Constrained ICA Algorithms

We have seen from the above analysis that with the zero-padding block transmission scheme, the BE problem becomes a standard BSS problem and can thus be solved with an ICA-based algorithm. Since we know that the “separating” matrix should have Toeplitz structure, this constraint can be enforced during iterations.

Forcing the Toeplitz structure on W will result in a C matrix that is also square Toeplitz with lower diagonal elements zero, but due to imperfect convergence the upper rows of the final C will generally contain small non-zero off-diagonal elements. As a result, the first several transmitted symbols, which correspond to the elements in the bottom of block

k

s , will be better recovered than the last ones. The effect of imperfect convergence gets more severe when Q is large compared with L. In this case, to limit the number of nonzero elements in the top rows of W, we can add a “length” constraint on w by forcing the uppermost diagonals of W to be zero. By doing so, we are adding a constraint that the coefficients of w be nonzero only up to a particular length. As a result, the number of possible non-zero elements in each row of matrix C will also be limited.

One general conclusion so far is that we can impose a Toeplitz structure constraint and length constraint on W in using a BSS adaptation scheme, with the expectation that it will allow improved performance. In fact, the performance of the constrained ICA-based algorithms depends on the characteristics of the channels. For minimum phase channel with the first tap having the largest magnitude, if we start from a good initialization it will not be difficult for the global system to converge to the identity; however, if the channel is

non-minimum phase or minimum phase with the largest tap behind the first tap, the inverse of H_T may have very large coefficients, which is hard for the algorithm to converge to. We will see examples based on simulations in Section 3.6.

We will next use the EASI algorithm explained in Chapter 2 as an example to show our idea of forcing Toeplitz structure and length constraint. Recall that with the EASI algorithm, the separating matrix has the adaptation as follows [8]:

Zblock Zblock Zblock Zblock Zbloc

1 ( ) ( )( ) k ( Zblock)

k k k k H k k H k k H k

W W y y I g y y y g y W , (3.7) where Zblock Zblock

k k k

y W x contains the separated symbols in the k-th block. Denoting the relative change in the brackets as U_k, the adaptation can be written as

1

k k k k

W W U W . (3.8)

There are two ways to enforce the Toeplitz constraint: one on the relative change matrix U_k, and the other on the whole perturbation U W_{k k}.

Constraint on relative change

Since the multiplication operation is closed in the space of Toeplitz matrices with lower elements zero, if we enforce Toeplitz structure on U_k, the perturbation U W_{k k} as a whole will still have the Toeplitz structure. With the Toeplitz constraint on the relative change, the adaptation can be written as

1 { }

k k Toeplitz k k

W W U W , (3.9)

where Toeplitz{ } means that the Toeplitz structure is enforced on Uk by taking averages along descending diagonals after forcing the lower left part to be zero. In fact, this Toeplitz

structure constraint can be proved to be the orthogonal projection of any square matrix onto the space of Toeplitz matrices with lower diagonals zero. In Chapter 4, the constraint will be enforced on non-square matrices. The proof of this orthogonal projection property will be given in the appendix of Chapter 4, and it can apply in a similar way for the square matrix case here.

From the adaptation in (3.9), we see that it has the form of “serial updating” introduced in Section 2.2.3. As a result, if the Toeplitz structure constraint is enforced on the relative change, the adaptation has the property of equivariance. In other words, the updating process does not depend on H_T as long as the global system C WH_T is the same.

For the length constraint, it is not possible to enforce the constraint on matrix U_k and keep the number of non-zero elements unchanged in the first several rows of W. The first row of Toeplitz{ }U Wk k gives the truncated version of the convolution of wk and the first row of Toeplitz{ }U_k . As a result, as long as Toeplitz{ }U_k is not identity, w cannot keep the number of nonzero elements in its tails unchanged from the previous iteration. However, the length constraint can be added after the matrix multiplication, i.e. on Toeplitz{ }U Wk k, which leads to the T-LC-EASI algorithm. The T-LC-EASI algorithm does not have the equivariance property.

Constraint on whole perturbation

The Toeplitz constraint can also be enforced on U W_{k k}, with the adaptation

1 { }

k k Toeplitz k k

However, the adaptation in (3.10) does not have the serial updating form, and thus is not equivariant. In addition to the Toeplitz constraint, the length constraint can be enforced on the term U W_{k k} after matrix multiplication.