## 1 Introduction  
In the last decade, interference alignment (IA) has received considerable attention due to the advantage that IA can achieve the theoretical optimal degree of freedom (DoF) performance in different kinds of multi-input multi-output (MIMO) interference channels (ICs), interfering multiple-access channels (IMACs) and interfering broadcast channels (IBCs) [1]. A complete cooperative IA transmission process includes several procedures, such as forward channel training and estimation, channel state information (CSI) quantization, quantized CSI feedback, IA solution calculation, decoding information (DI) distribution, and actual payload transmission [2]. Global CSI should be acquired by transmitters to calculate IA precoders and decoders, and the obtained decoders should be known by the receivers to recover their desired signal. Since the channel training sequences, feedback CSIand DI, and actual payload are all transmitted via wireless channels, the effective achievable sum-rate of IA is determined by the time fraction allocation for these procedures [3; 4]. When the channel coherence time is given, the more overhead allocated to CSI and DI feedback, the less time is left for actual payload transmission. Conversely, if the quantization accuracy of CSI and DI is reduced, the feedback overhead certainly shrinks, but the effective performance of IA may still be unsatisfactory since the precoders and decoders calculated according to low-accuracy CSI would result in considerable interference leakage and rate loss. Additionally, even though the precoders and decoders can be calculated from accurate CSI, the accuracy of the decoders' quantization and feedback is another bottleneck for the IA achievable rate. Therefore, a delicate trade-off should be obtained between the quantization accuracy and feedback overhead to optimize the parameter configuration of IA in different cooperative scenarios, and the key is to figure out the relationship between CSI/DI accuracy and effective performance of IA. In most works, it is a popular method to quantize CSI/DI and feed them back, which is called quantized feedback or limited feedback (LFB). Two common quantization strategies are usually adopted in LFB, i.e., scalar quantization (SQ) and vector quantization (VQ) [5]. In SQ, the real and imaginary parts of each entry in the complex channel matrix/vector are separately quantized. In VQ, a channel matrix or vector is mapped to a codeword from a known codebook, and thus the codeword index is sent instead of the matrix/vector itself. Generally, VQ results in fewer quantization bits and higher complexity than SQ when achieving the same quantization accuracy level. Thus, due to the low feedback overhead advantage, VQ-based LFB has been discussed theoretically in different wireless transmission scenarios, including single-user MIMO, multi-user MIMO and coordinated multiple points [6; 7; 8]. In practical applications, taking the 3 rd Generation Partnerships Project (3 GPP) standards for example, the VQ/SQ-based feedback strategies are also called the explicit feedback and implicit feedback, respectively, and the implicit feedback is actually adopted and applied in different closed-loop transmission modes of 4G Long-Term Evolution (LTE) and 5G New Radio (NR) [9; 10]. In VQ-based theoretical performance analysis works, the random vector quantization (RVQ) codebook [11] and the Grassmannian codebook [12; 13] are two of the most pervasive codebooks. The codewords in RVQ are independent isotropic vectors on a unit complex spherical surface. Due to its simplicity in codebook design, RVQ is usually adopted as the lower bound in performance analysis. The Grassmannian codebook is designed according to the Grassmannian line-packing problem solution, where the minimum value of chordal distances between any two codewords should be maximized [9]. The Grassmannian codebook achieves the upper-bound performance, but its off-line computation complexity for codebook design is high.