## 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 (HF2LI and UHFLI) can measure up to 8 frequencies simultaneously and include dedicated and patented arithmetics to facilitate sideband demodulation. This blog describes how to calculate AM and FM indices of modulation and phases based on 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 index and phase of th 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 three demodulators looking at the carrier and the two sidebands is defined at all times**. For that reason, only two oscillators are used to construct the references for the three bands. The required arithmetics for generating the sideband frequencies are part of the MOD option available for Zurich Instruments lock-in amplifiers (UHF-MOD and HF2LI-MOD). 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}

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

*Left: Screenshot of the LabOne user interface that shows the settings of the MOD option to generate an AM 100%-modulated signal and the time-domain view from the integrated scope. Right: Frequency-domain view of the same signal in the Spectrum Analyzer and the output of the Lock-in demodulated sidebands below.*

Sidebands can be found at f_{c}±f_{m}. Both sidebands are in-phase, which is in contrast to frequency modulated (FM) and phase modulated (PM) signals.

### 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 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 in order 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 Modulation

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 three complex values measured by three 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 uppper 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 the PLL a better suited tool rather 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}peak frequency deviation - f
_{m}frequency of the modulating signal - ΔΦ
_{p}peak phase deviation in radians.

So actually the modulation index m is of the unit 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 constraint to less than 0.2 rad = ~12 degree of modulation.

### Phase of the Modulation

The phase between the two sidebands for a pure phase modulation is always 180 degree – the two bands are aphasic. The same formulas as above for the AM case apply.

### Complex Calculation of Narrow-band FM Modulation

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} = J_{1}(Abs(**Z**_{mod_FM})) / J_{0}(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, also see the blog Measuring Bessel Functions.

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 uppper 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 for 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 affecting 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.

### Correct for 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 either

- 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.

### Correct Sign of Lower Sideband

A further effect to beware 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 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 in order to calculate the phase of a modulation

Φ_{mod_up} = Phase(**Z**_{up}) − Phase(**Z**_{c})

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
- Agilent Application Note 150-1, Spectrum Analysis Amplitude and Frequency Modulation
- ZI blog, Double-sideband Suppressed-carrier Modulation