Passive Electrical and Optical Methods of Ultra-Short Pulse Expansion for Event Timer-Based TDC in PPM Receiver

Abstract

The energy efficiency of a communication system using pulse position modulation (PPM) can be increased by reducing the duration of the pulses transmitted over the communication channel to several tens of picoseconds. The employment of an event timer device as a time-to-digital converter (TDC) for demodulation allows the use of PPM with many pulse positions and achieves competitive data transfer speeds. However, along with several-picosecond accuracy of modern event timers, they require a pulse duration of several hundred picoseconds for precise detection. This research is devoted to developing passive techniques for precise pulse expansion from tens of picoseconds to hundreds of picoseconds. We propose two methods: the electrical method, which employs a microstrip low-pass filter (LPF), and the optical method, which uses fiber Bragg grating (FBG). This research offers a detailed analysis of distortion-free pulse expansion requirements, the design of prototypes meeting these requirements, and experimental design verification. Theoretical background, mathematical models, and results of experimental validation of the proposed pulse expansion methods within the laboratory transmitted reference pulse-position modulation (TR-PPM) communication system are provided.

1. Introduction

Over the last decade, the amount of transmitted data and the data transmission speed have increased significantly. The urgency of the energy consumption associated with data transmission has increased correspondingly. Nowadays, communication systems designed to perform bandwidth-intensive high-speed data transmission extensively exploit microwave and optical frequency bands, thereby becoming increasingly demanding in energy consumption.

The overall energy consumption of a communication system consists of the energy lost in the system’s transmitter circuitry, transmission channel, and receiver circuitry. In long-distance communication, the energy lost in the communications channel exceeds the power consumed by the circuitry of the transmitter and receiver. On the other hand, in short-distance communication, the amount of energy absorbed by the circuitry is more significant than that associated with the transmission channel [1]. PPM is a modulation technique that allows theoretically unlimited energy savings at the cost of spectrum occupation, which makes the proposed modulation technique of particular interest to users with limited energy resources and significant signal power losses in the communication channel, for example, deep space communications, terrestrial and underwater optical wireless sensor networks, etc. It might also be important for other users since using the proposed modulation method can reduce overall energy consumption. However, the demodulation of a PPM signal by employing extremely short pulses is challenging as it requires fast electronic circuitry, such as comparators and triggers with bandwidths of tens of gigahertz. As PPM pulse duration does not carry information, the problem can be tackled by increasing the duration of the received pulses just before the demodulation.

This paper focuses on the expansion techniques of 50–100 ps short pulses using passive electrical and optical devices for subsequent use in PPM demodulators. As shown in the next sections, distortionless expansion of pulses with bandwidths above 10 GHz is a challenging task due to the high bandwidth and difficulties with the use of active methods. The main contribution of this research is to demonstrate that passive techniques can be successfully applied for high-bandwidth pulse expansion in both optical and electrical domains. A novel technique for developing requirements and calculating electrical LPF is validated by fabricating and testing filters in a real-time PPM communication system. To the authors’ best knowledge, the suitability of passive pulse expanders for work in event timer-based PPM demodulators was explored for the first time.

This paper is organized as follows: Section 2 provides a short introduction to PPM, Section 3 is devoted to the literature analysis and the state-of-the-art in the field of precise pulse duration expansion, and consists of Section 3.1, which provides a theoretical foundation for electrical pulse expansion, and Section 3.2, which is devoted to the optical pulse expansion, also known as stretching. Section 4 is focused on the experimental validation of the proposed solutions using data transfer in laboratory conditions. Finally, Section 4.2 and Section 5 summarize the research and conclude.

2. Overview of PPM Technique

Among the many different modulation techniques employed in communication systems, the PPM technique [2,3] is one of the most generally used ultra-wideband (UWB) modulation techniques and achieves energy savings via the use of very short pulses [4]. This efficiency comes at the cost of increased frequency bandwidths and higher bandwidth requirements for the receiver, transmitter frontend, and propagation media. The employment of PPM in optical communication systems is especially appealing as optical circuitry naturally operates at large frequency bandwidths, and optical fibers typically have very low signal attenuation. At the same time, traditional on–off keying modulation techniques have higher optical power utilization and lower sensitivity to nonlinear distortions.

The PPM technique encodes the information by the time of the transmitted pulse. The term “pulse position modulation (PPM)” was first coined in 1946 [5], and initially, it was mainly utilized in military applications. While PPM was originally employed in analog low-frequency radio and microwave systems [6], it has subsequently found use in digital communications [7,8,9] and microwave applications. In the optical frequency range, PPM is used in free-space optical communication and wired communications [10,11,12]. Furthermore, PPM is still in use in semi-analog radio control systems [13].

An overview of the benefits, disadvantages, and applications of PPM can be found in [14]. The main advantage of PPM is that it ensures higher energy efficiency than other modulation techniques and is tolerant of nonlinear distortions [15,16,17]. Those properties provide a solid foundation for future use of PPM, as pointed out in [6,18,19]. The development of the PPM technique has led to novel hybrid modulation types, such as multi-pulse-position modulation [20], where the symbol is composed of several pulses. It is worth mentioning that pulse-position modulation can be combined with pulse-amplitude modulation and pulse-width modulation techniques to increase the spectral efficiency of the transmission.

The operation principle of PPM is straightforward, as shown in Figure 1. The ${\log }_2\left(M\right)$ message bits are encoded by transmitting a single pulse with a duration of $\tau$ in one of M possible time positions that have a duration $\Delta$ over a symbol time interval of length $T=M·\Delta +T_s$, where $T_s$ is a guard time required for data processing and $M·\Delta$ is the payload interval in which the pulse is transmitted. In the PPM technique, there is no stringent requirement to maximally preserve the shape of the generated pulse being transmitted via a channel to the receiver unit, provided that the pulse shape variations do not vary considerably between the pulses; one only needs to ensure that the change in the pulse shape is not too significant to provide correct demodulation. These conditions are not unacceptably stringent and depend on the implemented PPM demodulation method.

There are several methods for PPM signal demodulation [6,12,19]. Early PPM demodulators employed energy-based detection [21,22]. This method can be used in communication channels where the received pulses’ exact arrival time and shape are unknown. However, the most commonly used technique in low-speed applications is first to convert pulse-position modulation to pulse-width modulation, then use an integrator circuit to produce an approximation of the message signal. When the number of positions is small, the Rake receiver can be employed [23]. In this case, the signal is demodulated using matched filtering in several branches that have different delays. This method is especially attractive in the case of non-coherent TR-PPM because the received signal provides the reference pulse for the matched filtering, mitigating the impact of multi-path propagation [24]. Other, more advanced approaches have been developed, including iterative demodulation methods [25] and even chaotic synchronization-based techniques [26].

One of the most elegant ways to implement a highly efficient digital PPM demodulator is to employ a TDC [27,28]. This method is based on the idea of direct time measurement using an event timer device [29,30]. The employment of accurate event timers [31] allows the use of a virtually unlimited number of positions in the symbol, achieving a very low duty cycle and exceptional energy efficiency. However, two major challenges exist when using an event timer to demodulate the PPM signal. Firstly, the timer imposes specific requirements concerning the minimal pulse duration at the input of the timer. If the pulse duration is too small, the event timer will not detect the event (incoming pulse). Secondly, time measurement by the event timers requires a relatively large guard time ($T_s$ in Figure 1), during which the timer estimates the time and does not react to the incoming signal. This time, sometimes called “dead time”, cannot be efficiently employed for data transfer and affects the achievable bitrate.

3. Electrical and Optical Pulse Expansion Methods

In this research, we employ one of the most precise commercially available event stream timers: Eventech Stream Time Tagger [32]. The event timer can reliably record the signal arrival intervals between pulses without much jitter if the PPM pulse duration is at least 150 ps. In addition, the pulse waveform must be smooth enough, with the first transition duration (rise time) as small as possible, so there is little measurement jitter and a low impact from the additive noise. In contrast, the second transition duration (fall time) is less critical; nevertheless, ripples in the pulse tail are highly undesirable. Additionally, to ensure an adequate event timer response, it is highly desirable for the overshoots and undershoots in the pre-transition and post-transition aberration regions of the incoming pulse to be as small as possible.

In this work, pulses with a duration of 50 ps are used in the PPM transmission channel, providing significant energy savings. As the particular event timer requires a pulse width of more than 150 ps, the pulses must be expanded or stretched at least three times. Therefore, it is necessary to develop a device that satisfies the following requirements:

• The full width at half maximum (FWHM) duration of the input pulse is 40–60 ps.
• The bandwidth of the input pulse is at most 25 GHz.
• The FWHM duration of the output pulse is at least 150 ps.
• The rise of the output pulse time is at most 50 ps.
• The fall time of the output pulse is at most 100 ps.
• The ripples and overshoots of the output pulse are at most 10% of pulse amplitude.
• The jitter of the output pulse rising edge is less than 5 ps.

It is worth stressing that in PPM, the pulse duration does not carry any information and can be manipulated freely. Figure 2 illustrates the structure of a PPM signal at the input of the event timer. The occupation of several time positions by the single incoming expanded pulse does not create any issues with PPM demodulation, as only the rising edge of the pulse carries the information.

There are many methods of pulse expansion using passive or active devices. In this paper, we address the employment of electrical LPF and optical techniques, namely, FBG, for pulse duration expansion. In the following sections, we will look at these technologies.

3.1. Electrical Pulse Expansion

Many modern electrical and optical systems—such as UWB communication devices; signal processing circuits; radar and chirped radar communication; indoor localization and radar systems, used for through-wall imaging and ground penetration; various detectors, including neutrino detectors, sensors, time measurement systems, etc.—are dealing with the transmission, measurement, and processing of pulses [33,34].

These operations often require the pulse to be expanded or compressed with minimal waveform distortion. Various delay lines [35] and other devices operating in the frequency domain (for example, filters) and making changes to it [36] and combinations of delay lines and filters are widely used for pulse expansion. Both digital and analog signal processing techniques are used, and each has advantages and disadvantages.

Here, we will mainly look at the methods used to expand analog signals without significantly affecting their waveform. The various dispersive delay lines have been handy devices in communication systems for decades, and the simplest of them are coaxial or microstrip delay lines [37]. There are several methods where two or more dispersive elements and a mixer are required. Delay and summation of delayed pulses allow the creation of a composite output signal whose duration is larger than the input signal. Using this technique, signals can be slowed down, sped up, or reversed in time with corresponding changes in the spectrum [38,39]. Today, active delay lines are also used to implement an on-chip time stretching system in the microwave band [40]. Some studies demonstrate the possibilities of influencing propagating pulse shape in the transmission line coupled a with split ring resonator in situations where negative group velocity is created; however, such devices are difficult to implement in practical applications [41].

Signals can be processed by passive filters whose magnitude and group delay responses are determined by the requirements for the waveform and spectrum of the pulse. All-pass filters (APFs) are widely used for passing through the signal without affecting its magnitude spectrum, providing phase shift and subsequently allowing the tailoring of the group delay response of the circuit. Digital APFs, which are used in digital signal processing (DSP) [42] and in many subsequent publications cited in this work, have gained widespread adoption in many applications. Various types of DSP filters have been developed and are widely used. Although DSP filters have taken an essential place in signal processing technologies, DSP suffers from some significant drawbacks, such as costly analog-to-digital conversion and digital-to-analog conversion, high power consumption, and inadequate performance at high frequencies.

As an alternative to DSP in the microwave range, microwave analog signal processing (ASP) technology has been proposed [43]. ASP devices are based on phaser devices, which ideally have an arbitrary group delay with a flat and lossless magnitude. Phaser synthesis aims to realize a response as close to ideal as possible for various applications. The practical implementation of the phaser is not easy and requires controllable dispersion for efficient group delay engineering. In ASP applications, various analog filters are widely used for signal spectrum transformation, which includes dispersive delay structures with controlled magnitude for analog signals. In [44], a fully differential second-order voltage-mode active APF topology was proposed. The filter consists of the generic two-transistor common-source differential pair, using all possible combinations of the surrounding impedances to obtain a second-order APF circuit. In [45], procedures are proposed for the design and synthesis of broadband filters with arbitrary group delay or phase response, allowing the implementation of the analog filters in integrated circuits. In [46], a closed-form synthesis method is proposed for the design of reflection-type phasers with arbitrary prescribed group delay response, and in [47], a systematic synthesis method is proposed for dispersive delay structures with controlled magnitudes for analog signal processing applications. In these two papers, a prototype waveguide filter was built and tested. In [48], a group-delayed, two-port, all-pass transmission line network application is presented, analytically modeled, and experimentally demonstrated. There are sections where all-pass network group delay synthesis examples and experimental results using genetic algorithm design are shown. In [49], all-pass network circuits have been synthesized and applied to on-chip technologies using a high-level architecture where the designed filter consists of eight second-order all-pass cells.

The circuits used in all APF implementations are complex, as they are primarily universal and offer many possibilities to manipulate pulses and apply them in many specific applications. For example, Ref. [50] demonstrates a completely on-chip implementation of the expansion and compression of continuous-time, wideband analog pulses. This work uses a cascade of a seventh-order APF coupled with a fifth-order LPF.

Another option for pulse shaping involves using filters that influence the pulse magnitude spectrum. Passive and active filters [51,52] can be utilized for this purpose. The simplest active filters are built around transistor and op-amp circuits as their fundamental components [53], although, in most cases, they tend to be more complex [54]. Active filters offer greater flexibility, allowing for reconfigurable bandwidths ranging from narrow/mid-band to ultra-wideband states, as described in [55]. Active filters can implement intricate filter functions such as low-pass, band-pass, and various filter types, and they can be designed with a reconfigurable magnitude response and phase-frequency dependence. However, active filters also come with some notable disadvantages. Due to the presence of active components, they tend to be more costly than passive filters. Additionally, they require an external power supply, which can introduce noise or interference. Furthermore, active filters have limited bandwidth since their performance is influenced by the characteristics and limitations of the active devices.

Much simpler designs than the use of APFs and conventional active filters can be successfully used for pulse expansion in many applications, as has been done in this research. As mentioned above, to realize the ideas of our work, i.e., ensure an unambiguous event timer response and the minimum possible jitter in time interval detection as well as the stable operation of the event timer and initial state recovery, it is necessary to ensure that the initial pulse is extended to about 150 ps, that the filtered pulse does not have noticeable ripples or a long tail, and that the rising edge of the filtered pulse is preferably as steep as possible.

LPFs perform the signal’s energy accumulation, leading to a situation where the signal at the output of the filter lasts longer than excitation. Therefore, when unipolar pulses are used in the PPM transmission channel, one of the most rational ways for pulse expansion is to utilize a LPF. As will be shown below, it is sufficient to design a filter whose attenuation in the passband (up to the cutoff frequency $f_{\mathrm{c}}$) is almost constant, whose stopband up to the frequency $3f_{\mathrm{c}}$ decreases by 10 dB, and whose group delay has minimal variations up to the frequency $2f_{\mathrm{c}}$.

Using the software presented in [56], several pulses with different waveforms have been modeled, and numerical calculations have been carried out for different types of filters with various cutoff frequencies and filter orders. Analytical derivations and numerical modeling in [56] have confirmed that, of all the filter types, the Bessel-type filter is the most suitable for pulse expansion, as it has a maximally flat group delay and a maximally linear phase response. Those characteristics ensure negligible ringing in the impulse response [57] and minimize overshoots in the pre-transition and post-transition aberration regions of the expanded PPM pulses. Before developing the custom pulse expander, our research group performed experiments with commercially available LPFs to determine if an off-the-shelf filter could be used for pulse expansion. An example of pulse expansion using the Mini-circuits^TM SLP-2400+ filter [58] is shown in Figure 3. As can be seen from the figure, the commercial filter introduced noticeable oscillations in the post-transition region. Moreover, it significantly flattened the pulse’s rising edge, increasing the timer’s sensitivity to the additive noise. Experiments with commercial filters showed that it would be difficult to find a suitable filter, so it was decided to design and build a custom filter. Microstrip technology was chosen because its fabrication is possible without the involvement of complicated and expensive technological processes.

3.1.1. Design of Custom LPFs

In this study, we designed two microstrip LPF filters to extend the pulse duration from 50 ps up to about 170 ps, which is slightly above the event timer’s minimum pulse duration threshold of 150 ps, to ensure that the timer will definitely respond. Analytical and numerical calculations, computer modeling, and measurements showed that the third-order Bessel LPF capacitor–inductor–capacitor filter, which is normalized by the group delay (hereafter called Classical), is the most suitable for the expansion task. First, the filter’s group delay was normalized, so that in the passband, it is $1/{\omega }_{\mathrm{c}}$, where $1/{\omega }_{\mathrm{c}}$ is the angular cutoff frequency, which was chosen to give the required pulse extension. The normalization of the second filter (hereafter called Magnitude-Normalized) was performed using the traditional magnitude normalization methodology, where the gain is −3 dB at the cutoff frequency).

To carry out simulations with Ansys HFSS software that calculate the scattering matrix elements and characteristics of a microstrip filter, it is necessary to specify at least the approximate physical lengths of the elements. The more accurately these lengths are calculated as initial values, the better and faster the result can be achieved in the Ansys HFSS simulations.

In this paper, a software package has been developed that consists of two modules, the first of which allows the effective permeability of the microstrip line used to be calculated. This software has been produced based on publications on the calculation of microstrip lines [59,60,61,62]. The second software module calculates the physical lengths of the selected microstrip filter’s capacitive and inductive elements, considering parasitic inductance and capacitance. Some important microstrip quasi-illuminated elements are discussed in [63].

Any electrically short (physical length much smaller than the wavelength) microstrip line discontinuity always has both capacitance and inductance. It can be replaced by one of two equivalent circuits: T-shaped or $\Pi$-shaped.

The inductance of the divider is proportional to the line’s characteristic impedance, and the capacitance is inversely proportional to it. Since the characteristic impedance increases with decreasing line width, inductive elements can be realized using narrow line sections. Of course, these elements also have capacitance, but this can be reduced by decreasing the line width. It is more convenient to describe an inductive line segment with an equivalent $\Pi$-shaped circuit, where the series element represents the inductance and the two shunt elements represent the parasitic capacitances (see Figure 4).

The inductance L and the two parasitic capacities $C^{\left(\mathrm{p}\right)}$ are calculated as

$$\begin{array}{cccc}\hfill L& =\frac{Z_{\mathrm{c},L}}{\omega }sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g},L}}{l}_{\mathrm{L}}\right),\hfill & \hfill C^{\left(\mathrm{p}\right)}& =\frac{1}{\omega Z_{\mathrm{c},L}}tan\left(\frac{\pi }{{\lambda }_{\mathrm{g},L}}{l}_{\mathrm{L}}\right),\hfill \end{array}$$

(1)

where $Z_0$ is line impedance (source and load impedances), $Z_{\mathrm{c},L}$ is inductance line’s characteristic impedance, ${\lambda }_{\mathrm{g},L}$ is wavelength of the inductance line, and $l_{\mathrm{L}}$ is the inductance line element’s physical length.

Capacitive elements are realized using wide microstrip line segments with very low characteristic impedance. It is more convenient to replace such an inductance with an equivalent T-shaped circuit, where the series elements represent the parasitic inductances and the shunt element represents the capacitance (see Figure 5).

The capacity of the element C and two parasitic inductivities is calculated as

$$\begin{array}{cccc}\hfill C& =\frac{1}{Z_{\mathrm{c},C}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g},C}}{l}_{\mathrm{C}}\right),\hfill & \hfill L^{\left(\mathrm{p}\right)}& =Z_{\mathrm{c},C}tan\left(\frac{\pi }{{\lambda }_{\mathrm{g},C}}{l}_{\mathrm{C}}\right),\hfill \end{array}$$

(2)

where $Z_{\mathrm{c},C}$ is capacitance line characteristic impedance, ${\lambda }_{\mathrm{g},C}$ is wavelength in capacitance line, and $l_{\mathrm{C}}$ is the capacitance line element’s physical length.

The calculation of the third-order Bessel filter is described below. The capacitor–inductor–capacitor topology was chosen because the inductor–capacitor–inductor topology requires inductive elements with a very high characteristic impedance, consequently, very narrow line segments with a width less than 0.4 mm, and successively, increased requirements for the tolerance. Such elements were difficult to fabricate using the available milling machine. The filters’ prototype values were selected by the methodology in [36,64], and the lumped values of the filters were calculated using the element transformations described in [36], chapter 3.

First, the element lengths are calculated, ignoring parasitic inductances and capacitances:

$$\begin{array}{cc}\hfill {\omega }_{\mathrm{c}}{C}_1& =\frac{1}{Z_{\mathrm{c}1}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}1}}{l}_1\right)\Rightarrow l_1=\frac{{\lambda }_{\mathrm{g}1}}{2\pi }{sin}^{-1}\left({\omega }_{\mathrm{c}}{Z}_{\mathrm{c}1}{C}_1\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\omega }_{\mathrm{c}}{L}_2& =\frac{1}{Z_{\mathrm{c}2}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{l}_2\right)\Rightarrow l_2=\frac{{\lambda }_{\mathrm{g}2}}{2\pi }{sin}^{-1}\left({\omega }_{\mathrm{c}}{Z}_{\mathrm{c}2}{L}_2\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\omega }_{\mathrm{c}}{C}_3& =\frac{1}{Z_{\mathrm{c}3}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}3}}{l}_3\right)\Rightarrow l_3=\frac{{\lambda }_{\mathrm{g}3}}{2\pi }{sin}^{-1}\left({\omega }_{\mathrm{c}}{Z}_{\mathrm{c}3}{C}_3\right),\hfill \end{array}$$

(5)

where guided wavelengths in striplines are:

$$\begin{array}{cccccc}\hfill {\lambda }_{\mathrm{g}1}& =\frac{c}{f_{\mathrm{c}}\sqrt{{\epsilon }_{\mathrm{eff},1}}},\hfill & \hfill {\lambda }_{\mathrm{g}2}& =\frac{c}{f_{\mathrm{c}}\sqrt{{\epsilon }_{\mathrm{eff},2}}},\hfill & \hfill {\lambda }_{\mathrm{g}3}& =\frac{c}{f_{\mathrm{c}}\sqrt{{\epsilon }_{\mathrm{eff},3}}},\hfill \end{array}$$

(6)

where c is the light speed in free space, $f_{\mathrm{c}}={\omega }_{\mathrm{c}}/\left(2\pi \right)$ is the cutoff frequency, and ${\epsilon }_{\mathrm{eff}}$ is the relative effective dielectric constant, which depends on dimensions of the lines and the parameters of the substrate. The relative effective permittivity ${\epsilon }_{\mathrm{eff}}$ and impedances of the lines $Z_0$ or $Z_{\mathrm{c}}$ are calculated using the calculation expressions given by [59,60,61,62].

Using the initial values of the element lengths $l_1$, $l_2$, $l_3$ calculated according to the Formulas (3)–(5), the parasitic inductances and capacities of the elements can be calculated:

$$\begin{array}{cccccc}\hfill L_1^{\left(\mathrm{p}\right)}& =\frac{Z_{\mathrm{c}1}}{{\omega }_{\mathrm{c}}}tan\left(\frac{\pi }{{\lambda }_{\mathrm{g}1}}{l}_1\right),\hfill & \hfill C_2^{\left(\mathrm{p}\right)}& =\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}2}}tan\left(\frac{\pi }{{\lambda }_{\mathrm{g}2}}{l}_2\right),\hfill & \hfill L_3^{\left(\mathrm{p}\right)}& =\frac{Z_{\mathrm{c}3}}{{\omega }_{\mathrm{c}}}tan\left(\frac{\pi }{{\lambda }_{\mathrm{g}3}}{l}_3\right).\hfill \end{array}$$

(7)

Although elements $C_2^{\left(\mathrm{p}\right)}$ and $C_1$ are not connected in parallel, the shunt inductance $L_1^{\left(\mathrm{p}\right)}$ is so small that they can be assumed to be connected in parallel and can therefore be combined. The same applies to elements $L_1^{\left(\mathrm{p}\right)}$ and $L_2$: they can be assumed to be connected in series.

In the next step, the lengths of the elements are calculated according to the parasitic parameters:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_1& =\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}1}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}1}}{\widehat{l}}_1\right)+C_2^{\left(\mathrm{p}\right)}=\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}1}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}1}}{\widehat{l}}_1\right)+\frac{Z_{\mathrm{c}2}}{{\omega }_{\mathrm{c}}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{l}_2\right)\Rightarrow \hfill \\ \hfill {\widehat{l}}_1& =\frac{{\lambda }_{\mathrm{g}1}}{2\pi }{sin}^{-1}\left({\omega }_{\mathrm{c}}{Z}_{\mathrm{c}1}\left[C_1-\frac{Z_{\mathrm{c}2}}{{\omega }_{\mathrm{c}}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{l}_2\right)\right]\right),\hfill \end{array}\end{array}$$

(8)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L_2& =\frac{Z_{\mathrm{c}2}}{{\omega }_{\mathrm{c}}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{\widehat{l}}_2\right)+L_1^{\left(\mathrm{p}\right)}+L_3^{\left(\mathrm{p}\right)}\hfill \\ & =\frac{Z_{\mathrm{c}2}}{{\omega }_{\mathrm{c}}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{\widehat{l}}_2\right)+\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}1}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}1}}{l}_1\right)+\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}3}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}3}}{l}_3\right)\Rightarrow \hfill \\ \hfill {\widehat{l}}_2& =\frac{{\lambda }_{\mathrm{g}2}}{2\pi }{sin}^{-1}\left({\omega }_{\mathrm{c}}{Z}_{\mathrm{c}2}\left[L_2-\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}1}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}1}}{l}_1\right)-\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}3}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}3}}{l}_3\right)\right]\right),\hfill \end{array}\end{array}$$

(9)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_3& =\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}3}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}3}}{\widehat{l}}_3\right)+C_2^{\left(\mathrm{p}\right)}=\frac{1}{{\omega }_{\mathrm{c}}{Z}_{\mathrm{c}3}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}3}}{\widehat{l}}_3\right)+\frac{Z_{\mathrm{c}2}}{{\omega }_{\mathrm{c}}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{l}_2\right)\Rightarrow \hfill \\ \hfill {\widehat{l}}_3& =\frac{{\lambda }_{\mathrm{g}3}}{2\pi }{sin}^{-1}\left({\omega }_{\mathrm{c}}{Z}_{\mathrm{c}3}\left[C_1-\frac{Z_{\mathrm{c}2}}{{\omega }_{\mathrm{c}}}sin\left(\frac{2\pi }{{\lambda }_{\mathrm{g}2}}{l}_2\right)\right]\right).\hfill \end{array}\end{array}$$

(10)

The last step of the procedure can be repeated to obtain even more accurate lengths. However, the results obtained this way are usually very little different from the previous ones. Therefore, it is not always practical to recalculate the length values.

The next and final step in calculating the physical dimensions of the LPF for pulse duration expansion was modeling using the Ansys HFSS software. The widths of the lines are selected beforehand, considering that the width of the inductive element is smaller than that of the capacitive element.

The simulations were carried out with respect to the physical sizes of the elements until the desired result was obtained (quasi-parametric modulation). It should be noted, however, that the relatively precise calculations carried out beforehand for the initial lengths of the elements are an essential condition for obtaining results in Ansys HFSS fast enough, as even with a good approximation of initial sizes, the optimization can take a few hours.

The calculation of the numerical values of elements and physical sizes of two microstrip filters, Classical (group delay normalized) and Magnitude-Normalized, was performed. The cutoff frequency $f_{\mathrm{c}}=1.8$ GHz was set to the same value for both filters, and the source and load impedances were requested at $Z=50$ $\Omega$. The results obtained after optimization and modeling by Ansys HFSS are shown in Figure 6.

The simulation result curves in Figure 6 show that the attenuation for the Classical filter and the Magnitude-Normalized filter differ minimally in the passband. Still, immediately after the passband and up to a frequency of 6 GHz, the attenuation for the Magnitude-Normalized filter is significantly higher, which means that the Classical filter cuts out fewer incoming pulse spectrum components in this band. Above 6 GHz, the attenuation of the Magnitude-Normalized filter decreases rapidly and fluctuates more, while for the Classical filter, the attenuation decreases and fluctuation is noticeable above the frequency of 9 GHz. The group delay curve for the Classical filter is significantly flatter than for the Magnitude-Normalized filter in the spectrum range of 0–6 GHz. As will be seen below, these differences are significant and determine the time duration and waveform of the filtered pulse.

Based on the results obtained by modeling with Ansys HFSS, two microstrip LPF were fabricated: Magnitude-Normalized and Classical third-order Bessel filters (see Figure 7). FR4-type printed circuit board laminate has been used for the substrate of the filters.

Simulated and measured curves of the transmission coefficient magnitude $\left|S_{21}\right|$ and group delay for both filters are shown, respectively, in Figure 8 and Figure 9. For the Classical filter, the simulation and the measured results agree well for the magnitude curves (see Figure 8). In contrast, for the Magnitude-Normalized filter, differences are visible, which could be explained by the fact that the actual permittivity of the FR-4 material did not correspond to the one used in the simulation (the manufacturer specifies the relative permittivity of FR-4 in the range from 3.8 to 5). The group delay curves obtained in the simulation and the experimental ones (see Figure 9) are in good agreement if the slopes of the curves are compared. The differences in group delay seen in the figures are almost constant across the whole frequency range, and they can be caused by different reference planes used in the simulation and experiments, as the actual board included the SMA connectors but the simulations did not.

3.1.2. Experimental Electrical Pulse Expansion

Figure 10 shows the block diagram of the LPF-based electrical pulse expansion test setup. Both are third-order Bessel filters with equal cutoff frequencies of 1.8 GHz. The results of a 50 ps input pulse expansion by the Classical LPF filter can be seen in Figure 11, whereas those of the Magnitude-Normalized LPF filter can be seen in Figure 12. It can be seen that the pulse expanded by the Classical filter to 146 ps is close to the desired pulse waveform as it meets the waveform requirements set out in this paper. The Magnitude-Normalized filter expands the pulse to 237 ps, and this pulse waveform has a first transition duration that is too large, a long tail, and ripples in the tail. Simulations with this type of filter showed that, by increasing the cutoff frequency, the duration of this pulse can be obtained at close to 150 ps. However, its waveform first transition duration will still have worse indicators (it is longer) than the pulse extended with the Classical filter. The explanation for these differences can be seen in the curves in Figure 8 and Figure 9. The group delay of the Classical filter is less variable across the frequency range, and the transfer coefficient magnitude is flatter and significantly larger at frequencies above 3 GHz compared to the Magnitude-Normalized filter. Those features affect the pulse waveform, especially the rise time, which should be small enough to comply with the parameters set out in this paper.

3.2. Optical Pulse Expansion

PPM pulses can be expanded not only in the electrical domain but also in the optical domain when signal transmission over optical fiber is realized. To increase optical pulse duration, so-called optical pulse stretchers or techniques are used [65,66]. Stretching optical pulses can be helpful in scenarios where it is necessary to increase the duration of a pulse without altering its waveform. These are crucial elements, for example, in ultra-fast lasers, to expand the pulse in the time domain and reduce the pulse’s peak power, eliminating adverse nonlinear effects [67]. Also, optical pulse stretching is a commonly employed technique to safeguard the input faces of optical fibers from damage [68]. In general, pulse stretchers can be classified into two categories: coherent and incoherent. Coherent pulse stretchers, such as grating pulse stretchers, stretch a pulse in a way that can later be undone. They provide positive or negative group delay dispersion and typically are based on dispersive elements like diffraction grating pairs, grisms (grating and prism), FBG, chirped Bragg gratings, or ring resonators. On the other hand, incoherent pulse stretchers irreversibly stretch a pulse [65,69]. For example, an incoherent pulse stretcher can be realized by circulating cavities, which divide an incident pulse into many smaller pulses spread over a time period [66]. In our application, we will apply tunable FBG to introduce positive dispersion and broaden the input optical PPM pulses.

Experimental Optical Pulse Expansion

Our target in this experiment was to expand 50 ps optical PPM pulses up to at least 150 ps. As mentioned in Section 2, this is necessary as the duration of the incoming PPM pulse on the input of the event timer should be at least 150 ps to provide stable operation. This experiment employed a tunable FBG dispersion compensation module (Teraxion Clear Spectrum TDCMX) for optical pulse expansion or stretching. The experiment setup consisted mainly of the transmitter and receiver sides.

The transmitter side included AWG Keysight M8195A, radio frequency (RF) amplifier SHF 810 with 29 dB gain, Photline 40 GHz Mach–Zehnder modulator (MZM) intensity modulator, Cobrite DX4 continuous wave (CW) C-band laser tuned at 193.1 THz central frequency (1552.52 nm) and $+3$ dBm out power, and power supplies used for the operation of the RF amplifier and setting the bias of MZM. The MZM is biased at 1.2 V to obtain the maximum amplitude and quality of the generated PPM pulses. The receiver part contained a linear variable optical attenuator Keysight FVA-3150 with 2 dB insertion loss at 1550 nm, optical power meter, Amonics PR10G 10 Gbit/s ac-coupled photo-receiver p-i-n photodiode (PIN) with a typical sensitivity of −20 dBm and maximum ac output voltage swing of around 350 mVpp, and Keysight DSAZ334A DSO (80 GSa/s, 33 GHz) with connected DC blocker. As one can see in Figure 13, tunable FBG was placed right between the transmitter and receiver sides of the experimental scheme. As this section exclusively focuses on the optical pulse expansion, more detailed parameters of the experimental setup are provided in Section 4 of the article.

Figure 14 shows the waveforms and spectra of electrical PPM pulses on the output of PIN when the amount of added chromatic dispersion (CD) by tunable FBG module is 0 ps/nm or $+1200$ ps/nm.

The PIN used in our experiment is typical for 10 Gbit/s intensity-modulated on–off keying signals with a high-frequency cutoff of around 8 GHz. The RF bandwidth of this PIN is lower than the RF bandwidth occupied by the original PPM signal. This limited PIN bandwidth also increased the PPM pulse duration from originally 50 ps (which occupies up to ≈20 GHz RF bandwidth) to around 76.76 ps (which occupies up to ≈13 GHz RF bandwidth).

CD, the value at $+1200$ ps/nm, was equal to around the dispersion value introduced by 75 km of standard ITU T G.652 single mode optical fiber. As a result, FBG set to this value of CD expanded the PPM pulses up to 154.03 ps. From the spectra shown in Figure 14, it can be concluded that FBG behaved as a slight band-rejection filter with a center frequency around 6 GHz.

We have shown the optical pulse expansion capacity of the tunable FBG module. The following section will provide deeper insight into the realized experimental setup and validate the use of the proposed pulse expanders in optical UWB communication systems.

4. Experimental Validation Using PPM Data Transmission

4.1. Experimental Setup

The TR-PPM transmission experiments have been carried out using optical and electrical pulse expansion methods. PPM pulses were kept as short as possible to demonstrate energy-efficient transmission. As stated in Section 2, if the event timer is used for the detection, the received pulse must be sufficiently long to ensure proper timer triggering.

The block diagram of the proposed and investigated PPM transmission system is illustrated in Figure 15. Key components of the system are a pulse expander module (optical FBG or electrical LPF) and a high-precision event timer used for the demodulation. AWG acts as a PPM transmitter, generating the desired signal waveform following our loaded sampled data. This study used unipolar PPM pulses with a time duration of 50 ps. The difference between the setup in Figure 15 and that previously shown in Figure 13 is that (a) the pulse expansion was also realized by our custom-designed LPF, and (b) both pulse expansion methods, namely, optical (using a FBG) and electrical (using a LPF), are assessed by the connected event timer device. When pulse expansion was realized by the LPF, the CD value of FBG was set to 0 ps/nm.

First, the ideal 50 ps pulses with a rectangular shape and intervals between pulses determined by the random data were generated by MATLAB. Then, the waveform was saved into an AWG-specific file format and loaded into its memory. Next, the prepared waveform was played on AWG’s output port at a maximal sampling speed of 65 GSa/s. The AWG output signal was amplified by a 38 GHz SHF 810 RF amplifier to drive the 40 Gbit/s MZM intensity modulator. The optical MZM output was connected to the Teraxion FBG module. The output of this module was connected to a variable optical attenuator and through a 10%/90% optical coupler to a 10 Gbit/s photoreceiver, where the optical signal was converted to electrical. The 10% port of the optical coupler was connected to the optical power meter. The PPM pulse duration on the output of the PIN was measured to be about 76.76 ps due to the limited bandwidth of the PIN, which was lower than the bandwidth of the original signal generated by AWG.

The next step was to test our custom-made electrical LPFs’ pulse expansion capability in this experimental setup. To do so, the microstrip LPF with Classical design (see Section 3.1) was connected to the PIN output. A DSO was connected to the output of the LPF to capture the broadened PPM pulse. As one can see in Figure 16, the PPM pulse at the output of LPF expanded up to 169.24 ps, which is similar to the pulse expansion result shown in Figure 11.

After validating both pulse expansion techniques, the next step was verifying the novel expansion filters in the PPM communication system. The bias tee (BiasT, 20–45 GHz, SHF BT45) circuit was connected before the event timer to adjust the PPM signal’s DC level so that the half-magnitude of the incoming pulses was approximately at the timer-triggering threshold of 1.2 V.

4.2. Results and Discussion

The experiment has been conducted using TR-PPM waveforms [70] with the parameters listed in

Table 1. Experiment parameters.

Number of Positions, N	Position Width $\mathbf{\Delta }$, ps	Pulse Width $\mathbf{\tau }$, ps	Expansion Method
512	50	50	Optical
256	100	50	Optical
128	200	50	Optical
512	50	50	Electrical
256	100	50	Electrical
128	200	50	Electrical

by employing optical or electrical pulse expansion methods. In both cases, bit error ratio (BER) measurements were respective to the average optical power received before the PIN, as shown in Figure 15. The BER versus the average received optical power with optical pulse expansion for multiple position width values can be seen in Figure 17, whereas in the case of electrical pulse expansion, the results can be seen in Figure 18. The range of the average received optical power values was selected to represent the region around which the data transmission transitions from error-free to completely unintelligible (BER value of 0.5). This range was experimentally determined for the given system and corresponds to the amplitude values of the electrical pulse on the edge of the detection threshold of the event timer.

The forward error correction (FEC) limit shown in Figure 17 and Figure 18 represents the BER threshold, at which the Reed–Solomon RS(255, 223) FEC algorithm is capable of correcting the errors in the received data pattern. This BER value is $6.27\times {10}^{-2}$ as the RS(255, 223) algorithm can correct up to 16 erroneous symbols in the block. It was decided to represent this FEC limit, as the Reed–Solomon FEC algorithm is widely used.

For the experiments conducted using a position width of 200 ps, the results reveal a distinct point where the data transmission becomes error-free. This happens when the incoming PPM signal at the input of the event timer significantly exceeds the threshold set by the DC bias voltage so that no adverse effects, such as noise and jitter, significantly impact the BER. In contrast, if the magnitude of the incoming pulses, which corresponds to the certain average received optical power, falls below the voltage threshold of the event timer, very low BER values are observed in all cases. The threshold effect is observable at a position width of 200 ps. In contrast, in the case of position widths of 50 and 100 ps, the experiments show an error floor, especially in the case of optical pulse expansion. This is expected behavior, as smaller position widths require more precise modulation in the transmitter and demodulation in the receiver. Adverse effects such as jitter and additive noise in all system components considerably impact the PPM signal with smaller position widths. As the position widths of PPM pulses decrease, the jitter can more likely cause the estimated pulse position in the receiver to shift to a different pulse position than the one generated by the modulator. The additive noise in the signal waveform can also cause errors by falsely triggering the event timer when the time corresponds to a different pulse position. The effect of the noise is especially pronounced when considering position widths smaller than the pulse width, as this provides a large time window (the entire pulse) where noise may falsely trigger the timer.

The authors have recognized that an automatic gain control system or a similar circuit could eliminate the effect of DC biasing in the experiment results. Ideally, the electrical pulse amplitude at the input of the event timer should be constant across a wide range of average received optical power values.

Nevertheless, the results validate the concept of both electrical and optical pulse expansion methods for use with the event timer, as the BER values even at very small position widths of 50–100 ps are acceptable. The lower BER values in the case of electrical pulse expansion at a similar average received optical power can be attributed to the excellent pulse shape at the output of the custom LPF, which leads to higher accuracy in the triggering of the event timer, resulting in less error-prone transmission. Employment of new pulse expansion techniques in conjunction with more capable TDC allowed us to achieve much lower BER compared to the results presented in [71].

5. Conclusions and Future Work

This research explores the employment of passive electrical or optical filters for picosecond-range pulse expansion to support demodulation in an event timer-based PPM communication system. The quality and characteristics of the proposed designs are examined through the acquisition of time-domain waveforms and the calculation of spectra. The suitability of the designed solutions for PPM signal processing is validated through data transmission tests in laboratory conditions.

The given research contributes to a detailed analysis of the requirements for distortion-free pulse expansion in the time domain, the design of microstrip filter prototypes according to requirements, and experimental verification of the proposed designs.

For expansion in the optical domain, an off-the-shelf FBG-based pulse stretcher is examined, whereas for electrical expansion, two custom-designed microstrip LPFs are proposed. The results have shown that both FBG-based optical and custom-designed LPF-based designs are capable of expanding the pulses to at least 150 ps and comply with the requirements of the Eventech Stream Time Tagger used as the TDC in the PPM demodulator.

The designed pulse expanders can satisfy requirements for at most 50 ps rising edge and oscillations after the main peak below 10% of pulse amplitude. The authors conclude that applied passive electrical and optical expansion are viable and cost-efficient methods to preprocess signals in UWB communication and sensing systems that employ event timer-based TDC.

The employment of advanced filters for the development of pulse expanders that preserve the steepness of the output pulse’s rising edge is one of the most prospective future development directions. In this case, the filter’s magnitude response needs to have several peaks, and such devices are no longer low-pass filters.

The expansion of pulses with a duration of 10 ps or less has great potential as shorter pulses save energy in PPM transmissions. This will become an important problem after the widespread adoption of terahertz technologies in transmission channels. Implementing such expanders in the electrical domain would require the employment of more advanced techniques to suppress spurious modes and cope with fabrication tolerances.