# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 3 Simultaneous Signal Processing Across Domains
### Tensor-Based Receiver Designs
#### 3.2.3 Applications of other Tensor Decompositions  
A receiver based on (18) assumes that the channel is known. However, even in the absence of complete channel state information, the structure of the received signal can be exploited through several tensor decomposition schemes for designing blind/semi-blind receivers [11, 12, 14]. Such receivers can perform joint symbol and channel estimation. In fact, one of the initial applications of tensors for wireless communication signal processing was considered in [11] for a blind receiver design where the signal in a DS-CDMA system is modeled as a third order tensor. Information is extracted from the received signal using tensor canonical polyadic (CP) decomposition, also known as parallel factors (PARAFAC) decomposition. The CP decomposition breaks a tensor into a linear combination of rank one tensors. By making use of the uniqueness of the three-way PARAFAC decomposition,  
Figure 5: TN representation of a multi-domain communication system model with sixth order channel tensor. ref. [11] presents a blind receiver for user separation, equalization, and detection in a DS-CDMA system. The work in [11] and several other blind receiver design schemes rely on the uniqueness of the PARAFAC decomposition of third order tensors, and therefore cannot be accomplished in a degenerate setting comprising vector or matrix formulations. Another common tensor tool is the Tucker decomposition which has primary applications in finding low rank structures, classification, and feature extraction in high dimension data [32, 38]. Some suitable variations of PARAFAC and Tucker decompositions lead to the development of many related tensor decompositions which can be used to model signals and develop coding techniques in multi-antenna communication systems as shown in [12, 13] and references within. For instance, a tensor space-time-frequency coding technique for a MIMO OFDM CDMA system is developed in [12], where the coding tensor is represented using a fifth order tensor. The baseband equivalent received signal is also represented as a fifth order tensor which under some suitable assumptions on the channel admits a generalized PARATUCK (a combination of PARAFAC and Tucker) model. Based on this model, semi-blind receivers are then used for joint symbol and channel estimation. A short summary of a few of the important tensor tools along with some applications can be found in Table 4. More details on various tensor decompositions and their numerous applications can be found in [14, 21, 38, 39]. A tensor singular value decomposition (SVD) is described in [15] which decomposes a tensor into two unitary tensors and a pseudo-diagonal tensor connected via the Einstein product. Note that both PARAFAC and Tucker decompositions can equivalently be represented using the Einstein product-based tensor SVD if the components of the unitary tensors are obtained using a product of the elements of the factor matrices in PARAFAC and Tucker decompositions as described in ([15] Lemma 3.20 and 3.22). All such tensor-based receiver designs rely on exploiting the received signal structure by simultaneously processing multiple domains. Simultaneous signal processing across domains reveals the trade-off between these domains, which can be used for efficient resource utilization, as discussed in the next section.