# NLMS algorithm for a MISO structure

I am trying to implement an NLMS algorithm for a multi-input single-output(MISO) structure.

We take a reference signal x, then we made a new set of P input signals from it as follows: x_op (k) = x(k)^p. k denotes the k-th sample of our reference signal x.

For the case where P = 1, our adaptive filter is just a vector of length N. But for P > 1, we have a matrix H (N x P) containing all the adaptive filters. (where N is the length of each filter).

Now I am confused as what would be dimenstion of the estimated output at each iteration (for each sample). For the case where P = 1, we can simply write:

y_hat = h_hat' * xk; where xk has the last N samples of x.

and y_hat is then a scalar. What about this case when H is a matric and not a vector anymore?

The way I understand it, y_hat for this case is no longer a scalar. But if it's not a scalar, then how I am supposed to define the error for each sample k, in order to write the adaption law equation?

• Widrow and Stearns in their Adaptive Filters book yield a nice example of such multiple input adaptation for LMS algorithm. Have you seen that ? Mar 7 '19 at 20:37
• No, and unfortunately I do not have access to that book right now. Mar 7 '19 at 20:55
• ok. It's much appreciated, and also useful, if you could upload a schematic diagram of your particular multiple input single output (MISO) system. Mar 7 '19 at 21:42