Basic Idea Behind Multiuser Alamouti STBC
In a MIMO (Multiple Input, Multiple Output) system, we use multiple antennas at both the transmitter and the receiver to improve performance (better data rate, reliability, etc.).
The Alamouti Space-Time Block Code (STBC) is a method used to send data in such a way that it becomes more robust to noise and fading.
Single-User Alamouti Example:
Let’s first recall the basic Alamouti code for one user with two antennas:
- At Time 1:
- Antenna 1 sends \( s_1 \) (the first data symbol).
- Antenna 2 sends \( s_2 \) (the second data symbol).
- At Time 2:
- Antenna 1 sends \( -s_2^* \) (the complex conjugate of \( s_2 \)).
- Antenna 2 sends \( s_1^* \) (the complex conjugate of \( s_1 \)).
This is the Alamouti STBC for one user.
Multiuser Alamouti STBC (For More Than One User):
Now, let’s imagine multiple users (e.g., 2 or more users) in the same system, each of them using Alamouti STBC.
Each user has two antennas and will transmit symbols using Alamouti's encoding scheme.
Let's Break it Down Step by Step:
1. Two Users Example:
Let’s say we have two users:
- User 1 with two antennas.
- User 2 with two antennas.
Each user will transmit two symbols using Alamouti STBC, as described earlier. Here's how the signals would look like:
- User 1:
- Time 1: Sends \( s_{1,1} \) from Antenna 1 and \( s_{1,2} \) from Antenna 2.
- Time 2: Sends \( -s_{1,2}^* \) from Antenna 1 and \( s_{1,1}^* \) from Antenna 2.
- User 2:
- Time 1: Sends \( s_{2,1} \) from Antenna 1 and \( s_{2,2} \) from Antenna 2.
- Time 2: Sends \( -s_{2,2}^* \) from Antenna 1 and \( s_{2,1}^* \) from Antenna 2.
2. Received Signals at the Base Station:
Now, assume the base station has multiple antennas (say 4 antennas). The received signal at each antenna will depend on the signals from both users and the channel between the transmitters and the receiver.
Let’s define:
- \( H_1 \) = Channel from User 1 to the base station.
- \( H_2 \) = Channel from User 2 to the base station.
- \( n \) = Noise at the base station.
The received signal at the base station will look like this (for each time slot):
\(
y(t) = H_1 x_1(t) + H_2 x_2(t) + n(t)
\)
Where:
- \( x_1(t) \) is the transmitted signal from User 1 at time \( t \).
- \( x_2(t) \) is the transmitted signal from User 2 at time \( t \).
This means the receiver sees a combination of signals from both users, along with some noise.
3. Breaking Down the System:
- The received signal at the base station contains signals from both users (each user using Alamouti STBC) and noise.
- The base station has to separate the signals of both users. This is where the multiuser detection comes into play.
In a real system, we might use algorithms like Zero-Forcing or MMSE (Minimum Mean Square Error) to separate the signals from different users. These algorithms try to remove interference between users and decode the original symbols.
4. Mathematical Representation (For Two Users):
If we have 2 users, the system equation for the received signal at the base station over two time slots will be:
\(
y_1 = H_1 x_1 + H_2 x_2 + n
\)
Where:
- \( y_1 \) is the received signal at the base station for the first time slot.
- \( x_1 \) and \( x_2 \) are the transmitted signals from User 1 and User 2.
- \( H_1 \) and \( H_2 \) are the channel matrices from User 1 and User 2 to the base station.
- \( n \) is the noise vector.
The same equation applies for the second time slot, just with the signals transmitted in the second time slot.
Summary:
- Each user (with two antennas) uses Alamouti STBC to encode two symbols.
- The base station receives a combination of signals from multiple users and needs to decode each user’s symbols.
- The decoding process involves multiuser detection to separate the users' signals.