# Joint Radar and Communications Waveform Design Based on Complementary Sequence Sets
## 1 Introduction
One typical example is the JRC waveform based on the orthogonal frequency division multiplexing (OFDM) waveform [23; 24; 25], which has a considerably high communications rate [26]. However, the OFDM waveform has a nonconstant modulus. In order to detect targets as far as possible, the amplifier of the radar transmitter usually operates in the nonlinear region [27]. However, this will cause serious distortion when the transmit waveform has a nonconstant modulus, and the radar and communications performance will be degraded [28]. In view of this, a JRC waveforms design method based on unimodular waveforms has been studied [29]. Nevertheless, the JRC waveforms proposed in [23; 24; 25; 29] have high autocorrelation sidelobe levels. In an automotive radar, if the autocorrelation sidelobe level of the transmit waveform is high, the returns from targets with small radar cross sections (RCSs) may be submerged by the sidelobes of targets with large RCSs at receivers [30]. In view of this, to ensure the radar performance of JRC systems, transmit waveforms with low autocorrelation sidelobe levels are required in JRC systems. To design unimodular JRC waveforms with low sidelobe levels, a JRC waveform combining the linear frequency modulation (LFM) waveform and the binary phase-shift keying (BPSK) waveform is proposed in [31; 32]. The JRC waveform is obtained by directly multiplying the BSPK waveform with the LFM waveform. For the BPSK in this method, the angular separation between the modulated bits \"1\" and \"0\" is set to be \(\phi_{\delta}(\phi_{\delta}<\pi)\) rather than \(\pi\) as with the traditional BPSK waveforms. Using this method, the designed JRC waveform in [31; 32] can achieve a better autocorrelation performance with the degradation in the bit error rate (BER). In [13], based on the traditional unimodular communications waveform, the author designs a JRC waveform by optimizing the phase perturbation in each chip, which is named the optimized phase perturbation (OPP) waveform. To keep the JRC waveform with a low BER, the phase perturbation of each chip of the JRC waveform is constrained by an upper bound. The aforementioned co-use JRC waveforms are all obtained by using single waveforms. However, the autocorrelation sidelobe of a single waveform cannot be decreased to as low as possible [33]. To further decrease the autocorrelation sidelobe level of JRC waveforms, a feasible way is to exploit the philosophy of the complementary sequence set (CSS) to design co-use JRC waveforms. A CSS contains multiple subsequences. The sum of the autocorrelation function (SACF) of subsequences in the CSS is the Kronecker delta function,i.e., all sidelobes of the SACF of subsequences are zero [34]. This property makes CSS a promising sequence to be used to design waveforms with low range sidelobe levels, which has been shown in radar applications [35; 36]. In our work, the philosophy of the CSS is exploited to design the JRC waveforms. In this paper, based on CSSs, two co-use JRC waveforms design methods are proposed. At first, unimodular JRC waveforms with low sidelobe levels are devised, which are called the JRC complementary waveform group (JRC-CWG). The devised JRC-CWG has a considerably low sidelobe level as well as good communications performance. Each JRC-CWG contains two kinds of unimodular waveforms. One kind of waveform carry out both radar and communications functions, and are termed waveforms with communications information (WCI). The traditional communications waveforms, such as the BPSK waveforms, are employed as the WCIs. The other kind of waveform in the JRC-CWG are called the waveforms to be optimized (WTO), which are optimized to suppress the sidelobe level of the JRC-CWG. Furthermore, considering that the Doppler resilience of the designed JRC-CWG is poor [37; 38; 39], a Doppler-resilient JRC complementary waveform (DR-JRC-CWG) design method is proposed. To design the DR-JRC-CWG, the Doppler sensitivity of the designed JRC-CWG is analyzed first.