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I have a system with no diversity and the channel's coefficients change every $T = 10\text{ bits}$ transmitted. The channel estimation system has a flaw and detects the channel's coefficients equal to $h$ while their real values are $h+c$ where $c$ is a constant equal to $1$.
Is there any method I can apply to achieve a better BER than simply using the faulty $h$?
I am using BPSK.
The strategy depends heavily your "realistic scenario", e.g. which estimator, which equalizer, which estimation error, etc.
In general scenario, you cannot trust the estimate of $h$, it means that you cannot use coheent detection. Thus try non-coherent detection (detection by energy) : to send bit $0$, use 2 channel uses (2 symbols) $x_0 = [x[0], x[1]] = [1, 0]$ and to send bit $1$, use $x_1 = [x[0], x[1]] = [0, 1]$. The receive symbols are $y = [y[0], y[1]]$
In the well-known Rayleigh flat fading channel, i.e. $y[m] = h[m]x[m]+w[m]$, the optimal detection rule will be \begin{align} \hat{x} &= x_0 \textrm{ if } |y[0]|^2 > |y[1]|^2\\ \hat{x} &= x_1 \textrm{ otherwise} \end{align}
