When a sinusoidal waveform is periodically modulated in either amplitude or frequency, sidebands are generated as a result of this carrier modulation. These sidebands can be measured using a lock-in amplifier. Zurich Instruments' lock-in amplifiers (such as the HF2LI and the UHFLI) can measure up to 8 frequencies simultaneously and include dedicated and patented arithmetics to facilitate sideband demodulation. This blog post describes how to calculate the AM and FM indices of modulation and phases based on the measurements from a lock-in amplifier. Applications include:
Principles of Sideband Measurements with Lock-in Amplifiers
The goal is to simultaneously measure all sidebands separately. This method is in contrast to measuring the entire signal band with a bandwidth sufficient to capture all three bands at once. The advantage of measuring the 3 bands with dedicated demodulators is that the entire set of information is thereby available to directly calculate the index and phase of the respective modulation without further signal processing. Further, AM and FM can discriminated and accurately measured at the same time.
Construction of reference signals
It is a fundamental requirement that the phase relation of the 3 demodulators looking at the carrier and the two sidebands is defined at all times. For that reason, only 2 oscillators are used to construct the references for the 3 bands. The required arithmetics for generating the sideband frequencies are part of the MOD option available for Zurich Instruments' lock-in amplifiers (see the HF2LI-MOD and UHF-MOD options). The first oscillator is running at the carrier frequency fc and the second oscillator is running at the modulation frequency fm. In a first step the respective reference phases Φref_c(t) and Φref_m(t) are calculated and in a second step the reference signals are calculated from these two phases as follows: Φref_c(t) = ∫0t fref_c(t) dt 2π + Φref_c0 and Φref_m(t) = ∫0t fref_m(t) dt 2π + Φref_m0 , where Φref_c0 and Φref_m0 are constant phaseshifts as can be set on a lock-in amplifier. In a second step the actual reference signals are generated for the center and sidebands such that refc(t) = sin( Φref_c(t) ) refup(t) = sin( Φref_c(t) + Φref_m(t) ) reflo(t) = sin( Φref_c(t) − Φref_m(t) ). Notice that in the last row, the reverse running −Φref_m(t) also implies that Φref_m0 is effectively negated.
Amplitude Modulation
