Matlab Graphics Tutorial

Matlab Graphics

Cheng-An Yang

September 22, 2013

2D Plots The Graphical User Interface Advanced Topics Animation 3D Plots More Plots Extra

Outline

1 2D Plots

2 The Graphical User Interface

3 Advanced Topics

4 Animation

5 3D Plots

6 More Plots

7 Extra

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Section 1

2D Plots

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Figure: create

Syntax

• To create an empty figure, callfigure();

• You can also number your figure likefigure(42);

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Figure: close

Syntax

• Close a specific figureclose(42);

• Close all figuresclose all;

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Plot

Syntax

• To create a line plot of vector y versus vector t, useplot(t,y);

• What happen if you don’t provide the x-axis data? Tryplot(y);

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Multiple Lines

Syntax

• You can input more vector pairs likeplot(t1,y1,t2,y2);

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Exercise: Multiple Lines

Visualize three elementary functions on 0 ≤ t ≤ 2π:

y1 = sin(t)

y2 = cos(t)

y3 = e−t

Plot them on the same figure.

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Adding Labels

Syntax

• Adding labels on axisxlabel(’x label’);ylabel(’y label’);

• Adding titletitle(’title for the figure’);

• Adding legendslegend(’first’,’second’,...);

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Grid

Syntax

• Turn on/off the grid linesgrid on;grid off;

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Changing the Axes Limits

Syntax

• Set the limits of each axisaxis([xmin xmax ymin ymax]);

• If you want to adjust only x-axis or y-axis,xlim([xmin xmax]);ylim([ymin ymax]);

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Customize Plots

• Using the string specifier to change the line style.• The string specifiers contains

• Line style: {’-’,’--’,’:’,’-.’,’none’}• Marker symbol: {’+’,’o’,’*’,’.’,’x’} and more.• Color: {’r’,’g’,’b’,’w’,’k’}.

• For example,plot(t,x,’--or’);plot(t,x,’r’,t,y,’-.xk’);

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Exercise: The Envelop

Please duplicate the figure shown below:

0 5 10−1

−0.5

0

0.5

1The blue line is

x = t2 cos(5t)e−t . (1)

The envelope of x(t) is

y = ±t2e−t . (2)

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Plot Matrix Data

Syntax

• By default plot(Y) will plot each column of Y.• When specifying the x vector,

plot(x,Y);

will try to match the dimension of x and Y.

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Exercise: Plot Matrix Data

LetY = [ y1; y2; y3 ];

• What is the dimension of Y?• Try plot(t,Y) and plot(Y). Can you anticipate the

outputs?

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Plot Complex Data

Syntax

• To plot complex array x, useplot(x);

• It is equivalent toplot(real(x),imag(x));

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Exercise: The Eigenvalues of Random Matrices

Gaussian Random MatrixA Gaussian Random Matrix H is a matrix with standard normalcomponents hij

d= N (0, 1).

h11

h22

h12 h21TX1

TX2

RX1

RX2

• Arises in many applications.• Wireless communication.• Channel gain hij is random.

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Exercise: The Eigenvalues of Random Matrices

Visualizing the distribution of theeigenvalues of random matrices.

• Generate H = randn(n,n).• [V,D] = eig(H).• Plot its eigenvalues as dots on the

complex plane.• Increase n form 10 to 1, 000. Can

you tell what’s the pattern of theeigenvalues?

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Plotting Multiple Lines On the Same Axes

• If you call plot twice, the first plot will be erased.• To retain current graph when adding new graph, tell Matlab to

hold on;

• If you want different lines to have different color, usehold all;

• The default ishold off;

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Exercise: Exponent and Convexity

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

xn

0.10.51210

• The function xα is convexwhen α ≥ 1.

• When 0 < α < 1 is itconcave.

• Use a for loop and hold toverify this.

• α = {0.1, 0.5, 1, 2, 10}.

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Section 2

The Graphical User Interface

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The GUI

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Saving and Loading

Some tips:• Save your file in vector formats, such as .eps and .pdf.• Use ‘copy figure’ between applications.• Keep a .fig copy so you can edit it later.

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Customization Using GUI

• Click the ‘Edit Plot’ icon.• Double click on the figure to

enter the editing mode.• Select an object, and the

property editor will appear.

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Exercise: Frequency Response

• The transfer function of a second order system has the form

H(s) =ω2

ns2 + 2ζωns + ω2

n,

where ζ is the damping ratio and ωn is the natural frequency.• The frequency response of a system is characterized by

• The magnitude |H(ω)|• The phase ∠H(ω)

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Exercise: Frequency Response

10−1

100

101−100

−50

0

50

Frequency (rad)

Mag

nitu

de (

dB)

10−1

100

101−π

−π/2

0

Frequency (rad)

Pha

se (

rad)

• Let ωn = 1, ζ = 0.2.• Frequencies from 10−1 to 10

(rad).• The unit imaginary number

in Matlab is 1j and 1i.• Note the magnitude

response is defined as20 log(|H(ω)|).

• You may want to useangle and logspace.

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Graded Task 1

The Bessel functions Jα(x) of the first kind and order α is thesolution to theBessel’s Equation

x2 d2ydx2 + x dy

dx + (x2 − α2)y = 0.

• Separable solution to many important PDE in cylindricalcoordinate.

• Useful in physics, signal processing and statistics.

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Graded Task 1

Bessel function has the following representation

Bessel Functions of the First Kind

Jα(x) =∞∑

m=0

(−1)m

m! Γ(m + α + 1)

(12x)2m+α

.

Matlab has built-in function for Jα(x)

Syntax

J = besselj(alpha, x);

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Graded Task 1

Visualizing Jα(x) with different α. Please create a figure like this:

0 1 2 3 10 15

0.0

1.0

x

Jα(x)

α = 0

α = 1α = 2

α = 3

α = 10

• α = {0, 1, 2, 3, 10}.• 0 ≤ x ≤ 15. Note that some

tick labels are omitted.• Use \alpha for α.• Insert a text box near the

maximum amplitude of eachfunction.

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Exploring Data

• Change view point, zoomin/out, pan.

• Create multiple datatips.• Select/Brush Data tool.

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Select And Linking Data

• Open the Variable Editor.• Click the ‘Link Plot’ icon.• Click the ‘Select/Brush

Data’ icon.• Select the data of interest.

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Exercise: Wave Editing

• Load the chirp.mat file.• Type sound(y,Fs) to

play it.• Use data selection tool to

select and ‘mute’ the chirps.• Play the modified recording.

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Section 3

Advanced Topics

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Why Learning Commands?

• Almost everything can be adjusted using GUI, why learningcommands?

• If you only need to edit a figure or two, use GUI.• If you have a dozen of figures, let computers do the work.• To create an animation, you have to know the commands.• Programmer’s pride.

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Hierarchy of The Graphic Objects

A Matlab plot is composed of (at least) three objects:

(a) Figure window. (b) Axis object. (c) Line series object.

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Get a Handle on Objects

• Each object has a unique identifier, called the handle.• You can ask Matlab to get and set the properties of the object

with its handle.• To get the handle of the current figure, type

h_fig = gcf;

• To get the handle of the current axis, typeh_ax = gca;

• To get the handle of the line series, useh = plot(t,x);

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Get and Set Property

• Get the value of the property:value = get(h,’PropertyName’);

• Set using property-value pair:set(h,’PropertyName’,value);

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Exercise: Batch Process

• The commandsaveas(h_fig, ’file_name’, ’fig’);

will save the figure h fig as file name.fig.• You can replace the third argument by other format. Type

help saveas for more information.• Write a script that does the following:

• For each frequency ω = {1, 2, . . . , 10}, plot sin(ωt) on0 ≤ t ≤ 2π in 10 separate figures.

• Save and name them as freq1.fig, freq2.fig,...freq10.fig files.

• Hint: the command sprintf(’freq%d.fig’,I) willgenerate the desired file name, where I is an integer.

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Subplot

Syntax

subplot(m,n,1);plot(t,x);

Will create a m-by-n subplot and place the plot of x versus t atlocation 1.

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Exercise: Sine Wave Matrix

Create 25 subplots of sine wave with increasing frequency.

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

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Section 4

Animation

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Making Animated Sequences

• The simplest way: continually erase and redraw your figure.• Key ingredients:

• Loop.• Update data.• Erase and redraw.• Pause.

Example

for I = 1:N % Loop.x = sin(t-I); % Update data.plot(t,x); % Erase and Redraw.pause(0.1); % Pause for 0.1 sec.

end

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Example: Vibrating String

Standing WaveAs oppose to a travelling wave, a standing wave oscillates in placewithout propagating.

Mathematically it is described by

y = A cos(ωt) sin(kx),

where ω is the angular frequencyand k is the wave number.

• Let ω = 5, k = 1.5,A = 2.• Let 0 ≤ x ≤ 2π.• Make an animation of such

a standing wave.

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Graded Task 2

Standing WaveA standing wave y can be seen as the result of interferencebetween two waves y1 and y2 travelling in opposite directions.

Mathematically,

y1 = A sin(kx − ωt)

y2 = A sin(kx + ωt)

and

y = A sin(kx−ωt)+A sin(kx+ωt).

• A: amplitude.• k: wave number.• ω: angular velocity.

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Graded Task 2

Create an animation of a standing wave.

0 2 4 6 8 10

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

• A = 1, k = 1, ω = 3.• Red wave: from left to right.• Blue wave: from right to

left.• Black wave: the resulting

standing wave.• Try update rate dt = 0.05.• Add a reference line x = 0.• Bonus problem: labelling the

nodes. (the red circles)

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Section 5

3D Plots

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3D Line Plot

For parametric data (x(t), y(t), z(t)), we can use the 3D versionof the plot function.

Syntax

• Plot lines in 3Dplot3(x,y,z);

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Exercise: Brownian Motion

−500

50

−20

0

20−20

0

20

The position vector xn of aBrownian particle at time n isgiven by

xn = xn−1 + v ,

where v is a standard normalrandom vector.

• Simulate the path of twoBrownian particles startingat the origin.

• Take N = 500 steps.

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Representing Matrix Data

−100

10

−10

0

10−0.5

0

0.5

1

• Line series are representedby vectors.

• Data with 2D indices arerepresented by matrices.

• Think of the values ofmatrix Z as “height”.

• The indices of x-axis andy-axis are stored in matrix Xand Y.

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Mesh Grid and Surface Plot

To visualize a function f (x , y) on (x , y) ∈ [a, b]× [c, d ],• First we need to create a rectangular grid.• Matlab provides a useful function to create such a grid:

[X,Y] = meshgrid(x,y);

where x and y are the sampling points on both axis.• Then we can use

surf(X,Y,f(X,Y));

to produce a surface plot.

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Exercise: The 3D Sinc Function

3D Sinc Function

sinc(r) :=sin(r)

r , where r2 = x2 + y2.

• Visualize the 3D sinc function on (x , y) ∈ [−8, 8]× [−8, 8].• One way to avoid the divided-by-zero error is to use

sin(r)./(r+esp) instead of sin(r)./r.• esp is the smallest representable floating point number in

Matlab.

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Contour Plot

• Surface plot is fancy but not suitable for putting in your paper!• Contour plot might be more appropriate.

Syntax

• The basic syntax iscontour(Z);

• You can specify the number of contour levelcontour(Z, n_level);

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Exercise: Scalar Field

Letf (x , y) = x2 − 3 sin(xy).

Use contour plot to visualize f (x , y) on [−2, 2]× [−2, 2].

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Vector Field

A 2D vector field is a vector-valued function

z(x , y) =

(zx (x)zy (y)

).

Matlab provides a function to visualize vector files.

Syntax

• Plot the vector field Zx and Zy as a function of X and Y

quiver(X,Y,Zx,Zy);

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Exercise: Scalar Field and its Gradient

• The gradient of a scalar field f (x , y) is defined as

∇f =

(∂∂x f∂∂y f

).

• Numerically, we can approximate the partial derivative by

∂

∂x f (x , y) ≈ f (x + h, y)− f (x , y)

h

for some small number h.• In Matlab, we can use gradient(F) to find the gradient of

the scalar field F. See help for more detail.

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Exercise: Scalar Field and its Gradient

−2 −1 0 1−2

−1

0

1

• Find the gradient of thescalar field

f (x , y) = x2 − 3 sin(xy).

• Plot f (x , y) usingcontour().

• Plot its gradient usinggradient() in the samefigure.

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Section 6

More Plots

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Histogram

Histogram is useful to visualize the distribution of univariate data.

Syntax

• Create a histogram of data vector xhist(x);

• To specify number of bins, usehist(x,m);

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Exercise: Random Number Generators

• Two important random number generators in Matlab:• Uniform between 0 and 1: rand().• Standard normal: randn().

• Exercise:• Generate N random samples from rand().• Use hist to visualize the distribution.• Experiment with different N and different number of bins.• How many samples are required to produce a good

approximation of it distribution?• Repeat for randn().

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Area, Bar and Pie ChartSyntax

• Stacks each data series and fill the underlying area withdifferent colors

area(X,Y);

• Create a bar chartbar(Y);

Each column of Y will have the same color and rows aregrouped together.

• Pie chartpie(Y);

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Exercise: Operating System War!

• Go to http://www.netmarketshare.com/ anddownload the operating system share trend data (Excel file).

• Import it into workspace.• Create the following plots:

• Line plot.• Area chart.• Bar chart.• Pie chart on April, 2012.

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http://www.netmarketshare.com/


Scatter Plot

Scatter plot is used to visualize the distribution of two dimensionaldata.Syntax

• To generate the scatter plot for the data vector X and Y

scatter(X,Y)

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Exercise: Basic Regression• Suppose we are given a data set x and y.• It is assumed that y = ax + b + noise.• Find the best linear fit y = ax + b.

0 0.5 12

3

4

5

6

x

y

y = 2.0x+ 3.0

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Section 7

Extra

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Solving Differential Equations

• Differential equation involves derivatives of a function in itsindependent variables.

• Almost everything is described by some differential equation.• Mostafa will talk about solving differential equations using

powerful tools provided by Matlab.• We can create a poor man’s differential equation solver.• The idea of Finite Difference Method (FDM):

∂u∂t ≈

u(t + h)− u(t)

h

for small h.

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The Finite Difference Method• Consider the wave equation

utt = c2∆u − but ,

where the Laplacian operator ∆ is defined as

∆u = uxx + uyy .

• Use finite difference to approximate derivatives:

ut ≈u(t + ∆t)− u(t)

∆t

utt ≈u(t + ∆t)− 2u(t) + u(t −∆t)

∆t2 .

Similarly for uxx and uyy .

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The Finite Difference Method

• In Matlab, we can use the discrete Laplacian function del2to approximate

∆u ≈ 4h2del2(u),

where h is the spacing in the spatial grid.• Substitute the derivatives by their finite difference

approximation, we get (verify this!)

u(t+∆t) ≈ 2u(t)−u(t−∆t)+h2∆u(t)−b∆t(u(t)−u(t−∆t)).(3)

• Provide the initial data of u and ut at t = 0 and the boundarycondition, we can approximate the solution of the waveequation by recursively solving (3).

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Demo: 2D Wave Equation

−4−2

02

4

−4

−2

0

2

40

2

4

6

8

10

Figure: The simulated solution to the wave equation.

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Exercise: Linear Congruential Generator (LCG)• LCG is a popular (and old) Pseudo-Random Numbers

Generator.• Simple and efficient to compute.• Poor choice of parameters lead to bad performance.

LCG

xn+1 ≡ (axn + b) (mod m), (4)

• m > 0: the modulus• a > 0: the multiplier• b ≥ 0: the increment• x0: the seed

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Histogram

0 10 20 300

5

10

15• Try different parameters and use

hist to evaluate its performance.• If you can’t figure out what

combination of parameters wouldwork, try

• a = 3• b = 0• m = 31• x0 = 1

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Polar Plot• Sometimes it is easier to express coordinate in the polar form.• Let (x , y) be the coordinates in the Cartesian coordinate

system, its corresponding polar coordinates is given by

r =√

x2 + y2 (5)θ = tan−1(y/x). (6)

Syntax

• Plot r versus theta in the polar coordinatepolar(theta,r);

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Exercise: Butterfly CurveThe butterfly fly curve, discovered by Temple H. Fay, is generatedby the equations

r = esin θ − 2 cos(4θ) + sin5(2θ − π

24

). (7)

2

4

6

30

210

60

240

90

270

120

300

150

330

180 0

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Demo: Lorenz Attractor

Lorenz system is a simplified model for atmospheric convection, itis modeled by the ordinary differential equations

dxdt = σ(y − x), (8)dydt = x(ρ− z)− y , (9)dzdt = xy − βz , (10)

where x , y and z are the coordinate of the state, t represents time,ρ, σ and β are parameters.

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3D Line Plot

Example Lorenz Attractor

−20 0 20

−500

50

0

20

40

60

xy

z

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Exercise: Visualizing Gibbs’ Phenomena

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Exercise: Complex Data as 2D Representation

Plot the Hypocycloid.

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Education

Matlab Graphics Tutorial