Alamouti Scheme Explained

Alamouti Scheme

Alamouti's Space-Time Block Coding (STBC) is a fundamental technique in MIMO wireless communication used to achieve transmit diversity. In a 2x1 system, it allows the receiver to recover symbols effectively even under severe fading conditions without requiring channel state information at the transmitter.

1. The Precoding Matrix

The Alamouti precoding matrix defines the symbols transmitted from two antennas over two consecutive time slots:

X = [ s1 , -s2* ; s2 , s1* ]

(Columns represent Time Slots, Rows represent Antennas)

2. Channel Model & Matrix Equations

In a 2x1 MIMO system, let h1 and h2 be the complex channel gains from Transmit Antenna 1 and 2 to the receiver, respectively. The received signals y1 and y2 over two time slots are:

Time Slot 1: y1 = h1*s1 + h2*s2 + n1
Time Slot 2: y2 = -h1*s2* + h2*s1* + n2

Taking the complex conjugate of the signal in the second time slot, we can represent the system in matrix form:

[ y1 ; y2* ] = [ h1 , h2 ; h2* , -h1* ] [ s1 ; s2 ] + [ n1 ; n2* ]

The effective channel matrix H_eff is orthogonal.

3. Orthogonality Property

The beauty of the Alamouti scheme lies in its orthogonality. Let x1 and x2 be the vectors transmitted in time slots 1 and 2:

x1 = [ s1 ; s2 ] and x2 = [ -s2* ; s1* ]

The inner product x1^H * x2 is:
x1^H * x2 = [s1* , s2*] * [-s2* ; s1*] = -s1*s2* + s2*s1* = 0

This zero result proves that the transmitted sequences are perfectly orthogonal, allowing the receiver to decouple s1 and s2 using a simple linear combiner.

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