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:
- Cascaded or tandem 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 [1]. 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 (also known as MOD 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 MOD option of Zurich Instruments. The fundamental limitations of tandem demodulation will be investigated and we will demonstrate how the AM/FM option 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) [2] and narrow-band FM (frequency modulation) [3] 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 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. Figure 1 shows the block diagram of cascaded demodulators used in the tandem method.
