The Alamouti scheme is a simple and effective space-time block coding (STBC) technique used in wireless communications to achieve diversity gain. It's designed for systems with two transmit antennas and one or more receive antennas, providing transmit diversity.
Alamouti's Space-Time Block Coding (STBC) is a technique used in MIMO wireless communication systems to achieve diversity gain without requiring channel knowledge at the transmitter.
Alamouti 2 X 1 Matrix Equation Representation
The Alamouti precoding matrix is constructed based on the Alamouti code, which defines the phase relationships between the symbols transmitted from different antennas over two consecutive time slots. For a 2x1 MIMO system (two transmit antennas and one receive antenna), the Alamouti precoding matrix is as follows:
Precoding Matrix=[s1 −s2∗; s2 s1∗]
Where:
s1 and s2 are the symbols to be transmitted from the two antennas in the current time slot.
s1∗ and s2∗ are the complex conjugates of s1 and s2 respectively.
This matrix ensures that the symbols transmitted from the two antennas in the current time slot have the necessary phase relationships to achieve diversity gain at the receiver.
Here's how the Alamouti precoding matrix works:
In the first time slot, symbols s1 and s2 are transmitted from the two antennas without any phase manipulation.
In the second time slot, symbols −s2∗ and s1∗ are transmitted from the two antennas. The negative sign and complex conjugate ensure the correct phase relationship required for diversity gain at the receiver.
At the receiver, combining the signals from the two time slots using Alamouti decoding allows for effective recovery of the transmitted symbols, even in the presence of fading.
By using Alamouti's STBC and the corresponding precoding matrix, the MIMO system can achieve diversity gain and improve performance without requiring explicit channel knowledge at the transmitter.
Orthogonality Property
Alamouti's Space-Time Block Coding (STBC) scheme ensures that symbols transmitted from different antennas in successive time slots are orthogonal to each other. This orthogonality property is essential for enabling simple decoding at the receiver and achieving diversity gain without requiring channel knowledge at the transmitter.
Now, let's calculate the inner product (dot product) between two encoded symbols transmitted from different antennas in successive time slots.
Let x1x1 and x2x2 be the encoded symbols transmitted from the two antennas in the first and second time slots respectively.
x1=[s1 ; s2]
x2=[−s2∗ ; s1∗]
The inner product x1' * x2 is given by:
x1' * x2 = [s1 ; s2] * [−s2∗ ; s1∗]
=−∣s2∣^2 + ∣s1∣^2
Since the symbols s1 and s2 are independent and identically distributed (IID) random variables with equal power, their magnitudes are equal, i.e., ∣s1∣=∣s2∣. Therefore, the inner product x1' * x2 simplifies to:
x1' * x2 = −∣s2∣^2 + ∣s1∣^2 = 0x1Tx2= −∣s1∣^2 + ∣s1∣^2 = 0
This shows that the inner product between the encoded symbols transmitted from different antennas in successive time slots is zero, indicating orthogonality.
This orthogonality property allows the receiver to effectively decode the transmitted symbols by taking advantage of the diversity provided by the multiple antennas without interference between symbols transmitted from different antennas.
(Get MATLAB Code for Alamouti's Precoding Matrix for 2 X 2 MIMO in MATLAB)
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