Does zero-forcing equalizer need known channel impulse response?

Question

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 1

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.

Answer 2

You need an estimate of the channel to receive the sequence but the zero-forcing equalizer does not need the channel response as an input. The zero forcing equalizer estimates the channel response. This can be done either with a training sequence, or can be decision directed when signal to noise ratios are high enough.

Given the received signal is the convolution of the transmitted signal and the channel, if we know what the transmitted signal was for a known received signal (as is the case with a training sequence), we can then mathematically solve for the channel, this is the channel estimation process. The type of equalizer is defined by the channel estimation process used.

Some further details:

A zero-forcing equalizer forces the inter-symbol-interference (ISI) to be zero at the decision sampling locations (hence "zero-forcing") but does not take into account the effects of noise so it is usually only considered for use in high SNR conditions on static channels. In contrast, a Minimum Mean Square Error (MMSE) equalizer does not result in zero-ISI, but minimizes the total noise between ISI and noise contributions. In high-SNR the MMSE equalizer will converge to be the same as a zero-forcing equalizer, so has better performance over all conditions. For the case of frequency selective channels (deep nulls in the frequency passband), a Decision-Feedback Equalizer is often a better choice.