# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 1 Introduction
In [20], it proposes a tensor-based approach to address the dual problems of channel estimation and target sensing in massive MIMO-ISAC systems. Through a shared training pattern, both channel and target parameter estimation are handled in a unified manner. In [21], the utilization of time modulated array (TMA) is proposed to enhance the age of information (AoI) in covert communication and derive a communication covertness metric based on Kullback-Leibler (KL) divergence. Compared to methods that handle these processes separately, this unified approach enhances sidelobe suppression. Meanwhile, in [22], the waveform and filter are co-designed using alternating minimization and majorization-minimization (MM) algorithms, achieving an effective reduction in radar pulse compression sidelobes while managing the associated gain attenuation. In [23; 24; 25; 26], they all achieve sidelobe suppression and interference minimization by introducing mismatched filters, which enhance the radar system's dynamic range and detection accuracy. Moreover, they maintain low losses and consistent high performance even with changes in filter length. The methods for joint optimization of waveforms and filters mentioned above aim to reduce radar pulse compression sidelobes, improve detection accuracy, and minimize interference. By jointly designing waveforms and mismatched filters, these methods achieve better system performance compared to separate processing while attempting to minimize signal loss. Some methods accept a certain degree of SNR loss in exchange for better sidelobe suppression. They employ various optimization techniques, such as alternating projection, space-frequency modulation, and iterative algorithms, and are optimized for specific issues like jamming mitigation. However, common drawbacks include SNR loss and increased computational complexity. In this letter, we utilize continuous phase modulation (CPM) signals employing non-linear frequency modulation (NLFM) within the DFRC system. To further mitigate sidelobe interference, we propose a design methodology for the environment-based weighted mismatched filter (EWMF). Maximizing radar efficiency requires attaining an elevated signal-to-interference-and-noise ratio (SINR) and minimizing the leakage power amplification at the output of the crafted EWMF. To realize this objective, we formulate a weighted optimization problem and derive its optimal solution through the alternating direction multiplier method (ADMM). The key advantage of ADMM is its ability to deconstruct complex problems into simpler subproblems, thereby boosting solution efficiency and minimizing computational complexity. We then analyze the LPG caused by mismatched filtering and derive its lower bound using the Young inequality. Finally, we validate the algorithm's performance and the impact of communication modulation parameters through experimental simulation data. The results show that the EWMF algorithm effectively reduces NRSI. In contrast to the WMMF introduced in [11], the innovations presented in this letter can be outlined as follows:  
1. The weighted time domain correlation matrix is introduced into the sidelobe suppression of integrated waveforms. 2. The impact of communication modulation is considered in the EWMF design, and the mismatch filter is optimized using ADMM. 3. For the EWMF design, we consider the impact of NRSI, while it is not considered in [11]. The subsequent sections of this letter are structured as follows: In Section 2, the DFRC waveform model is derived. Section 3 formulates and resolves the optimization problem. Section 4 presents numerical simulations. Lastly, Section 5 offers the conclusions. The notation and corresponding representation of the symbols used in this work are summarized in Table 1.