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.
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  • $\begingroup$ Nice answer. It would be great if you could accompany it with code to show different approaches. $\endgroup$ – Royi Nov 18 '19 at 15:53
  • $\begingroup$ Sure, the problem is that (semi-)blind receivers are no simple thing, there is a plethora of techniques, most of them are of iterative nature. It would require a lot more to explain such receivers and thus make any corresponding code readable. I chose not to go into detail about specific algorithms, since the question was refering to the concept of semi-blind receivers generically. $\endgroup$ – Florian Nov 18 '19 at 16:50

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