# 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 usual approach is to include a "training sequence" along with the data. The training sequence is known to both transmitter and receiver.
– MBaz
Oct 25, 2016 at 16:38
• Thank you for your comment. This means that since such quantities are estimated once from the training sequences, the equalizers in practice does not time varying coefficients. Right? Because in research literature it is often commented that the equalizer coefficients are needed to be computed for every index $k$ which adds to receiver complexity Oct 25, 2016 at 16:43
• Usually you re-train the equalizer periodically. The wireless channel will stay constant for a duration known as the coherence time; after that time, you'll need to train the equalizer again.
– MBaz
Oct 25, 2016 at 17:06
• ok. Does this mean that for time duration $[T_1,T_2]$ when channel is considered static, the equalizer coefficients are computed only once using the training sequence ? Then in next time slot $[T_2,T_3]$ the whole procedure is repeated for new channel information. Oct 25, 2016 at 17:10

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".