Signals in MatLab.pdf

7/27/2019 Signals in MatLab.pdf

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E E 2 7 5 Lab January 15, 2012

LAB 1. Signals in Matlab

Introduction

This lab will describe how to use Matlab for some basic signal representation and manipu-lation: Creating and importing signals Sampling and resampling Signal visualization

Modeling noise Modulation

You will be needing to run some special programs for all the MATLAB Labs in EE289.Go to

http://www.cems.uvm.edu/~mirchand/classes/EE275/2012/Software/

and download sg01 and sg02 and save it somewhere on your disc. When using Matlab, seta path to it, so that all functions in sg01 and sg02 are accessible.

Discrete Signals

Time base: t = [0.0 0.1 0.2 0.3]Signal data: x = [1.0 3.2 2.0 8.5]

The central data construct in Matlab is the numeric array, an ordered collection of realor complex numeric data with one or more dimensions. The basic data objects of signal

processing (one-dimensional signals or sequences, multichannel signals, and two-dimensionalsignals) are all naturally suited to array representation.

Matlab represents ordinary one-dimensional sampled data signals, or sequences, as vectors.Vectors are 1-by-n or n-by-1 arrays, where n is the number of samples in the sequence.

One way to introduce a sequence into Matlab is to enter it as a list of elements at the com-mand prompt. The statement

x = [1 2 3 4 5]


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creates a simple five-element real sequence in a row vector. It can be converted to a columnvector by taking the transpose:

x = [ 1 2 3 4 5 ]

Column vectors extend naturally to the multichannel case, where each channel is representedby a column of an array.

Another method for creating vector data is to use the colon operator. Consider a 1-secondsignal sampled at 1000 Hz. An appropriate time vector would be

t = 0:1e-3:1;

where the colon operator creates a 1001-element row vector representing time from zero toone second in steps of one millisecond.

You can also use linspace to create vector data:

t = linspace(0,1,1e3);

creates a vector of 1000 linearly spaced points between 0 and 1.

Try:

t1 = [0 .1 .2 .3];

t2 = 0:0.1:0.3;

t3 = linspace(0, 0.3, 4);

T = [t1 t2 t3];

X = sin(T)

Q1: What does this code show?

Sampling Signals

Analog signal sources include electromagnetic, audio, sonar, biomedical and others. Analogsignals must be sampled in order to be processed digitally.

Samplingx(n) = xa(nTs)x is a discrete signal sampled from the analog signal xa with a sample period of Ts and asample frequency of Fs = 1/Ts.

Try:

Fs = 100;

N = 1000;

stoptime = 9.99;


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t1 = (0:N-1)/Fs;

t2 = 0:1/Fs:stoptime;

x1 = sin(2*pi*2*t1);

x2 = sin(2*pi*3*t2);plot(x1)

figure, plot(x2)

An alternative to creating signals is to use a toolbox function. A variety of toolbox functionsgenerate waveforms . Each of them requires that you begin with a vector representing a timebase. Some of these functions will be described later in this lab.

Aliasing

Digital signals are often derived by sampling a continuous-time signal with an analog-to-digital (A/D) converter. If the continuous signal, xa(t), is bandlimited, meaning that it doesnot contain any frequencies higher than a maximum frequency fM, the Shannon samplingtheorem says that it can be completely recovered from a set of samples if the samplingfrequency fs is greater than two times the maximum frequency of the signal to be sampled:

fs > 2fM

This maximum frequency fM is known as the Nyquist frequency. If the sampling frequency isnot greater than two times the Nyquist frequency, the continuous signal cannot be uniquely

recovered and aliasing occurs. You heard examples of aliased signals in homework number1.

fs > 2fM: Original signal and sampled signal have the same frequency.fs 2fM: Sampled signal is aliased to half the original frequency.

Try:

t = 0:0.001:2;

xa = sin(2*pi*5*t);

plot(t,xa)

hold onfs = 15;

ts = 0:1/fs:2;

xs1 = sin(2*pi*5*ts);

plot(ts,xs1,ro-)

fs = 7.5;

ts = 0:1/fs:2;

xs2 = sin(2*pi*5*ts);

plot(ts,xs2,ro-)


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hold off

Q2: What is the frequency of xs2? (That is, were you to reconstruct the signal, whatfrequency would you see?) You have to think about this one. (Note: you can use the GUIsptool, import the signal, set the sampling frequency to 7.5 and do a fft; that will show youthe answer).Q2A. What does the hold function do?

Signal Visualization

View signal amplitude vs. time index Functions: plot, stem, stairs, strips

Listen to data: sound

Note: the sound and soundsc commands will not work if your computer hardware isnt setup. If that is the case, view the signals instead of listening to them.

Try:

t = [0.1 0.2 0.3 0.4];

x = [1.0 8.0 4.5 9.7];

plot(t,x)

figure, stem(t,x)

figure, stairs(t,x)

fs = 1000;

ts = 0:1/fs:2;

f = 250 + 240*sin(2*pi*ts);

x = sin(2*pi*f.*ts);

strips(x,0.25,fs)

sound(x,fs)

plot(ts,x)

plot(ts(1:200),x(1:200))

Q3: What does the strips command do? (Type help strips on the command line.)Q4: What does the .* operator do in the equation for x above? (In Matlab, go to Help Product Help; then search for vector multiplication).


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Signal Processing Tool

The Signal Processing Toolbox application, SPTool, provides a rich graphical environment

for signal viewing, filter design, and spectral analysis.

You can use SPTool to analyze signals, design filters, analyze filters, filter signals, and an-alyze signal spectra. You can accomplish these tasks using four GUIs that you access fromwithin SPTool:

The Signal Browser is for analyzing signals. You can also play portions of signals usingyour computers audio hardware.

Using the Signal Browser you can

View and compare vector/array signals Zoom in on a range of signal data to examine it more closely Measure a variety of characteristics of signal data Play signal data on audio hardware

To open/activate the Signal Browser for the SPTool, Click one or more signals (use the Shift key for multiple selections) in the Signals list ofSPTool. Click the View button in the Signals list of SPTool.

The Filter Designer is for designing or editing FIR and IIR digital filters. Note that theFDATool is the preferred GUI to use for filter designs. FDATool is discussed in later labs. The Filter Viewer is for analyzing filter characteristics. The Spectrum Viewer is for spectral analysis.

Open SPTool by typing sptool at the command prompt >>.

Try:

>>sptool

You see 3 panes - Signals, Filters Spectra. View all the signals in these panes. You can

play the signals in Signals through using the speaker icon in the Signals browser. (Note:The speaker will only play the signal between the two cursors on the Signal Browser).

When viewing spectra, note that many methods of determining spectra, including FFT areavailable. At this point, just use the FFT. In the Filters pane just View all three filters(LSip, PZip, FIRbp) at this time and get a sense of the capabilities of Filters.


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Importing a Signal

You can use SPTool to analyze the signals, filters, or spectra that you create at the Matlab

command line.

You can import signals, filters, or spectra from the Matlab workspace into the SPToolworkspace using the Import item under the File menu.

Try:

fs = 1000;

ts = 0:1/fs:0.5;

f = 250 + 240*sin(2*pi*ts);

x = sin(2*pi*f.*ts);

Import these signals (f

andx

) into the SPTool and use the tool to examine them.

Q5: Explain in one sentence the signal x that you are generating.Q6: What are the icons to use for horizontal zoom and unzoom?Try zooming and unzooming using the mouse.

Changing Sample Rates

To change the sample rate of a signal in SPTool,

1. Click a signal in the Signals list in SPTool.

2. Select the Sampling frequency item in the Edit menu.

3. Enter the desired sampling frequency and cliick OK. That is all there is to it.

Try changing the sampling rate of the imported signal. You can play it and see the differencein the sound.

Signal Generation

Signals

Create a time base vectort = [0:0.1:2];


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Create a signal as a function of timex = sin(pi*t/2);

plot(t,x)

Useful Matlab functions

Nonperiodic functionsones, zeros

Periodic functionssin, cos, square, sawtooth

Nonperiodic Signals

t = linspace(0,1,11)

Step:y = ones(1,11);

stem(y)

Impulse:y = [1 zeros(1,10)];

stem(y)

Ramp:y = 2*t;

plot(y)


step, impulse, gensig

Try:

Step function:fs = 10;

ts = [0:1/fs:5 5:1/fs:10];

x = [zeros(1,51) ones(1,51)];

stairs(ts,x)

Impulse function with width w:


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fs = 10;

w = 0.1;

ts = [-1:1/fs:-w 0 w:1/fs:1];

x = [zeros(1,10) 1 zeros(1,10)];plot(ts,x)

Delta function:ts = 0:0.5:5;

x = [1 zeros(1,length(ts)-1)];

stem(ts,x)

axis([-1 6 0 2])

Sinusoids

Sinusoid parameters Amplitude, A Frequency, f Phase shift, Vertical offset, B

The general form of a sine wave is

y = Asin(2f t + ) + B

Example: generate a sine wave given the following specifications: A = 5 f = 2 Hz = /8 radians

t = linspace(0,1,1001);

A = 5 ;

f = 2 ;

p = pi/8;

sinewave = A*sin(2*pi*f*t + p);plot(t, sinewave)

You can also edit a Matlab .m file that is in your path. Try:

edit sine_wave

sine_wave


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The first command lets you look at the program. The second command runs the program.You can also try:

edit sinfun

[A T] = sinfun(1,2,3,4)

Square Waves

Square wave generation is like sine wave generation, but you specify a duty cycle, which isthe percentage of the time over one period that the amplitude is high.

Example: duty cycle is 50% (the Matlab default) frequency is 4 Hz.

t = linspace(0,1,1001);

sqw1 = square(2*pi*4*t);

plot(t,sqw1)

axis([-0.1 1.1 -1.1 1.1])

Example: duty cycle is 75% frequency is 4 Hz.

t = linspace(0,1,1001);

sqw2 = square(2*pi*4*t,75);

plot(t,sqw2)

axis([-0.1 1.1 -1.1 1.1])

Sawtooth Waves

Sawtooth waves are like square waves except that instead of specifying a duty cycle, youspecify the location of the peak of the sawtooth.

Example: peak at the end of the period (the Matlab default) frequency is 3 Hz.


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t = linspace(0,1,1001);

saw1 = sawtooth(2*pi*3*t);

plot(t,saw1)

Example: peak is halfway through the period frequency is 3 Hz.

t = linspace(0,1,1001);

saw2 = sawtooth(2*pi*3*t,1/2);

plot(t,saw2)

Complex Signals

Periodic signals can be represented by complex exponentials:

x(t) = ej2ft = cos(2f t) + jsin(2f t) = cos(t) + jsin(t)

If t is measured in seconds, then f will have units of sec1, and will have units of radi-ans/second.

In signal processing, we associate the unit circle with one sampling cycle, so that a samplingfrequency of Fs is associated with 2 radians, and the Nyquist frequency Fs/2 is associatedwith radians. Values of in the upper half-plane, in units of Hz, then correspond tofrequencies within the sampled signal.

In Matlab, type:

x = exp(2*pi*j*f*t);

plot(x)

Matlab recognizes either j or i as the square root of -1, unless you have defined variables jor i with different values.


real, imag, abs, angle

Try:

edit zsig

zsig(5)


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Look at both figures and describe what you see.

Importing Data

An important component of the Matlab environment is the ability to read and write datafrom/to external sources. Matlab has extensive capabilities for interfacing directly with datafrom external programs and instrumentation.

In this lab, we concentrate on reading and writing data that has already been stored inexternal files.

Files come in a variety of standard formats, and Matlab has specialized routines for workingwith each of them. To see a list of supported file formats, type:

help fileformats

To see a list of associated I/O functions, type:

help iofun

Matlab provides a graphical user interface, the Import Wizard, to the various I/O functions.You access the Wizard by choosing File Import Data or by typing:

uiimport

The Matlab command importdata is a programmatic version of the Wizard, accepting all ofthe default choices without opening the graphical user interface. You can use importdata inM-files to read in data from any of the supported file formats.

Matlab also has a large selection of low-level file I/O functions, modeled after those in the Cprogramming language. These allow you to work with unsupported formats by instructingMatlab to open a file in memory, position itself within the file, read or write specific formatteddata, and then close the file.

Try:

You can skip the next 12 lines

jan = textread(all_temps.txt,%*u%u%*[^\n],headerlines,4);

[data text] = xlsread(stockdata.xls);

plot(data(:,2))


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legend(text{1,3})

Explain how the colon operator works in the preceding plot command.

I = importdata(eli.jpg);

image(I)

which theme.wav

uiimport

Browse for:

theme.wav

soundsc(data,fs)

Replace the above 12 lines with the following:

Use uiimport to import eli.jpg and theme.wav. Look at the image and play the sound.

Save and Load

Two data I/O functions are especially useful when working with Matlab variables.

The save command writes workspace variables to a binary Matlab data file (MAT-file)with a .mat extension. The file is placed in the current directory.

The load command reads variables from a MAT-file back into the Matlab workspace.

Although quite specialized, save and load can be used for day-to-day management of yourMatlab computations.

Try:

doc save

doc load

t = 0:0.1:10;

x1 = sin(t);

x2 = sin(2*t);

x3 = sin(3*t);


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save myvars

clear

load myvars t x3

Note the list of variables in the workspace tab in the upper left of the Matlab window.

Modeling Noise

To model signals in space, in the atmosphere, in sea water, or in any communications channel,it is necessary to model noise.

Matlab has two functions for generating random numbers, which can be added to signals to

model noise.

Uniform random numbers

A = rand(m,n);

generates an mxn array of random numbers from the uniform distribution on the interval[0,1]. To generate uniformly distributed random numbers from the interval [a,b], shift andstretch:

A = a + (b-a)*rand(m,n);

Specifically:un=-5 + rand(1,1e6);

hist(un,100);

Gaussian random numbers

A = randn(m,n);

generates an mxn array of random numbers from the standard normal distribution withmean 0 and standard deviation 1. To generate random numbers from a normal distributionwith mean mu and standard deviation sigma, shift and stretch:

A = mu + sigma*rand(m,n);

Specifically:un=10 + 5*randn(1,1e6);

hist(un,100);


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Random numbers from other distributions

Random numbers from other distributions can be generated using the uniform random num-ber generator and knowledge of the distributions inverse cumulative distribution function.Random number generators for several dozen common distributions are available in theStatistics Toolbox.

Adding Noise to a Signal

noisy signal = signal + noise

y1 = x + rand(size(x)) % uniform noise

y2 = x + randn(size(x)) % Gaussian noise

Example:Add Gaussian noise to middle C.

fs = 1e4;

t = 0:1/fs:5;

sw = sin(2*pi*262.62*t); % middle C

n = 0.1*randnsize(sw);

swn = sw + n:

Try:

edit noisyC

noisyC

strips(swn, .1,1e4)

Zoom in on the strips plot. (Note: you might have to cut and paste from the noisyC scriptto generate swn.)

Pseudorandomness

This number:

0.95012928514718

is the first number produced by the Matlab uniform random number generator with itsdefault settings. Start up Matlab, set format long, type rand, and you get the number.

If all Matlab users, all around the world, all on different computers, keep getting this same


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number, is it really random? No, it isnt. Computers are deterministic machines andshould not exhibit random behavior. If your computer doesnt access some external de-vice, like a gamma ray counter or a clock, then it must really be computing pseudorandom

numbers.A working definition of randomness was given in 1951 by Berkeley professor D. H. Lehmer,a pioneer in computing and, especially, computational number theory:

A random sequence is a vague notion ... in which each term is unpredictableto the uninitiated and whose digits pass a certain number of tests traditional with

statisticians ...

Random number generators proceed deterministically from their current state. To view thecurrent state of rand, type:

s = rand(state)

This returns a 35-element vector containing the current state.

To change the state of rand:

rand(state,s) Sets the state to s.rand(state,0) Resets the generator to its initial state.rand(state, sum(100*clock)) Sets to a new state each time.

Commands for randn are analogous.

Try:

s = rand(state)

format long

rand

rand(state,sum(100*clock))

s = rand(state)

format longrand

Resampling

The Signal Processing Toolbox provides a number of functions that resample a signal at ahigher or lower rate.


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y = downsample(x,n)

decreases the effective sampling rate ofx by keeping every nth sample starting with the firstsample. x can be a vector or a matrix. If x is a matrix, each column is considered a separatesequence.

y = upsample(x,n)

increases the effective sampling rate ofx by inserting n 1 zeros between samples. x can bea vector or a matrix. If x is a matrix, each column is considered a separate sequence. Theupsampled y has x n samples.

y = resample(x,p,q)

resamples the sequence in vector x at p/qtimes the original sampling rate, using a polyphasefilter implementation. p and q must be positive integers. The length of y is equal to

ceil(length(x) p/q). Ifx is a matrix, resample works down the columns of x.

y = interp(x,r)

increases the sampling rate of x by a factor ofr. The interpolated vector y is r times longerthan the original input x.

y = decimate(x,r)

reduces the sampling rate of x by a factor of r. The decimated vector y is r times shorterin length than the input vector x. By default, decimate employs an eighth-order lowpassChebyshev Type I filter. It filters the input sequence in both the forward and reversedirections to remove all phase distortion, effectively doubling the filter order.

Try:

load mtlb

sound(mtlb,Fs)

mtlb4 = downsample(mtlb,4)

mtlb8 = downsample(mtlb,8)

sound(mtlb8,fs/8)

What are the sizes of mtlb, mtlb4, and mtlb8?

(If sound doesnt work, plot the signals.)

t = 0:0.00025:1;

x = sin(2*pi*30*t) + sin(2*pi*60*t);

y = decimate(x,4);

subplot(211), stem(x(1:120))

axis([0 120 -2 2])

title(Original Signal)

subplot(212), stem(y(1:30))


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title(Decimated Signal)

Modulation and Demodulation

Modulation varies the amplitude, phase, or frequency of a carrier signal with reference to amessage signal.

The Matlab modulate function modulates a message signal with a specified modulationmethod. The syntax is

y = modulate(x,fc,fs,method)

where: x is the message signal. f c is the carrier frequency. f s is the sampling frequency. method is a flag for the desired modulation method (see table below).

Method Descriptionamdsb-sc or am Amplitude modulation, double side-band, suppressed carrier

amdsb-tc Amplitude modulation, double side-band, transmitted carrieramssb Amplitude modulation, single side-band

fm Frequency modulationpm Phase modulation

ppm Pulse position modulationpwm Pulse width modulationqam Quadrature amplitude modulation

The demod function performs demodulation, that is, it obtains the original message signalfrom the modulated signal. The syntax is:

x = demod(y,fs,fs,method)

demod uses any of the methods shown for modulate. The signal x is attenuated relative toy because demodulation uses lowpass filtering.

Exercise: High and Low

1. Create a signal equal to the sum of two sine waves with the following characteristics: 3-second duration


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Sampling frequency = 2 kHz Sinusoid 1: frequency 50 Hz (low), amplitude 10, phase = 0 Sinusoid 2: frequency 950 Hz (high), amplitude 1, phase = 0

2. View and listen to the signal using an M-file

3. Import the signal into SPTool and view it. Listen to the signal.

HAND IN ANSWERS TO ALL QUESTIONS STARTING WITH THE LET-

TER Q IN FRONT OF IT.

(Contribution from Paul Beliveau for helping put this material together is acknowledged. The mate-rial derives principally from the MathWorks training document MATLAB for Signal Processing,

2006.)

c2012GM

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Signals in MatLab.pdf