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white_noise_project [2014/05/05 18:51]
wikimanager [Software and data analysis]
white_noise_project [2014/06/04 03:46] (current)
wikimanager [Software and data analysis]
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 //Feel free to expand and contribute//​ //Feel free to expand and contribute//​
 ====General background==== ====General background====
-Pure [[wp>​white noise]] can be observed when no signal is transmitted into the receiver, for example if the radio or TV is not tuned to any particular station, or no microphone is plugged into the audio amplifier. As a matter of fact, any recording or measurement process ​has some degree of white noise superimposed onto the recorded or measured signal. For example, any resistor at a given temperature will generate a white-noise voltage, called [[wp>​Johnson-Nyquist noise]]. In some applications,​ such as [[wp>​MRI]] or [[wp>​ultrasound]] imaging, the persistence of white noise in the images is the dominant factor negatively affecting the scan duration ​and the image quality, or both. The goal of this project is to get an insight into the statistical properties of white noise, and to distinguish between random and non-random origins of certain types of noise. ​+Pure [[wp>​white noise]] can be observed when no signal is transmitted into the receiver, for example if radio or TV set is not tuned to any particular station, or if no microphone is plugged into an audio amplifier. As a matter of fact, any recording or measurement process ​exhibits ​some degree of white noise superimposed onto the recorded or measured signal. For example, any resistor at a given temperature will generate a white-noise voltage, called [[wp>​Johnson-Nyquist noise]]. In some applications,​ such as [[wp>​MRI]] or [[wp>​ultrasound]] imaging, the persistence of white noise in the images is the dominant factor negatively affecting the scan duration ​or the image quality, or both. The goal of this project is to get an insight into the statistical properties of white noise, and to distinguish between random and non-random origins of certain types of noise. ​
  
 ... ...
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 ====Computational goals==== ====Computational goals====
 The main goal of this project is to get a feeling of how unrelated non-random signals, such as sine waves, are randomly combined to produce much more random (and ultimately, normally-distributed) white noise. Specifically,​ we want to demonstrate The main goal of this project is to get a feeling of how unrelated non-random signals, such as sine waves, are randomly combined to produce much more random (and ultimately, normally-distributed) white noise. Specifically,​ we want to demonstrate
-  - The distribution of readings in a single sine-wave is very different from the Gaussian (bell-shaped curve of Fig. 4). +  - The distribution of readings in a single sine-wave is very different from the Gaussian (the bell-shaped curve of Fig. 4). 
     * It is much closer to a [[wp>​bimodal distribution]].     * It is much closer to a [[wp>​bimodal distribution]].
   - As several non-related sine waves of different frequencies are combined, the distribution of readings becomes lumpier.   - As several non-related sine waves of different frequencies are combined, the distribution of readings becomes lumpier.
-  - Eventually, when many sine waves are combined, the readings of their sum signal are distributed very closely to a Gaussian.+  - Eventually, when many sine waves are combined, the readings of their sum (or average) ​signal are distributed very closely to a Gaussian.
   - //feel free to suggest more//   - //feel free to suggest more//
 ====Software and data analysis==== ====Software and data analysis====
-Microsoft Excel software will be used. The following ​functions are needed: +Microsoft Excel software will be used. The following ​functionality is needed: 
-  - Creating a "​grid"​ of numbers, e.g. a row or column of {0, 0.001, 0.002, ... 1} for the model the times in which recordings take place+  - Creating a "​grid"​ of numbers, e.g. a row or column of {0, 0.001, 0.002, ... 1} for modeling ​the time points at which recordings take place
     * typing up such an array is tedious, but it's quite easy using the "drag by the corner"​ trick:     * typing up such an array is tedious, but it's quite easy using the "drag by the corner"​ trick:
       * type ''​0''​ into a cell       * type ''​0''​ into a cell
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       * sigma can be just a number (e.g. 1), or a cell someplace else containing the desired value       * sigma can be just a number (e.g. 1), or a cell someplace else containing the desired value
   - //<color blue>​**Exercise 1**</​color>//:​ generate a column of 200 (and/or, separately, 20000) normally-distributed random numbers with sigma=1   - //<color blue>​**Exercise 1**</​color>//:​ generate a column of 200 (and/or, separately, 20000) normally-distributed random numbers with sigma=1
 +    * //<color red>​**Question**</​color>//:​ How do we plot the input versus the output. ​ What is the input and what is the output?  ​
 +      * //​**Answer**//:​ In this exercise, just create a column of random numbers. No need to plot yet.  --- //​[[nkuzma@pdx.edu|Nicholas Kuzma]] 2014/05/18 22:34//
   - Plotting (using scatter chart) the output column versus the input column   - Plotting (using scatter chart) the output column versus the input column
     * with main title      * with main title 
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           * $n_i$ is the observed (or predicted) number of occurrences in the //​i//<​sup>​ th</​sup>​ bin           * $n_i$ is the observed (or predicted) number of occurrences in the //​i//<​sup>​ th</​sup>​ bin
           * $N$ is the total number of observations           * $N$ is the total number of observations
 +    * //<color red>​**Question**</​color>//:​ I have the two histograms plotted. ​ For the quantitative comparison should I calculate the probability density for each bin?  for the exercise with $N=200$ would this be done by using the following equation: ${\text{probability density}}=$ $\frac{\text{#​ observations in bin}}{200\times 0.3}$ ?  I am having a hard time understanding if we need to calculate the predicted number in each bin.  Should I do this?  If so, how would I figure out how to do this? 
 +      * //​**Answer**//: ​ to compare "​experiment"​ with theory, you need either to convert your bin counts to the probability density (by dividing the counts by the total # of observations and by the bin width), and compare that to the theoretical curve, or, alternatively,​ convert the theoretical probability density to the predicted bin count, that is by multiplying the theoretical probability density by the total # of observations (200) and by the width of your bins (I guess, 0.3).  Then plot the two curves on the same plot, the theoretical curve using lines and the experimental bin counts using dots or other symbols. ​ --- //​[[nkuzma@pdx.edu|Nicholas Kuzma]] 2014/05/23 17:47//
   - Save the exercises above for the "​Intro",​ "​Theory",​ and "​Methods"​ sections of your report   - Save the exercises above for the "​Intro",​ "​Theory",​ and "​Methods"​ sections of your report
     * You can convert any screen content into an image that can be pasted into your report:     * You can convert any screen content into an image that can be pasted into your report:
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         * Switch to the editing software (e.g. Word or Pages), and paste at the desired spot         * Switch to the editing software (e.g. Word or Pages), and paste at the desired spot
       * on a PC, press ''​Alt''​ and ''​PrtScn''​ ("​Print Screen"​) at the same time, then release       * on a PC, press ''​Alt''​ and ''​PrtScn''​ ("​Print Screen"​) at the same time, then release
-        * Switch to the editing software (e.g. WordPowerpoint, or Paint)+        * Switch to the editing software (e.g. WordPowerpoint, or Paint)
         * Paste at the desired spot         * Paste at the desired spot
         * Crop the excessive margins as needed         * Crop the excessive margins as needed
-  
-====Coding tasks==== 
-This is the detailed list of tasks to be accomplished:​ 
-  -  
  
-| ... | +{{ :​projects:​noise:​noisefig5.png?nolink |...}} 
-^ Figure 5. ...  ^  +^ Figure 5. Examples of histogram plots in Excel.  ^ 
- +| In the top figure, the theoretical curve has been scaled (multiplied by $N\Delta x$) to yield the predicted numbers of counts in each bin.  |
-=====Data=====+
  
  
 +====Coding tasks====
 +This is the detailed list of tasks to be accomplished:​
 +  - Generate a single sine wave of amplitude $A=5$, phase $\phi=1.5\,​$rad,​ and frequency $\omega=2\pi\times 3.5\,$Hz:
 +    * $V(t)=5\sin\,​(2\pi\cdot 3.5\,​t+1.5)$ ​
 +      * First, generate a column containing a grid of time values from 0 to several seconds with a step of 0.001 s
 +      * In the next column, type a formula containing the above equation and drag it by the corner all the way down
 +        * Use ''​PI()''​ or ''​3.1415926''​ for $\pi$, numbers from the preceding column for $t$
 +        * You can plot this sine wave (or some part of it) versus time for illustrative purposes for your report
 +  - Calculate and plot the histogram of the observed values in this pure sine wave, using Exercise 3 above as a guide
 +    * comment on the apparent "​[[wp>​bimodal_distribution|bimodality]]"​ of this histogram in your "​Discussion"​ section
 +    * what are the most-frequently observed values in a sine wave?  Why is that so (explain based on your intuition)?
 +    * what are the least-often observed values in a sine wave? Why?
 +  - Using the same grid of time values, generate several more sine waves with the same $A$ and $\omega$, but different $\phi$
 +    * Place these new columns next to the original sine-wave column
 +    * Finally, compute another column corresponding to the average of these waves:
 +      *  For each time point, the wave average = (sum of recorded voltages in all the sine waves at this time point)/​(number of waves)
 +    * Plot the resulting average wave. What properties does it have? Is it a good model for "​random"​ noise?
 +  - Using the same grid of time values, now generate 200 sine waves (by dragging formulas horizontally as well!):
 +    * Set all amplitudes to 1, all phases to 8
 +    * The frequencies $\omega$ should be on a grid from $2\pi\times 0.3\,$Hz to $2\pi\times 199.3\,$Hz with a step of $2\pi\times 1\,$Hz
 +  - Compute the average of these sine waves in another column
 +    * you can use the ''​=AVERAGE(H3:​GY3)''​ formula to average the numbers in the cell range from ''​H3''​ to ''​GY3''​ in this example) ​
 +  - Plot some part of this average of 200 sine waves versus time.
 +    - Does it look like "​random noise"? ​
 +    - If not, what is the pattern that you observe?
 +  - Generate and plot the histogram of the measured values in this average sine wave.
 +    * Make sure to zoom in on the interesting part of the plot
 +      * How does the distribution of measured values, represented by this histogram, look like?
 +      * What are the most frequently observed values?
 +      * What are the least-frequently observed values?
 +      * Compare and contrast this to the histogram of a pure sine wave obtained above. ​
 +        * Try to offer your intuition as to why adding (averaging) several pure sine waves results in such a drastically different histogram
 +        * If intrigued, try to experiment with fewer than 200 sine waves. At what point does the dramatic change of the histogram happen?
 +    * //<color red>​**Question**</​color>//:​ I need to figure out what is wrong with my frequency graph (average of 200 sine waves). I have re-plotted it again, checked my formulae, and I still am unable to find any error.  ​
 +      * //​**Answer**//:​ I looked at your file, and the data is actually correct. The problem is with your plot.  Do you know how to change the axis range? As it is now, you are "​zoomed in" too much on your figure: the //x// axis is only from 0.1 to 0.3 s somehow, and the //y// axis is from $-0.02$ to $+0.02$. Basically it is blowing up a tiny little aspect of the plot, and not showing the whole picture. ​
 +        - Select your plot.
 +        - On the "​ribbon"​ in excel, select "Chart Layout"​
 +        - Then click the box "​Axes",​ "​Horizontal axis", "Axis options"​
 +        - Make sure the first two boxes ("​minimum"​ and "​maximum"​) are checked, to stretch the scale over the whole range of your data
 +        - Click OK
 +        - Do the same for the vertical axis. 
 +        - Your plot will be showing the whole data then, and it will be correct.
 +  - Finally, generate a number (you decide how many) of pure sine waves with different amplitudes and frequencies. ​
 +    * Try random frequencies and/or random amplitudes/​phases (using a scaled version of the ''​RAND()''​ function).
 +    * Can you "​synthesize"​ a noise trace that looks like Fig. 1?
 +    * How does a histogram of measured values of such a "​noise"​ signal look like? 
 +    * Can you estimate the standard deviation $\sigma$ of your noise signal?
 +      * you can either compare to the theoretical curve 
 +      * or use the ''​=STDEV(GZ2:​GZ5001)''​ function on the range (e.g. ''​GZ2:​GZ5001''​) of your generated voltages
  
 =====References===== =====References=====
  
  
white_noise_project.1399315862.txt.gz · Last modified: 2014/05/05 18:51 by wikimanager