﻿ MIMO in LTE for 2X2 Downlink DL and Uplink UL antennas. Multiple Input Multiple Output. Mathematical equations
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LTE supports data rates upto 100 Mbps for downloads and 50 Mbps for uploads. The the high throughput needs to be transmitted over the air  without causing packet corruption or packet loss.  This is also used in WiMax antenna as an option
LTE MIMO - Multiple Input Multiple Output
MIMO is discussed below for 2x2 antennas
Before we discuss the mathematical equations for MIMO, lets first discuss the theoretical setup of a 2X2 MIMO.
LTE MIMO.

LTE downlink by default requires MIMO 2x2 antenna configuration (In WiMax its optional). By MIMO 2x2 we mean 2 transmit antennas at the base station and 2 receive antennas at the mobile. Antenna configurations with MIMO 4x4 are also discussed in the standard.

MIMO schemes for LTE Uplink are different in comparison to Downlink. Uplink (UL) is a data link in which Mobile is transmitter and eNodeB is receiver.  In order to reduce complexities in mobile UL path, a different antenna scheme is used. This scheme is called MU-MIMO., where  MU stands for Multiple Users.
In MU-MIMO, multiple user terminals can transmit simultaneously on the same resource block.  Only one transmit transmit antenna is required for mobile. This techniques is only possible with use of orthogonal SC-FDMA modulation.
MIMO MIMO DL Frame LTE Uplink MIMO AAS Antenna Diversity LTE HARQ HARQ Type II ARQ vs. HARQ
h11
h12
h21
h22
Matrix channel and parameters
h11, h12, h21 and h22
Transmitter
From the setup of 4G LTE 2x2 MIMO, we can observe that each receiver is getting two streams of data one from Transmitter 1 and other from transmitter 2.  The straight line is a direct path whereas angle path is an indirect path. The direct connections are always denoted with same numbers like h11 or h22. The indirect components are denoted with combinations of different transmitters, like h12 or h21.
If we characterize the Transmitters as x and Receiver components as Y, we can get our equation for MIMO as
This technique supports high data rates. This approach of combining signals allows better resistance against interferers like - multipath, channel fading etc.
Mathematical formula for the theoretically equivalent of channel capacity is referred from Shannon- Hartley theorem, discussed below.

Capacity = BW x log2 (1 + SNR)

For MIMO the same equation is going to allow the Capacity to be multiplied by a factor equal to the number of transmitters in the MIMO system.

Capacity = N x BW x log2 (1 + SNR)
Serving System (S) Architecture (A) Evolution (E) Gateway or Packet Data Network (PDN) SAE Gateway
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Multi-antenna scheme - 2x2 MIMO:

MIMO techniques are multi-antenna techniques used in wireless devices to support higher data rates into radio links or RF. MIMO stands for Multiple Input Multiple Output antenna system. A set of mathematical equations are used to determine performance improvements.  We will be discussing the 2x2 MIMO antenna architecture below.
Interview Questions. Main, FPGA, Digital Fundamentals
More details on LTE MIMO and frame format from Next
When we combine all the components in one matrix we get Matrix channel or transmission matrix H. Can also be extended for 4x4 LTE MIMO
Y = Hx
Adding Guassian Noise and G we will get our final equation.
Y = Hx +G
H =   h11  h12

h21  h22