The channel matrix in wireless communication is a matrix that describes the impact of the channel on the transmitted signal. The channel matrix can be used to model the effects of the atmospheric or underwater environment on the signal, such as the absorption, reflection or scattering of the signal by surrounding objects.
When addressing multi-antenna communication, the term "channel matrix" is used. Let's assume that only one TX and one RX are in communication and there's no surrounding object. Here, in our case, we can apply the proper threshold condition to a received signal and get the original transmitted signal at the RX side. However, in real-world situations, we see signal path blockage, reflections, etc., (NLOS paths [↗]) more frequently. The obstruction is typically caused by building walls, etc.
Multi-antenna communication was introduced to address this issue. It makes diversity approaches possible, greatly increasing the likelihood of the signal being received.
Let me show an example to describe the channel matrix. Assume that the TX and RX communication antennas each have two antenna elements. T1, T2, and R1, R2 are the corresponding TX and RX MIMO antennas.
The complex channel gain between T1 and R1, T1 and R2, T2 and R1, and T2 and R2 is represented by the channel matrix, H.
In a channel matrix, for example, the elements h11 and h21 each represent the complex channel gain between R1 and T1 antennas, R2 and T1 antennas, and so on.
Example of a 4 X 16 Channel Matrix:
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The sample shown above is a 4 x 16 channel matrix demonstration. In this illustration, there are 16 TX antennas and 4 Rx antennas. We diagonalize the channel matrix to allow communication between T1 and R1, T2 and R2, and so on, in order to enable practical MIMO antenna communication. Interference is any signal that is received at R1 from T2, T3, and so on, etc. By diagonalizing data, it is possible to minimize signal interference between many simultaneous data streams.
The Importance of Channel State Information (CSI)
For systems to effectively utilize the channel matrix, especially for diagonalization, the transmitter often needs to know the Channel State Information (CSI). CSI refers to the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of scattering, fading, and power decay with distance. With accurate CSI, sophisticated signal processing techniques can be applied at the transmitter (e.g., precoding) and receiver (e.g., spatial multiplexing or beamforming) to optimize data rates and reliability. Without CSI, or with outdated CSI, the benefits of MIMO systems are significantly reduced, often limiting performance to simple diversity gains rather than the full capacity enhancements possible with spatial multiplexing.
What is rank of a channel matrix?
The rank of the channel matrix is evolving into a crucial wireless communication parameter as we move steadily toward MIMO and higher frequency transmission. The number of the stronger independent data streams that can travel between the TX and RX in MIMO communication is indicated by the rank of the channel matrix.
Implications of Channel Rank:
- Spatial Multiplexing Capacity: The rank directly determines the maximum number of parallel data streams (or spatial multiplexing gain) that can be supported by the MIMO channel. A higher rank means more independent paths, allowing more data to be transmitted simultaneously, thus increasing data throughput.
- Impact of Environment: In rich scattering environments (e.g., urban areas with many reflections), the channel matrix tends to have a higher rank, which is beneficial for MIMO performance. In line-of-sight (LOS) scenarios or environments with very few scatterers, the rank can be lower, limiting the spatial multiplexing gain, even with many antennas.
- Antenna Selection: Understanding the rank helps in optimizing antenna configurations and selecting the most effective transmit and receive antenna pairs to maximize the number of usable data streams.
Procedure of finding rank of channel matrix in MATLAB [click here]
Python code to find rank of a matrix [click here]
What is condition number of a channel matrix:
We can determine the strength of a channel matrix's maximum singular value by comparing it to its lowest singular value using the condition number.
Implications of the Condition Number:
- Channel Robustness: The condition number is a measure of the "robustness" or "well-behavedness" of the channel. A low condition number (closer to 1) indicates a well-conditioned channel where all independent data streams (eigenmodes) have similar strengths. This means the channel is stable, and small perturbations or noise won't drastically affect the received signal.
- Sensitivity to Noise and Interference: A high condition number implies an "ill-conditioned" channel. In such a channel, some data streams are significantly weaker than others. Attempting to transmit data over these very weak streams makes the system highly susceptible to noise and interference, potentially leading to significant errors or requiring much higher transmit power for those specific streams. This also impacts the effectiveness of signal detection algorithms at the receiver.
- Practical System Design: System designers often aim for channels with lower condition numbers to ensure stable and reliable communication. Strategies like antenna placement, adaptive modulation and coding, or even adding artificial scattering (though less common) can indirectly influence the channel's condition number to improve performance.
MATLAB code to find condition number of a channel matrix. [go]