Physics IA / Ver. / 0.3.4
Last updated / 11.01.02
Pink and White Noise
Syllabus reference / 4Assessment Criteria / Data Collection, Data Processing and Presentation, Conclusion and Evaluation
Date delivered out
Date for handing in
Aim / Beside getting a qualitative understanding of power spectra the aim is to demonstrate how it is possible to measure characteristics of nonharmonic waves by Cooley-Tukey fast Fourier transform.
Assessment criteria
Planning A: Based on the theory in the section Theory below, what kind of power spectra do you expect from the three sources given in the section Experimental Procedure ?
Evaluation: Evaluate your planning
Equipment
TI Calculator with PHYSICS program Radio A sealable box
CBL with michrophone probe Running water
Theory
A pure sinusoidal wave with a definite frequency and amplitude is an extreme idealization of most waves. In stead a "real" wave will consist of many (even infinite) number of sinusoidal waves, each with its own frequency and amplitude. The power, energy pr time, radiated from a source will therefore in general get different contributions from different frequencies. The plot of the power versus frequency for a given wave source will then give information about the composition of the various sinusoidal waves from the source - the power spectrum. Beside beeing a standard technique in acoustics and electromagnetism, the large scale matter distribution of the universe itself has recently been studied successfully in this way[1].
We will in this investigation consider only two[2] characteristic power spectra of backgroud noise: pink and white noise. White noise[3] is a source with a power spectrum where all frequencies contributes equally (think by analogy to light: if all frequncies are present we obtain white light). Therefore the
power spectrum is a horizontal line. For pink noise the power spectrum is inverse proportional to the frequency and therefore the lowest frequencies contribute most.
Experimental procedure
Each group member should collect one sample of data from each of the three following sources:
1. A badly tuned radio
2. Running water
3. "No source" - i. e. microphone placed in a sealed box
Data analysis
For each of the data samples the following analysis should be done in Graphical Analysis:
First of all we want the amplitudes and frequencies of the harmonic waves:
1. Copy the data set from your calculator to Graphical Analysis.
2. Plot a graph of the amplitudes of harmonic waves versus their frequencies by choosing the menue command Window/New Window/FFT Graph. The result is two new coloumns in the data window, frequency and amplitude.
We want now to display the power spectrum, but due to a bug in the program we can't neither selecting the frequency coloumn along the first axes nor define a derived coloumn based on any of these two new coloumns. Therefore we have to do this plotting in a more cumbersome way:
3. Copy the resulting two coloumns by first selecting them in the data window, then applying the command Edit/Copy Data.
4. Get a new working area by selecting File/New. Answer no to the question of saving the old file.
5. Select the x and y coloums and paste the data by the Edit/Paste Data.
6. Make a new coloumn called "Prop power" (proportional to power) and define it to be equal to the square of the amplitude coloumn.
7. By choosing the axes plot "Prop power" versus frequency.
8. Use the Analyze menue to check whether the graph or a portion of it (in that case take note of the boundaries of the region) can be considered on the form y=b (white noise) or on the form y=1/x (pink noise).
References
P. Bloomfield, Fourier Analysis of Time Series: An Introduction, Wiley 1976
Hida, Kuo, Potthoff and Streit, White noise: An Infinite Dimensional Calculus, Kluwer 1993
B. B. Mandelbrot, Multifractals and noise: Wild Self-Affinity in Physics, Springer 1998
M. Schroeder, Fractals, Chaos, Power Laws: Minutes from An Infinite Paradise, Freeman 1991
Ed. N. Wax, Noise and Stochastic Processes
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[1]1Stephen D. Landy "Mapping the Universe", Scientific American June 1999 pp 30-37.
[2] Brown noise, characteristic for brownian motion and with power spectrum inversely proportional to the square of the frequency, will not be considered.
[3]Also called Johnson noise.