## 2 System Framework  
The signal sampled passes through the channel and can be articulated as:  
\[y[l]=A[l]e^{j(\omega l+\varphi)}x[l]+n[l],l=1,\ldots,L \tag{1}\]  
where the complex Additive White Gaussian Noise (AWGN) is denoted by \(n[l]\) and \(x[l]\) is the signal that the transmitter has modulated using a particular modulation method. The channel gain is represented by \(A[l]\), where \(\omega\) and \(\varphi\), respectively, stand for the frequency and phase offset. The number of symbols in a signal sample is represented by \(L\), which can be represented as follows. In-phase/quadrature (I/Q) form, which is represented as follows, can be recorded for the received signals:  
\[y=[\Re\{y[1]\},\ldots,\Re\{y[L]\};\Im\{y[1]\},\ldots,\Im\{y[L]\}] \tag{2}\]  
The offset interference received by the signal in the channel mainly includes frequency offset and phase offset. Different signals contain different frequency components, and different arrival times will lead to a phase offset. Frequency deviation between transmitter and receiver and instability of frequency oscillator result in frequency offset. The constellation diagram of QPSK signal makes it easy to observe changes directly. Select a group of constellation diagrams of the signal at 18 dB to see the impact of frequency offset and phase offset on the signal. It can be seen from Figure 1 that when the signal is not affected, it is correctly represented on the constellation diagram. When the phase offset is 30\({}^{\circ}\) and the frequency offset is 400 Hz, the rotation dispersion changes obviously; when the phase offset is 60\({}^{\circ}\) and the frequency offset is 800 Hz, the changes in the constellation diagram are more obvious. The above changes can be approximately summarized as offset, rotation, and scaling. GRU is a variant of RNN, which mainly combines the forget gate and input gate into a single update gate. The GRU model is simpler than the standard RNN model. Its effect is similar to RNN, but the parameters are reduced, such that it is not easy to overfit. To eliminate errors of the channel reception, as shown in Part A of Figure 2, the predictive correction module is introduced. The sampled I/Q data are altered by channel noise and interference, leading to phenomena like temporal shifting, linear mixing, and spinning of the received signal. In order to eliminate channel reception errors and reduce the training cost, a learnable predictive correction model is shown in Figure 2, which includes Part A and double-layers GRU convolutional neural network noted by Part B. By co-training with the subsequent model, the Part A predictive correction model may estimate the parameters. These parameters are learned from the effects of frequency offset, phase offset, time drift, and noise on the signal, which can be represented as position information carried by frequency offsets and phase offsets. Two dense layers and a flatten  
Figure 1: The constellation diagram of a QPSK signal under the influence of frequency and phase offset at 18 dB. layer can each learn phase and frequency offset information for every 128 \(\times\) 2 I/Q sample. After the flatten layer flattens the input signal \(y\) into a vector to match the input dimension of the dense layer, the data traveling through the layer give the offset parameters. The dense layer's linear activation function yields and estimates the offset parameter across an unlimited, continuous range. The original signal y can be multiplied to produce the corrected signal by the use of the offset parameter and inverse transformation, denoted as follows:  
\[\hat{y}[l]=y[l]e^{-j\hat{y}}= \begin{bmatrix}\Re\{y[l]\}\cos\hat{\phi}+\Im\{y[l ]\}\sin\hat{\phi}\\ \Im\{y[l]\}\cos\hat{\phi}-\Re\{y[l]\}\sin\hat{\phi}\end{bmatrix}  \tag{3}\]  
\(\hat{y}=[g[1],\dots,g[L]]\), which represents the correction signal. Part A learned positional information containing 6 parameters, including translation, rotation, and scaling transformation information, namely, the relationship between phase offset and frequency offset.