Experiment 2: Electronic Enhancement of S/N and Boxcar Filtering

Transcription

1 Experiment 2: Electronic Enhancement of S/N and Boxcar Filtering Synopsis: A simple waveform generator will apply a triangular voltage ramp through an R/C circuit. A storage digital oscilloscope, or an ADC + computer will be use to co-add sweeps. Students will compute S/N and rise time as a function of co-added spectrum. Students will compute a Fourier transform to determine the frequency range required to achieve an acceptable rise time. READING Read pages of this book. Equipment 1 signal generator set at: 1 Hz 1 signal generator set at 150 Hz Resistor: 1000 ohms Capacitor: 10 microfarads Oscilloscope ADC converter/computer Wire the system as shown in the below Figure. 37

2 A. Data Acquisition 1. Check that the system in front of you conforms to Figure Turn on all the active components and describe what you see on the oscilloscope. By hitting the hold next button you can stop the trace and may in fact plot out the oscilloscope data. 3 Follow instructions for the ADC to get capture this waveform (signal + noise) digitally. Instructions for data collection and manipulation with the DAS 1800 Start the Keithley Metrabyte software by double clicking the icon labeled Data Collection in the Instrumental 311 folder. 1. Click on the OK button of the open configuration file module. 2. Select the File menu of the DAS 1802-HR Test & Control module. 3. Select the Open option of the File menu. 4. Open the file labeled CHEM310.PAR 5. Click on Setup 6. Enter 6 for Start and Stop channels; leave gain at 1(one). 7. Click the start button of the DAS 1802-HR Test and Control module 8. Wait for the acquisition status to become 100% done. (Note: Although the status may not change, it is not necessarily done). 9. Select Panels and DDE 10. Go into Setup DDE, and add msoffice\after c:\ and before \excel on both the Program and Sheet lines. 11. Click on Connect and wait for status bar to say Connected ; click on Close. 12. Click on Transfer 13. Save on diskette. 14. Click on the Next button of the Text Import Wizard step - 1 of 3 module. 15. Select the box marked - comma by clicking on either the box or the word comma. (Note: you will notice in the data below that the comma s will disappear and a line will appear.) 16. Click on the Finish button of the Text Import Wizard step - 2 of 3 module. 17. Select columns A and B by placing your mouse cursor on the A button of the spreadsheet and click and hold the left hand mouse button. Drag the cursor across to the B button. 18. Cut the data by selecting the Cut option of the Edit menu. 19. Select columns B and C in a similar manner as you did previously. 20. Paste the data by selecting the Paste option of the Edit menu. 21. Enter a zero into cell A Enter the formula = A1+1 into cell A Copy the formula in cell A2 by selecting cell A2 by clicking on it with the mouse, 38

3 selecting the Copy option of the Edit menu, then selecting cells A3 to A500 with the mouse, and paste the formula as you did previously. 24. Select A, B, and C columns in the same manner as you did previously. 25. Select the Chart option of the Insert menu. 26. Select the As new sheet option of the Chart option. 27. Click on the Next button of the Chartwizard - Step 1 of 5 module. 28. Select the XY Scatter button of the Chartwizard - Step 2 of 5 module. 29. Click on the Next button of the Chartwizard - Step 2 of 5 module. 30. Select the Option 6 button of the Chartwizard - Step 3 of 5 module. 31. Click on the Next button of the Chartwizard - Step 3 of 5 module. 32. Select the Finish button of the Chartwizard - Step 3 of 5 module. 33. Edit the graph as you please. 34. Print a copy of the graph. 35. Save the data on your disk for you records. 4. Disconnect temporarily the noise signal generator and capture via the ADC the waveform (signal). This completes data acquisition for the boxcar filter. Save the data to be filtered later. Save three separate waveforms. 5. Turn off the noise signal generator. 6. On the oscilloscope make a measurement of peak to peak amplitude of the waveform and tabulate it as a function of the frequency. 7. Increment the frequency in units of 10 Hz and repeat the measurement. 8. Continue until the measured peak to peak amplitude is 1/3 of the original signal. B. Data Manipulation for the Analog Filter 1. Make a plot (Bode diagram) of V peak to peak vs log Hz V peak to peak, maximum This will be a plot of the channel A peak to peak voltage divided by the channel B peak to peak voltage as a function of logh. 2. Create a theoretical plot of Vout/Vin vs log Hz and Hz and plot on same graph. The two graphs should be equivalent. Do these plots conform to each other? Why or why not? 3. Calculate the cutoff frequency from the value of the R and C used in the experiment and 39

4 compare that to the measured cutoff frequency. C. Data Manipulation for the Boxcar 1. Import data into a spreadsheet, such as excel. 2. Average points 1-3 and place the average value in a column adjacent to the raw data next to the raw point Average points 2-4 and place the averaged value in a column adjacent to the raw data next to the raw point Average points 3-5 and place the averaged value in a column adjacent to the raw data next to raw point Continue to the n-3 to nth point of the raw data. This completes the data manipulation for a 3 point sliding boxcar. 6. For a 5 point sliding boxcar the averaged value of points 1-5 will be designated as point 3 and so on. 7. For a 7 point sliding boxcar the averaged value of points 1-7 will be designated as point 4 and so on. 8. Produce a graph showing the waveforms obtained with a) no boxcar smoothing b) 3 point sliding boxcar c) 5 point sliding boxcar d) 9 point sliding boxcar e) 25 point sliding boxcar D. Data Smoothing by Waveform Summation In this exercise you will take the three replicate noisy waveforms you acquired and sum them. If the three wave forms are in phase (have the same frequency and starting point) they will sum the base frequency constructively and destructively sum the noise. The net result should be to increase the S/N. 1. In the excel worksheet line up your three noisy sin waves in adjacent columns (for example a, b, and c). 2. Plot the three waveforms on the same graph. You do this to determine if they have the same starting point. If they do proceed to D.3 immediately below. 3. Sum the wave forms by typing into column c the formula =sum(a1.c1) 4. Copy the formula down the spreadsheet. Plot the resulting summed waveform. It should have an amplitude three times as large as any of the original waveforms. 5. Determine S/N of one of the original waveforms and the resulting waveform. The S/N is calculated by taking the total voltage from peak to valley of the base sin wave and calling that the signal. The noise is calculated by sitting near the peak of the base waveform. At some distance either side of the flat portion of the peak find the highest voltage and find the lowest voltage that represents the noise added to the peak. This peak to peak distance should be ~6s. Calculate N for the S/N by dividing this peak to peak noise by 6. 40

5 6. Compare the S/N obtained from the original data to the summed waveform data. Which is better and why? 7. What will happen if you sum the sin waves out of phase? Demonstrate. 8. If your original three waveforms were not in phase (same starting point) you will need to shift the data in column b up or down in order to move it into phase with column a. You will, of course, lose some points this way. To determine how far to shift up or down go to your graph and place the cursor on the first peak for waveform a. The computer will give you the (x,y) coordinates of that peak (t,v). Move the cursor to the peak associated with the waveform in column b. Find the coordinates (t,v) associated with the peak of b. You will want to move column b such that the point in column b associated with the maximum voltage you just found is next to the maximum voltage found in column a. You can check that you have accomplished this by looking back at the graph and seeing if you have managed to shift sin wave b in phase with sin wave a. When you have all three sin waves in phase go to steps 3-7 above. E. Fourier Transform Exercise The point of this exercise is to transform the unfiltered data. Since you have artificially constructed the unfiltered data from two different frequency sin waves a transform into F.T. space should yield back information about the two different frequencies. The F.T. can be performed in Excel, but takes a little bit of trickery. Fourier Transform By Excel To perform a fast fourier transform by excel your time varying data must be a string (in a column) that is 2 n long (2, 4, 8, 16, 32, ). The digital FT performed by excel does not make reference to your measured time, but treats your data as a string of data from n to 2 n. Open an excel spread sheet. At the top create a base frequency as f source = 2ð/512. We are using 512 as our total data string because 512 = 2 7. Create a frequency for your noise that is not equal to that of the source, nor should it be some integer multiple. I chose f noise = 1.3f source. In column A create a time string from 1 to 512. In column B create the signal associated with the source (=sin{(2ð/512)*(time in column A}). The sin function should include (2ðtime/512). In column C create a similar set of data for the noise. In column D sum the source and the noise. Open the tools pack and go to Fourier transform. Specifiy the 512 data points associated with the detector in column D. Tell it to place the transformation in column E. The data placed into column E will be complex number representation of the amplitude of the frequency 41

6 components for 0 (f source ), then 1(f source ), then 2(f source) and so on. (An aside: we have deliberately created our source frequency to pass one cycle in 512 second. If you use the FFT in other applications, the fft considers the fundamental period of the frequency to be what happens in the 512 data points you scan). The first point in column E will represent 0 frequency, or DC. Each sequential box will represent the amplitude of frequency 1 (1cycle/512 second), frequency 2 and so forth up to point 256 at which point the numbers display the amplitude of each decreasing frequency. The data displayed is a complex number. To convert to something you can graph create a column F (=imabs(columne). Create a histogram of column F. You do not need to use the Histogram package in Excel or to create bins here. Simply highlight the data in column F including only the 2 nd through the ~ 30 data points (base frequency to frequency multiple 30). Using that single column insert a bar graph. The computer will automatically assign numbers 1, 2... to your x axis. The data you highlighted will be the y axis or height of the bar graph. In this bar graph you should see at least two frequency components. One component is the base frequency or signal and the second is the high frequency noise sitting on top of it. To filter the data create column G Every number in column H should be zero except for the point representing the base frequency (point 2 in column E). Copy the complex number in column E over. Note, for some unexplained reason you will find a duplicate of this number at the end of the data string. Copy the complex number at the end of the column that is identical to the one you just copied from column E to column G. Now re-create a time string of data by performing the inverse fft on column H. To do this go to tools, Fourier Transform. In the Fourier Transform command box you click inverse. Tell the computer to place the data in column H. This data needs to be truncated. To truncate in column I type =value(column H). Now plot the data in column H as an xy scatter plot. You should see a beautiful noise free sin wave identical to the original noise free sin wave you generated or acquired. Report: In addition to introduction, materials, methods, and data, results, and data analysis use all of the following questions to construct an essay. Do not write the answers out question by question. Integrate them into a coherent essay. You may wish to consider other issues not listed below. 1 How does this type of filtering of sine waves relate to concepts discussed in the statistics lab? 42

7 2. How the circuit be reconfigured to create a high pass filter? 3. What would be the cut-off frequency of this high pass filter? 4. When you observed the Signal summed to the Noise did you observe a "beat" frequency? 5. Why does the boxcar averaging lower the amplitude of the high frequency component? 6. In what way does the amplitude of the high frequency component vary with the size of the boxcar. Be very specific both in words and in math and graphs. (Hint compute s from the peak to peak variation of the high frequency component). 7. Do you think that it is appropriate to average five points together even when the signal is clearly variable with time? Would it be appropriate to average 100 points together? Why or why not? What might be a better way of averaging the data?. 8. How does the S/N change when you sum waveforms? What might be a problem in summing a waveform? 9. Which worked best: FT or summing or boxcar? 10. What would be a problem with doing a mathematical FT? 43