Hello, The usual setup: Suppose that I can model a channel as FIR filter h such that the received signal is y=h*x+w (*: convolution; x: transmitted signal; w: measurement noise). The goal is to find an FIR equalizer g such that xhat=g*y is close to x. Writing x/y as vectors, the relation can be written as y=Hx+w. The equalizer xhat=Gzf y with Gzf = (H^T H)^-1 H^T is called the ZF equalizer. It's just inverting the channel via Least Squares. Similarly, I know that xhat=Gmmse y with Gmmse = Cxx H^T (H Cxx H^T + Cww)^-1 is the MMSE solution. (Cxx: auto correlation matrix of input; Cww of measurement noise). Question 1: According to Wikipedia, the alternative form is Gmmse = (A^T Cww^-1 A + Cxx^-1)^-1 A^T Cww^-1 from which can be seen, that it amounts to weighted least squares combined with Tikhonov regularization. Cww=I and Cxx=I results in ordinary Least Squares. But in my case, Cxx is not invertible. Hence both forms cannot be equivalent. What is the difference between them and how would the first form result in ordinary Least Squares? I would like to understand the edge case in which the first form becomes ordinary Least Squares (and hence ZF solution). Question 2: I can also write the equation xhat=g*y+w as xhat=Yg+w where g is the coefficient vector of the equalizer and can be directly estimated without knowledge of the channel using LS. What kind of equalizer is this? ZF? MMSE? Something else/in between? (In my simulations, it gives different results... ). Question 3: Can I obtain the MMSE equalizer only from y and x? I.e. using the form xhat=Yg (rather than xhat=Gy)? Question 4: Can the LMS algorithm be interpreted as a sample based approximation to any of the above? Which equalizer would it correspond to? (I would think MMSE equalizer. But this is inconsistent because (a) the matrix version of MMSE requires the channel estimate h. And (b) I think RLS can be interpreted as Weighted Least Squares solution.) Question 5: What is the relation between LMS, RLS and Kalman filter? Can they be interpreted as sample based approximations of LS, WLS, Tikhonov Regularization? If I am completely wrong I would be happy if someone could explain the relationship between those. Peter

# Linear equalizers and similarities

Started by ●July 18, 2017