(2*pi*t/T2);
>> stem(t,s)
>> t=0:.01:50; % Time vector with a .01 ms interval
>> s=sin(2*pi*t/T1)+sin(2*pi*t/T2);
>> hold % hold the last plot%Current plot held
>> plot(t,s,'k') % The signal s is plotted with a black
line (k flag)>> hold off % Release the plot so that it is not
re-used by later plots
Topic 5 Polynomials and their rootsIn engineering mathematics,
finding the roots of polynomials is an important mathematical
procedure. In general, a polynomial in the variable z can be
written as:
The set of numbers are the coefficients of the polynomial while
the number n is the order of the polynomial. The roots of the
polynomial are those values of z for which p(z) = 0. According to
the Fundamental Theorem of Algebra, the number of roots of a
polynomial is equal to the order. For example, a cubic equation (n
= 3) will always have three roots. The roots may be either real or
complex or both, as the examples below will show. With the possible
exception of quadratic equations (n = 2), it is usually difficult
to find the roots of a polynomial, so a calculator or MATLAB is
necessary. MATLAB uses the command roots to find the roots of a
polynomial.
Example 1: Find the roots of the polynomial . Since this
polynomial has order 3, we expect to find three roots.>>
c=[1,2,3,4] % The coefficients in descending powers of z
c =
1 2 3 4
>> roots(c) % The roots command only needs the vector of
coefficients.
ans =
-1.6506 % As expected, 3 roots, one real, two complex
-0.1747 + 1.5469i % The complex roots are complex conjugates of
each other
-0.1747 - 1.5469i
Example 2. Find the roots of the polynomial . Notice that this
polynomial is equivalent to:
We expect to find eight roots, all of which will have the
magnitude of one.
>> c=[1,0,0,0,0,0,0,0,-1];
>> r=roots(c)
r =
-1.0000
-0.7071 + 0.7071i % Notice that all complex roots come in
complex conjugate pairs -0.7071 - 0.7071i
0 + 1.0000i
0 - 1.0000i
1.0000
0.7071 + 0.7071i
0.7071 - 0.7071i
>> abs(r) % The magnitude of all the roots is 1ans =
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
A picture of the location of the roots in the complex plane can
be generated by the command zplane. >>
c=[1,0,0,0,0,0,0,0,-1];
>> zplane(c) % In this case zplane uses the vector of
polynomial coefficients
In the above figure, the roots of are shown as small open
circles. The dotted circle defines the unit circle in the complex
plane where the magnitude of complex numbers is equal to one.
Topic 6 MATLAB help
MATLAB has extensive built-in help and examples. It is a good
idea to always read the help file for a command or an M-file
(custom MATLAB program) to understand the syntax and options for
the command.
Finding the commands you want can sometimes be challenging.
Several useful tips are the following:1. Try help elfun and help
specfun for lists of the many mathematical functions in MATLAB.
2. Try help general for general purpose commands.
3. Try help graph2d for 2 dimensional plotting commands
4. If you think the command name or M-file may start with a
particular set of letters, try help with your guess, followed by
tab. This will list all the commands that start with the letters
you entered.
5. The command lookfor will find matches to a word or phase in
the first line of the help documentation for all commands.
Example: Try to find a command that will carry out numerical
integration. Our first guess would be help integ . Trying this, you
will only find a match for integerMath. Next, try the lookfor
command:>> lookfor integral % Search for the word integral in
the first line of a help file.ELLIPKE Complete elliptic
integral.
EXPINT Exponential integral function.
DBLQUAD Numerically evaluate double integral.
QUAD Numerically evaluate integral, adaptive Simpson
quadrature.
QUAD8 Numerically evaluate integral, higher order method.
QUADL Numerically evaluate integral, adaptive Lobatto
quadrature.
TRIPLEQUAD Numerically evaluate triple integral.
COSINT Cosine integral function.
SININT Sine integral function.
COSINT Cosine integral function.
FOURIER Fourier integral transform.
IFOURIER Inverse Fourier integral transform.
SININT Sine integral function.Notice that the lookfor command
found the commands involving QUAD, which, it turns out, are the
commands for carrying out a numerical integration.
Topic 7 Symbolic Math Toolbox
When we discuss Laplace transforms, Z-transforms, and Fourier
transforms, we will show how MATLAB can help find transforms and
inverse transforms using the Symbolic Math Toolbox. This is a much
more convenient method than the traditional table look-up of
transforms. The Symbolic Math Toolbox has other capabilities, but
we will mainly restrict our attention to finding transforms and
their inverses. The commands for transform analysis are the
following:1. laplace(f) finds the Laplace transform of the function
f.
2. ilaplace(F) finds the inverse Laplace transform of the
transform F.
3. ztrans(x) finds the Z-transform of the sequence x
4. iztrans(X) finds the inverse Z-transform of the transform
X
5. fourier(f) finds the Fourier transform of the function f
6. ifourier(F) finds the inverse Fourier transform of the
transform F.When using the Symbolic Math Toolbox, those characters
that are to be used as symbols must be declared as symbols by the
syms command. A few examples will illustrate the usage.
Example: Find the Laplace transform of the function . This is a
sinusoid with a frequency of radians/sec. Since MATLAB does not use
Greek characters, we will use w instead of .>> syms t s w %
This declares the symbols
>> f=sin(w*t); % The function for which we will find the
Laplace transform
>> F=laplace(f) % The command to find the Laplace
transform
F =
w/(s^2 + w^2)This result is equivalent to:
Reference to a table of Laplace transforms will verify that this
is the correct answer.
Example: Find the inverse Laplace transform of
>> syms t s % Declare the symbols>> F=1/(s+3); % The
Laplace transform for which we want the inverse>>
f=ilaplace(F) % Finds the inverse Laplace transformf =
1/exp(3*t)
This result is equivalent to:
_1306859799.unknown
_1306866148.unknown
_1384155236.unknown
_1384155608.unknown
_1384167146.unknown
_1384155406.unknown
_1384154369.unknown
_1306863799.unknown
_1306865509.unknown
_1306863646.unknown
_1306779803.unknown
_1306859650.unknown
_1306777566.unknown
