This session is used to introduce basic (MATLAB) programming concepts, including
• Variable definitions
• Arrays and array manipulations
• 2D plotting
• Script M-files
Basic (Engineering / Science) concepts, including
• Parameter estimation from noisy data
• Matching a mathematical model to data
The driving application involve estimation of the speed of sound in air 1 and acoustic distance measurement to remote
objects. Specific tasks include
• Measuring the travel time between ultrasound transducers
• Estimating the speed of sound in air from measurement data
• Acoustic distance measurement
The components & equipment used in this experiment include
• Two sound wave transducers: an emitter (essentially, a small, directional loudspeaker) and a receiver (essentially, a
directional microphone); a mark on the basis of each transducer assembly indicates which is the emitter and which
is the receiver.
• The emitter is connected to a pre-programmed signal generator
• Both transducers are connected to a digital oscilloscope
• A measuring tape (metric), glued to the lab bench
• A clip board
• MATLAB, installed on the computer.
Experiment Outline: The emitter transducer emits a short burst of several cycles of a sound wave at a 40kHz frequency2.
The receiver, located at a selected distance from the emitter, picks up the sound wave. The two transducers are connected
to the two input channels of a digital oscilloscope. The process is automatically repeated, periodically, every 10 ms. 3
The oscilloscope displays the recorded sound waves from both transducers. Using features of the oscilloscope display
you will be able to measure the time it takes travel time of the sound wave from the emitter to the receiver. In principle, a
single measurement suffices to compute an estimate of the speed of sound:
speed of sound =
travel distance
travel time
(1)
or equivalently
travel distance = speed of sound · travel time (2)
1 Actually, the speed of sound varies with temperature and air pressure, and the results obtained will provide an estimate for the altitude and
temperature of our lab environment.
2The frequency of 1 Hz (one Hertz) means one cycle per second; 1 kHz (one kilo-Hertz) stands for 1000 cycles per second. Thus the sound wave
used in our experiments is of 40,000 cycles per second.
3The notation “ms” stands for “milisecond”, or 0.001 second. We shall later encounter also the “μs”, standing for “microsecond”, which is 1μs =
1 · 10−6 = 1e − 6 sec.
2
In practice, multiple measurements are needed to overcome measurement errors of both the travel time and the distance
between the transducers. Moreover, due to the physical structures of the transducers, distance measurements will be
biased, meaning that even without any other inaccuracies, the actual distance between transducers is different from what
we can measure by a fixed amount:
measured travel distance = speed of sound · travel time + bias (3)
This last equation is of the general form of a linear graph
y = a x + b (4)
You will conduct ten experiments and obtain ten values for distance / travel time pairs. You will then use a MATLAB
program to optimally fit a linear graph of the form (4) to your data. The quantity “a” in that fit will be your estimate of
the speed of sound, as in (3).
Having estimated the speed of sound and the distance measurement bias, acoustic distance measurement to a remote
object will again use the formula (3): Now the two transducers will be placed side by side. The emitter’s sound wave will
travel to the target object, bounce back and be recorded by the receiver. The travel distance will thus be twice the distance
to the remote object. Having measured the travel time, we can thus conclude
distance to remote object = 0.5 (speed of sound · travel time + bias) (5)
The representation of the relationship between travel distance and travel time in the basic formula (3) is a mathematical
model. Your final task in this lab will be find an alternative mathematical model, relating the distance to the peak
amplitude of the return sound wave, instead of to the travel time.
Programming: The data recording and processing required in this experiment require an introduction to some of the
essential building blocks of programming, in any programming language. We shall use this lab as an opportunity to
introduce these concepts, as explained below.
2 Homework Assignment
¿From Lab 2 and on, our practice will be to go through relevant components of the text book before the lab meeting, and
include assigned preparatory work by then. Since this is our first meeting, the primer component of Lab 1 is included in
the work for the final lab report. In preparation of your report
• Read this entire handout carefully.
• Do the following in a Pre-Lab assignments
– Read4 in the course text MATLAB: An Introduction with Applications
_ Chapter 1
_ Chapter 2
_ Optional Section 4.3.1 (pp. 91-93) (We shall discuss the necessary material in class)
_ Optional: 2D Plots, pp. 119-124 (We shall discuss the necessary material in class)
– Do the following problems (include a printout of the MATLAB command window with the solution of each
problem):
_ Problems 1-5 and 1-15 on pages 28, 30
_ Problems 2-1, 2-3, 2-14 on pages 53, 54
• Following completion of the Lab, write a Post-Lab report, summarizing the experiments and your findings, as explained
below.
4Here and throughout, preparatory reading is essential for the understanding of class material and the ability to perform tasks in class or in the lab.
3
3 Measuring The Speed of Sound: The Experiment
You will conduct ten experiments and record your data in your notebook. Later you will analyze these data using MATLAB.
Attention: Make sure you record all your measurements in the standard meters, seconds and Volts units!
1. Position the emitter so that its face will be directed along the measuring tape and the front of its base is flush with
the 0 position.
2. Position the receiver at a point of your selection along the measuring tape, anywhere between 5 - 50cm, facing the
emitter. (The position is measured from the front of the transducer’s base.)
The oscilloscope displays both the emitter signal (Channel 1) and the receiver signal (Channel 2). You will notice the
record of the short interval of a high amplitude emitter waveform, early on (on the left), followed by an essentially
(close to) zero reading and then, an area of increased amplitude waveform, more to the right. This second waveform
is the record of the sound wave sensed by the receiver.
3. Read the time it takes sound to travel from the emitter to the receiver:
(a) Press the curser bottom
(b) Under the display window, select t1
(c) Use the “entry” knob to bring one vertical cursor in the oscilloscope window to the point where, by your
visual judgement, the emitter burst begins
(d) Under the display window, select t2
(e) Use the “entry” knob to bring the second vertical cursor in the oscilloscope window to the beginning of the
elevation in the response, in Channel 2.
(f) The corresponding time difference - the sought travel time - appears as “_t” in the oscilloscope display
window. (Pay attention to the time units used: 1 ms = 1e-3 s; 1μs = 1e-6 s.)
4. Record the data pair, consisting of the sound travel time (in seconds!) and the distance between the fronts of the
bases of the two transducers (in meters!)
Using a calculator, estimate the speed of sound from each of the ten experiments, using the formula (1).
5. For later use, record also the peak-to-peak amplitude of the received sound wave
(a) Press the curser bottom
(b) Under the display window, select v1
(c) Use the “entry” knob to bring one horizontal cursor in the oscilloscope window to the lowest peak of the
receiver signal
(d) Under the display window, select v2
(e) Use the “entry” knob to bring the second horizontal cursor in the oscilloscope window to the highest peak of
the receiver signal
(f) The corresponding voltage difference appears as “_v” in the oscilloscope display window. (Pay attention to
the voltage units used! Record the peak-to-peak value in Volts.)
4
4 Acoustic Distance Measurement: The Experiment
Here we shall demonstrate the principles of commercially available acoustic distance measurement devices (with diverse
applications ranging from monitoring chemical process reactors to real estate), with our simple setup.
• Place the standing clipboard at the zero position of the measuring tape, with its face facing the length of the measuring
tape.
• Place both the receiver and the emitter, side by side, at a selected position along the measuring tape (5 - 30 cm),
facing the clipboard. (The front of their bases is used as the transducers’ position, just as in the previous experiment.)
• Use the oscilloscope to measure the travel time of the sound wave from the emitter to the receiver. Record the
distance and travel time in your notebook.
• Repeat the experiment 2 more times, at different distances.
In §5.6 you will compare the actual distance with a distance estimated in terms of the travel time.
5
5 Data AnalysisWith MATLAB
MATLAB basics will be learned, based on your assigned reading and class practice. §5.1 and §5.2 are included as
immediate reminders. The discussion of data analysis begins in §5.3.
5.1 Creating & Saving a New M-File on a Memory Stick
MATLAB computations can be executed directly from the MATLAB command window, or by use of program files.
MATLAB programs are saved as M-files in an accessible directory, as explained below. Since lab computers will be used
by others we shall never save programs on the hard disk. Here and throughout, programs will be saved as M-files on a
memory stick and run from the appropriate directory. In what follows we refer to the directory of the memory stick as
“E”. Make sure to check which directory is actually occupied by the memory stick and substitute the correct reference for
“E”. Files saved on the hard disk might be erased and lost without prior notice!
• Insert a memory clip
• In the MATLAB command window, change directory to the floppy drive
>> cd E:\
• Verify the current directory
>>pwd
>> E:\
• Under “File”, select “New M-file”
• Save the new M-file under a name of your choice, with the suffix “.m” (for example SOS Lab1.m is a good name
for a program used to measure the speed of sound in Lab 1). The following rules apply to names used for both
MATLAB variables and M-files:
– the name must begin with a letter
– the name may contain letters, numbers and the underscore sign
– the name may not contain other characters, or a space
– the name may not be equal to the name of a recognized MATLAB function (such as sin, cos, log, etc.).
When in doubt, do not use a suspected name for your variables and programs!
To run an M-file as a script program, type its name in the MATLAB workspace and hit return. Alternatively, select
the run icon in the editor window.
5.2 Defining MATLAB Variables
This part is covered in the required Lab 1 book reading and will be practiced in class. The following is are few highlights:
• Variables are used to store data values. The name of the variable is a reference to a point in the memory.
• MATLAB will not recognize a variable before it is expressly defined.
To see that type boom in the command window. Assuming a variable bum was not previously defined, the response
will be
>>boom
>>??? Undefined function or variable ’boom’.
6
• The equal sign “=” is used in MATLAB, as in most programming languages, for the assignment operation, acting
from left to right. Thus
>>moom=5
assigns the value 5 to the variable moom. If moom did not previously exist, it also creates this variable in the
memory. Contrary to the usual usage of “=” in mathematics, the legal operation
>>moom=moom+1
assigns to the existing variable moom a new value, comprising its old value plus one.
• By default, each value assignment command is followed by a display of the assigned value in the MATLAB command
window. This is mostly undesirable, and in case of large array, will dramatically slow down you programs!
To prevent this and as a matter of standard practice, end each assignment command with a semicolon “;”
• Variables may be scalars (the default), taking a single number (or character) as a value. They may also be one, two
or three dimensional arrays of values. Details are provided in the book (pp. 40-48).
5.3 Experimental Data Recording and Plotting
The following commands should be included in an M-file used to record and plot the data from the speed of sound
experiment. Your Lab report should include the M-file and a printout of the MATLAB command window were the M-file
is executed5
• define three row vectors of size 1 × 10 (remember to type a semicolon “;” at the end of each vector definition!):
– A vector tdistance whose entries are the 10 distances between the two transducers in your experiments
– A vector ttime whose entries are the corresponding measured travel times
– A vector amplitude whose entries are the corresponding measured maximal peak-to-peak amplitudes of
the receiver signal
• Create a 3 × 10 matrix named sosdata whose first row comprises the vector tdistance, the second row is the
vector ttime and the third row is amplitude.
• Use the disp command to display the tabulated data in the matrix sosdata.
• Include a command to discrete plot of the travel distance (vertical axis) as a function of the travel time (horizontal
axis), using black asterisks for data points
plot(ttime,tdistance,’k*’)
• Following the plot command, include the title “Speed of Sound Data: <your team names>”, and axis titles “travel
time” and “travel distance”:
>>title(’Speed of Sound Data: ...’)
>>xlabel(’travel time’)
>>ylabel(’travel distance’)
Save and run theM-file. Your lab report will include the printedM-file and printed output of the final version of theM-file.
5The simplest way to print a portion of the command window is to use the mouse to highlight that section, then selecting Print Selection
under the File menu.
7
5.4 Polynomial Curve Fitting - General
Our goal here is to fit the data in the vectors tdistance and ttime with an equation of of the form (3):
tdistance = speed of sound · ttime + bias (6)
Referring to this form, we are trying to estimate the values of the coefficients a=speed of sound and b=bias of a
first order polynomial that best fit our recorded data. Here we shall use two MATLAB functions that are introduced only
much later in the book (search the index):
_ TheMATLAB function polyfit is used to fit data with a polynomial curve. We shall be interested in the spacial case of
a first order (linear) polynomial of the form (4)-(6). Following is a general description of polyfit and the way it is used:
Given same size vectors x=[x(1),...,x(N)] and y=[y(1),...,y(N)] and a prescribed polynomial order M,
the objective of polyfit is to find the polynomial
p(u) =
N
X
k=0
akuk (7)
that solves the approximation problem
min
a0,...,aM
N
X
l=1
(y(l) − p(x(l)))2 (8)
That is, the graph of the polynomial p(u) has to best approximate the graph of the data pairs (x(k),y(k)) in the least
mean squares sense. The general syntax of polyfit is:
>>p=polyfit(x,y,M)
where, in the notations of (7), we identify “p” with the coefficients’ vector p = [aM, aM−1, . . . , a1, a0].
_ The MATLAB function polyval is used to evaluate a polynomial v=p(u) at a point, or list of points x. The syntax
is
>>y=polyval(p,x)
where the interpretation of p = [aM, . . . , a1, a0] is as above, and where x is a single variable or an array. The returned
value(s) y=p(x) are stored in an array y whose size is the same size as that of x.
5.5 Linear Fitting of The Speed of Sound Experiment
As stated above, our goal is to fit the data pairs (ttime,tdistance) by a straight line, as in (6). You should therefore
include in your M-file the command
>>est=polyfit(ttime,tdistance,1)
In accordancewith the notations of (6), the returned vectorwill be the best estimate of est=[est(1),est(2)]=[speed
of sound, bias]. If we want to see how good this estimate is, we should substitute the estimated coefficients and
the entries of ttime in the right hand side of (3). This is done with the command
>>polyval(est,ttime)
For a visual comparison you will therefore include in your program a plot command that compares the raw data with the
linear estimate
8
>>plot(ttime,tdistance,’k*’,ttime,polyval(est,ttime))
Notice that here we allow a continuous plot of the linear approximation. Add a title (including name of team members)
and axes labels commands.
To evaluate the quality of your estimate of the speed of sound, search the web for “speed of sound” and compare your
estimate to what you find in the web. The site http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe.html is one option.
5.6 Acoustic Distance Measurement
Here you will compute the estimated distance from the transducers to the clipboard in the experiments carried in §4. The
estimated distance is given by Equation 5, where you have to use your estimates of the speed of sound and the bias term.
That is, you have to use theMATLAB function polyval and the linear polynomial est=[est(1),est(2)]=[speed
of sound, bias]. In your report compare the distance estimate with the actual distance, as recorded in the experiment.
6 An Alternative model
As an alternative to the model (3), you will seek a representation of the distance as a function of the maximal peak-to-peak
amplitude of the receiver signal. We expect a reverse relation: the longer the distance, the smaller the amplitude. Therefore
the sought model will be a function of the variable x = 1
amplitude . You are allowed to use the polyfit function to test
polynomial approximations of the form
travel distance _
M
X
l=0
alxl (9)
with various levels of the degree M. At this point you may rely on visual inspection to evaluate the quality of the
approximation. Clearly, the higher the degree M you allow, the better the fit can be. However, we want the simplest
possible formula, and will stop increasing M if there is no noticeable improvement. The findings in your report should
include:
• The values of the optimal coefficients and polynomial degree
• A plot, comparing the discrete data pairs ttime,amplitude with the plot of the approximating function (9) that
you identified. The plot must include a title (with your name) and axes labels.
Hint: To create the data vector x = 1
amplitude from the data vector amplitude, you use the place-by-place vector
division command
>> x=1./amplitude;
If you expect the amplitude to vary with the distance according to a power law, (amplitude = constant _ distancen)
you can try another way to fit the data: Taking the logarithm of both sides of the equation above, you get the following
equation:
>>log (amplitude) = log (constant) + n log(distance)
If you then define new variables:
u=log(amplitude)
v=log(distance)
L=log(constant)
you get the equation:
u = L + n * v
9
This is a linear equation and you can do a least squares fit to the data to get he best fit value for n (the slope) and L (the
y-intercept). Using the data you took for amplitude vs. distance, use MATLAB to create the new variables u and v. Then
use polyfit to get the best fit values for the slope and y intercept. Finally, convert your results back to get the best fit
values for constant and n in the equation above: amplitude = constant _ distancen.
Question (answer in your lab report):
1. How does your best fit values for constant and n compare with the values from your Power Law fit in Excel?
2. How do you think that Excel gets its Power Law fit?
3. How could you use polyfit and a linear straight line fit to find the best fit values of yo and a from (x, y) data that fits
the following exponential equation:
y = y o * exp (a *x)
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