Appendix P. Modulation and synchronous detection: It's not magic, but it's close.

In some experimental designs it may be beneficial to apply the technique of modulation, in which one of the controlled independent variables is oscillated in a periodic fashion, and then to detect the resulting oscillation in the measured signal. With the right instrumental design, some types of noise and drift may thereby be reduced or eliminated.

A simple example applies to optical measurement systems like the one pictured on the right. A light source illuminates a test object (DUT="Device Under Test") and the resulting light from the test object is measured by photo detector. Depending on the objective of the experiment and the arrangement of the parts, the detector might measure the light transmitted by, reflected by, scattered by, or excited by the light beam. A optical chopper rapidly and repeatedly interrupts the light beam falling on the test object, so that the photo detector sees an oscillating signal, and the following electronic system is designed to measure only the oscillating component and to ignore the constant (direct current) component. The advantage of this arrangement is that any interfering signals introduced after the chopper - such as constant light that comes from the test object itself, or ambient light that leaks in from the outside, or any constant background signal generated by the photo detector itself - are not oscillating and are thus rejected. This works best if the electronics is synchronized to the chopper frequency. That's actually the function of the lock-in amplifier, which receives a synchronizing reference signal directly from the chopper to guarantee synchronization even if the chopper frequency were to vary (c.f. the interactive simulation on https://terpconnect.umd.edu/~toh/models/lockin.html and T. C. O'Haver, "Lock-in Amplifiers," J. Chem. Ed. 49, March and April (1972). The lock-in amplifier is sometimes viewed as a "black box" with almost magical abilities, but in fact it is actually performing a rather simple (but very useful) operation, as shown in this simulation.

AmplitudeModulation.m is a Matlab/Octave script simulation of modulation and synchronous detection, in which the signal created when the light beam scans the test sample is modeled as a Gaussian band ('y'), whose parameters are defined in the first few lines of the script (updated for recent version of Matlab). As the spectrum of the sample is scanned, the light beam is amplitude modulated by the chopper, represented as a square wave defined by the bipolar vector 'reference', which switches between +1 and -1, shown in the top panel on the left. The modulation frequency is many times faster than the rate at which the sample is scanned. The light emerging from the sample therefore shows a finely chopped Gaussian ('my'), shown in the second panel on the left. But the total signal seen by the detector also includes an unstable background introduced after the modulation ('omy'), such as lighted emitted by the sample itself or detector background, which in this simulation this is modeled as a random walk (Appendix O), which seriously distorts the signal, shown in the third panel. The detector signal is then sent to a lock-in amplifier that is synchronized to the reference waveform. The essential action of the lock-in is to multiply the signal by the bipolar reference waveform, inverting the signal when the light of off and passing it unchanged when the light is on. This causes the unmodulated background signal to be converted into a bipolar square wave, whereas the modulated signal is not effected because it is "off" when the reference signal is negative. The result ('dy') is shown in the 4^th panel. Finally, this signal is low-passed filtered by the last stage in the lock-in amplifier to remove the modulation frequency, resulting in the recovered signal peak 'sdy' shown in the bottom panel. In effect, the modulation transforms the signal to a higher frequency ('frequency' in line 44) where low-frequency weighted noise on the baseline (line 50) are less intense.

These various signals are compared in the figure on the right. The original Gaussian signal peak ('y') is shown as the blue line, and the contaminating background {'baseline') is shown in black, in this case modeled as a random walk. The total signal ('oy') that would have been seen by the detector if modulation was not used is shown in green. The signal distortion is evident, and any attempt to measure the signal peak in that signal would be greatly in error. The signal 'sdy' recovered by the modulation and lock-in system is shown in red and overlayed with the original signal peak 'y' in blue for comparison. The fact that the blue and red line are so close to each other indicates the extent to which this method is successful. To make a more quantitative comparison, this script also uses the peakfit.m function, which employs a least-squares method to measure the peak parameters in the original unmodulated total signal (green line) and in the modulated recovered signal (green) and to compute the relative percent error in peak position, width, and width by both methods:

SignalToNoiseRatio = 4
Relative % Error:   Position         Height         Width
               Original:      8.07                23.1              13.7
           Modulated:      0.11                0.22              1.01

Each time you run it you will get the same signal peak but a very different random walk background. The S/N ratio will vary from about 4 to 9. It's not uncommon to see a 100-fold improvement in peak height accuracy with modulation, as in the example shown here. (If you wish, you can change the signal peak parameters and the noise level in the first few code lines of this simulation. For an even greater challenge, change line 47 to "baseline=10.*noise+cumsum(noise);" to make the noise a mixture of white and random walk drift, which results in a really ugly raw signal; you can see that the white noise makes it through the synchronous detector but is reduced by the smoothing lowpass filter in the last stage). In effect, the low-pass filter determines the frequency bandwidth of the lock-in system, but it also increases the response time to step changes (as in the Morse Code example). 

This improvement in measurement accuracy works only because the dominant random error in this case is

(a) introduced after the modulation, and
(b) a mostly low-frequency noise.

If the noise were white, there would be no improvement, because white noise is the same at all frequencies; in fact there would be a slight reduction in precision because of the fact that the chopper blocks half of the light on average. If the sample (device under test) generates an "absorption" peak that starts a some positive value and then dips down to a lower value before returning, the demodulated output will the a negative-goring  peak rather than a positive peak (see AmplitudeModulationAbsorption.m).