Frequency Hopping Joint Radar-Communications with Hybrid Sub-pulse Frequency and Duration Modulation

The frequency-hopping (FH) joint radar-communications systems (JRC) [1, 2, 3, 4, 5] have become more and more prevalent in both military and civil applications, e.g., airborne, shipborne, ground-based combat systems, the connected autonomous vehicle (CAV). The most popular FH JRC systems include multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM), multi-Carrier AgilE phaSed Array Radar (CAESAR), and FH-MIMO[2].

A key challenge in FH JRC is to embed/modulate data (onto frequency sub-pulses) at the transceiver and demodulate the signal at the receiver. For data embedding, the most dominant technique is the index modulation (IM), which utilizes frequency selection/combination and/or antenna selection/permutation for data representation. Specifically, the data bits are embedded by selecting different sets of sub-pulse frequency (i.e., frequency combination) and allocating them to different antennas (i.e., antennas permutation). For data demodulation, the optimal demodulator can be based on the maximum likelihood principle [3], while sub-optimal methods, which require lower computation complexities, are based on compressive sensing (CS) or discrete Fourier transform (DFT) [2]. However, the performance of these methods depends on the accuracy of channel estimation. Unfortunately, a long training sequence necessary for accurate channel estimation is not always feasible. This is because a long training sequence reduces the communication fraction over the whole time frame, thus decreasing the data bit rate. In particular, the data embedding schemes using both sub-pulse frequency combinations and antenna permutations require complex demodulation techniques that are prone to demodulation error. On the other hand, the data embedding scheme using only the sub-pulse frequency combination has a limited data transmission rate.

This paper proposes novel techniques to embed and demodulate data in an FH JRC system to increase the data rate and reduce the demodulation error. For data embedding, we use both sub-pulse frequency and duration, therefore increasing the data transmission rate compared to only using the sub-pulse frequency. For data demodulation, we propose a novel scheme based on the signal’s time-frequency image (TFI) and a YOLO-based detection system. This demodulation scheme does not require channel estimation and is robust to the Doppler shift and the timing offset between the transceiver and the communication receiver. Moreover, the proposed data embedding and demodulation schemes are spatially flexible and not limited to the sidelobe of the transmit beampattern, since the data is not embedded by utilizing the phase or amplitude of the beampattern sidelobe.

In this work, we use the Costas arrays to generate the FH sequence of each signal pulse in one of the data embedding schemes. Using the Costas array, an FH signal possesses a narrow peak at the origin of its ambiguity function (AF) and low sidelobes elsewhere, which is desirable for the radar measurement (i.e., range and speed) accuracy. Details of the Costas arrays, their construction, and characteristics can be found in [6, Ch. 5].

We propose two data embedding techniques, namely Random and Costas-based schemes. Unlike existing studies that only utilize sub-pulse frequencies to convey information, both sub-pulse frequencies and durations are used in our proposed schemes. We first design the codebooks $\mathcal{S}$ and $\mathcal{S}_{\rm C}$ of the Random and Costas-based schemes, respectively. Then, for each embedding scheme, data bits are embedded by mapping each element of the corresponding codebook to a symbol. Let ${\bf{\mathcal{F}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}[(1,2,...,N_{\rm f})\times f_{\rm f}]$ be the selection set of the sub-pulse frequency, where $f_{\rm f}$ is the fundamental frequency. Let ${\bf{\mathcal{D}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(\triangle_{1},\triangle_{2},...,\triangle_{N_{{\rm{T}}}})$ be the selection set of the sub-pulse duration, where $N_{{\rm{T}}}$ is the number of sub-pulse duration selections. The codebooks $\mathcal{S}$ and $\mathcal{S}_{\rm C}$ are designed as follows.

• •

  Random scheme: $\mathcal{S}$ is designed by selecting each sub-pulse frequency from the frequency set ${\bf{\mathcal{F}}}$, and selecting each sub-pulse duration from the duration set ${\bf{\mathcal{D}}}$ as

  	$\displaystyle f_{i}$	$\displaystyle\in{\bf{\mathcal{F}}},\triangle t_{i}\in{\bf{\mathcal{D}}},\forall i\in(1,2,...,N_{\rm f}),$		(3)
  	$\displaystyle f_{i}$	$\displaystyle\neq f_{i+1},\forall i\in(1,2,...,N_{\rm f}-1).$		(4)

  Here, condition (4) means every two consecutive sub-pulse frequencies are not equal, which is necessary for the data demodulation technique that will be presented in Section V.

• •

  Costas-based scheme: Unlike the Random scheme, for designing $\mathcal{S}_{\rm C}$, each sub-pulse frequency is not individually selected. Instead, the whole FH sequence of each signal pulse is selected from the Costas-based FH sequence defined as follows.

  Let ${\bf{\mathcal{A}}_{\rm C}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}({\bf{A}}_{1},{\bf{A}}_{2},...,{\bf{A}}_{N_{\rm C}})$ be the set of Costas array of length $N_{\rm f}$, where ${\bf{A}}_{i},\forall i\in(1,2,...,N_{\rm C})$ is an $\mathbb{N}^{N_{\rm f}\times 1}$ Costas array, and $N_{\rm C}$ is the number of Costas sequence with length of $N_{\rm f}$. Then, let ${\bf{\mathcal{F}}_{\rm C}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}[({\bf{A}}_{1},{\bf{A}}_{2},...,{\bf{A}}_{N_{\rm C}})\times f_{\rm f}]$ be the selection set of the Costas-based FH sequence. Then, the FH sequence of each signal pulse is selected from ${\bf{\mathcal{F}}_{\rm C}}$. On the other hand, similar to the Random scheme, each sub-pulse duration is selected from the sub-pulse duration set ${\bf{\mathcal{D}}}$ as

  $\displaystyle{\bf{F}}$

  $\displaystyle\in{\bf{\mathcal{F}}_{\rm C}},\triangle t_{i}\in{\bf{\mathcal{D}}},\forall i\in(1,2,...,N_{\rm f}).$

  (5)

Note that for the Costas-based scheme, because the FH sequence is generated using the Costas array, condition (4) is satisfied. The dimensions of $\mathcal{S}$ and $\mathcal{S}_{\rm C}$ are given by

As shown in (6), the number of selections of sub-pulse frequencies is $N_{\rm f}(N_{\rm f}-1)^{(N_{\rm f}-1)}$ instead of $N_{\rm f}^{N_{\rm f}}$ due to the requirement in (4).

It follows from (6) and (7) that the maximum number of bits $C$ and $C_{\rm C}$ that can be embedded in each pulse of the two schemes are given by

where $\lfloor.\rfloor$ denotes the floor function, determining the closest smaller integer. As can be seen, the Random scheme achieves a higher transmission rate than the Costas-based scheme.

This section demonstrates that by using the sub-pulse frequencies and duration for data embedding as presented in the previous section, the detection probability of the sensing function is not affected. Let $P_{\rm D}$ and $P_{\rm FA}$ denote the detection and false alarm probabilities of the JRC transceiver, respectively. Let $E_{\rm{p}}$ be the signal energy per each pulse of the JRC signal and ${\sigma_{\rm w}^{2}}$ be the noise variance. The energy-to-noise ratio (ENR) of the JRC signal is given by

To detect the transmitted signal $s[k]$ buried in the reflected RF signal $r[k]$, the JRC transceiver performs the matched filtering. Let $R_{s}$ denotes the output of the matched filter. Based on Neyman-Pearson criterion, the transceiver decides a detection if $R_{s}$ exceeds a threshold $\gamma$

is the complement of the cumulative distribution function (CDF) of the standard Gaussian distribution, and $Q^{-1}(x)$ denotes the inverse function of $Q(x)$. The characteristic of $P_{\rm D}$ is given in Theorem 1 below.

Theorem 1: The value of $P_{\rm D}$ is independent from the JRC signal waveform and only depends on $P_{\rm FA}$ and ENR.

Proof: From [7, Ch. 4], $P_{\rm D}$ can be expressed by $P_{\rm FA}$ and ENR as

Accordingly, $P_{\rm D}$ only depends on $P_{\rm FA}$ and ENR, and is independent from the JRC signal waveform. $\blacksquare$

Therefore, varying the sub-pulse frequencies and durations does not affect the detection performance of the sensing function, as long as the signal energy per each pulse is fixed. Note that, however, varying the sub-pulse durations may cause some increase in the peak-to-average power ratio (PAPR), which reduces the efficiency of the power amplifier at the front end of the JRC transceiver and communication receiver. Therefore, the sub-pulse selection set ${\bf{\mathcal{D}}}$ should be carefully designed to avoid an excessive PAPR value.

This section presents the detailed procedure of the proposed data demodulation schemes and its advantages, including being unaffected by channel estimation error (i.e., the channel estimation is not required), and robustness against the Doppler shift or timing offset between the transceiver and the communication receiver.

Table. I shows the parameter values used to produce the simulation results. Note that the number of Costas array $N_{\rm C}$ is not a freely set value, but depends on $N_{\rm f}$, the number of sub-pulses per pulse. For instance, $N_{\rm C}=40$ for $N_{\rm f}=5$. The list of available Costas arrays and their construction can be found in [6, Ch. 5]. To train the YOLO detection system, for each data embedding scheme, we generate $2700$ signals with SNR ranging from $-6$ dB to $10$ dB and a step size of $2$ dB. The $2700$ signals are divided into a training set of $2160$ signals ($80\%$ of the total) and a validation set of $540$ signals (20% of the total). For the testing data, we generate $3300$ signals for each embedding scheme, with SNR ranging from $-10$ dB to $10$ dB and a step size of $2$ dB.

Fig. 5 shows the detection probability of the JRC sensing function for different signal waveforms (i.e., Random scheme, Costas-based scheme, and fixed sub-pulse length), $P_{\rm FA}$, and ENR values. The results are obtained by averaging $10,000$ independent trials. As can be seen, for each $P_{\rm FA}$ value, the three curves almost overlap, meaning the detection probability only depends on $P_{\rm FA}$ and ENR but not the signal waveform. Therefore, varying the sub-pulse durations does not affect the detection capability of the sensing function of the FH RJC system, as long as the signal energy per each pulse is fixed.

Fig. 6 demonstrates the maximum number of bits that can be transmitted over a signal pulse of the two proposed embedding schemes and those in [3, 5]. As can be seen, the Random scheme can embed a significantly higher amount of data than the Costas-based scheme, while the Costas-based scheme can convey a slightly higher number of data bits than the scheme in [3, 5]. That is because the schemes in this paper use both sub-pulse durations and frequencies to embed signals. On the other hand, the technique in [3, 5] focuses on using antenna permutations for data embedding. Therefore, the proposed techniques are more suitable when the number of antennas is limited, while the technique in [3, 5] is suitable in the case of an extensive number of antennas is available.

Fig. 7 illustrates the symbol error rate (SER) of the two proposed schemes and those of the techniques in [5] as functions of the signal-to-noise ratio (SNR). As can be seen, the two proposed schemes have lower SERs compared to those of the techniques in [5] for all SNRs. Distinctively, at low SNRs (i.e., $\leq-2$ dB), the two proposed schemes have similar SERs, and are about $4$ dB and $10$ dB better than those of the FH and BPSK techniques in [5], respectively. This is because the proposed techniques use the CWD, which contains the auto-correlation operation, to generate CWD-TFIs of the signals. As such, the proposed techniques are more robust against noise compared to the other techniques in [5].

We have proposed novel approaches for embedding and demodulating information bits in FH JRC systems. For data embedding, we have used both sub-pulse durations and frequencies to increase the data transmission rate. For data demodulation, to reduce the error rate, we have proposed a CWD-TFI and YOLO-based demodulation scheme that does not require channel estimation and is robust against Doppler shift and timing offset between the JRC transceiver and communication receiver. Simulation results have shown that our proposed techniques achieve higher data transmission and lower symbol error rates than the existing ones.