The dependence of the dynamic range of the phase generated carrier (PGC) technique on low-pass filters passbands is investigated using a simulation model. A nonlinear character of this dependence, which could lead to dynamic range limitations or measurement uncertainty, is presented for the first time. A detailed theoretical analysis is provided to verify the simulation results and these results are consistent with performed calculations. The method for the calculation of low-pass filters passbands according to the required dynamic range upper limit is proposed.

Fiber optic interferometric sensors have been intensively developed for the past few decades. Fiber-optic interferometers are used as a sensitive element in fiber optic gyroscopes, hydrophones, and so forth [

This technique has been used in many applications such as hydroacoustics [

However, implementation methods of demodulation techniques have changed with the advent of high performance field programmable gate arrays (FPGAs). FPGAs allow realizing complex real-time digital signal processing algorithms, and therefore they are ideally suitable for the multichannel PGC technique implementation to demodulate signals from fiber-optic sensor arrays [

This paper is devoted to the detailed analysis of the PGC technique taking into account its practical implementation problems and digital design restrictions. The dependence of the dynamic range of the PGC technique on low-pass filters passbands is investigated for the first time and the nonlinear character of this dependence is shown and explained.

The PGC demodulation scheme is presented in Figure

PGC demodulation scheme.

Several parameters were constants in this model:

the duration of simulated signals was 1 second;

the carrier frequency was set to 10 kHz;

the sampling frequency of the model was 100 kHz.

Low-pass filters passbands were set to 500 Hz hence the input signal at the frequency 500 Hz passed through these filters without attenuation.

Results of performed simulations are shown in Figure

Simulation results of the PGC demodulation scheme: dependencies of the output signal on the measured phase signal amplitude (

During this simulation the amplitude of the measured phase signal (at the frequency 500 Hz) and the carrier modulation depth were changing from 0 to 6.28 rad.

According to obtained results (see Figure

The demodulated result of the PGC scheme (see Figure

As shown in Figure

According to (

In order to explain obtained results, signals after low-pass filters in the PGC demodulation scheme (see Figure

According to (

But during the practical implementation of the digital PGC scheme it is impossible to provide infinite passbands of digital low-pass filters. Anyway, all passbands will be limited by the digital scheme sampling frequency. Therefore, some high-order harmonics of signals

To determine the output signal dependence on the amplitude of the measured phase signal, further PGC technique operations on signals

To simplify these calculations, low-pass filters are supposed to be ideal and thus all filtered out harmonics will not be present in filtered signals. The initial phase of the interferometer is considered to be constant.

Then, in the simplest case, when low-pass filters passbands are equal to the fundamental frequency of the measured phase signal (i.e., low-pass filters passbands include the first harmonic of the measured phase signal and the input signal at the frequency of this harmonic could pass through low-pass filters without attenuation), signals (

Derivatives of (

Then cross-multiplication between signals (

Subtracting the signal (

After the integration, the output signal can be expressed as

Performing the same mathematical manipulations with signals

To verify the obtained result (

Theoretical and simulation results: output signal amplitude dependencies on the measured phase signal amplitude (

The plot in Figure

This dependence is of significant interest in terms of the practical implementation of the digital PGC technique, because it determines the dynamic range upper limit of the demodulation scheme. The wider the low-pass filters passbands are, the higher the dynamic range is.

It should be noted that the demodulated signal is generally contaminated by high-order harmonics and their amplitude dependencies on the measured phase signal amplitude are also nonlinear. However, a detailed description of their exact behavior is beyond the scope of this paper.

There is one more important result of low-pass filters passbands restrictions, which should be taken into account during the PGC technique implementation. It is the dependence of the dynamic range on the input signal frequency. This dependence (normalized according to the dependence on

Simulation results: the dynamic range of the PGC demodulation scheme increases with the decrease in the input signal frequency (low-pass filter passbands: 500 Hz).

As shown in Figure

In real demodulation schemes the dynamic range upper limit is determined by the required maximum amplitude of the measured phase signal. Therefore, low-pass filters passbands should be calculated according to this maximum amplitude using (

In the present work it was shown for the first time that the dependence of the dynamic range of the PGC technique on low-pass filters passbands is nonlinear for large input signal amplitudes because of low-pass filters passbands restrictions. The nonlinear character of this dependence could cause dynamic range limitations or measurement uncertainty of input phase signals (see (

According to obtained results, the additional calculation of low-pass filters passbands should be performed in accordance with requirements to the amplitude dependence of the output signal and the necessary dynamic range of the PGC demodulation scheme.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was done in NRU ITMO and supported by the Ministry of Education and Science of the Russian Federation under Project 02.G25.31.0044.