Filtering and Detection of Ultra-Wideband Chaotic Radio Pulses with a Matched Frequency-Selective Circuit

Abstract

An approach is proposed to the filtering of an additive mixture of ultra-wideband chaotic signals and white Gaussian noise, in order to retrieve the useful signal in the receiver. The role of the filter is performed by a passive frequency-selective circuit, identical to the one involved in the formation of oscillations in the chaos generator. A mathematical model of a modulating chaos generator, detecting and receiving a sequence of ultra-wideband chaotic radio pulses in a noisy channel is designed. For the receiver of sequences of symbols encoded by chaotic radio pulses with 2- and 4-position modulation, the bit error ratio as a function of the noise level and the pulse duration is estimated numerically.

1. Introduction

Since the end of the 20th century, the use of ultra-wideband (UWB) signals [1,2] in civil wireless systems has been the focus of attention of the scientific community. Today, many UWB technologies have already been put into practice. In particular, implementation of UWB solutions intended for mass use has allegedly begun with the introduction of the corresponding spectral mask by the US FCC [3] and subsequent development of a number of international standards for UWB wireless communications, such as IEEE 802.15.3a [4,5] (not adopted in the end), IEEE 802.15.4a [6], IEEE 802.15.6 [7], and IEEE 802.15.4z [8,9,10]. A number of large manufacturers of consumer electronics have already included UWB solutions in their devices [11,12]. In addition to wireless communication, a promising field of UWB signal application is localization and positioning, where UWB microwave systems demonstrate high accuracy [13,14].

A special place among UWB wireless communication systems is hold by the communication systems based on chaotic oscillations. A review of such systems is given in [15,16].

One of the approaches to organizing data communication using chaotic signals involves microwave chaotic radio pulses as a carrier. The systems based on microwave chaotic radio pulses were called direct chaotic communications (DCC) [17,18]. They are used in personal- and local-area wireless communication networks [19,20,21,22].

From the very beginning, it was clear that the development of methods for modulating and receiving UWB signals would face three classic problems of microwave wireless communications: noise in the signal frequency band, multipath propagation, and narrowband interference. Each of these problems represents a separate research direction.

One of the main approaches to solving the problem of increasing the signal-to-noise ratio (SNR) in the channel with additive white Gaussian noise (AWGN) is the use of various signal filtering methods in the receiver.

From a practical point of view, when receiving microwave UWB chaotic signals, it would be good to have a filtering method capable of extracting a useful signal both after the AWGN channel and after the channels with multipath propagation, which play a significant role in wireless UWB communications [23].

For chaotic signals, the use of matched filtering (or coherent reception methods) is very difficult, since it is fundamentally impossible to reproduce the signal waveform in the transmitter and the receiver without the use of special synchronization methods or methods of digital generation/processing of chaotic signals.

Therefore, in the works devoted to the problem of filtering chaotic signals, this problem is posed mostly in the context of applying various nonlinear filtering methods to solve the problem of chaotic system synchronization.

Much attention is attracted by approaches based on various modifications of the Kalman filter. Thus, in [24,25,26], the use of the extended Kalman filter was studied to achieve synchronization of chaotic oscillations in the transmitter and receiver. Lorenz system [24], Ikeda map [25], tent map, Markov map, and logistic map [26] were used as model systems.

For example, to synchronize chaotic systems or maps, nonlinear filtering algorithms based on a particle filter and an unscented Kalman filter were proposed in [27]. A model of a nonlinear predictive filter was studied in [28]. The simulation was carried out on the example of synchronization of the Lorentz or Mackey–Glass systems, as well as the Ikeda map.

In [29], a neural filter was proposed to provide chaotic synchronization; the model system was the Ikeda map.

A comparative study of several approaches to denoising and filtering was given in [30], where a fourth-order Butterworth filter, a median filter, a wavelet-based denoising method, and a method based on empirical mode decomposition were considered, and a new filtering algorithm based on the cascade of driven chaotic oscillators was proposed. These methods were simulated on the Roessler oscillator as a source of chaos [30].

In [31], a linear filtering method was proposed and analyzed, which essentially exploits the properties of a dynamic system with chaotic oscillations. The method is based on removing the noise that lies outside the subspace in which the phase trajectory of the dynamical system is concentrated; this procedure increases the signal-to-noise ratio in the receiver of chaotic signals.

In addition to the problem of reproducibility of the shape of chaotic signals on the transmitting and receiving sides, we note another aspect: a short autocorrelation time of noise-like UWB signals, including dynamic chaos. This leads to incoherent summation of beams at the receiving point, arriving with different delays due to multipath propagation, which is a typical situation in real microwave communication systems.

A negative effect of incoherent summation is that the signal at the receiving point acts as an interference on itself, which raises the question of developing various methods for equalizing the channel response. Actually, equalization is also a kind of digital filtering [32,33,34,35,36,37,38,39,40,41].

At the same time, incoherent summation can also have a positive effect due to the almost complete absence of fading, which makes the performance of modulation methods in a multipath channel identical to the AWGN channel, since the power of the pulse is not destroyed by interference, but is accumulated (multipath amplification [19,42]). This phenomenon can also be exploited with the use of RAKE receivers, which are also a type of digital filter.

Summarizing this brief overview of the state of the art in the field of filtering chaotic oscillations, we can conclude that there is a certain lack of the methods of filtering chaotic signals that could be practically applicable in microwave devices for wireless UWB communication.

This work attempts to fill this gap.

Practical applicability here means avoiding digital signal processing methods and avoiding the necessity to know the chaotic dynamic system. This is important, because the dimension of real technical devices that generate microwave chaotic oscillations is practically and theoretically large, and the exact equations of such a dynamic system are unknown. This, for example, limits the applicability of methods that require a priori knowledge of the dynamic system (a Kalman filter). An a priori unknown channel response is also a serious restriction when trying to compensate for multipath propagation by means of designing filters inverse to the channel filter.

Taking into account the above numerous fundamental and technical aspects, we propose here a method of linear filtering. In the context of this work, the linear filtering can be considered as a compromise that allows us to increase the signal-to-noise ratio at the receiving point, regardless of the number of UWB chaotic signal paths that come to the receiver as well as of the knowledge of chaos generator equations.

With such a problem statement, we can use commercially available general-purpose UWB filters designed for microwave devices of various applications. However, often, the frequency response of such filters does not correspond to the frequency properties of UWB signals generated by chaotic generators. Moreover, the task of developing miniature UWB filters for the required frequency range is in itself a challenge in the context of mastering new microwave bands [43,44,45].

The aim of this work is to propose and investigate a linear filtering method that, first, would have selective properties with respect to “its own” chaotic signal, and second, could be practically implemented without digital signal processing in an ultra-wide microwave frequency band. The problem is posed for the AWGN channel, since this is the basic model also for the multipath channel. The statement of the problem is based on the filtering properties of an FSC involved in the forming of the chaotic signal.

The novelty of the paper is a new approach to the filtering of chaotic signals. We propose not just any band-pass filter (BPF), but a filter whose inner structure is intrinsically related with a concrete chaotic signal, because this filter is actually a frequency-selective system of a generator that produces this chaotic signal, and the frequency response of such a filter complements the frequency response of the chaotic signal. This is not used in the traditional scheme for receiving regular signals, in which the signals are filtered at the input of the receiver, regardless of the way they are formed. The efficiency of this approach was verified in a pulse-position modulation (PPM) scheme.

The paper is organized as follows. First, a filtering scheme with its schematic and a mathematical model of a chaos generator and a filter are described. Then, a numerical simulation scheme is presented that allows us to evaluate the performance of the proposed filtering scheme in terms of the error ratio per bit as a function of the noise level $E_B/N_0$ at the filter input. Finally, the simulation results are described and discussed.

2. Materials and Methods

2.1. Filtering Models

As is known [46], filtering noisy signals is an inseparable feature in the schemes of coherent reception (Figure 1a,b). By the coherent reception of modulated regular signals (Figure 1a,b), the unmodulated signal shape is assumed known, and the received signal is processed with a matched filter, and no additional operations aimed at increasing $SNR$ of the signal-and-noise mixture before demodulation are necessary. Theoretically, this type of receiver is suitable for any signal type, including chaotic (noise-like) signals, if it is possible to reproduce an exact copy of a chaotic or noise-like signal in the receiver.

With incoherent reception of narrowband signals, if the signals can be represented by a combination of two orthogonal functions (e.g., sine and cosine), an algorithm of signal reception with an unknown phase, but a known shape, is used (Figure 1c). Preliminary filtering of the noisy channel for signal detection is also not necessary in this case.

For incoherent energy reception (Figure 1d), if the signal shape is unknown and if only its frequency-time characteristics (frequency range and its envelope parameters, such as pulse duration, modulation method) are known, then the filtering within the signal band can increase $SNR$ at the input of the energy detector.

To date, energy reception is one of the few practical methods for detecting chaotic signals [19,20,21,22,47].

The filtering scheme, proposed below, corresponds to the case of incoherent energy reception shown in Figure 1d.

2.2. Block Diagram

Below, a method for filtering a UWB microwave chaotic signal is proposed and investigated, which uses the filtering properties of a linear FSC that forms the feedback loop of the chaos generator and participates in shaping the frequency spectrum of the generator signal.

The basic idea of this method is realized then, i.e., selective extraction of the own signal against the noisy background without knowing the exact form of chaotic oscillations.

In this paper, the problem of filtering a UWB chaotic signal is considered in the context of a scheme that simulates modulation of such a signal by a stream of discrete symbols, its transmission through a noisy channel, filtering, restoring the symbol stream, and finally comparing the transmitted stream with the received one (Figure 2).

The scheme includes a pulse-position modulator (PPM) that implements position modulation, a UWB chaotic signal source, an AWGN channel, a band-pass filter, a demodulator, and a comparison operator to compare the received symbol stream with the symbol stream at the input of the modulator.

The band-pass filter is based on a frequency-selective system that forms the feedback loop of the UWB chaotic generator. The frequency response of such a filter is identical to that of the frequency-selective system that forms the chaotic spectrum of the signal.

According to the pulse-position modulation method, the symbols are encoded by chaotic radio pulses (the 4-PPM scheme is shown in Figure 2), i.e., fragments of a chaotic signal at prescribed time positions. The energy detector extracts the pulse envelope. Therefore, it is not necessary to know the carrier signal waveform in order to receive and demodulate it. The transmitter and the receiver are assumed synchronized at the symbol level, i.e., the receiver knows the starting moments of the symbol positions. This is a reasonable theoretical assumption necessary to assess the impact of the proposed filtering method on the BER.

As usual, this filter is necessary to eliminate out-of-band noise to increase the signal-to-noise ratio at the input of the energy detector that extracts the envelope of the radio pulses.

According to the block diagram in Figure 2a, the mixture of the signal and additive Gaussian noise is filtered by the proposed FSC bandpass filter, passes through a quadratic detector, and is integrated over a time interval corresponding to the duration of the chaotic radio pulse. As a result, at the output of the energy detector, the energy of a chaotic radio pulse located at a predetermined time position is estimated.

For clarity, typical waveforms at different points of the transmission path are shown in Figure 2b, i.e., video pulses modulating the chaos generator with a random bit sequence according to the 4-PPM scheme, a modulated chaotic signal, a mixture of chaotic radio pulses with additive Gaussian noise before and after the filter, and an envelope of chaotic radio pulses at the detector output.

This scheme allows us to estimate the filtering efficiency in terms of bit error ratio ($BER$) as a function of the channel signal-to-noise ratio $E_B/N_0$. As a result, we have a direct comparison of the efficiency of the proposed filtering scheme with the “ideal” Fourier filtering (Figure 1).

2.3. Model of the Chaotic Source and the Modulation Scheme

The shaper of UWB chaotic radio pulses is a non-autonomous generator of chaotic oscillations (Figure 3a). According to the method of [48,49,50], UWB chaotic radio pulses are formed by means of turning the power supply of the chaotic oscillation generator ($V_E$) on and off at specified time slots.

The power supply is turned on and off by an external video signal, i.e., a sequence of rectangular video pulses that encode the data bit stream. The operation mode of the generator is determined by the amplitude of video pulses $V_E$.

The generator model used here belongs to a family of models of single-transistor generators; theoretical and experimental approaches to the development of such generators have been developed over several decades. The starting point was the Colpitz oscillator, the chaotic modes of which were shown in [51].

To bring the basic idea to practical applications, a series of studies was carried out [52,53,54] in which sources of chaotic oscillations were created that are suitable for practical engineering applications and provide chaotic oscillations in a required frequency band with wide zones of chaotic modes in the parameter space, which ensure generation of chaotic oscillations under conditions of the spread of radio component ratings and power supply instability in real technical systems.

In these studies [52,53,54], the path from mathematical models to the implementation of generators with preset characteristics in various frequency ranges, both on discrete elements [52,53] and in the form of integrated circuits [54], was passed.

The key approach, which allowed us to solve this range of problems, was the synthesis of an FSC with the appropriate parameters in the generator feedback loop, due to which the spectral characteristics of the generated chaotic signal were formed.

Successful use of these generators in direct chaotic communications in various microwave bands gave the reason for applying the proposed approach to filtering a mixture of a chaotic signal and noise.

The system of nondimensional equations describing the dynamics of the oscillator (a source of chaotic oscillations, Figure 3a) is based on Kirchhoff’s laws:

$$\begin{array}{cc}\hfill \frac{R{C}_0}{\alpha }{\dot{v}}_0& =j_1-j_C\left(v_2\right)\hfill \\ \hfill \frac{R{C}_1}{\alpha }{\dot{v}}_1& =j_2-j_1\hfill \\ \hfill \frac{L_1}{\alpha R}{\dot{j}}_1& =v_1-v_0-\frac{R_1}{R}{j}_1\hfill \\ \hfill \frac{R{C}_2}{\alpha }{\dot{v}}_2& =\frac{R}{R_E}(V_E{s}_V\left(t\right)-v_2)-j_2-\frac{1}{\beta }{j}_C\left(v_2\right)-s_M\hfill \\ \hfill \frac{L_2}{\alpha R}{\dot{j}}_2& =v_2-v_1-\frac{R_2}{R}{j}_2\hfill \\ \hfill \frac{RC}{\alpha }{\dot{s}}_M& =\frac{RC}{\alpha }{\dot{v}}_2-s_M\hfill \end{array}$$

(1)

where $j_1$ is the current through the inductor $L_1$; $j_2$ is the current through the inductor $L_2$; $v_0$, $v_1$, $v_2$ are the voltages across the capacitors $C_0$, $C_1$, $C_2$, respectively; $s_V\left(t\right)$ is the signal that modulates the supply voltage of the generator that forms chaotic radio pulses; $s_M\left(t\right)$ is the generator signal fed into the channel.

Dimensional time t is related to nondimensional $t_d$ as $t=t_d\alpha$, where $\alpha =2\pi \sqrt{L_1{C}_1}$. Voltage $V=V_{T}v$; current $J=(V_T/R)j$. Current–voltage response of the transistor is $J_C\left(V_2\right)=I_0(exp(V_2/V_T)-1)$, where $I_0=0.22$ fA, $V_T=25.3$ mV, $\beta =425$, or, in nondimensional form, $j_C\left(v_2\right)=I_0/(V_T/R)(exp\left(v_2\right)-1)$.

The mode of the generator modulated by rectangular video pulses is determined by pulse amplitude $V_E$.

During the simulation, we used the value of the generator supply voltage $V_E=1.55$ V to form chaotic oscillations.

The model presented here demonstrates a wide range of oscillation mode characteristics [52,53,54]: from periodic to advanced chaotic modes, which is illustrated by the bifurcation diagram and the plot of the highest Lyapunov exponent as the function of the supply voltage (Figure 4a).

Examples of the dependence of the generated power spectral density on the frequency in regular and chaotic oscillation modes, as well as the amplitude-frequency response of the FSC in the generator feedback circuit are shown in Figure 4b.

The pulse formation mechanism is based on the transistor opening: the supply voltage is applied to the base-emitter junction, which opens the transistor, turns the positive feedback on, and starts oscillations [52,53,54].

In the numerical experiment, the sequence of video pulses that modulates the generator is formed by a random bit stream grouped into symbols; K bits per symbol. Position modulation was considered; the time interval $T_S=2^K{T}_P$ corresponding to the $S_i$ symbol was divided into $M=2^K$ slots ${\tau }_m$, $m=0\dots 2^K-1$, one of which contained a chaotic radio pulse. The situation was analyzed for $K=1$ and $K=2$.

In the case of position modulation (PM) (Figure 2), the chaotic radio pulse encodes one symbol; the interval $T_S$ of the symbol is divided into $M=2^K$ positions of duration $T_P$ each, $T_P$ is the duration of the chaotic radio pulse.

The position number at which the pulse is located encodes the character being transmitted.

For an external modulating signal $s_V\left(t\right)$, which is a sequence of video pulses (Figure 2b), the chaos generator produces a signal $s_M\left(t\right)$, which is a stream of chaotic radio pulses, each of duration $T_P$.

2.4. The Channel Model

We consider an AWGN channel, in which noise $n\left(t\right)$ is added to the sequence of chaotic radio pulses $s_M\left(t\right)$

$$s_C\left(t\right)=\sqrt{\frac{K}{E_P}}{s}_M\left(t\right)+{\sigma }_{N}n\left(t\right)$$

(2)

where $n\left(t\right)$ is the noise signal (sequence of delta-correlated samples with unit variance); the variance ${\sigma }_N^2=\frac{1}{2}{10}^{(-E_B/N_0)/10}$ is determined by the given value of the ratio of energy per bit to noise spectral density $E_B/N_0$ (in dB); K is the number of bits per symbol, the symbol being encoded by the chaotic radio pulse. Here, $E_P$ is the average energy of the chaotic signal at the pulse duration $T_P$:

$$E_P={\overline{\left(\frac{1}{T_P}{\int }_{t-T_P}^t{s}_M^2\left(\tau \right)d\tau \right)}}_{t\to \infty }$$

2.5. Model of Filtering

The main role of this model is to show that the use of the filtering properties of the FSC of the chaos generator at the receiver input allows us to retrieve the useful signal and to increase $SNR$.

Signal after the channel $s_C\left(t\right)$ comes to filter (3) (Figure 3b), and an estimate signal $s_F\left(t\right)$ is formed at the filter output.

The filter repeats the FSC in the feedback loop of the source of chaotic oscillations (Figure 3a). It is described by a system of differential equations:

$$\begin{array}{cc}\hfill \frac{R{C}_0}{\alpha }{\dot{u}}_0& =i_1+\frac{R}{R_S}(s_C\left(t\right)-u_0)\hfill \\ \hfill \frac{R{C}_1}{\alpha }{\dot{u}}_1& =i_2-i_1\hfill \\ \hfill \frac{L_1}{\alpha R}{\dot{i}}_1& =u_1-u_0-\frac{R_1}{R}{i}_1\hfill \\ \hfill \frac{R{C}_2}{\alpha }{\dot{u}}_2& =-i_2-s_F\hfill \\ \hfill \frac{L_2}{\alpha R}{\dot{i}}_2& =u_2-u_1-\frac{R_2}{R}{i}_2\hfill \\ \hfill \frac{RC}{\alpha }{\dot{s}}_F& =\frac{RC}{\alpha }{\dot{u}}_2-s_F\hfill \end{array}$$

(3)

Resistances R and $R_S$ at the output and input simulate 50-ohm loads.

The following set of parameters was used in the simulation for (1) and (3): $L_1=3.5$ nH, $L_2=3.5$ nH, $C_0=1.2$ pF; $C_1=1.2$ pF; $C_2=1.2$ pF; $R_E=200$ Ohm; $R_1=8$ Ohm; $R_2=8$ Ohm; $C=2.5$ pF, $R=50$ Ohm, $R_S=50$ Ohm, $\alpha =2\pi \sqrt{L_1{C}_1}$.

The frequency response of the filter corresponds to the shape shown in Figure 4b for the frequency response of the FSC of the generator.

2.6. Signal Representation in the Simulation Model

Chaotic signals, as one of the types of wideband signals, have a complex waveform that cannot be represented within the narrowband formalism as a product of the signal envelope by the carrier frequency. Therefore, there are two possible ways of modeling such signals: to use methods for wideband signal approximation based on the representation of the signals as a set of independent samples based on Kotelnikov–Nyquist–Shannon theorem [18]; or to represent a continuous waveform by discrete samples with the sampling rate greater than twice the upper limit of the signal bandwidth.

The first method is convenient from the viewpoint of minimizing the amount of calculations, since to represent a signal, a minimal set of samples is used, which is determined by the Kotelnikov–Nyquist–Shannon theorem. Unfortunately, this method is not appropriate if it is necessary not only to evaluate the statistical characteristics of the signals, but also to see the dynamics of the system or to find out how the properties of the dynamic system affect statistical estimates. For example, this is necessary in the case of modeling communication schemes based on chaotic synchronization, where the temporal dynamics of the system is essential.

The second method allows us to track the system dynamics over time, but requires more computing resources.

The signals in model (1), (2), (3) are presented with the sampling frequency $f_s=1/\Delta t$, where $\Delta t$ is the step of the dynamic system integration, which is several times higher than the upper bound of the chaotic signal frequency.

The system is integrated by the 4th-order Runge–Kutta method with a fixed step. As is known, the Runge–Kutta method requires calculation of the signal values in the intermediate nodes $(t_q+\Delta t/2)$, where $t_q=0,\Delta t,2\Delta t,..,M\Delta t$; M is a number of samples. When integrating non-autonomous systems (1) and (3), it is necessary to know the values of the signals $n\left(t\right)$ and $s_V\left(t\right)$ at the intermediate nodes $(t_q+\Delta t/2)$. These values were calculated as the arithmetic averages of the neighboring samples: $n(t_q+\Delta t/2)=(n\left(t_q\right)+n(t_q+\Delta t))/2$ and $s_V(t_q+\Delta t/2)=(s_V\left(t_q\right)+s_V(t_q+\Delta t))/2$, respectively.

2.7. Model of Detection and Reception of Chaotic Radio Pulses

The detection model is based on the energy detector of chaotic radio pulses.

After filtering (3), the signal $s_F\left(t\right)$ is fed to the envelope detector. At moments $t_k=k{T}_S$, which correspond to the end of the symbols, the signal energy estimates $E^{(k,m)}\left({\tau }_k^{\left(m\right)}\right)$ are formed on time intervals ${\tau }_k^{\left(m\right)}=[t_k-T_P-m{T}_P,t_k-m{T}_P]$; here, m is an integer (slot number within the symbol position of length $T_S=2^K{T}_P$), ranging from 0 to $2^K-1$.

$$E^{(k,m)}\left({\tau }_k^{\left(m\right)}\right)={\int }_{t_k-T_P-m{T}_P}^{t_k-m{T}_P}{s}_F^2\left(t\right)dt$$

(4)

The decision rule that m-th symbol $S_m^{\prime }$ is received is based on the search for the position number m corresponding to the maximum value of $E^{(k,m)}\left({\tau }_k^{\left(m\right)}\right)$:

$$S_m^{\prime }:E^{(k,m)}\left({\tau }_k^{\left(m\right)}\right)>E^{(k,j)}\left({\tau }_k^{\left(j\right)}\right))$$

(5)

for all $j=0..2^K-1$ except $j=m$.

2.8. Method to Evaluate the Filtering Efficiency

There are many ways to evaluate the effectiveness of filtering. For instance, by comparing $SNR$ before and after the filter. However, in the context of digital communications, this parameter does not indicate the filter quality directly. Therefore, it is reasonable to evaluate the effect of filtering by the rate of erroneously received symbols at different values of the channel noise.

In this paper, the efficiency of the proposed filtering method was evaluated for the position modulation scheme, which belongs to the class of orthogonal modulation/demodulation methods. For the position modulation scheme for narrowband signals, there are theoretical dependencies of $BER$ on $SNR$ [46], which can be used to compare and verify the results obtained in this numerical model.

The filtering efficiency was evaluated (Figure 2) by comparing bit error probabilities $P_B$, $P_B^F$, and $P_B^G$ for three different situations:

• No filtering, i.e., $s_F\left(t\right)=s_C\left(t\right)$;
• A filter matched to the signal band: $s_F\left(t\right)=G\left(s_C\left(t\right)\right)$, when filter G passes $99\%$ of the useful signal energy;
• A filter based on the FSC of the chaos source: $s_F\left(t\right)=F\left(s_C\left(t\right)\right)$.

In all three cases, $BER$ for a given value of $E_B/N_0$ (in dB) was evaluated as follows:

–

A fragment of the noise signal $n\left(t\right)$ was formed;

–

A fragment of the modulation signal $s_V\left(t\right)$ was formed from a random sequence of $N_S{2}^K$ bit: $b_i=(0,1,1,0,..,0)$, $i=1..N_S{2}^K$, with equal probability of zeros and ones; here, $N_S$ is the number of symbols;

–

Systems (1), (2), (3) for the signals $n\left(t\right)$, $s_V\left(t\right)$, and $s_M\left(t\right)$ were integrated numerically;

–

For each symbol, a set of pulse energy estimates 4 was formed $E^{(k,m)}\left({\tau }_k^{\left(j\right)}\right)$, $m=0..2^K-1$, $k=1..N_S$, among which the maximum value and the pulse position number for this value were determined, which was converted into the symbol number $S_m^{\prime }$;

–

By comparing the calculated sets of transmitted and received symbols, the number of symbol reception errors was determined, as well as $BER$ by bitwise comparison of the original sequence $b_i$ with the received sequence $b_i^{\prime }$, so that the pairs of values $P_B(E_B/N_0)$, $P_B^G(E_B/N_0)$, and $P_B^F(E_B/N_0)$ were formed.

Typical waveforms obtained during the simulation are shown in Figure 5. In Figure 5a, a waveform encoding symbols 01 at the channel input is given; Figure 5c shows the same signal after adding white Gaussian noise with $E_B/N_0=23$ dB; in Figure 5e, the mixture of the useful signal and noise after the filter is shown.

3. Results

First, we give results that have theoretical verification for the model in Figure 1c.

3.1. Bit Error Ratio for a Narrowband Signal

To verify the proposed numerical model and normalization (2), instead of the modulated chaotic signal $s_M\left(t\right)$, a modulated harmonic signal $s_M\left(t\right)=s_V\left(t\right)sin(2\pi f_{0}t+{\varphi }_0)$ was used, from which the pulses of duration $T_P$ were formed; then, their amplitude was normalized in agreement with (2), noise was added to the signal, the mixture was filtered according to the scheme in Figure 1c, and, finally, the signal was demodulated using the decision rule (5).

For narrowband signals, for this type of modulation and reception, a theoretical result is known [46] that relates the bit error ratio with $E_B/N_0$ in the case of $M=2^K$ position modulation:

$$P_B=\frac{2^{K-1}}{2^K-1}\sum _{n=1}^{M-1}{(-1)}^{n+1}\left(\begin{array}{c}M-1\\ n\end{array}\right)\frac{1}{n+1}exp\left(-\frac{nK{E}_B}{(n+1)N_0}\right)$$

(6)

As a result, using the considered numerical model, a dependence of the bit error ratio on $E_B/N_0$ was obtained, which completely coincided (Figure 6a) with the theoretical value (6).

3.2. Bit Error Ratio for UWB Chaotic Radio Pulses

The bit error ratio as a function of $E_B/N_0$ is calculated for a UWB chaotic signal with a continuous power spectrum (Figure 3c, blue line) generated by system (1). Following the method described in Section 2.8, three dependencies were obtained: $P_B(E_B/N_0)$ in the absence of filtering, $P_B^G(E_B/N_0)$ for Fourier filtering, and $P_B^F(E_B/N_0)$ for the filter of FSC (3).

In Figure 6a, theoretical dependencies and the results of numerical simulations are shown for 2-position and 4-position manipulation schemes and various radio pulse durations (5 to 40 ns) for the case of no filtering, i.e., $s_C\left(t\right)=s_F\left(t\right)$.

In Figure 6b, the results of simulations using Fourier filtering are shown, the Fourier filter bandwidth coinciding with the signal bandwidth. The signal bandwidth is defined as the frequency range $(f_L,f_H)$ that contains $99\%$ of the signal power. Fourier filtering was carried out by means of transforming the signal to the frequency domain, discarding all the frequency components lying outside the frequency band $(f_L,f_H)$, and transforming the obtained frequency image back to the time domain.

Finally, Figure 6c shows the results of simulations for the case of filtering based on the FSC of the chaos generator (3).

4. Discussion

In all three cases, with increasing pulse duration, there is an increase in the bit error ratio. This phenomenon [18] is explained by the fact that as the pulse duration increases, the signal dimension also increases and, accordingly, more noise power is acquired by the receiver. This is the charge for the noise power when the energy receiver is used. There is no such effect in the coherent receiver.

Since the chaotic radio pulse is a signal with a large base $B=2T_P\Delta F>>1$, then, as is shown [18], in the case of an incoherent receiver, the bit error ratio depends on the signal base nonlinearly: there is a minimum of the bit error ratio for a certain base value; for radio pulses with a larger or smaller base value, the probability of incoherent detection is greater than this minimum value.

Thus, with an equal duration of radio pulses formed from a UWB chaotic signal and from a harmonic signal, for the chaotic signal, the bit error ratio will be higher because of its larger base.

In the case of no filtering and energy receiver, for chaotic radio pulses, the bit error ratio is significantly worse than the theoretical limit values for narrowband signals. For example, a bit error ratio of $P_B={10}^{-3}$ is achieved at $E_B/N_0=20$ dB.

When the noise bandwidth is narrowed down to the signal bandwidth by incoherent reception, i.e., when the relation between the noise and the signal levels is optimal, the resulting bit error ratio significantly decreases, other parameters being equal, and approaches the theoretical limit value.

With the filtering, the error probabilities for the Fourier filtering and the filter based on the generator’s FSC become almost indistinguishable. For example, with the pulse length of 40 ns, the filtering reduces the symbol-error ratio by two orders at $E_N/N_0=18$ dB for both filters.

The use of a filter based on the FSC of the chaos generator eliminates the need to use filter systems that are not consistent with the parameters of the chaotic signal. In addition, linear filtering with the energy detector solves the problem of receiving signals in a channel with multipath propagation, because it does not require knowledge of the phase of the signal and the transient response of the channel.

Thus, the assumptions that served as motivation for this research were fully confirmed.

In our opinion, the proposed method simplifies the development of UWB receivers of chaotic signals since it could save designers from the necessity to look for or develop a filter with a bandwidth matched to the chaotic signal bandwidth. The method could also eliminate the need for adjusting the parameters of the designed chaos generators to the filters available on the market.

5. Conclusions

A method for linear filtering a chaotic signal in a noisy channel is proposed and analyzed. As is shown, in a noisy channel, there is a natural way to incoherently detect chaotic signals, based on the filtering properties of the dynamic system that forms these signals.

It has been found that the proposed filtering method is as effective as an ideal filter that removes the noise from the frequency band containing 99% of the signal power. It has been shown that the BER achieved with this filter is the same as with ideal filtering.

The efficiency of this approach was verified in a PPM scheme.

A direct comparison of the error rate for regular and chaotic signals shows that the use of the FSC of the chaos generator as a filter draws the performance of quadratic reception to the theoretical limits of energy detection.

The obtained results can be taken as a basis for upcoming experiments.