Feel free to expand and contribute
Pure white noise can be observed when no signal is transmitted into the receiver, for example if a radio or a 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 Johnson-Nyquist noise. In some applications, such as MRI or 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.
|Figure 1. White noise, observed by N.K. in an NMR spectrometer (tuned to 129Xe nuclear-precession frequency). The vertical scale is the detector voltage, and the horizontal scale is time in milliseconds.||Figure 2. In contrast to Fig. 1, this recording is dominated by the 60-Hz interference from the power line (N.K.)|
John B. Johnson, while working at Bell Labs in 1926, was the first to quantify the white noise in resistors. He described his results to Harry Nyquist, a Bell Labs theorist, who was able to come up with the explaination.
Figure 3. John B. Johnson and Harry Nyquist of Bell Labs, who came up with the first quantitative theory of white noise in resistors (Courtesy Wikipedia). Note that John's photo exhibits quite a bit of noise superimposed onto his image.
White noise (a.k.a “static”) is mentioned in …
By definition, white noise is a sequence of statistically independent random measurements with the same distribution centered on 0. A more special, albeit perhaps more commonly encountered type of white noise is the Gaussian white noise, with the additional requirement that each sample in the recording is “normally” distributed (i.e. its statistical distribution has a “bell-shaped” curve,
|Figure 4. Normal (gaussian) probability distribution given by Eq. (1), where 68% of observations fall between $-\sigma$ and $\sigma$. About 95% of observations fall between $-2\sigma$ and $2\sigma$.|
The idea is, while each individual source of noise might not be random (for example, electromagnetic emissions from nearby microprocessors, electric motors next door, distant lightnings and radio stations all have very specific frequencies and time-domain signatures), when multiple unrelated sources are combined at the receiver input, their sum signal is, to a much greater degree, random. Moreover, as the central limit theorem of statistics would suggest, such sum of many unrelated non-random signals is not just random, but is also normally distributed according to Eq. (1). We shall experimentally test this hypothesis in this project.
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
Microsoft Excel software will be used. The following functionality is needed:
0into a cell
0.001into the next cell below
=into a cell, followed by a formula content (e.g.
SQRT(2)), then hitting “Enter”:
=SQRT(2)will calculate the square root of 2
=SQRT(A2)will calculate the square root of the number in the cell
A2(i.e. in the column
=sigma*NORM.S.INV(RAND())function, where sigma is the desired standard deviation (width) of the curve
Bof your excel sheet that are greater or equal to 3.2 but less then 3.3, use
COUNTIFcounts the number of cells that are greater or equal to 3.2
COUNTIFcounts the number of cells that are greater or equal to 3.3
B) will be counted, but the number 3.3 will not make it to the difference count
C2:C1001that exceed the number in
D4but no greater than the number in
=0.5*(E2+E3)formula, assuming the first bin boundary is in
E2, the second in
E$3to avoid vertical shifting
$E3to avoid horizontal shifting when dragging horizontally
$E$3to avoid any shifting
4keys at the same time.
PrtScn(“Print Screen”) at the same time, then release
|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.|
This is the detailed list of tasks to be accomplished:
3.1415926for $\pi$, numbers from the preceding column for $t$
=AVERAGE(H3:GY3)formula to average the numbers in the cell range from
GY3in this example)
=STDEV(GZ2:GZ5001)function on the range (e.g.
GZ2:GZ5001) of your generated voltages