Chapter 2: Channel Estimation

MIMO Channel¶

In a classical wireless MIMO system, the true channel is the channel matrix $\mathbf{H} \in \mathbb{C}^{N_r \times N_t}$, which describes the propagation characteristics between $N_t$ transmit and $N_r$ receive antennas, affected by noise, fading, and interference. The received signal is:

where $\mathbf{X} \in \mathbb{C}^{N_t \times T}$ is the pilot matrix, $\mathbf{N}$ is additive white Gaussian noise, and $\mathbf{I}$ is interference. The estimated channel $\hat{\mathbf{H}}$ is derived from $\mathbf{Y}$, often using methods like least squares ($\hat{\mathbf{H}}_{\text{LS}} = \mathbf{Y} \mathbf{X}^\dagger$) or advanced techniques like diffusion models, as discussed in your prior questions.

You propose a metaphorical "entangled channel" that exists between the true channel $\mathbf{H}$ (real) and the generated channel $\hat{\mathbf{H}}$ (fake), inspired by the GAN framework. In a GAN, a generator produces fake data to mimic real data, and a discriminator tries to distinguish between them. As training progresses, the generated data becomes so similar to the real data that the discriminator cannot reliably tell them apart. You are asking whether a similar concept can apply to channel estimation, where an "entangled channel" (a hybrid or intermediate representation) is so close to both $\mathbf{H}$ and $\hat{\mathbf{H}}$ that distinguishing whether it is real or fake is challenging.