Soft-output decision-feedback equalization with a priori information

Soft-output equalizers that exploit a priori information on the channel inputs play a central role in turbo equalization. Such equalizers are traditionally implemented with the forward-backward or BCJR algorithm, whose complexity is prohibitive for channels with large memory. Many reducedcomplexity alternatives to the BCJR algorithm have been proposed that use a linear equalizer and use the a priori information to perform soft intersymbol interference cancellation. In this work, we propose a soft-feedback equalizer (SFE) that combines the equalizer output and the a priori information to improve interference cancellation. Also, by assuming a statistical model for the a priori information and the SFE output, we obtain an equalizer with linear complexity, as opposed to the quadratic complexity of some similar structures. Simulation results show that the SFE may perform within 1 dB of a system based on an BCJR equalizer, within 0.3 dB of quadratic complexity schemes, and consistently outperforms other linear complexity schemes. I. I NTRODUCTION Soft-output equalizers that exploit a priori information on the channel inputs are useful in a variety of applications. Most notably, such equalizers play a central role in turbo equalization, where soft equalizer outputs are fed to a softinput channel decoder, and where soft decoder outputs are used by the equalizer as a priori information in subsequent iterations [1]. Traditionally, the soft outputs take the form of a posteriori probabilities (APP) for each transmitted symbol, given the channel outputs and the a priori information. The APP may be computed exactly by the forwardbackward or BCJR algorithm [2]. However, the computational complexity of BCJR is exponential in the channel memory, so it is not practical when the channel memory is large. This has motivated the development of reduced-complexity alternatives to the BCJR algorithm, such as the equalizers proposed in [38]. These structures use a linear filter to equalize the received sequence. The output of this linear filter contains residual intersymbol interference (ISI), which is estimated based on the a priori information, and then cancelled. In this work, we propose the soft-feedback equalizer (SFE), which combines the soft equalizer outputs and the a priori information to form more reliable estimates of the residual ISI. A similar system is proposed in [7] that uses hard decisions on the equalizer output to help estimate the residual ISI. However, because hard decisions are used and because the a priori information is not combined with the equalizer output before a decision is made, the system with feedback of [7] performs worse than schemes without feedback. As in [5-8], the SFE coefficients are computed to minimize the mean squared error (MSE) between the equalizer output and the transmitted symbol. By assuming a statistical model for the a priori information and the equalizer output, we obtain a linear complexity equalizer, i.e. , the complexity is proportional to the number of taps. A similar statistical model is assumed in [9] to obtain a linear complexity hard-input hardoutput equalizer with ISI cancellation (IC). The minimumMSE (MMSE) schemes in [5-8] have quadratic complexity. We will see that in special cases, the SFE reduces to a MMSE linear equalizer (LE), an MMSE-decision-feedback equalizer (DFE) or an IC. We will show that the SFE performs reasonably well when compared to the BCJR and quadraticcomplexity algorithms, while it consistently outperforms other linear-complexity structures proposed in the literature. II. C HANNEL M ODEL AND P ROBLEM S TATEMENT This paper considers the transmission of a sequence of symbols a = [ a 0 , ... , a L –1 ] through a channel with output r k = h m a k–m + n k , (1) where the channel has memory μ and impulse response h = [ h 0 , ... , h μ ] , and where n k ~ N (0, σ 2 ) is real white Gaussian noise. For notational ease, we restrict our presentation to a BPSK alphabet, with a k ∈ { ± 1} . The results can be extended to other alphabets using the techniques of [8]. In contrast to a classical equalizer, which assumes that the channel inputs are uniformly distributed, we assume that the receiver has a priori information about the channel inputs. For binary alphabets, this information is captured by the logarithm of the ratio of the a priori probabilities: