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Signals and noise

Experimental measurements are never perfect, even with sophisticated modern instruments. Two main types or measurement errors are recognized: systematic error, in which every measurement is either less than or greater than the "correct" value by a fixed percentage or amount, and random error, in which there are unpredictable variations in the measured signal from moment to moment or from measurement to measurement. This latter type of error is often called noise, by analogy to acoustic noise. There are many sources of noise in physical measurements, such as building vibrations, air currents, electric power fluctuations, stray radiation from nearby electrical apparatus, interference from radio and TV transmissions, random thermal motion of molecules, and even the basic quantum nature of matter and energy itself. The quality of a signal is often expressed quantitatively as the signal-to-noise ratio, which is the ratio of the true signal amplitude (e.g. peak height) to the standard deviation of the noise. Signal-to-noise ratio is inversely proportional to the relative standard deviation of the signal amplitude.

One of the fundamental problems in signal measurement is distinguishing the noise from the signal. The thing that really distinguishes signal from noise is that the noise is not reproducible, that is, it is not the same from one measurement of the signal to the next, whereas the genuine signal is at least partially reproducible. So if the signal can be measured more than once, use can be made of this fact by measuring the signal over and over again as fast as practical and adding up all the measurements point-by-point. This is called ensemble averaging, and it is one of the most powerful methods for improving signals, when it can be applied. An example is shown in Figure 3.

Figure 3. Window 1 (left) is a single measurement of a very noisy signal. There is actually a broad peak near the center of this signal, but it is not possible to measure its position, width, and height accurately because the signal-to-noise ratio is very poor (less than 1). Window 2 (right) is the average of 9 repeated measurements of this signal, clearly showing the peak emerging from the noise. The expected improvement in signal-to-noise ratio is 3 (the square root of 9). Often it is possible to average hundreds of measurement, resulting is much more substantial improvement.

Video Demonstration. This 17-second video (EnsembleAverage1.wmv) demonstrates the ensemble averaging of 1000 repeats of a signal with a very poor signal-to-noise ratio. The signal itself consists of three peaks located at x = 50, 100, and 150, with peak heights 1, 2, and 3 units. These signal peaks are buried in random noise whose standard deviation is 10. Thus the signal-to-noise ratio of the smallest peaks is 0.1. The video shows the accumulating average signal as 1000 measurements of the signal are performed. At the end, the noise is reduced (on average) by the square root of 1000 (about 32), so that the signal-to-noise ratio of the smallest peaks ends up being about 3, just enough to detect the presence of a peak. Click here to download the video (2 MBytes) in WMV format. (This demonstration was created in Matlab 6.5. If you have access to that software, you may download the original m-file, EnsembleAverage.zip).

SPECTRUM, the freeware signal-processing application that accompanies this tutorial, includes several functions for measuring signals and noise, plus a signal-generator that can be used to generate artificial signals with Gaussian and Lorentzian bands, sine waves, and normally-distributed random noise.
Matlab has built-in functions that can be used for measuring and plotting signals and noise, such as mean, max, min, range, std, plot, hist. You can also create custom functions to automate commonly-used algorithms. Some examples that you can download and use are these custom functions to calculate typical peak shapes commonly encountered in analytical chemistry, gaussian and lorentzian, and typical types of random noise (whitenoise, pinknoise), which can be useful in modeling and simulating analytical signals and testing measurement techniques. (Click on these links to inspect the code, or right-click to download for use within Matlab). Once you have created or downloaded those functions, you can use them to plot a simulated noisy peak such as in Figure 3 by typing x=[1:256];plot(x,gaussian(x,128,59)+whitenoise(x) .
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This page is maintained by Prof. Tom O'Haver , Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at toh@umd.edu.