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- Open Access

# Adaptive blind timing recovery methods for MSE optimization

- Wonzoo Chung
^{1}Email author

**2012**:9

https://doi.org/10.1186/1687-6180-2012-9

© Chung; licensee Springer. 2012

**Received: **18 June 2011

**Accepted: **13 January 2012

**Published: **13 January 2012

## Abstract

This article presents a non-data-aided adaptive symbol timing offset correction algorithm to enhance the equalization performance in the presence of long delay spread multipath channel. The optimal timing phase offset in the presence of multipath channels is the one jointly optimized with the receiver equalizer. The jointly optimized timing phase offset with a given fixed length equalizer should produce a discrete time channel response for which the equalizer achieves the minimum mean squared error among other discrete time channel responses sampled by different timing phases. We propose a blind adaptive baseband timing recovery algorithm producing a timing offset close to the jointly optimal timing phase compared to other existing non-data-aided timing recovery methods. The proposed algorithm operates independently from the equalizer with the same computational complexity as the widely used Gardner timing recovery algorithm. Simulation results show that the proposed timing recovery method can result in considerable enhancement of equalization performances.

## Keywords

- Mean Square Error
- Timing Phase
- Multipath Channel
- Decision Feedback Equalizer
- Mean Square Error Performance

## 1 Introduction

A different sampling timing phase produces different channel responses in the presence of multipath channels. For finite length equalizers, which are always insufficiently long in practice for wireless multimedia broadcasting systems such as advanced television systems committee (ATSC) receivers, the mean squared error (MSE) performance of a fixed length minimum MSE (MMSE) equalizer depends on the sampled channel. Certain timing offsets yield channels relatively easy to equalize with baud-spaced equalizers and, consequently, the MSE performance of the MMSE equalizer of a given length is limited by the choice of timing phase offset. The problem of finding the optimal timing phase in the presence of long delay spread multipath distortion has been considered resolved with the introduction of fractionally spaced (FS) equalization [1]. FS equalizers not only equalize multipath channel distortion more effectively, but also plays a role of interpolation filter for the timing phase to produce the best MSE performance [1]. However, for long delay spread channels such as the ones ATSC digital television (DTV) receivers are facing, FS equalizers covering the entire range of multipath delays are often impractical due to hardware limitations. Therefore, most receivers prefer a baud-spaced linear equalizer combined with a decision feedback equalizer (DFE) operating at the baud rate. Consequently, the timing phase problem has resurfaced in ATSC receivers.

Most widely used timing recovery schemes are Gardner algorithm [2] and band-edge algorithm, or known as Godard algorihtm, [3]. The band-edge algorithms has originated from the output energy maximization (OEM) of sampled received signals, i.e., finding timing phase maximizing the energy of the sampled signals. Since the sampled signals is mixed with inter-symbol-interference terms, the timing phase based on OEM is optimized for infinite length equalizers but not for a finite length equalizer. As we will show in this article, Gardner algorithm also belongs to this OEM category and, consequently, cannot produce optimized timing offset for a finite length equalizer. In general, it is difficult and costly task to optimize timing phase for a given finite length equalizer: joint optimization of timing and equalization has inherent latency problem and often requires frequent training signals.

Especially for ATSC receivers, the most important application area for baud timing recovery algorithms, several timing phase optimization techniques have been developed and applied. Most of these approaches use repetitive data segment syncs or periodically apply a timing phase correction computed from the field sync, in parallel with commonly used timing acquisition algorithms such as Gardner, band-edge or variant of Gardner algorithms [4–6] algorithm. For example, a correlation function of three symbols (1 0 1) [7] or four symbols (1 1 -1 -1) [8, 9] in segment training signals is used to generate the timing phase information, or the field sync sequence is used to generate the timing phase correction [10]. However, these data-aided timing phase acquisition approaches use only a fraction of the data (e.g., a four-symbol segment sync among 832 symbols in the data segment) to optimize timing offset.

In this article, we propose a non-data-aided (blind) timing acquisition method designed to approximate the optimal timing phase in the presence of multipaths. The timing phase offset generated by the proposed symbol timing recovery algorithm is located close to the optimal timing phase offset compared to the Gardner [2] or band-edge algorithms [3] without help of the equalized data without feedback from the equalizer. Hence, the proposed algorithm can be used with the data aided approaches in the place of the Gardner algorithm for ATSC receivers.

The purpose of this algorithm is to find the timing phase optimized for a single tap equalizer, the opposite extreme of the infinite length equalizer. This approach is called dispersion minimization (DM) approach [11] and produces better MSE performance for most finite equalizers than OEM timing, but an adaptive algorithm version of this DM algorithm has not been studied yet. We developed a baseband blind adaptive timing recovery algorithm that is closely related to this DM approach as Gardner is closely related to the OEM approach. Simulation results show that the proposed timing recovery algorithm enhances the performance of MMSE DFEs in comparison with Gardner timing.

In Section 2 we introduce OEM timing recovery approach and the relation to Gardner timing. In Section 3 a new blind timing recovery algorithm based on DM approach is proposed with a tutorial example showing the enhanced performance. Section 4 presents simulation results and Section 5 provides the conclusion.

## 2 Symbol timing offset of symbol timing recovery algorithms

*s*

_{ k }} is converted to analog signal by a pulse shaping filter

*p*(

*t*)

*w*(

*t*)

*r*(

*t*) is matched filtered with

*g*(

*t*) and

*h*(

*t*) is overall channel response combining the multipath channel

*c*(

*t*), pulse shaping filter

*p*(

*t*), and the matched filter

*g*(

*t*),

*y*(

*t*) is sampled at the baud rate

*T*with a timing phase offset

*τ*generated from a timing offset generation mechanism. Depending on the timing phase offset

*τ*, we have a different discrete time domain channel. Denoting the discrete time impulse response sampled from

*h*(

*t*) with respect to the sampling phase

*τ*as a vector

*h*

_{ τ },

where *w*_{
k
}is sampled noise term.

*τ*are developed. OEM approach to timing phase recovery involves choosing the timing phase to maximize the power of the sampled data, i.e.,

This approach consequently optimizes the MSE of the equalizers with infinite length, since the output energy usually contains inter-symbol interference (ISI) terms $\left({\sum}_{i\ne k}{s}_{i}{h}_{\tau}\left[k-i\right]\right)$, in the presence of multipath channels. An infinite length equalizer will deal with the ISI component to convert the ISI component to the signal component perfectly. For a finite or a relatively short equalizer, the OEM timing fails to achieve MMSE, since the remaining ISI degrades the MSE performance [11]. Godard's band-edge algorithm [3] is a passband domain implementation of this approach.

*μ*is a step size and

*τ*

_{ k }is the timing phase at time

*kT*, can be viewed as an approximated gradient descent implementation [12] of the OEM approach (7). The stochastic update equation [13] to achieve (7) is given by

Combining (10) and (11), we obtain Gardner algorithm (9) as an approximation of OEM algorithm. Hence, we can conclude that the Gardner algorithm, which is commonly used in symbol timing recovery circuits of ATSC DTV receivers, falls into the OEM timing recovery category. Consequently, as reported in [10], the Gardner algorithm does not perform optimally for ATSC receivers, in which the length of equalizers is always short when dealing with widely spread multipath channels.

where *γ* is the dispersion constant [14] computed from the source signal ($\gamma =8/\sqrt{21}$ for 8-PAM). This DM timing phase is optimized for one tap equalizer and located closer to the best timing phase offset for a finite length equalizer, minimizing equalizer output MSE better than other timing methods based on OEM [11]. In general, the baud-spaced channel produced by DM timing is easier to equalize with finite equalizers than the one produced by OEM timing. In the following section, we consider the adaptive solution of DM timing in the baseband.

## 3 Proposed timing recovery method

*y*(

*kT*+

*τ*

_{ k })|

^{2}-

*γ*. We expect this new timing algorithm to inherit the optimized MSE performance of DM timing. Figure 2 illustrates a possible implementation structure of a timing recovery circuit using the proposed timing algorithm.

*peaky*channel. This difference produces a difference in the MSE performance of the finite length MMSE equalizer, as shown in Figure 6 in the following simulation section.

Figure 6 plots the MSE performance of a finite length MMSE linear equalizer for normalized timing phase offsets spanning -0.5 to 0.5, i.e., [-*T*/2,*T*/2]. Since the effect channel lengths are about 12 taps in Figure 5, we have set equalizer length to 20 under 30 dB SNR. None of those timing offsets have achieve the MMSE, but DM timing and the proposed timing perform relatively better than OEM approaches (about 1 dB).

*c*(

*t*) =

*δ*(

*t*) +

*δ*(

*t*- 0.51

*T*) under 30 dB SNR. The proposed algorithm outperforms Gardner algorithm only for the equalizer length less than about 130. Unfortunately, the exact filter length determining the boundary is hard to obtain in general. However, we believe equalizers are always short in most practical situations.

## 4 Simulation results

Channel profiles

Profile | Path 1 | Path 2 | Path 3 | Path 4 | Path 5 | Path 6 | |
---|---|---|---|---|---|---|---|

Channel 1 single echo | Delay [ | 0 | 0.5 | ||||

Gain [dB] | 0 | -3 | |||||

Phase [deg] | 0 | 0 | |||||

Channel 2 Brazil B | Delay [ | 0 | 0.3 | 3.5 | 4.4 | 9.5 | 12.7 |

Gain [dB] | 0 | -1.2 | -4 | -7 | -15 | -22 | |

Phase [deg] | 0 | 0 | 0 | 0 | 0 | 0 |

## 5 Conclusion

In this article, we described a blind timing method for ATSC DTV systems that produces better equalizer output MSE performance than other OEM-based timing methods such as Gardner timing. The proposed timing recovery algorithm can be considered as a baseband adaptive implementation of the DM timing approach. Simulation results confirmed the MSE enhancement of DFE output when equipped with the proposed timing algorithm.

## Declarations

### Acknowledgements

This work was supported by Basic Science Research Program through the NRF funded by the MEST (NRF- 2010-0025437) and BK21.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.