The primary objective of these experiments is to familiarize you with digital data acquisition of time-varying signals. This lab covers concepts in frequency analysis of time varying signals and sampling theory. It also provides an introduction to digital data acquisition (DAQ) systems. You wil use a personal DAQ to sample signals produced by waveforms stored in mp3 formats and converted to analog electrical signals. You will explore issues in: sampling, including the Nyquist limit and aliasing; filtering and its use for noise reduction; and digitization errors. You will also use your DAQ and a microphone to record and analyze sounds in an experiment chosen by yourself.
Most experimental measurements involve the dimension of time. Experimental data is acquired over the course of some time, and during this time the actual physical parameter of interest (the measurand) may change. This could be due to a transient, such as the stress induced in a material by a sudden impact, a periodic phenomena, like the bending and twist of a helicopter blade due to flutter, or random or chaotic fluctuations, like the turbulent velocity in a wind tunnel. Even when the measurand is nominally constant in time, other parameters that influence the measurement may vary, for example drifts in the measurement device. Thus, the experimenter is often interested in measuring a variable that can be described by the general function (or waveform),
(1)
One of the simplest time-dependent functions we encounter is the sine (or cosine*),
*Either function is acceptable, since , i.e., the two functions are identical except for a phase difference of π/2 or 90°, meaning that shifted by one-fourth of a cycle, cosine looks just like sine.
(2)
(3)
For example, Figure 1 shows a simple waveform composed of two frequencies, a fundamental mode at 50 Hz and its 9th harmonic. Thus the complete waveform is repeated every 20 ms (period=1/fundamental frequency =1/50 s). The waveform shown in the figure also has a DC component. In other words, the signal has a nonzero value when averaged over its period. In general, we can write the DC amplitude as
(4)
The other coefficients of the Fourier expansion are given by
(5)
(6)
(7)
(8)
A second example that shows the ability of a combination of sine waves to create an arbitrary periodic function is shown in Fig. 2. Five sine waves and a DC component (see Fig. 3) were combined to create a function approaching a square wave. While the constructed function resembles a square wave, it is clear that more sine waves would be needed to produce a sharp square wave.
(9)
(10)
For example, Figure 4 graphically shows the Fourier transforms of various functions, including sine and cosine waves, a rectangle function (Π), a triangle function (Λ) and a constant, or DC, function. The sine, cosine and DC waveforms result in Fourier transforms that are nonzero at a single frequency**;
(11)
Extensions of the Fourier Transform method have been developed for non-continuous functions, specifically for signals that have been discretely sampled by a computer, data acquisition system, or produced by digital means. These are generally known as Discrete Fourier Transforms (DFT), and a computationally efficient approach is known as the Fast Fourier Transform (FFT). These concepts are described in detail in References 2 and 4. The digital data acquisition system you will use employs these techniques to compute power (and phase) spectra.
An example PSD is shown in Fig. 5 for an 80 Hz sine wave based on a DFT. Unlike the Fourier Transform of the sine wave in Fig. 4, the PSD for a sine wave is no longer infinitely thin, i.e., the DFT shows power spread over a small range of frequencies around 80 Hz. This occurs because the DFT has a limited frequency resolution (it can only report values at discrete frequencies).
Noise
As a simple example, consider a single sine wave. If we know we are dealing with a single frequency sine wave, it takes at least two measurements per period to determine its frequency, which means we must sample at twice the sine wave’s frequency. If we sample any slower, we actually infer a lower frequency than the actual frequency of the sine wave (you will see this in the lab). This process, by which information at a higher frequency shows up at a lower frequency is known as aliasing.
Data will be acquired with a standalone digital data acquisition system (DAQ), LabJack™ T4, that communicates with your computer through a USB connection and using a LabVIEW™ software interface. Most DAQs can be connected to more than one input source; each signal (e.g., a voltage) is connected to one channel of the DAQ. A typical DAQ consists of a multiplexer, a sample-and-hold device, an amplifier, an analog-to-digital converter, a memory buffer, a microcontroller, and an interface to a computer (see Figure 7).
The multiplexer** ** (MUX) is a switch that connects one of a number of input channels (usually numbered starting at 0) to the sample-and-hold** ** (S/H). The input voltage on the channel switched by the MUX “charges up” the sample-and-hold during some time interval, which is a fraction of the sampling period (the time between samples). This circuit is then disconnected from the input voltage, and some of the stored charge is drained from it. The amount of charge leaving during this time is proportional to the original input voltage. The output of the S/H is amplified and then converted to a digital value by the analog-to-digital converter** ** (ADC). The digital result is then moved to the buffer memory, and communicated to the computer.
(12)
Multiple signal inputs are recorded by using the MUX to cycle through each of the input channels at a rate that must be faster than the overall sampling rate (how often a given channel is read) times the number of input channels being read. In the sequential sampling system illustrated in Fig. 6 (and which is representative of the system you will be using), note that the channels are not read at exactly the same time. There is a time delay (skew) between when one channel and the next is read. The skew is determined by the maximum switching and reading rates of the MUX, S/H and ADC. This is illustrated in Fig. 8. Simultaneous data acquisition systems, which have negligible skew, typically employ multiple, synchronized S/H systems just upstream of the MUX (see Fig. 9).
The following terminology is commonly used in DAQ systems, and you should become familiar with these terms.
Sample = a single measurement (i.e., at an "instant" in time) captured by the DAQ from one channel
Sampling period = the time between two successive samples
Sampling rate = 1/sampling period, with typical units of Samples/sec (S/s) or Hz
Record = a group of successive samples acquired by the DAQ
Record length = the number of samples in a record, typical units of Samples (S)
Record duration = the time between the first and last sample in a record
The software for this lab currently only works on Windows, and cannot be made to work for Mac (or any other OS). This leaves Mac users with the following options that we have tested and verified:
Borrow a Windows machine from a peer, or at the Georgia Tech library
Install and use Parallels
This works for both Intel and Apple chips.
It is not free but a student version costing $39.99/year is available
This is a "virtual machine" that creates an instance of Windows running within your Mac OS. As such, no reboot is required, and only 500Mb for the installation file is required.
When running, Parallels will take around 4Gb of RAM.
You will also need a copy of Windows to install on the partition; see here for instructions on how to get a free student copy of Windows.
Windows 10 Education (not the N version) is recommended since W11 requires a lot more memory.
Use the free in-built Bootcamp software
This only works if you have the Intel chip.
This "dual boots" your machine which involves partitioning your drive with >64Gb of your drive's space permanently allocated to Windows. You will also need to reboot every time you want to switch between operating systems.
You will also need a copy of Windows to install on the partition; see here for instructions on how to get a free student copy of Windows.
Windows 10 Education (not the N version) is recommended since W11 requires a lot more memory.
Install and use the free Oracle VirtualBox
This works only for Intel chips.
This is a "virtual machine" that creates an instance of Windows running within your Mac OS. As such, no reboot is required, and only ~125Mb for the installation file is required.
When running, VB will take at least 0.5Gb of RAM.
You will also need a copy of Windows to install on the partition; see here for instructions on how to get a free student copy of Windows.
Windows 10 Education (not the N version) is recommended since W11 requires a lot more memory.
If you need a Windows download and a license key, Microsoft's Azure For Education program gives you access to Education versions of Windows OS for free. Head over to that link, sign in with your GT credentials, and download from there.
If you don't have the ability to borrow a Windows machine from a peer, you will need to check out a loaner Windows laptop from the library. Follow this guidance:
Library laptops can only be checked out for 4 hours at a time. However, they can be renewed at the library for another 4 hours once per day.
GT-OIT has programmed the machines to fully reset to their nominal state if it is shut down or restarted. Therefore, ignore any instructions to restart the machine after software installation. Instead, select RESTART LATER.
When you return the laptop to the library but immediately check out again to extend your time, try and check back out the same machine, else you will need to repeat the installation.
Remember to send yourself any saved data before returning the laptop.
Log in to the laptop using the default user account credentials supplied on the device, DO NOT use your GT credentials otherwise you won't have admin rights to install software
Once you've acquired the laptop, follow the instructions below as normal.
The following tasks should be accomplished prior to meeting for the first week of lab. The intention of this section is to ensure you have working software on your computer before coming to the lab. The hardware can only be tested if the software is installed so make sure to complete this step to prevent delays during the lab.
Software preparation:
Go to the LabJack software installation page and download the "Release" version for your operating system and screen size
Run the installer and follow all steps to install in your preferred location
Download and unzip the GTAE Simple DAQ installation files (found in the Software and Files section below)
Run the installer and follow all steps to install in your preferred location
Restart your machine
The following tasks should be accomplished during the lab. The intention of this lab is to make sure you verify all items in your kit functions correctly and that you have a basic understanding of how they function.
Kit preparation:
Obtain a Digital Sampling kit from your TAs
Check that the contents of your Digital Sampling kit match the box label; if anything is missing, obtain any replacement parts from a TA. The kit should contain:
LabJack T4 DAQ
USB-A to USB-B Power Cable
LabJack Screwdriver
Miniature Eyeglass Screwdriver
Microphone
3.5mm Audio Breakout Adapter
At least 3 male-male Jumper Wires
Female 3.5mm Audio Jack to Male USB-C Adapter
Register your LabJack T4 serial number with your TAs
Verify your LabJack T4 DAQ functions correctly:
Connect your LabJack T4 DAQ to your computer using the provided USB cable
Open LabJack's Kipling software that you previously installed, and connect to the T4 by clicking on the green USB button (refresh devices if it wasn't found)
!!! NOTE - If your search bar doesn't find it, navigate to Kipling under the LabJack folder after pressing the Windows Start icon !!!
Go to the Dashboard tab, where you will see a schematic of the T4 with live display of the inputs and output pins. Perform the following "loopback" tests to make sure your DAQ is functioning correctly:
By default, AIN0 to AIN3 should be reading around 1.4V
With one of the jumper wires and larger screwdriver from your kit, connect AIN0 to a nearby GND pin; the AIN0 voltage should now go to around 0V (the pin is being pulled to ground)
Now connect AIN0 to VS (supply voltage); since VS is coming from your USB port you should be reading around 5.1V
Now set DAC0 and DAC1 to 2 different voltages less than 5V (DAC means Digital-to-Analog-Converter i.e. these pins are analog voltage outputs); connect AIN0 to each of them, verifying that the measured voltages match the output voltages
Disconnect wire and close Kipling
Verify your peripherals function correctly:
Ensure your T4 is plugged in
Open GTAE Simple DAQ, choose a save data folder, and the application should now be running. !!! NOTE - Sometimes the SimpleDAQ will throw an error on startup or mid-acquisition. If this happens, restart the software and power cycle (unplug and plug back in) the DAQ !!!
If your monitor is clipping the edges of SimpleDAQ, try adjusting the display resolution and scaling (zoom) until it fits. In Windows this is done by right-clicking on the Desktop and selecting 'Display Settings'. If no options work for you, try plugging in an external monitor.
Turn on Autoscale for both axes of both plots; Time History and Power Spectrum. You should now see a real time plot of channel AIN0 with a voltage of around 1.4V plus or minus a small amount of noise
Test your 3.5mm audio breakout adapter:
Retrieve the breakout adapter and, with one jumper wire, connect either its L or R terminal to your DAQ's AIN0 port (note that you may need to open up the screw terminal on the DAQ before being able to insert the wire)
With another wire, connect the ground terminal (far right) to your DAQ's GND port (the one next to AIN0)
Ensure you have good wired connections by gently tugging on them to ensure they don't come loose. Be careful not to bend the pins as they can snap fairly easily.
Plug the audio breakout into your laptop/tablet/smartphone (any device that can play audio and access YouTube).
In Windows you will likely get a popup asking which device you just plugged in. Here you should select whichever option gives the best signal quality with low noise/distortion... this should be done iteratively and using your best judgement.
Ensure that your operating system is set to play over the headphone jack (i.e. not your bluetooth headphones or in-built speakers).
Set the YouTube video player to maximum volume and your device volume to roughly 50%. You may need to turn it up higher if your waveform has low amplitude.
Go to YouTube and find a "human hearing test" video such as this one
With the video playing, observe what's happening in GTAE Simple DAQ; you should see a sine wave of increasing frequency both in the time history and as a shifting spike in the power spectrum that should match that shown in the YouTube video (until around 11 kHz when it will start to diverge)
Adjust your device volume to observe the shifting amplitude of the signal
Once you have verified correct functionality after a minute or two (don't play the whole video), stop the video, unplug the audio adapter and remove the wires from the adapter (don't remove them from the DAQ)
Test your microphone:
!!! WARNING - the microphone has bare electronics on the back of the board which can result in damage if shorted with a metal tool or surface; be sure to handle carefully and away from metal when powered !!!
Retrieve the microphone and micro screwdriver from your kit
Connect the GND port of the microphone to the GND port on your DAQ which should already have a wire connected (note that you may need to open up the screw terminal on the microphone before being able to insert the wire)
Connect the OUT port of the microphone to the AIN0 port on your DAQ which should already have a wire connected
Connect the VCC port of the microphone to the VS port of the DAQ with a new wire
You should now see a rough waveform in GTAE Simple DAQ centered around roughly 2.5V. To stop the time history jumping around so much, turn off Y-Axis Autoscale and set the scale's maximum and minimum directly. Do this by double-clicking on the highest and lowest and numbers on the axis (6V and 0V might be a good start but adjust as necessary)
Test the microphone by generating some sounds of interest such as whistles, taps on the desk, talking, playing a tone from headphones, etc. Don't get the microphone too close to the noise source, it is quite sensitive. Bear in mind there will be lots of background noise so you may want to go somewhere quiet. !!! Note - the front of the mic is the black pad inside the metal can... point this towards your noise source !!!
Once you are satisfied the microphone is functioning correctly, remove all wires from all terminals.
!!! WARNING - the DAQ, and especially the microphone are fragile items. MAKE SURE they do not get tossed around or they will break and you will be unable to finish your lab !!!
Before leaving the lab, understand the objectives and resources for the next 3 weeks:
Read the following steps in this manual that outline what you must have completed prior to each lab session. Check with your TAs if you are unsure of anything.
A more in-depth explanation of GTAE Simple DAQ is given at its dedicated webpage, including a short video on how to use the software.
The LabJack website is a great resource for understanding more about how the DAQ works, and what you can do with it.
By Week 2 lab, at a minimum, you must complete the following tasks, but you are free to try other things to learn about digital data acquisition, sampling theory, and frequency content of signals. Keep notes on what you observe or find as you do each task and keep good, clean, legible, and organized notes. Answer any questions that are posed. All of your observations, notes, and answers to questions posed this will be checked by the TAs during Week 2 lab and graded for accuracy/effort!
You can work on this wherever and whenever you want (but before your regularly scheduled lab session). Also, you must work on these task by yourself. If you have trouble with the equipment or do not understand the required tasks, you can reach out to the TAs during office hours or using Piazza.
You will still attend your lab session at your regularly scheduled time, this week - bring your DAQ/computer system with you. You will be asked to demonstrate certain things to the TAs and answer some questions based on having done these tasks and learned the underlying concepts. If you can not do the required tasks or successfully answer the TAs questions, you can work during the lab time to work on the material, and be re-assessed by the TAs when you think you are ready. You have an unlimited number of attempts to pass the assessment, but only until the lab session ends. Your TA's can and WILL take points off if you come into Week 2 lab with nothing finished or it seems like very little effort was put into it. Furthermore, you will be required to finish everything during lab.
Setup DAQ system
Following the steps from Week 1, configure your DAQ and 3.5mm audio breakout adapter so that your laptop/tablet/PC/smartphone can capture the waveforms in the above YouTube video
!!! WARNING - Some devices (usually PCs/laptops) do not have good sound cards, resulting in significant distortion of the waveform. You will know this is happening if you don't recognize any square/triangle/ramp waveforms in the subsequent steps, and even the sine waves have significant distortion. If this occurs, switch to a device designed more for sound playing such as a smartphone or tablet (check with other members of your group or friends if you don't have one). You could also download the tracks and import them to an MP3 player. Please use the USB-C to 3.5mm adapter provided since this has historically yielded better results. If all else fails, contact a TA who will help you find an appropriate device !!!
In GTAE Simple DAQ, set the following settings:
Sampling rate = 25,000 S/s
Record length = 1,000 S
Autoscale = initially ON for all axes (you will need to turn this off to adjust the time scale to "zoom in" during waveform identification)
All other settings should be okay based on the defaults when you start the application
Use DAQ to perform waveform identification
In the Waveforms section below you will find 11 different audio tracks. Each audio track contains a different periodic signal. These signals include: single sine waves (at different frequencies), a sum of three sine waves (each at a different frequency), a product of two sine waves (e.g., sin(At) x sin(Bt), also known as amplitude modulation), a sine wave of a sine wave (e.g., sin(sin(At)), also known as frequency modulation), and periodic waveforms that are not sine waves: square waves, triangle waves, and ramps. Some tracks also have "noisy" versions of some of these waveforms.
For each track, observe the time plot and power spectrum (adjust the output/volume level as required to see the waveforms clearly)
Tip: With the waveform displayed as you like, you can toggle the Continuous/Hold switch to the Hold position so that the display just shows the last data captured (doesn't keep taking new samples)
Examine complex waveforms and interpret power spectra
Play the track you identified as product of sines (amplitude modulation)
Play the track you identified as the triangle wave
Alternate playing the triangle wave track and the square wave track
Examine quantization error
Set your computer volume to max. Play the track with the square wave.
Make sure the settings are: Continuous, Sampling Rate = 10,000, Record Length = 1000, all axes have Autoscale=on except time plot, set maximum limit on time plot x-axis to 0.01 seconds. Turn off time plot y-axis autoscale.
Observe both the time plot and power spectrum as you turn the amplifier down
Explore effects of record length and sampling rate on power spectrum
Play the track containing the 1 kHz sine wave
Set the Sampling Rate = 4000 S/s and the Record Length = 4 S
Make sure the power spectrum is set to Autoscale for both x and y axes
FIND:
Repeat the above 3 FINDs for a few longer Record Lengths (keep the number of samples low, less than 16 and always pick an even number of samples)
Set the Sample Rate = 8000 S/s, Record Length = 4 S and repeat the FINDs
Observe aliasing
Find and play the track containing the 3 sine waves
Set the Sampling Rate = 5000 S/s, the Record Length = 5000 S, and toggle the switch to Continuous
Set the Autoscale=off switch for the x-axis on the power spectrum, and set the maximum frequency on the power spectrum axis to be 11kHz
Observe the 3 frequencies of the 3 peaks in the power spectrum
Increase the Sampling Rate to 7500 S/s
Continue increasing the Sampling Rate by 2500 S/s until you get to at least 25,000 S/s
Examine noise
Set the Sampling Rate = 5000 S/s, the Record Length = 5000 S, and toggle the switch to Continuous
Play each of the two tracks you identified as sine waves with noise
Explore the implementation of a low pass filter
Play the track containing the sum of three sine waves
Set the Sampling Rate = 22,000 S/s and Record Length = 1000 S
Set the Spectrum Display Settings = Amplitude-Log
In the time plot, Autoscale = off for the x-axis and set the maximum on the axis to 0.005 s
Set the Filter = on, select from the pull-down menu Low Pass, select from the next pull-down menu Butterworth, set Low Cutoff Frequency = 11,000 Hz by typing in the value
Paying attention to both the time plot and power spectrum of both filtered and unfiltered signals. Click the checkboxes next to "Chan 0 Filt" to turn on the plotting for the filtered signal. Keep reducing the cutoff frequency until you obtain a filtered signal that has eliminated the highest frequency sine wave
Shutdown procedure
When you are through, hit the STOP APPLICATION button
If you are done using the DAQ please disconnect the cables from it before you transport or store it away
Remember the DAQ and associated peripherals are FRAGILE items!
Bring your laptop, audio player if you used one (MP3 player, etc.), andDigital Sampling Kit to lab.
Your TA's will be grading you based on your observations, notes, and answers to the all the items in the "Prior to Week 2" section of the procedure. Make sure you take good, clean, and diligent notes so the TAs can look over your work quickly and accurately.
You will not lose points for incorrect inferences or answers to questions. However, you WILL lose points if the TAs deem you did not put in the appropriate amount of effort into doing the "Prior to Week 2" procedure or if you are outright missing data.
As a general guideline, below is how you will be graded. Please note that there may be intangibles that affect your grade apart from what is listed here based on how your TA deems you performed your take home experiments:
Full points: "Prior to week 2" data was captured diligently and notes reflect appropriate amount of effort. Most of the observations are correct and incorrect data/observations were fixed during Week 2 lab.
Half points: "Prior to week 2" data was sloppy/incomplete and notes reflect inadequate effort. A non-negligible amount of the observations are incorrect, but these were fixed during Week 2 lab.
Zero points: "Prior to week 2" data was not captured at all or the most of observations/notes taken were incorrect. Very little to no effort was put into completing this portion of the lab. Student is forced to finish as much of this portion of the lab as possible during week 2 lab.
You will have until the end of Week 2 lab to pass the in-lab assessment if any of your inferences or answers are wrong. The TAs can help you here if you are stuck, confused, or need any other help.
After passing your Week 2 in-lab assessment, think about and decide what experiment(s) you want to perform with your DAQ/microphone system. You are required to use a microphone and DAQ but you may also use the 3.5mm audio jack breakout in your experiment if you please. You will want to find something that interests you, but also allows you to explore some interesting issues regarding the frequency content or frequency analysis of your signals. So for this part of your experiment, there are no procedures supplied.
However, as part of this experiment - like any real-world test or experiment, you will need to first validate/characterize your equipment - in this case the microphone system. The idea is to measure how well the microphone responds at various frequencies. The following procedure describes this latter process.
The following procedure is ONLY for validating your microphone. This is NOT the procedure for your experiment! That is something you must come up with and perform on your own after you have validated your microphone.
Setup Tone Generator
Find a device (computer, tablet, etc.) with a wired audio output jack
Locate any online Tone Generator app on the web
Hint: a Tone Generator will output a sine wave of a single frequency on the audio channel of your device
Set the volume on the Tone Generator app to at least 75% of maximum
Setup DAQ and Function Generator
Connect DAQ to the computer using the USB cable
Connect audio output of the device you are using to play the Tone Generator to the DAQ using the 3.5mm audio jack and jumper wires
Open the GTAESimpleDAQ.exe application
Start application (by hitting Run button)
Record and save single tone output
Use the Tone Generator to output a tone at 200 Hz
Set the Sampling Rate and Record Length to appropriate values
Set the Continuous/Hold button to Hold to capture a record of this tone
In the window titled Data Saving, enter a filename in the Filename text box, then click on the Power Spectrum Save button
Tip: the data will be saved in a folder that you had chosen/created when you started the application; also the filename will have a date-time-stamp prepended at the beginning
Tip: do not include a file extension in your file, it will be saved as a ".txt" file
Repeat Step 3 for at least 4 more tone frequencies ranging from 20 Hz to 10,000 Hz
Setup DAQ/microphone
Remove 3.5mm audio jack and jumper wires from both your device and the DAQ
Connect three jumper wires to the screw terminal on the microphone and to the GND, VS and AIN0 input connections on the DAQ (as instructed in the video you watched)
Record microphone data
Find a device (computer, tablet, etc.) with a good speaker system
Locate any online Tone Generator app on the web
Hint: a Tone Generator will output a sine wave of a single frequency
Set the volume on the Tone Generator app and on your speaker system to at least 75% of maximum
Use the Tone Generator to play each tone frequency from Steps 3 and 4 through your device's speakers
For each tone, record the microphone output with the DAQ using the same sampling setting used for that tone in Steps 3-4
Save the record for each tone by first entering a filename in the Filename text box (or you can use the name already there), then click on the Power Spectrum Save button
Return equipment to the lab
Disconnect the jumper wires from the microphone and DAQ, the jumper wires from the 3.5mm audio jack
Carefully replace all the equipment (including wires and screwdrivers) in the original DIY kit boxes
Return your DIY kit boxes to the lab at your designated lab time during Week 3
MAKE SURE YOUR TA SIGNS YOUR LABJACK KIT BACK IN WITH THE SERIAL NUMBER. If not, you will be held responsible for it even if you have turned it in!
Notes taken while you were doing the required tasks (digital notes). The notes should contain the data, observations and answers to the questions posed in the Procedure section. Your notes do need to be legible, but there is no special format requirement.
Time histories and power spectra at 5 (or more) single frequencies (from a Tone Generator) recorded by connecting the audio output of the computer device directly to the DAQ.
Time histories and power spectra at the same single frequencies as in Item 1, but recorded by playing them through the device speakers and capturing them with the microphone connected to the DAQ.
Appropriate data, notes, observations, etc. for your own experiment.
Including such topics as experimental motivation, procedure, shortcomings, problems/issues during experimentation and mitigation strategies, etc.
Take GOOD notes! You WILL forget things if you do not!
From "Data to be Taken" --> "Prior to Week 3": Items 1 and 2:
Calculate the relative response of the microphone-speaker system (power from microphone at each tone frequency divided by power at same tone frequency recorded from direct connection to DAQ).
Whatever data reduction is appropriate for your experiment.
Note: This will be presented as an Slides Report, so you must follow instructions on how to prepare the Slides Report on the Canvas course page.
From your "Prior to Week 2" data/observations/notes, include a table containing waveform identification information:
The track number
Type of signal (ramp. product of sines, etc.)
Peak principal frequency
Any other peak frequencies you observed in the power spectrum
Any other Tables and Figures you created from "Prior to Week 2" data/observations/notes that you feel are important to present
Answers to any supplemental questions (None for Spring 2024)
From your experiment:
Experimental motivation
What problem you are trying to solve and/or what question you are trying to answer
Experimental goals
Experimental procedure
Raw (if necessary) and reduced results you obtain from the experiment you designed
Success criteria / conclusions
Complications encountered and mitigation strategies
Anything else you feel is important
Think about what sort of stuff we thought was important enough for you to put into your lab reports for the previous 3 labs
This is NOT an exhaustive list!
R. V. Churchill and J. W. Brown, Fourier Series and Boundary Value Problems, 3rd ed., McGraw-Hill, 1978.
R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., McGraw-Hill, 1978.
T. G. Beckwith, R. D. Marangoni and J. H. Lienhard V, Mechanical Measurements, 5th ed., Addison-Wesley, 1995.
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flanner_, Numerical Recipes - The Art of Scientific Computing_, 2nd ed., Cambridge University Press, 1992.
Navigate to the software release folder here, logging in with your GT credentials, and download the correct version for your operating system. You can also download and use the LabView source files but you will need a full LabView install on your machine for this.
Click to open in a new tab. Download if desired by then clicking on the ellipsis.
where is the amplitude, is the circular frequency (e.g., rad/s), is the cyclic frequency (e.g., cycles/s, Hertz or ), and is the phase, which represents the time-shift of the sine-wave from some reference time that defines . Such a function is often denoted as a simple harmonic waveform.
More general periodic waveforms, which repeat themselves with a period T and thus have a frequency , can be written as a linear combination of simple harmonic modes. There is the basic, fundamental mode (with frequency), and harmonics of the fundamental mode, with integer multiples of its frequency (, , …). For example, we could describe the vibrations of a tuning fork or the acoustic oscillations in a pipe this way. Mathematically, this linear combination of modes is expressed as a Fourier series expansion,
where represents the frequency of the mode ( for the fundamental, for the first harmonic, etc.), represents the steady component of the waveform, and the , are the harmonic coefficients (or amplitudes) of each mode. The steady amplitude, , is often called the DC component of the waveform, in reference to classical electrical power systems, which are either Direct Current (steady) or Alternating Current (sinusoidal with a zero average).
and they can be combined into a complex number (since, ),
The power, , contained in single mode is given by the square of the amplitude
and the phase (or phase angle) of a mode is given by
The procedure outlined above for periodic functions can be extended to general functions, which are not necessarily periodic, by considering any arbitrary function to be periodic with an infinitely long period. This approach leads to the Fourier Transform. Given a function , its Fourier Transform is a complex function defined by
in parallel to the complex Fourier function of equation (6). The function represents the information given by transformed from the time domain to the frequency domain. The transformation is nearly identical in the reverse direction, with simply a change in the phase (note the sign of the exponent), i.e.,
**The negative frequencies relate to phase information for the sine and cosine and do not actually represent different frequencies, i.e., for real functions , it can be shown that . That means that if you take the absolute value of V, the part of V below 0 frequency looks like a reflection of the part for .
in other words, they contain information at only one frequency (the DC function, which does not change in time, is associated with a frequency of zero). The rectangle and triangle functions shown are not repeating like the sine waves; the functions shown here represent "single pulses." The Fourier transforms of these single rectangle and triangle functions result in and functions, where , which contains information at many frequencies, but with multiple frequency “peaks”.
Instead of looking at the Fourier transform, we often are interested in the power spectrum (or power spectral density, PSD) of a waveform. This represents the amount of power or energy in a region between and . For real (noncomplex) functions , this is given by
where it is sufficient to consider only since the PSD of a real function is symmetric about .** Thus the PSD of the rectangle function, as shown in Figure 4, is the square of its Fourier transform, or (also shown in Figure 4).
Measured signals in ground experiments and flight tests (as well as other applications such as communications and controls) include noise from various sources. In the frequency domain, noise can have a very complicated structure. There are some simple noise models, however, that can be appropriate in many situations. For example, white noise has a flat power spectrum, meaning it has the same power at every frequency over some wide range. Another type of noise observed in many systems, including electronics, music and many biological systems, is called noise (or pink noise). In this case, the power spectrum (again over some wide frequency range) scales as the inverse of the frequency, i.e., the power of the noise at each frequency is inversely proportional to the frequency. Both of these types of noise can be observed at the same time (as well as other noise types). For example, Fig. 6 shows a power spectrum (power spectral density, PSD vs frequency) with both and white noise components. Using these models, one can interpolate the noise at a frequency that has "real" signal, and estimate the signal-to-noise ratio (PSD of signal at a given frequency divided by the PSD of the estimated noise).
For most situations involving either computer generated or computer acquired data, the continuous function is sampled at evenly spaced, discrete intervals in time, separated by an amount . The sampling frequency (or data acquisition rate) is thus .
For a given sampling rate, we might ask how accurately the discretely acquired data can reproduce the actual waveform being sampled. The answer depends on the frequency content of the waveform and a special frequency, called the Nyquist frequency , which is half the sampling frequency, i.e., . If the waveform contains no components above the Nyquist frequency, then the waveform can be completely determined by the sampled data (assuming no errors in the measurement).*** This is known as the Nyquist/Nyquist-Shannon Sampling Theorem.
***A waveform that has information in only a limited range of frequencies is called bandwidth limited. Due to phase ambiguity, the sampling frequency should actually be more than twice the maximum frequency in the waveform. For example, a sine wave sampled at , , , etc. would always have a 0 result and could be confused with a null function.
Aliasing occurs for any sampled waveform having components with frequencies above the sampling system’s Nyquist frequency, i.e., . One way to remove this problem is to filter the data before it is sampled. This can be accomplished by a low pass filter, a filter that only passes frequencies below some cut-off frequency. One would set the cut-off at or below the Nyquist frequency. The high frequency information is thus removed before it can be aliased. In essence, the filter produces a bandwidth limited waveform.
The digital value produced by the ADC (sometimes referred to as a “word” of data) depends not only on the input voltage, but also on the voltage range and number of bits of the ADC/amplifer system. The range is given by the minimum and maximum voltages that the ADC/amplifier can read (e.g., 0 and 5 V). The number of bits () in the ADC determines its digital dynamic range (= ). Thus the relation between the digitizer output and the voltage input is given by
where output has to be an integer value. As an example, for a 2.05 V input into a DAQ with a 0-10 V range, and an 8-bit digitizer (possible digital values of 0-255), the output value would be 52 (not 52.275). Any signal amplitude variations below the difference between two adjacent quantized levels are lost; this is known as the quantization error =. In the example above, we can only say the input value was 2.039V0.0196 V (assuming the example ADC rounds rather than truncates). One would normally choose an ADC with a number of bits sufficiently high that the quantization error is less than the dominant sources of error in the measurement. Other factors, though, may influence the choice of ADC bits, including cost and data storage requirements, both of which increase with the added number of bits.
In this lab, you control the data acquisition process through a software interface called a LabVIEW virtual instrument (VI). The VI creates a display on the computer screen that lets you think of the data acquisition system as a box with “knobs”, “dials”, and other displays. For this experiment, the VI allows you to control parameters such as the minimum and maximum voltages read by the DAQ, the sampling rate, and the number of samples recorded.
