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What are Precoding and Combining Weights / Matrices in a MIMO Beamforming System


 

Figure: configuration of single-user digital precoder for millimeter Wave massive MIMO system

Precoding and combining are two excellent ways to send and receive signals over a multi-antenna communication process, respectively (i.e., MIMO antenna communication). The channel matrix is the basis of both the precoding and combining matrices. Precoding matrices are typically used on the transmitter side and combining matrixes on the receiving side. The two matrices allow us to generate multiple simultaneous data streams between the transmitter and receiver. The nature of the data streams is also orthogonal. That helps decrease or cancel (theoretically) interference between any two data streams.

The channel matrix is first properly diagonalized. Diagonalization is the process of transforming any matrix into an equivalent diagonal matrix, where all of the diagonal elements are non-zero and all of the other elements are zeros. Let me explain an example,

H =
     2     0     2
     0     1     2
     0     1     0
(H = Channel Matrix)

For a typical wireless communication system,
y = h*x + n

We can now calculate the corresponding diagonal matrix for H and also assume the diagonal matrix is D.
D =
h11       0          0
0         h22        0
0           0        h33

Now, 
y = D*x + n

You will now find it much simpler to retrieve all independent data streams sent from TX.
y1        h11       0          0     x1 
y2 =     0         h22        0   *x2 + n
y3        0           0        h33   x3 

or,
y1 = h11*x1 + n
y2 = h22*x2 + n
y3 = h33*x3 + n

If the combining matrix is W and the precoding matrix is F.

F*H*W = D

When the precoding F matrix is used on the TX side and the W matrix is combined on the RX side.

Summary:

In an environment with many scatterers, modern wireless communication systems use spatial multiplexing to increase data flow within the system. To transmit multiple data streams over the channel, a set of precoding and combining weights is derived from the channel matrix. Then, each data stream can be independently retrieved. Magnitude and phase terms are included in these weights, which are frequently utilized in digital communication.


Also read about

[1] Beamforming in MIMO

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