Subspace Tracking Algorithms

It is seen from Section 2.5 that the linear multiuser detectors are obtained once the signal subspace components are identified. The classic approach to subspace estimation is through batch eigenvalue decomposition (ED) of the sample autocorrelation matrix or batch singular value decomposition (SVD) of the data matrix, both of which are computationally too expensive for adaptive applications. Modern subspace tracking algorithms are recursive in nature and update the subspace in a sample-by-sample fashion. An adaptive blind multiuser detector can be based on subspace tracking by sequentially estimating the signal subspace components and forming the closed-form detector based on these estimates. Specifically, suppose that at time (i - 1), the estimated signal subspace rank is K[i - 1] and the components are U_s[i - 1], L_s[i - 1], and s²[i - 1]. Then at time i, the adaptive detector performs the following steps to update the detector and to detect the data.

Algorithm 2.6: [Blind adaptive linear MMSE detector based on subspace tracking—synchronous CDMA]

• Update the signal subspace: Using a particular signal subspace tracking algorithm, update the signal subspace rank K[i] and the subspace components U_s[i], L_s[i], and s²[i].

• Form the detector and perform detection:

  

  
  

Various subspace tracking algorithms are described in the literature (e.g., [43, 87, 97, 406, 412, 461, 493, 586]). Here we present two low-complexity subspace tracking algorithms: the PASTd algorithm [586] and the more recently developed NAHJ algorithm [412].