# Understanding the concept of semi-blind receivers

In the book "Signals and Images: Advances and Results in Speech, Estimation, Compression... ", an overview of using tensor analysis in signal processing is given. On page 243 it is explained that an algorithm for semi-blind receivers can be obtained by combining two tensors.

Can someone provide me with an explanation for semi-blind receivers? What is it?

Let us consider a very simple example of a frequency-flat quasi-static MIMO communication system. In such a setup, we can write the received signal from, say, $$M_{\rm R}$$ receive antennas as $$\mathbf y(t) = \mathbf H \cdot \mathbf x(t) + \mathbf w(t),$$ where $$\mathbf x(t) \in \mathbb{C}^{M_{\rm T}}$$ is the vector of transmitted symbols from the $$M_{\rm T}$$ transmit antennas, $$\mathbf w(t)$$ represents the additive noise, and $$\mathbf H \in \mathbb{C}^{M_{\rm R} \times M_{\rm T}}$$ is the MIMO channel matrix.
Now, if we know the channel matrix $$\mathbf H$$ at the receiver, it is simple to decode the transmitted information, for $$M_{\rm R} \geq M_{\rm T}$$, you can use a linear receiver. But how do we know $$\mathbf H$$? Several approaches:
• Pilot-aided receivers: Here we rely on a training phase, where a set of known vectors $$\mathbf x(t)$$ is transmitted. If we observe the signal from $$N_{\rm P} \geq M_{\rm T}$$ pilots, we can uniquely recover $$\mathbf H$$ and, as long as it doesn't change to rapidly, use this information to decode following unknown symbol vectors $$\mathbf x(t)$$. The obvious drawback is the pilot overhead since the $$N_{\rm P}$$ pilot vectors occupy channel resources but carry no information.
• Blind receivers try to alleviate the problem by separating a sequence of received vectors $$\mathbf y(t)$$ into $$\mathbf H$$ and $$\mathbf x(t)$$, which are both assumed unknown. This requires special techniques (tensor techniques are among them) since in general the problem is not unique per se (e.g., a low multilinear rank over time, space, and frequency or some other assumption). Note that some ambiguities just cannot be resolved, since, e.g., $$\mathbf H \cdot \mathbf x = (-\mathbf H)\cdot (-\mathbf x)$$. Therefore, such receivers must typically be combined with designated modulation schemes (e.g., things like differential modulation).
• Semi-blind receivers are inbetween: They use a few pilots (meaning, in this example, $$N_{\rm P} < M_{\rm T}$$) to resolve the ambiguities, yet try to estimate $$\mathbf H$$ also directly from the data. They are hence more resource efficient than purely pilot-based techniques and yet less prone to ambiguities than blind receivers.