I am having difficulty in understanding how the Authors in this paper, An EM based method for semi blind identification of linear systems driven by Chaotic signals
have used the expressions derived from applying an EM_EKF estimation method when the driving process is a non-linear input. They have considered the noise model as AWGN. They then apply their method for in equalization. They apply the estimators derived in Eq(12)--Eq(15) to estimate the MA process with Rayleigh fading coefficients
The channel is a linear filter with taps whose magnitudes are modeled as Rayleigh random variables. Assuming, the input $(\mathbf{z}_n)$ take in 8 symbols from QAM constellation instead of two symbols $\pm1$ as in BPSK. The model is
$$ y_n = \mathbf{H} (\mathbf{z}_n) + w_n \tag{1} $$ expanding $$y_n = h_1 z_{n}+ h_2 z_{n-1} + h_3 z_{n-2} + w_n \tag{2}$$
with channel coefficients as $\mathbf{h} = [h_1,h_2,h_3]$.
$$\mathbf{z}_n = [z_n,...,z_{n-p+1}] = [z_{n},z_{n-1},z_{n-2}] \tag{3}$$
$\mathbf{y}= [y_1,y_2,y_3, ...]$ is a one dimensional array (scalar valued) noisy output of n = 1,....,N
data points.
The number of multipaths = 3. The Authors say that they have used a spreading sequnce in equalization. If so, would (1) still be the model and the expression for the estimators derived in the paper would not change? It is not clear form the paper how they have applied their method in equalization.
I am not aware what would be the functional form of the model in equalization application when a spreading sequence is used. Can somebody please explain how the method is applied in equalization using spreading sequence, the model (FIR) equation and how to apply the estimators using the spreading sequence. (An explanation with implementation would also be useful, if possible).
Thank you.