## Introduction

When analyzing signals with multiple frequency components, it is important to measure the amplitude and phase of each frequency component accurately so that a change in the characteristics of one component does not affect the measurement of another frequency component. There are two different approaches to multi-frequency signal demodulation:

• Synchronized parallel demodulation.

An obvious way to demodulate a signal with a carrier and two equidistant sideband components is to use two lock-in amplifiers in series such that the first device demodulates at the carrier frequency with wide bandwidth and the second one demodulates at the sideband frequency. This method induces delay, distortion and noise due to cabling and extra analog-to-digital conversion (ADC) and digital-to-analog conversion (DAC) stages between the two instruments. This drawback was later addressed by implementing a tandem demodulation within a single lock-in amplifier instrument to avoid additional ADC and DAC stages . Nevertheless, tandem demodulation still suffers from several drawbacks which can be overcome using synchronized parallel demodulators.

Although the MFLI Lock-in Amplifier and UHFLI Lock-in Amplifier from Zurich Instruments are capable to perform tandem demodulation internally, the AM/FM Modulation option enables Zurich Instruments lock-in amplifiers to carry out synchronized parallel demodulation. In this post, we compare the method of tandem demodulation with the AM/FM modulation option of Zurich Instruments. The fundamental limitations of tandem demodulation will be investigated and we will demonstrate how the AM/FM Modulation option of Zurich Instruments can easily overcome the barriers and facilitate accurate and efficient measurements without cumbersome post-processing corrections.

## Modulated Signals

A modulated signal at the carrier frequency $$\omega_c = 2\pi f_c$$ and the sideband frequency $$\omega_m = 2\pi f_m$$ is represented in the following form:

$s(t) = A_c\cos(\omega_c t + \phi_c) + A_u\cos((\omega_c+\omega_m)t + \phi_u) + A_l\cos((\omega_c-\omega_m)t + \phi_l)$

This form can represent both AM (amplitude modulation)  and narrow-band FM (frequency modulation)  in which a carrier at $$f_c$$ is modulated by two sidebands at $$f_c+f_m$$ (upper sideband) and $$f_c-f_m$$ (lower sideband). In fact, AM and narrow-band FM signals are the following special cases of the modulated signal $$s(t)$$:

• AM: $$A_u = A_l$$ and $$\phi_u + \phi_l = 0$$
• FM: $$A_u = A_l$$ and $$\phi_u + \phi_l = 180°$$

It should be noted that phase modulation (PM) can be expressed in terms of frequency modulation equivalently. Not only is the modulation option of Zurich Instruments capable to generate AM and FM signals but also it can generate the general form of modulation given by $$s(t)$$ with arbitrary amplitudes $$A_c$$, $$A_u$$ and $$A_l$$ as well as arbitrary phases $$\phi_c$$, $$\phi_u$$ and $$\phi_l$$.

In the following, the modulated signal $$s(t)$$ will be demodulated with both tandem technique and MOD option using a UHFLI Lock-in Amplifier equipped with the UHF-MF Multi-frequency and UHF-MOD AM/FM Modulation options.

## Tandem Demodulation

In order to demodulate a double-modulated signal such as $$s(t)$$, at least two demodulators are necessary. Using the tandem technique, the in-phase or X-component of demodulated signal at the carrier frequency is internally routed to the second demodulator at sideband frequency. The following figure shows the block diagram of cascaded demodulators used in the tandem method. Fig. 1. Scheme of tandem demodulation.

According to the above configuration, a severe limitation of tandem demodulation lies in the speed of the first demodulator. The sideband modulation at $$f_m$$ needs to pass through the low-pass filter (LPF) of Demodulator 1, ideally without any reduction in amplitude and any alteration in phase. Therefore, the first LPF needs to have a large enough bandwidth or equivalently a small time constant. In other words, the sideband modulation frequency is limited to the bandwidth of the first stage LPF. Such a limitation is completely removed by the AM/FM modulation option of Zurich Instruments because the carrier and sidebands are demodulated at their corresponding frequencies directly.

Here in Fig. 1 we assume that the filter bandwidth is much larger than the sideband frequency $$f_m$$ so that we can neglect its impact on the signal. Now, the main question is that how to relate the demodulated amplitude $$A$$ and phase $$\phi$$ in Fig. 1 to the parameters of signal $$s(t)$$. Using trigonometric identities, we reach the following fundamental expressions for the demodulated amplitude and phase:

$A =\frac{1}{2}\sqrt{A_u^2 + A_l^2 + 2A_uA_l\cos(\phi_u+\phi_l)}$

$\phi = \tan^{-1}\bigg(\frac{A_u\sin\phi_u – A_l\sin\phi_l}{A_u\cos\phi_u + A_l\cos\phi_l}\bigg)$

In order to verify the above formulas, a tandem demodulation measurement is performed using the UHFLI lock-in amplifier. The instrument generates a carrier at 10 MHz with two equal sidebands 100 kHz away from the carrier. Then, we sweep the phase of one sideband (the upper sideband phase $$\phi_u$$ in this case) and record the amplitude and phase of tandem demodulation. The following screenshot of LabOne user interface shows the settings highlighted by red boxes as well as the recorded signals in the sweeper tab. Fig. 2. LabOne settings for a tandem demodulation as well as the recorded amplitude (blue) and phase (green).

It should be noted that in Fig. 2, Aux Out 1 is the in-phase component of demodulator 4 which is set in the Aux tab of LabOne. To validate the tandem demodulation formulas, we simply save the measured data and compared them with the analytical expressions of $$A$$ and $$\phi$$ using MATLAB. Fig. 3 depicts such a comparison demonstrating a perfect agreement between theory and measurement. Fig. 3. Comparison of measurement and theory in tandem demodulation.

Figures 2 & 3 clearly show that the measured amplitude using a tandem demodulation system depends on the phase of each sideband. Therefore, by measuring the amplitude $$A$$ and phase $$\phi$$, it is not possible to obtain the amplitude of each sideband because according to the mathematical expressions of $$A$$ and $$\phi$$, there is no unique solution for $$A_u$$ and $$A_l$$ unless we know $$\phi_u$$ and $$\phi_l$$. Even if both sidebands have equal amplitude, i.e. $$A_u = A_l = a$$, there is still ambiguity in obtaining $$a$$ in terms of $$A$$ and $$\phi$$. For a modulated signal with equal sidebands, the measured amplitude by tandem demodulation is given by:

$A = a\sqrt{\frac{1+\cos(\phi_u+\phi_l)}{2}}$

According to the above expression, only for an AM signal ($$\phi_u + \phi_l = 0$$) with equal sidebands, i.e. $$A_u = A_l = a$$, the outcome of tandem demodulation is the same as the modulating signal amplitude, i.e. $$A = a$$.  Otherwise, the measured signal is distorted from the original one. For instance, if the signal is a narrow-band FM ($$\phi_u + \phi_l = 180°$$), the outcome of tandem demodulation is simply null, i.e. $$A = 0$$ which cannot indicate whether there is a modulation on top of the signal or the signal is unmodulated.

## AM/FM Modulation Option

The MOD option of Zurich Instruments can generate and detect modulated signals directly using a single-stage configuration. Fig. 4 shows a simplified block diagram of the MOD option demonstrating 3 dual-phase demodulators at $$f_c$$, $$f_c + f_m$$ and $$f_c – f_m$$. As depicted in the figure, each component is demodulated directly using a single-stage demodulator at its corresponding frequency. This way, the bandwidth of LPFs can be arbitrarily small and there will be no bandwidth constraint. In contrast to tandem demodulation, all the 6 components of the 3 frequency lines, i.e. $$(A_c,\phi_c)$$, $$(A_u,\phi_u)$$ and $$(A_l,\phi_l)$$ can be extracted using the MOD option. Fig. 4. Scheme of MOD option.

Fig. 4 shows that the MOD option can generate three coherent tones at $$f_c$$, $$f_c+f_m$$ and $$f_c-f_m$$ to demodulate the input signal at these 3 frequencies simultaneously and coherently. In order to compare the performance of tandem demodulation technique and MOD option, a modulated signal is generated at a carrier frequency of 10 MHz and a sideband frequency of 100 kHz. As shown in Fig. 5, the carrier, upper and lower sidebands have the amplitudes 400 mV, 100 mV and 80 mV, respectively. Fig. 5. AM/FM modulation option of Zurich Instruments.

## Tandem vs. MOD Option

The generated signal using the MOD option in Fig. 5 is once fed to a tandem demodulator depicted in Fig. 1 and once to the MOD option illustrated in Fig. 4. For comparison, we sweep the phase of upper sideband component of the signal for a whole cycle from 0° to 360° and obtain the output of two demodulation methods using the Sweeper tool of the LabOne user interface. The output of tandem demodulation includes only one amplitude $$A$$ and one phase $$\phi$$; while the output of MOD option contains the amplitude and phase of all the 3 frequency components.

Fig. 6 illustrates the demodulated amplitude and phase of the tandem technique in terms of the upper sideband phase. As shown in the figure, by modifying the phase of one sideband component, both demodulated amplitude and phase change substantially which imposes a systematic limit on the tandem method. Fig. 6. Output of tandem demodulation: Amplitude (blue) and phase (green) versus the phase of sideband component.

We can easily verify the above measurement in Fig. 6 with the mathematical expression of $$A$$ and $$\phi$$. Fig. 7 shows such a comparison using MATLAB which implies a complete agreement between theory and experiment. Fig. 7. Comparison of measurement and theory in tandem demodulation.

It should be noted that the phase difference in horizontal axis of Fig. 6 and Fig. 7 is due to the compensation of phase shift induced by the ADC/DAC stages and coaxial cables. Regarding the parallel method, Fig. 8 plots the output of MOD option measured by the Sweeper tool of LabOne.

According to Fig. 8, all the demodulated components remain unchanged except the one which corresponds to the same component in the modulated signal at input. In this case, the phase of upper sideband at the modulated signal is swept and exactly the same phase (Demod 6 Sample Phase in Fig. 8) is detected while all the other 5 components are not affected. The same scenario happens when an amplitude or multiple components are altered. This shows an important advantage of MOD option with respect to the tandem technique which indicates that no phase or amplitude information is lost during the demodulation process of MOD option, whereas in the tandem method a part of both amplitude and phase information is lost depending on the type of modulation.

## Conclusion

Zurich Instruments offers the AM/FM modulation option for all its lock-in amplifiers to generate and detect arbitrary modulated signals with one carrier and two sidebands including AM and FM signals. Compared to the tandem modulation technique, it has several advantages including:

• The MOD option offers accurate measurements over the entire range of amplitude, phase and frequency while the tandem technique suffers from signal distortion.
• Only the MOD option provides the full set of amplitude and phase information for all frequency components.
• The MOD option is capable of extracting signals with a sideband frequency over the entire frequency range of the instrument (600 MHz in case of UHFLI), while the tandem method is limited to the bandwidth of the first-stage demodulator.
• With the MOD option, users are not concerned with the carrier phase drift which exchanges the X and Y components over time; while the cascaded technique is sensitive to the carrier phase shift.
• In addition to demodulation, the MOD option can also generate multi-frequency signals with arbitrary phase relations, amplitudes and frequencies.

## References

1. Wikipedia: “Amplitude modulation.”
2. Wikipedia: “Frequency modulation.”
3. Signal Recovery: “7270 General Purpose DSP Lock-in Amplifier.”