8 - Chapman MATLAB Programming for Engineers

hold on; plot(z1,’b.’); plot(z2,’go’); plot(z3,’rx’); plot(z4,’m*’); plot(z5,’c+’); plot(z6,’kd’); plot(z7,’kp’);

text(real(z1)+0.1,imag(z1),’z1’);

text(real(z2)+0.1,imag(z2),’z2’);

text(real(z3)+0.1,imag(z3),’z1+z2’);

text(real(z4)+0.1,imag(z4),’z1*z2’);

text(real(z5)+0.1,imag(z5),’z1*’);

text(real(z6)+0.1,imag(z6),’j*z2’);

text(real(z7)+0.1,imag(z7),’z2/z1’); xlabel(’Real Part’); ylabel(’Imaginary Part’); title(’Complex Numbers’);

The plot that results is shown in Figure 4.9.

Figure 4.9: Complex number plot

Vector Form Plot

Complex numbers can also be represented graphically as arrows, with the base at the origin of the complex plane and the arrow head at the location of the complex number. This represents the

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complex number as a vector, which will be defined and described in greater detail later in the course.

Matlab supports the plotting of complex numbers as vectors through the compass command. Consider the following example, in which two complex numbers z1 and z2 are added to form z3 and the three complex numbers are plotted using compass.

z1 = 1.5 + 0.5j;

z2 = 0.5 + 1.5j;

z3 = z1 + z2; axis([-3 3 -3 3]); axis square;

grid on; hold on;

plot(z1,’b.’);

plot(z2,’go’);

plot(z3,’rx’);

compass(z1);

compass(z2);

compass(z3);

text(real(z1)+0.1,imag(z1),’z1’);

text(real(z2)+0.1,imag(z2),’z2’);

text(real(z3)+0.1,imag(z3)+0.1,’z1+z2’); xlabel(’Real Part’);

ylabel(’Imaginary Part’); title(’Compass Plot’);

The plot that results, shown in Figure 4.10, clearly shows the parallelogram property of the sum of two complex numbers.

Polar Form Plot

The magnitudes and angles of complex numbers can also be plotted in polar coordinates using the polar command. Consider the following example, in which two complex numbers z1 and z2 are multiplied to form z3 and the three complex numbers are plotted using polar.

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Figure 4.10: Compass plot of complex number addition

polar(phi3,r3,’*’); hold on

text(real(z1)+0.1,imag(z1),’z1’)

polar(phi2,r2,’+’)

text(real(z2)+0.1,imag(z2),’z2’)

polar(phi1,r1,’o’) text(real(z3)+0.1,imag(z3)-0.2,’z1*z2’)

The plot that results, shown in Figure 4.11, clearly shows that the magnitudes multiply and the angles add under complex multiplication. Note that while Matlab usually uses radian units for angles, degrees are used in this plot.

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90

270

Figure 4.11: Polar plot of complex number multiplication

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Section 5

Arrays and Array Operations

Scalars: Variables that represent single numbers, as considered to this point. Note that complex numbers are scalars, even though they have two components, the real part and the imaginary part.

Arrays: Variables that represent more than one number. Each number is called an element of the array. Rather than than performing the same operation on one number at a time, array operations allow operating on multiple numbers at once.

Row and Column Arrays: A row of numbers (called a row vector) or a column of numbers (called a column vector).

Two-Dimensional Arrays: A two-dimensional table of numbers, called a matrix.

Array Indexing or Addressing: Indicates the location of an element in the array.

5.1Vector Arrays

First consider one-dimensional arrays: row vectors and column vectors.

Consider computing y = sin(x) for 0 ≤ x ≤ π. It is impossible to compute y values for all values of x, since there are an infinite number of values, so we will choose a finite number of x values. Consider computing

y = sin(x), where x = 0, 0.1π, 0.2π, . . . , π

You can form a list, or array of the values of x, and then using a calculator you can compute the corresponding values of y, forming a second array y. These can be written down as:

x and y are ordered lists of numbers, i.e., the first value or element of y is associated with the first

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value or element of x.

Elements can be denoted by subscripts, e.g. x1 is the first element in x, y5 is the fifth element in y. The subscript is the index, address, or location of the element in the array.

Vector Creation by Explicit List

A vector in Matlab can be created by an explicit list, starting with a left bracket, entering the values separated by spaces (or commas) and closing the vector with a right bracket.

>> x=[0 .1*pi .2*pi .3*pi .4*pi .5*pi .6*pi .7*pi .8*pi .9*pi pi]

x=

Columns 1 through 7

Matlab functions can be applied to vectors to compute a resulting vector:

>> y=sin(x)

y=

Columns 1 through 7

The two vectors produced above are row vectors, each containing one row and 11 columns. They are each also known as a one-by-eleven array, a 1 × 11 array, or a row vector of length 11.

Vector Addressing

A vector element is addressed in Matlab with an integer index (also called a subscript) enclosed in parentheses. For example, to access the third element of x and the fifth element of y:

>>x(3) ans =

0.6283

>>y(5) ans =

0.9511

Colon notation: Addresses a block of elements

The format for colon notation is:

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(start:increment:end)

where start is the starting index, increment is the amount to add to each successive index, and end is the ending index, where start, increment, and end must be integers. The increment can be negative, but negative indices are not allowed to be generated. If the increment is to be 1, a shortened form of the notation may be used:

(start:end)

• 1:5 means “start with 1 and count up to 5.”

>> x(1:5) ans =

• 7:end means “start with 7 and count up to the end of the vector.”

Thus, a negative index can be used, but the indices generated must all be positive integers.

• 2:2:7 means “start with 2, count up by 2, and stop at 7.”

•3:1 means “start with 3, count up by 1, and stop at 1.” This clearly doesn’t make sense, so Matlab generates the error message:

>> x(3:1) ans =

Empty matrix: 1-by-0

A vector can be used to access elements of an vector in a desired order. To access elements 8, 2, 9, and 1 of y in order:

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Vector Creation Alternatives

Explicit lists are often cumbersome ways to enter vector values.

Alternatives:

•Combining: A vector can also be defined using another vector that has already been defined. For example:

>>B = [1.5, 3.1];

>>S = [3.0 B]

S =

•Changing: Values can be changed by referencing a specific address. For example,

>>S(2) = -1.0;

>>S

S =

changes the second value in the vector S from 1.5 to −1.0.

•Extending: Additional values can be added using a reference to a specific address. For example, the following command:

>>S(4) = 5.5;

>>S

S =

extends the length of vector S from 3 to 4.

Applying the following command

>>S(7) = 8.5;

>>S

S =

extends the length of S to 7, and the values of S(5) and S(6) are automatically set to zero because no values were given for them.

•Colon notation: A generalized version of the colon notation used to index array elements can be used to generate array elements. The format for this notation is:

(start:increment:end)

where start, increment, and end can now be floating point numbers and there is no restriction that the values generated must be positive.

Thus, to create x in the example above:

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>> x=(0:0.1:1)*pi

x=

Columns 1 through 7

As in addressing, (0:5) creates a vector that starts at 0, increments by 1 (implied), and ends at 5.

>> x = 0:5 x =

A particular choice of start and increment values may not lead to the generation of a last value equal to end. The last value generated must be less than or equal to end. For example:

>> x = 0:2:9 x =

In this case, the next value to be generated would be 10, but this is not included in the output since it would be greater than 9, the value of end.

•linspace: This function generates a vector of uniformly incremented values, but instead of specifying the increment, the number of values desired is specified. This function has the form:

linspace(start,end,number)

The increment is computed internally, having the value:

increment = end − start number − 1

For example:

>> x=linspace(0,pi,11)

x=

Columns 1 through 7

In this example:

π

increment = 11 − 1 = 0.1π

• logspace: This function generates logarithmically spaced values and has the form

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logspace(start_exponent,end_exponent,number)

To create a vector starting at 100 = 1, ending at 102 = 100 and having 11 values:

Vector Length

To determine the length of a vector array:

length(x): Returns the length of vector x.

Example:

Vector Orientation

In the preceding examples, vectors contained one row and multiple columns and were therefore called row vectors.

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