Sideband Analysis with Lock-in Amplifiers

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:

• Kelvin probe force microscopy (KPFM)
• Ultra-sound microscopy and imaging
• Gyroscopes (MEMS- and laser-based)

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 f_c and the second oscillator is running at the modulation frequency f_m. In a first step the respective reference phases Φ_{ref_c}(t) and Φ_{ref_m}(t) are calculated and in a second step the reference signals are calculated from these two phases as follows: Φ_{ref_c}(t) = ∫₀^t f_{ref_c}(t) dt 2π + Φ_{ref_c0}  and Φ_{ref_m}(t) = ∫₀^t f_{ref_m}(t) dt 2π + Φ_{ref_m0} , where Φ_{ref_c0} and Φ_{ref_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 ref_c(t) = sin( Φ_{ref_c}(t) ) ref_up(t) = sin( Φ_{ref_c}(t) + Φ_{ref_m}(t) ) ref_lo(t) = sin( Φ_{ref_c}(t) − Φ_{ref_m}(t) ). Notice that in the last row, the reverse running −Φ_{ref_m}(t) also implies that Φ_{ref_m0} is effectively negated.

Amplitude Modulation

Modulation amplitude or modulation depth

The amount of an amplitude modulation can be expressed in a relative and an absolute way. The absolute measure is the amplitude of the signal envelope E_max−E_min. The relative measure is the modulation depth m, defined as m = (E_max−E_min) / (E_max+E_min) When m=1, the modulation depth is 100% so that E_min=0. Each sideband will then have half of the amplitude as the centerband.

Phase of the amplitude modulation

The phase between the two sidebands for a pure amplitude modulation is always 0 - the two sidebands are in phase. The phase of an amplitude-modulated signal is the phase between the modulation signal applied to the DUT and the phase of the modulation of the measured signal. In order to determine the phase of the modulation, it is important to understand the relationship of the phases of the 3 reference signals ref_c(t), ref_up(t) and ref_lo(t). All three signals include any phase of the carrier band at f_c. If you are interested in phases of the modulation, you need to subtract the phase of the carrier band. In addition, note that to obtain the lower sideband, Zurich Instruments lock-in amplifiers subtract the modulation phase Φ_m(t). Therefore the lower sideband has a reverse running modulation reference, which is reflected in the negative sign below: Φ_modUp = Phase(Z_up) − Phase(Z_c) Φ_modLo = −(Phase(Z_lo − Phase(Z_c)) A pure AM signal has Φ_modUp − Φ_modLo = 0, if the subtraction systematically deviates from 0, this indicates that there is an additional FM of the signal.

Complex calculation of AM

A periodic amplitude modulation can be described by a complex number, Z_{mod_AM}. The index of modulation m and the phase of the modulation Φ_mod relate to this complex number like: m = Abs(Z_{mod_AM}) Φ_mod = Phase(Z_{mod_AM}) where Abs(x) is the absolute value and Phase(x) is the argument of the complex value x. The relationship to the 3 complex values measured by 3 demodulators of a Zurich Instruments lock-in amplifier is: Z_{mod_AM} = Z_up/Z_c + (Z_lo/Z_c) where

• Z_c is the complex value of the demodulation of the carrier signal.
• Z_up is the complex value of the demodulation of the upper sideband.
• Z_lo is the complex value of the demodulation of the lower sideband.
• (x) denotes the complex conjugate of x, which negates the imaginary part of a complex number.

Narrow-Band Frequency Modulation

Here I will only look at narrow-band FM and I will leave aside wide-band FM. Narrow-band is defined as an FM with an modulation index of m < 0.2. For small indices of modulation the FM generates only two significant sidebands. For wide-band FM this is different and many sidebands are generated, which makes a phase-locked loop (PLL) a tool better suited than direct sideband analysis.

Degree of modulation

While there is a 100% degree of modulation for AM, there is no such limit for FM signals. The modulation index m is m = Δf_p / f_m = ΔΦ_p where

• Δf_p is the peak frequency deviation.
• f_m is the frequency of the modulating signal.
• ΔΦ_p is the peak phase deviation in radians.

So the modulation index m is in units of radian (rad) and tells us how many rad we deviate per modulation period from the carrier signal with the frequency f_c. Therefore, narrow-band FM is constrained to less than 0.2 rad = ~12 deg of modulation.

Phase of the modulation

The phase between the 2 sidebands for a pure phase modulation is always 180 deg - the 2 bands are aphasic. The same formulas as above for the AM case apply.

Complex calculation of narrow-band FM

A PM or narrow-band FM can be described by a complex number, Z_{mod_FM}. The index of modulation m and the phase of the modulation Φ_mod relate to this complex number like: Δf_p = J₁(Abs(Z_{mod_FM})) / J₀(Abs(Z_{mod_FM})) * f_c / 2 m = Δf_p / f_m Φ_mod = Phase(Z_{mod_FM}) where J_n(x) is the Bessel Function of the first kind, as discussed in this blog post.

The relationship to the 3 complex values measured by three demodulators of a Zurich Instruments lock-in amplifier is: Z_{mod_FM} = Z_up/Z_c − (Z_lo/Z_c) where

• Z_c is the complex value of the demodulation of the carrier signal.
• Z_up is the complex value of the demodulation of the upper sideband.
• Z_lo is the complex value of the demodulation of the lower sideband.
• (x) denotes the complex conjugate of x.

Considerations Regarding Phase Relations

The complex formulas given above provide a robust and compact method to calculate modulation parameters. However, sometimes it is helpful to have a better understanding of the phase readings of the demodulated signal bands. There are indeed two particularities about the phase readings. The first is that the centerband or carrier phase also affects the phase of the sidebands, and the second is that the lower sideband has a reversed phase. Here we describe these two cases in more detail.

Correcting for the centerband phase

It is important to see that the phases of Z_{up,lo} include any phase of Z_c. For FM and AM, this means that any phase added to the carrier is also added to the measured sidebands. To get rid of this phase and measure the pure sideband phase, you can:

• Subtract the phase measured at the center demodulator, phase(Z_c), from the measured sideband phases;
• Use the "autozero" functionality on the centerband demodulator to obtain 0-phase reading, phase(Z_c)=0, on the centerband demodulator; or
• When working with complex values, divide by the complex value, Z_c, of the carrier band.

Correcting the sign of the lower sideband

A further effect to be aware of is that for the lower sideband, the modulation phase is inverted. Since we subtract the output of the modulation phase accumulator from the output of the center-band phase accumulator, the modulation is running at a negative frequency for the lower sideband. Therefore, we need to invert the phase of the lower modulation after subtraction of the carrier phase from the lower sideband. In summary, keep these relations in mind to calculate the phase of a modulation: 

Φ_{mod_up} = Phase(Z_up) − Phase(Z_c) and Φ_{mod_lo} = −(Phase(Z_lo) − Phase(Z_c)). 

For AM: Φ_{mod_up} − Φ_{mod_lo} = 0. For FM: Φ_{mod_up} − Φ_{mod_lo} = 180.

References

• Wikipedia, Amplitude modulation
• Wikipedia, Frequency modulation
• Double-sideband Suppressed-carrier Modulation