Robustness of least-squares and subspace methods for blind channel identification/equalization with respect to effective channel undermodeling/overmodeling

Abstract

The least-squares and the subspace methods are two well-known approaches for blind channel identification/ equalization. When the order of the channel is known, the algorithms are able to identify the channel, under the so-called length and zero conditions. Furthermore, in the noiseless case, the channel can be perfectly equalized. Less is known about the performance of these algorithms in the practically inevitable cases in which the channel possesses long tails of "small" impulse response terms. We study the performance of the m m mth-order least-squares and subspace methods using a perturbation analysis approach. We partition the true impulse response into the m m mth-order significant part and the tails. We show that the m m mth-order least-squares or subspace methods estimate an impulse response that is "close" to the m m mth-order significant part. The closeness depends on the diversity of the m m mth-order significant part and the size of the tails. Furthermore, we show that if we try to model not only the "large" terms but also some "small" ones, then the quality of our estimate may degrade dramatically; thus, we should avoid modeling "small" terms. Finally, we present simulations using measured microwave radio channels, highlighting potential advantages and shortcomings of the least-squares and subspace methods.

Blind identification of multipath channels: a parametric subspace approach

“…The true value of the channel order is . First, this simulation confirms the result of [20], which is recalled in Section III, i.e., performance of the "plain" subspace method is better for an underestimated value of the channel order. On the other hand, as , from Theorem 3, we know that the channel remains identifiable, whatever the order estimation.…”

“…The set contains all the delays appearing in (36). We will show that under I3, the set of delays verifies the condition given by (20 On the other hand, standard results on the perturbation of the eigenprojectors (see [17]) show that (43) where is the pseudo-inverse of . By collecting the previous relations, we obtain where On the other hand, from (4) and (15) …”

“…Thus, if , the response is identified up to a multiplicative constant. The case where the channel order is underestimated has been studied in [20]. This point is particularly relevant when applying the unstructured subspace method to channels with small leading and trailing terms.…”

“…We propose to study this point by simulation, as in [31]. We consider rays equispaced in time between 0 and such that is the maximum number of delays verifying (20). The modulation waveform is a raised-cosine pulse with excess bandwidth , truncated to zero outside the interval ( ).…”

“…In particular, they are highly sensitive to modeling errors: an overdetermination of the channel length leads to inconsistent estimates. In practical situations, the exact channel order should be estimated from the data, which is proved to be a rather involved task (moreover, most of the time, the concept of channel order is ill defined [20]; see also [10] and [34] for some possible adaptations of subspace ideas leading to robust algorithms).…”