I'm studying some of the basic equalizing structres and I understand how Zero-forcing works, but it seems to me that a known channel impulse response is needed. Am I right? If so, what's the point? I mean, you're not likely going to know how the channel is, so how Zero-forcing is in any way useful?
Answer
If the received signal can be written as
$$\mathbf{y} = \mathbf{H}\,\mathbf{x} + \mathbf{n}$$
where $\mathbf{H}$ is the channel matrix, $\mathbf{x}$ is the transmitted vector, and $\mathbf{n}$ is the AWGN of the channel, then a zero forcing equalizer is simply (assuming that the channel matrix is square, and it's estimated perfectly at the receiver)
$$\mathbf{H}^{-1}\mathbf{y} = \mathbf{x}+\mathbf{H}^{-1}\mathbf{n}$$
Obviously, you need the channel impulse response, which is captured in the channel matrix. This channel matrix is estimated in practice using any channel estimation technique, but the estimation is usually not perfect, and thus the aforementioned ZF equalizer serves as a theoretical limit.
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