Blind equalization

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Blind equalization is a digital signal processing problem in which the channel impulse response and the transmitted signal are inferred from the received signal. In contrast to the equalization problem, the input signal is not assumed to be known, save its statistics.

Blind equalization is essentially blind deconvolution formulated as a digital communications problem. None the less, the emphasis in blind equalization is on online computation of the equalizer, which is the inverse of the channel impulse response, rather than computation of the channel impulse response itself. This is due to blind equalization common mode of usage in digital communications systems.

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[edit] Mathematical models

[edit] Noiseless model

The channel is modeled as an LTI system whose impulse response will be denoted \{h[n]\}_{n=-\infty}^{\infty}. The noiseless model of the blind equalization problem relates the received signal r[k] to the transmitted signal s[k] via

r[k]=\sum_{n=-\infty}^{\infty}h[n]s[k-n]

The blind equalization problem can now be formulated as follows; Given the received signal r[k], find a filter w[k], called an equalization filter, such that

s[k]=\sum_{n=-\infty}^{\infty}w[n]r[k-n]

The solution s[k] to the blind equalization problem is not unique. In fact, it may be determined only up to a signed scale factor and an arbitrary time delay. That is, if \{\hat{s}[k],\hat{h}[n]\} are estimations of the transmitted signal and channel impulse response, respectively. Than \{c\hat{s}[k+d],\hat{h}[n-d]/c\} give rise to the same received signal r[k] for any choice of a real scale factor c and an integral time delay d. In fact, the role of s[k] and h[k] in the above formula is symmetric.

[edit] Noisy model

In the noisy model includes an additional additive term n[k] representing additive noise. The model is therefore

r[k]=\sum_{n=-\infty}^{\infty}h[n]s[k-n]+n[k]

[edit] Algorithms

Many algorithms have been suggested over the years for the solution of the blind equalization problem. As in practice, one usually has access to only a finite number of samples from the received signal r(t), further restrictions must be imposed over the mathematical models described above, in order to render the problem tractable.

[edit] Bussgang methods

The Bussgang methods further assume that the channel has a finite impulse response. Specifically, the channel impulse response is assumed to be of the form \{h[n]\}_{n=-N}^{N}, where N is an arbitrary natural number. This assumption may be justified over physical ground, as any physical signal must have finite energy, and consequently, its impulse response must tend to zero. Thus, it may be assumed that beyond a certain index, all coefficients are negligibly small.

[edit] Polyspectra techniques

Polyspectra techniques utilize higher order statistics in order to compute the equalizer.

[edit] See also

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