## 2 Wireless Signal Modulation Recognition Model  
In digital communications in real radio frequency (RF) environments, the transmitter forms a binary bit stream of information. After that, the bits stream is encoded and modulated, and finally transmitted via the wireless channel. During the modulation process, the baseband signal is moved to the shifted by the carrier frequency to produce the wireless signal. A typical flow of modulation recognition signal processing is presented in Figure 1. At the receiving end, we consider a typical communication receiver in the noncooperative mode. In general, there is only a partial or even minimal prior knowledge of communication. The receiver monitors the activity of the radio frequency (RF) signal in the electromagnetic spectrum of interest through a spectrum analyzer and gives an estimated value of the centre frequency, the bandwidth of the signal. Based on these parameters, the receiver through the bandpass filtering and downconverts to flit the signal capture to appropriate intermediate frequency (IF). For the digital communication signal, the phase-locked loop is used to further accurately extract the carrier frequency and phase parameters of the intermediate frequency signal. Then, using the orthogonal downconversion technique, the baseband signal of the unknown modulation mode is outputted at the output end of the matched filter. Then, parameters such as baud rate and symbol timing are obtained by symbol synchronization technology, matched filtering, and synchronous sampling are performed to obtain a symbol sequence of an unknown modulation mode. The signal is then subjected to subsequent demodulation. Our approach focuses on AMR on complex baseband signals, which is a coherent, asynchronous AMR method. During signal reception and processing, due to imperfections in the detection and isolation stages, carrier frequency estimation errors, bandwidth estimation errors, etc. are introduced during the RF signal to the IF signal acquisition process. In the processing of the intermediate frequency signal to the complex baseband signal, carrier frequency and phase synchronization errors are also introduced. Then, given the channel impairments and hardware defects of the receiver, the baseband signal we obtained is the corrupted version of the originally transmitted signal. In the AMR application for the original baseband signal, the definition and evaluation of these impairments is very important. We define typical signal impairments:  
* Frequency Offset: Frequency offset is introduced during the RF signal isolation phase and IF signal parameter estimation. The receiver local oscillator (LO) introduces frequency deviation due to hardware impairments. * Phase offset: Phase offset is caused by the frequency offset of LO, which causes an instantaneous phase drift of signal captures. * Timing drift: Timing drift is caused by unmatched sampling rates. It is introduced due to the estimated bandwidth deviation of the RF signal isolation phase. Besides, the sampling rate error is introduced due to hardware impairment of the receiver LO. * Noise: Noise introduced by components such as the antennas and receivers. We usually model this noise as additive white Gaussian noise (AWGN). Signal capture and processing flow of the receiver includes: amplifying, mixing, low-pass filtering, and analog-to-digital conversion. Then, the raw baseband IQ signal is obtained. \(N\)-th point sampling are performed on the baseband IQ signal. We present the values as \(r_{raw}\in\mathbb{C}^{N}\). This matrix is the input sequence of the AMR, which contains information if signal captures such as the type of wireless technology, the type of modulation method, interferer, etc. The task of AMR is to obtain the type of modulation through this segment of the signal capture. Therefore, the output of the end-to-end system can be expressed as \(l\). The output of the system output is the judgment of AMR, which is the highest confidence of modulation type in the result set.