[Introduction]  [Signal arithmetic]  [Signals and noise]   [Smoothing]   [Differentiation]  [Peak Sharpening]  [Harmonic analysis]   [Fourier convolution]  [Fourier deconvolution]  [Fourier filter]  [Wavelets]   [Peak area measurement]  [Linear Least Squares]  [Multicomponent Spectroscopy]  [Iterative Curve Fitting]  [Hyperlinear quantitative absorption spectrophotometry] [Appendix and Case Studies]  [Peak Finding and Measurement]  [iPeak]   [iSignal]  [Peak Fitters]   [iFilter]  [iPower]  [List of downloadable software]  [Interactive tools]

     

Appendix J: How Low can you Go? Performance with very low signal-to-noise ratios.


This is a simulation of several techniques described in this paper to the quantitative measurement of a peak that is buried in excess of random noise, where the signal-to-noise (S/N) ratio is below 2. (Ordinarily, a S/N ratio of 3 is desired for reliable detection).


The Matlab/Octave script LowSNRdemo.m performs the simulations and calculations and compares the results graphically, focusing on the behavior of each method as the S/N ratio approaches zero.  Four methods are compared:

(1) smoothing, followed by the peak-to-peak measure of the smoothed signal and background;
(2) a peak finding method based on the  findpeakG function;
(3) unconstrained iterative least-squares fitting (INLS) based on the peakfit.m function; and
(4) constrained classical least squares fitting (CLS) based on the cls2.m function. 

The measurements are carried out over a range of peak heights for which the S/N ratio varies from 0 to 2. The noise is random, constant, and white. Each time you run the script, you get the same set of underlying signals but independent samples of the random noise.

 

  Results for the initial values in the script are shown in the plots on the left and in the table printed below, both of which are created by the script  LowSNRdemo.m. The graphs on the left show correlation plots of the measured peak height vs the real peak height, which should ideally be a straight line with a slope of 1, an intercept of zero, and an R-squared of 1.  As you can see, the simplest smoothed-peak method (upper left) is completely inadequate, with a low slope (because smoothing reduces peak height) and a high intercept (because even smoothed noise has a non-zero peak-to-peak value). The findpeaks function (upper right) works OK for height for higher peak heights but fails completely below a S/N ratio of 0.5 because the peak height falls below the amplitude threshold setting. In comparison, the two least-squares techniques work much better, reporting much better values of slope, intercept of zero, and R-squared. But if you look closely at the low end of the peak height range, near zero, you can see that the values reported by the unconstrained fit (lower left) occasionally stray from the line, whereas the constrained fit (lower right) decrease gracefully all the way to zero every time you run the script. Essentially the reason why it's even possible to make measurements at such low S/N ratios is that the data density is very high: that is, there are many data points in each signal (about 1000 points across the half-width of the peak with the initial script values).

 The results are summarized in the table below. The height errors are reported as a percentage of the maximum height (initially 2). (For the first three methods, the peak position is also measured and its relative accuracy is reported. The constrained classical least squares fitting does not measure peak position but rather assumes that it remains fixed at the initial value of 100). You can see that the CLS method has a slight edge in accuracy, but you have to consider also that this method works well only if the peak shape, position, and width are known. The unconstrained iterative method can track changes in peak position and width.


Number of points in half-width of peak: 1000
Method         Height Error     Position Error
Smoothed peak    21.2359%        120.688%
findpeaksG.m     32.3709%        33.363%
peakfit.m        2.7542%         4.6466%
cls2.m           1.6565%  

 You can change several of the factors in this simulation to test the robustness of these methods. Search for the word 'change' in the comments for values that can be changed. Reduce MaxPeakHeight (line 8) to make the problem harder. Change peak position and./or width (lines 9 and 10) to show how the CLS method fails. As usual, the more you know, the better your results. Change the increment (line 4) to change the data density; more data is always better.

(Surprisingly, as it will be shown in Appendix U: Measurement Calibration, it is not even necessary to have an accurate peak shape model in order to get a good correlation between measured and actual height).
 
LowSNRdemo.m also computes the power spectrum of the signal and the amplitude (square root of the power) of the fundamental, where most of the power of a broad Gaussian peak falls, and plots it in Figure(2). The correlation to peak height is similar to the CLS method, but
the intercept is higher because there is a non-zero quantity of noise even in that one frequency slice of the power spectrum.

  We are now, in the 21st century, into the era of "big data", where high-speed automated data systems can acquire, store, and process greater quantities of data than ever before. As this little example shows, greater quantities of data allow researchers to probe deeper and measure smaller effects that previously.

This page is part of "A Pragmatic Introduction to Signal Processing", created and maintained by Tom O'Haver, Professor Emeritus, 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. Updated July, 2022.