MMSE Detection for Spatial Multiplexing MIMO (SM-MIMO)

• General solution of minimum-mean square error detection (MMSE)
– Problem description: Let us assume we have desired signal d[n] and d[n] is corrupted somehow into u[n].
Then observing u[n] and using the linear filter W we want to detect d[n]. The problem description is
illustrated in Fig. 2.
d[n] ; desired signal
u[n] ; input signal to MMSE (or Wiener) filter which contains ”desired signal d[n]”
y[n] ; estimated d[n], i.e., ˆd[n]
As an example, u[n] = d[n] + z[n] for AWGN and u[n] = h[n]d[n] + z[n] for fading channel.
Figure 2: Problem description of Wiener filter.
Now we want to find the optimum weight vector W = [w 0 , w 1 , · · · , w K−1 ] T such that the mean-squared
error is minimized. Denote the error is defined as
where
e[n] = d[n] − y[n]
y[n] =
K−1
∑
k=0
w ∗ ku[n − k]
with w k = a k + jb k . Now let us define the cost function J as
J = E[e[n]e ∗ [n]] = E[|e[n]| 2 ].
The MMSE solution for W is given as
W ∗ = min
W [J]
– Solution:
e[n] = d[n] − y[n]
=
K−1
∑
d[n] − (a k − jb k )u[n − k]
k=0
Let us define a gradient operator ∆ k with respect to the real input a k and the imaginary part b k such as
∆ k =
Apply the operator ∆ k to the cost function J yielding
Now the MMSE solution is obtained when
∂ + j ∂ , k = 0, 1, · · · , K − 1
∂a k ∂b k
∆ k J = ∂J
∂a k
+ j ∂J
∂b k
, k = 0, 1, · · · , K − 1
∆ k J = 0 for all k = 0, 1, · · · , K − 1
2