# Degrees of freedom in coding (for a communication channel)

I was reading on my lecture slides about how to achieve diversity (i.e. transmit more copies of the same message in order to avoid its corruption; this technique may be realized by repeating the message in different instants, by sending it at different frequencies, and by using multiple antesas) in a communication system.

Consider now a MIMO channel with two transmit and two receive antennas:

One can also design codes specifically for the transmit diversity system (space‐time codes). The simplest of these, and yet one of the most elegant, is the so‐called Alamouti scheme. This is the transmit diversity scheme proposed in several third‐generation cellular standards and is designed for two antennas (but can be extended to more antennas to some extent):

With BPSK symbols, it can be shown that the repetition scheme requires a factor of 2.5 (4 dB) more power than the Alamouti scheme. The repetition scheme suffers from an inefficient utilization of the available degrees of freedom in the channel: over the two symbol times, bits are packed into only one dimension of the received signal space, namely along the direction . In contrast, the Alamouti scheme spreads the information onto two dimensions – along the orthogonal directions

The degrees of freedom (DoF) of a channel can be defined as the dimension of the received signal space. In a channel with two transmit and a single receive antenna, this is equal to one for every symbol time. The repetition scheme utilizes only half a degree of freedom per symbol time, while the Alamouti scheme utilizes all of it.

Now, I need a basic and simple explanation of what is happening. Precisely, what does it mean in practice "degrees of freedom" of a channel?

Consider for instance the Alamouti scheme, in which the message u is described through a 2x2 matrix (with columns (u1, u2) and (-u2*, u1*). Degrees of freedom is the rank of this matrix, if I have understood it correctly. But what does it mean in practice? Which properties should the message have to achieve this?

• Yes the degree of freedom is the rank of channel matrix, which is also the dimension of its column space ("the dimension of the received signal space" as in your lecture slides). What it means in practice is that if we can model MIMO propagation as a matrix linear operation, this DoF is the number of independent paths provided by the MIMO channel with linear processing at TX and/or RX. One of space-time code design goal is to achieve full DoF. Jun 3, 2020 at 13:27