2.4 Equalization | DSL Advances

Equalization is a term generally used to denote methods employed by DSL receivers to reduce the mean-square ISI. The most common forms of equalization for DSLs appear in this subsection.

2.4.1 Linear Equalization

Linear equalizers are very common and easy to understand, albeit they often can be replaced by nonlinear structures that are less complex to implement and that work better, but are harder to understand. Nonetheless, this study begins with linear equalization and then progresses to better equalization methods in later subsections. The basic idea of linear equalization is to invert the channel impulse response so that the channel exhibits no ISI. However, straightforward filtering would increase the noise substantially, so the design of an equalizer must simultaneously consider noise and ISI. Noise enhancement is depicted in Figure 2.11. The equalizer increases the gain of frequencies that have been relatively attenuated by the channel, but also simultaneously boosts the noise energy at those same frequencies. The boosting of noise is called noise enhancement.

Figure 2.11. Illustration of noise enhancement with linear equalizer.

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The Minimum-Mean-Square Error Linear Equalizer (MMSE-LE) appears in Figure 2.12. A matched filter for the channel pulse response and sampler precede a linear filter whose transfer function is selected to minimize the mean-square difference between the channel input and the equalizer output. This filter's response has setting has D-transform

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Figure 2.12. MMSE linear equalizer.

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where SNR = E _x / s ² and R ( D ) is the discrete-time transform of the samples of r(t) = p(t)*p*(-t). ^[4] The equalizer need only be symbol- spaced (implemented at the symbol sampling instants in discrete time), but some implementations often combine the matched filter and linear filter into a single filter called a fractionally spaced equalizer. This single combined filter must be implemented at a sampling rate at least twice the highest frequency of the channel pulse response. The MMSE value in either implementation is

^[4] E _x and s ² must consistently be for the same number of real dimensions.

where w is the center tap (time zero sample) in the discrete-time symbol-spaced response w _k . A symbol-by-symbol detector processes the equalizer output. The scaling just prior to the detector removes a small bias inherent in MMSE estimation [12]. The performance of the MMSE-LE is characterized by an unbiased signal-to-noise ratio at the detector

This receiver signal-to-noise ratio is equivalent to that characterizing an AWGN channel's ML receiver for independent decisions about successive transmitted symbols. Thus, a probability of error can be approximated by assuming the error signal is Gaussian and computing P _e according to well established formula for PAM and QAM. For instance, square QAM on a bandlimited channel using an MMSE-LE with SNR _LE,U computed above has probability of error

Other formulae for other constellations with an MMSE-LE can similarly be used by simply substituting SNR _LE,U for SNR in the usual AWGN channel expression.

2.4.2 Decision Feedback Equalization

Decision feedback equalization ( DFE ) avoids noise enhancement by assuming past decisions of the symbol-by-symbol detector are always correct and using these past decisions to cancel ISI. The DFE appears in Figure 2.13.

Figure 2.13. Decision-feedback equalizaton.

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The linear filter characteristic changes so that the combination of the matched filter and linear feedforward filter adjust the phase of the intersymbol interference so that all ISI appears to have been caused by previously transmitted symbols. Because these previous symbols are available in the receiver, the feedback filter estimates the ISI, which is then subtracted from the feedforward filter output in the DFE. The DFE always performs at least as well as the LE and often performs considerably better on severe-ISI channels where the noise enhancement of the LE is unacceptable. Clearly, the LE is a special case of the DFE when there is no feedback section.

The best settings of the DFE filters can be obtained through spectral ("canonical") factorization of the channel autocorrelation function,

where g > 0 is a positive constant representing an inherent gain in the channel and G(D) = 1 + g ₁ · D + g ₂ · D ² + g ₃ · D ³ = ... is a monic ( g = 1 ) , causal (exists only for non-negative time instants), and minimum-phase (all poles/zeros outside the unit circle) polynomial in D . ^[5] Also, is monic, anti-causal, and maximum-phase. The feedforward filter has setting

^[5] The factor G(D) can be obtained by finding roots of magnitude greater than one, or can be directly computed for finite lengths in practice through a well-conditioned matrix operation known as Cholesky factorization.

and the combination of the matched-filter and feedforward filter is known as a mean-square whitening matched filter (MS-WMF). The output of the MS-WMF is the sequence X(D)G(D)+E(D) , a causally filtered version of the channel input plus an error sequence E(D) that has minimum mean square amplitude among all possible filter settings for the DFE. Because this sequence is causal, trailing intersymbol interference may be eliminated without noise enhancement by setting B(D) = G(D). The sequence E(D) is "white," with variance s ² _DFE = s ² / g . Thus the signal to noise ratio at the DFE detector is

The SNR of a DFE is necessarily no smaller than that of a linear equalizer in that the linear equalizer is a trivial special case of the DFE where the feedback filter is B(D) = 1. The DFE MS-WMF is approximately an all-pass filter (exactly all-pass as the SNR becomes infinite), which means the white input noise is not amplified nor spectrally shaped ”thus, there is no noise enhancement. As such, the DFE is better suited to channels with severe ISI, in particular those with bridge taps or nulls. Also, recalling that the channel pulse response may be that of an equivalent white noise channel, narrowband noise appears as a notch in the equivalent channel. Thus the DFE can also mitigate narrowband noise much better than a linear equalizer. The DFE is intermediate in performance to a linear equalizer and the optimum maximum- likelihood detector.

The removal of bias is also evident again for the MMSE-DFE in Figure 2.13. Error propagation is a severe problem in decision-feedback equalization and occurs when mistakes are made in previous decisions fed into the feedback section. Reference [1] investigates the problem and solutions.

In practice, the filters of either a LE or a DFE are implemented as finite-length sums-of-products (or FIR filters). Such implementations are better understood and less prone to numerical inaccuracies, while simultaneously lending themselves to easy implementation with adaptive algorithms. See [1] for a development of finite-length equalizers.

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