# A Scalable Spatial-Temporal Correlated Non-Stationary Channel Fading Generation Method
## 1 Introduction  
Multi-input multi-output (MIMO) technology is widely employed in wireless communications to effectively enhance channel capacity and frequency band utilization [1; 2; 3]. With advancements in antenna technology leading to miniaturization, MIMO communication systems are being applied in various communication scenarios [4; 5; 6]. However, diverse and complex propagation environments have a significant impact on system performance [7; 8]. Research into optimizing the performance of MIMO communication systems has garnered significant attention [9; 10; 11]. Channel emulators provide a means to emulate the effects of real-world propagation environments on signals within a laboratory setting, aiding in the design, evaluation, and optimization of communication systems [12]. As the number of antennas increases and dynamic propagation environments become prevalent, existing channel emulators struggle to support massive and non-stationary MIMO channel emulations [13; 14; 15]. There is a growing need for the development of more efficient and accurate fading generation methods to address these challenges. Currently, commercial channel emulators include Agilent's N5106A PXB [16], Keysight's propsim F64 [17], and Azimuth's ACE 400 WB [18]. Despite their good performance in generating standardized channel models, these emulators are both intricate in structure and financially burdensome. Furthermore, the simulation methods they used had limitations,which researchers had studied. The authors in [19] developed a non-stationary channel generation method on a software radio platform. However, this method was unsuitable for scenarios with fast-moving transceivers, because the generated Doppler frequencies were imprecise compared to the measurements. The sum-of-sinusoids (SoS) method proposed in [20; 21; 22] was suitable for the implementation of FPGA hardware platforms, but it suffered from a problem of phase discontinuity. In [23; 24], the authors proposed the sum of frequency modulation (SoFM), which solved the problem of Doppler frequency inaccuracy and phase discontinuity but did not account for the spatial-temporal correlation function (ST-CF) of channel fading. In [25; 26], the authors introduced a MIMO generation method that incorporated a spatial-temporal correlation function (ST-CF) based on the Kronecker model. However, no hardware simulation implementation was included in their work. In [27], the authors provided a hardware implementation scheme, but its structure was complex, and its scalability was limited. Hardware implementation of correlated channel fading has also posed challenges in terms of resource consumption and generality. The conventional approach for generating the cosine waveform for channel fading was the look-up table (LUT) method, which offered satisfactory real-time performance, but incurred significant consumption of random-access memory (RAM) resources [28; 29; 30]. The coordinate rotation digital computer (CORDIC) was an alternative for the LUT method [31]. In [32], the authors reduced random access memory (RAM) consumption through an asynchronous pipeline architecture. However, this method resulted in many iterations and a long delay. In [33; 34; 35], the authors proposed a high-radix CORDIC algorithm to reduce delay, but the hardware circuit structure was complex and difficult to implement. To solve this problem, the authors of [36] proposed a greedy CORDIC algorithm which greatly reduced the usage of RAM resources and improved real-time performance. However, these CORDIC algorithms were not appropriate for the e-exponent, logarithm, and square root functions, which are required for channel fading. The methods in [37; 38] were able to calculate the e-exponent, square root, and logarithmic functions, but the input convergence domain was limited. To address this limitation, the authors of [39] proposed a method to expand the convergence domain. However, the hardware architecture was inflexible and not suitable for multi-channel systems.