# MIMO Signal Detection Based on IM-LSTMNet Model
## 2 Methods
### MIMO-OFDM Signal Detection Model Based on FrFT  
OFDM technology has some defects, such as frequency bias sensitivity and uniform waveform parameters of the full frequency band, which cannot fully support the various scenarios in future communication, e.g., new multi-carrier technology--OFDM based on fractional Fourier transform [18]. The processes of modulation and demodulation are completed by fractional Fourier transform. In the FrFT-OFDM system, the subcarriers' frequency changes linearly with time and is no longer a fixed value. Therefore, the optimal  
Figure 2: Flow chart of the IM-LSTMNet model detecting the MIMO-OFDM signal based on FrFT. transformation order can be selected to ensure that the system has the best performance in the corresponding fractional domain and to improve the ability of the system to resist time-selective fading. In this sense, the traditional OFDM system is a special case of FrFT-OFDM when the order is 1, and it is an extended form of OFDM technology. The structure block diagram of the MIMO-OFDM system based on FRFT is shown in Figure 3. It can be seen that the modules and functions of the system are almost the same as those of the traditional MIMO-OFDM system IFrFT and that FrFT replaces the IFFT and FFT function blocks in the OFDM system. Subcarrier modulation and demodulation are accomplished in different ways. The FrFT-OFDM system increases the flexibility of the system and ensures the BER performance by adding the supplementary parameter of order p.  
Figure 3 clearly illustrates that, in the MIMO-OFDM system model based on FrFT, the input data symbol stream is first divided into sub-symbol streams by string/parallel transformation, and the encoded data are modulated by the QPSK modulator. The QPSK-modulated signal realizes inverse fractional Fourier transform orthogonal frequency division multiplexing (IFrFT-OFDM) modulation processing in the IFrFT circuit, and completes the conversion of frequency domain data into time domain data. This process is  
\[s_{i}(m)=\sum_{k=0}^{N-1}S_{i}(k)F_{-\alpha}(m,k),0\leq m<N-1 \tag{1}\]  
where \(S_{i}(k)\) is the QPSK-modulated signal and \(F_{-\alpha}(m,k)\) is the IFrFT transform. The FrFT is given by  
\[F_{\alpha}(m,n)=A_{\alpha}exp[j\frac{cot\alpha}{2}(m^{2}t^{2}+n^{2}u^{2})-j \frac{2\pi nm}{N}] \tag{2}\]  
where \(A_{\alpha}=\sqrt{(\textit{s}\textit{i}\textit{n}-\textit{j}\textit{c}\textit{ s})/N}\). \(t\) and \(u\) are sampling intervals in the time domain and the fractional Fourier domain, respectively, which satisfy \(t\cdot u=2\pi\textit{s}\textit{i}\alpha/N\). CP is added to each FrFT-OFDM symbol to reduce the effect of channel delay expansion. These processed FrFT-OFDM signal streams are transmitted parallelly to each other, and each signal stream is transmitted according to a specified transmitting antenna. Figure 3: Block diagram of the FrFT-based MIMO-OFDM system. The signal passes through the channel, and the received signal can be represented as  
\[\mathbf{y}_{N_{r}\times 1}=\mathbf{H}_{N_{r}\times N_{l}}\mathbf{x}_{N_{l}\times 1 }+\mathbf{n}_{N_{r}\times 1} \tag{3}\]  
where \(\mathbf{y}=[y_{1},y_{2},\ldots,y_{N_{r}}]\) represents the received signal; \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{N_{r}}]\) represents the sending signal vector; \(H\) represents the channel matrix; and \(\mathbf{n}=[n_{1},n_{2},...,n_{N_{r}}]\) is the Gaussian white noise vector at the receiver. As shown in Figure 3, the receiver performs a signal processing opposite to the transmitter, first receiving signals through the \(N_{i}\) root receiving antenna at the receiver; these signals are demodulated by FrFT, and then the CP is removed. At this step, the time domain data are transformed into frequency domain data, and then the data are restored in the frequency domain according to the channel matrix. The \(\mathrm{IFrFT}/\mathrm{FrFT}\), the addition and removal of CP, the string/parallel transformation, and the parallel/string transformation of the signal are all carried out in each separate transmission and reception module.