How to equalize an ISI channel when the transmitted symbols are unknown? I am using MATLAB to simulate transmission of binary data sequence, $\mathbf{x}={x_k\in\{+1,-1\}\forall 0\leq k\leq N}$, over an ISI channel whose impulse response $\mathbf{h}=[h_0 h_1 \cdots h_{\mu}]$ is perfectly known. My channel observation sequence is $\mathbf{y}$ is a noisy version of filtered $\mathbf{x}$ that can be expressed mathematically as

$$y_k=Hx + w_k,$$

where $w_k\sim\mathcal{N}(\mu,\sigma_w^2)$.

For such system, equalizers can be designed to mitigate the effect of ISI. This often includes computing quantities like autocorrelation matrices $R_{xx}=E[x^{}x^T]$ and $R_{yy}=E[y^{}y^{T}]$, and cross correlation $R_{yx}=E[y^{}x^{T}]$

In MATLAB, I have access to $\mathbf{x}$ and I know noise variance $\sigma_w^2$ so computing $R_{xx},R_{yy},$ and $R_{yx}$  is straight forward.

My question is how do we compute such quantities in practice when we only have access to $\mathbf{y}$, our channel observation and  we don't know what input sequence $\mathbf{x}$ is?
The classic approach is to include a "training sequence" along with the data. The training sequence is known to both transmitter and receiver, so it can play the role of $\mathbf{x}$ in your question.

Now, in practice most wireless channels fluctuate over time. This means that the vector $\mathbf{h}$ is not fixed, but changes randomly in time. If the symbol rate is not too high, then the channel will remain essentially fixed during the transmission of several symbols. This time is called the "channel coherence time". When the symbol time is much smaller than the coherence time, we say the the channel exhibits "slow fading".

Over a slow fading channel, the training sequence is repeated approximately once every coherence time. During one coherence time interval, the equalizer remains fixed.