## Optimal Precoding for Millimeter wave Massive MIMO Systems

In case of MIMO system we deploy multiple transmitter antennas at receiver side and multiple receiver antennas at receiver side. MIMO technology was introduced to support multiple simultaneous data streams between transmitter and receiver to multiply the capacity of a system. But there is also interference between multiple data streams. Precoding technique minimizes the interference between multiple data streams.

## What Exactly Precoding Technique is

We all are familiar with the channel matrix of an MIMO system, that looks like, **H **=

**\ R1 R2 R3 R4**

**T1 **h11 h12 h13 h14

**T2 **h21 h22 h23 h24

**T3 **h31 h32 h33 h34

**T4** h41 h42 h43 h44

Here, in the above figure channel matrix, **H **is shown. In channel matrix it shown different gains between different antennas. Now, we see in the above matrix for example, h11 represents the channel gain between transmitter antenna, T1 and receiver antenna, R1 and h11 also means connection between the antennas as well. R1 also receives the signals from T2, T3, and T4 antennas too. So, there is some kind of interface between multiple data streams when we process the signal at receiver side. Here, precoding help us to reduce interference between multiple data streams.

## Optimal Precoding in MIMO

Typically, received signal at receiver side is represented as,

**y = Hx + n **.....(i)

Where, **H **is channel matrix gain

**y = **Received signal vector

**x **= Transmitted signal vector

**n **= Additive white Gaussian noise

Here, in the above equation you can image channel matrix, **H **as same as above channel matrix where we've shown channel gains between TX side antennas T1, T2, T3, T4, and receiver side antennas, R1, R2, R3, R4, respectively. We've also talked about interference with T1's signal at R1 antenna due to transmission from T2, T2, and T3.

Now, let imagine your channel matrix looks like that, **H **=

\ **R1 R2 R3 R4**

**T1 **h11 0 0 0

**T2 **0 h22 0 0

**T3 **0 0 h33 0

**T4 **0 0 0 h44

Now in equation (i), if you the put the above channel matrix value then you see there is no interference with T1' signal with T2, T3, and T4's transmission at receiver R1.

Similar approach is performed for optimal precoding technique we channel matrix is decomposed in to two unitary matrix **U, V, **and one diagonal eigen value matrix, **Î£**. We've already talked about "**Singular Value Decomposition in MIMO Channel" **in a separate article.

There is matrix, Î£**, **we operate row and column matrix in a such way that Î£** **becomes diagonal matrix where elements are in descending order. We do that by operating multiple operations in **U **& **V **matrix as shown in the above mentioned article.

Generally, **H **matrix is decomposed into, **H = U**Î£**V****H**

As **U **and **V **are unitary matrix so, multiplication of those matrix with its hermitian matrix itself are identity matrix. Alternatively, **UU****H**** ****= V****V****H**** = I**

**Signal Processing at Receiver Side for Optimal Precoding**

During transmission we multiply original message signal vector with unitary matrix, **V. **So, now transmitted signal vector becomes, V**x**. On the side at receiver side, received signal vector is multiplied with vector **U****H**. So, as per above equation (i), received signal vector at receiver side as follows

**y = ****U****H**** (U**Î£**V****H**) Vx + n

**y= I**Î£Ix + n

y = Î£x +n

Now, you see Î£ is a diagonal matrix and signal vector, **x **is multiplied with that diagonal matrix. So, you can observe there the simultaneous data streams between MIMO transmitter and receiver antennas without interference among them. Now we further do optimal power allocation to each antennas to maximize sum-rate or overall throughput as shown in a separate article. There is the URL link above.

Why OFDM precoding modulation used in uplink?