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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 matrices 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, which helps decrease or cancel (theoretically) interference between any two data streams.
For a MIMO system, the channel matrix can be effectively **diagonalized through techniques like Singular Value Decomposition (SVD)**. This process transforms the complex multi-path channel into an equivalent set of independent sub-channels. Let's consider a general channel matrix H:
H =
2 0 2
0 1 2
0 1 0
(H = Channel Matrix, in a realistic scenario, H would typically be filled with non-zero complex numbers representing channel gains between transmit and receive antennas)
For a typical wireless communication system, the signal equation for a single stream is:
y = h * x + n
However, for a multi-antenna MIMO system, this becomes a matrix equation where y, x, and n are vectors:
y = H * x + n
When appropriate precoding and combining matrices are applied, the effective channel becomes diagonal. Let the effective diagonal channel be D:
D =
d11 0 0
0 d22 0
0 0 d33
Now, the signal equation for the received data streams, after combining, can be viewed as:
y_eff = D * x_tx + n_eff
Where y_eff, x_tx, and n_eff are the effective received signals, transmitted symbols, and noise vectors, respectively, after applying precoding and combining.
This makes it much simpler to retrieve all independent data streams sent from the transmitter (TX):
y_eff1 = d11 * x_tx1 + n_eff1
y_eff2 = d22 * x_tx2 + n_eff2
y_eff3 = d33 * x_tx3 + n_eff3
If the combining matrix is W and the precoding matrix is F, then the relationship illustrating the diagonalization of the channel can be expressed as:
W^H * H * F = D
This occurs when the precoding F matrix is applied on the TX side and the W matrix is applied (as a conjugate transpose or inverse, denoted by W^H) on the RX side.
Let's look at a simplified 2x2 example for clarity:
Assume a channel matrix H and ideal precoding F and combining W^H matrices are found such that:
H =
1.0 0.5
0.5 1.0
And through a process like SVD, we derive F and W^H that perfectly "diagonalize" this channel. For illustrative purposes, imagine the result is:
W^H * H * F = D =
1.5 0.0
0.0 0.5
Here, d11 = 1.5 and d22 = 0.5. If we transmit two data streams, x_tx1 and x_tx2, the effective received signals will be approximately:y_eff1 = 1.5 * x_tx1 + n_eff1
y_eff2 = 0.5 * x_tx2 + n_eff2
As you can see, the two data streams are now isolated and can be independently decoded without interference from each other, even though they passed through the original complex channel H.
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 (often via SVD). These weights transform the original channel into an equivalent set of independent sub-channels, allowing each data stream to be independently retrieved. Magnitude and phase terms are included in these weights, which are frequently utilized in digital communication.
Further Reading
[1] Optimal Precoding in MIMO using SVD