This chapter introduced the carrier frequency and phase synchronization. We consider both the open-loop and the closed-loop structures. In the open-loop structure, we study the Fitz’s algorithm for the signal frequency estimation and then propose a multi-resolution algorithm and the modified Fitz’s algorithm according the analysis of the Fitz’s algorithm. Analytical and simulated results illustrate the proposed algorithms can reach near CRB in a low cost manner.
The X4 SFE is also studied to solve the frequency estimation in QAM system.
In the other part of this chapter, we study the closed-loop synchronize. The main topic in this section focuses on the design of blind PDs and the dynamic loop bandwidth control scheme.
We first propose a set of the PD based on the reduced constellation concept; then, a hybrid combination of the RCPD and the DDPD is used to provide both the better acquisition ability and the tracking stability. Besides, the dynamic loop bandwidth control is also presented using the lock indication triggered manner. Final simulations demonstrate that the CRL using HPD and the dynamic loop bandwidth control provide superior capability in both the acquisition ability and tracking stability.
Chapter 3
Blind Adaptive Decision Feedback Equalization
The equalizer is used to combat the inter-symbol-interference (ISI) channel distortion. There are many equalization structures, such as the linear equalization, the maximal likelihood sequence estimation, the turbo-equalization. However, the implementation cost and the performance have made the decision feedback equalizer the most commonly used structure for the application over static channel environment.
A typical equalizer contains two properties: the structure and the coefficients. In the example of the DFE, its structure includes a feed forward filter (FFF), a feedback filter (FBF) and a decision circuit, as illustrated in Fig. 3.1. The FFF is to shorten the channel response and the FBF is to remove the residual ISI term. The responses of the filters are the filter coefficients which are determined by the channel responses. As shown in Fig. 3.1, the coefficients are generated by the coefficient adapter. The realization of the coefficient adaptor depends on the available materials and the implementation costs.
Figure 3.1: The structure of the decision feedback equalizer that is composed by a feed forward filter, a feedback filter, a decision circuit and a filter coefficient adapter.
This chapter mainly focuses on the coefficient adaptation. We adopt the blind adaptive filter [29] to realize the adaptor. The key issue in blind adaptive process is tradeoff between the steady-state and training speed. We suggest the variable stepsize (VSS) algorithm to compromise this tradeoff. After the blind start operation, the DFE is worked in the tracking mode to provide small mean-square error when the tentative decisions are near error-free.
Switch time of the operation mode is another critical issue in adaptor design. For this, we introduce a soft-switching concept that the operation mode smoothly transfers from blind acquisition to decision-directed tracking. To achieve this soft-switching, we suggest the hybrid
blind algorithm that combines directly the decision-directed least-mean square algorithm [29]
and the blind VSS adaptive algorithm.
3.1 Optimal Coefficient and LMS Algorithm
Before the description of the blind mode operation of the DFE, we first consider the minimal mean-square error (MMSE) coefficients of the equalizer. We first consider the coefficient adaptor in linear equalizer and extend to the adaptor of the DFE.
Let H be the Toeplitz matrix of the “composite” channel (which includes the responses of the transmitting filter, the channel and the receiving filter), ak be the k-th input signal, and rk be the k-th output sample of the channel. The received signal rk can be expressed as
rk = Hak+ nk (3.1)
where rk = [rk rk−1 · · · rk−L+1]T is received vector, ak = [ak ak−1 · · · ak−M +1]T is the trans-mitted vector, and nk is AWGN vector.
Now, consider the equalizer operation. If w is the L × 1 coefficient vector of the equalizer and x is the input vector of the equalizer, we have that
yk= hw, xki =
L−1X
l=0
w(k − l)xk−l. (3.2)
If the linear equalization structure is used, we have that xk = rk. The target of the channel equalization is to inverse the channel effect. In MMSE manner, we want to minimize the cost function that
J = E |yk− ak+d|2, (3.3)
in which d is the process delay of the equalizer. If the channel response and the noise variance are available, from the Wiener filter, the coefficients of linear MMSE equalizer are given by [29]
w =¡
HHH + σ2nI¢−1
hd (3.4)
where hd is the Hermitian of the d-th row of H.
In practice, we can estimate the channel response and the noise variance for the coefficient calculation. Alternatively, if considering low-cost implementation, we can adopt the adaptive signal process to realize the coefficient adaptor. When the training data or the correctly tentative decisions exist, the least-mean-square (LMS) algorithm [29] is feasible and widely adopted one to train the coefficients, which adaptation is given by
wk+1= wk− µekx∗k (3.5)
where µ is the stepsize, and
ek= ak− yk (3.6)
is the error function. In steady-state of the LMS adaptor, the performance of the LMS adaptor approaches the MMSE equalizer when eliminating the excess error due to the LMS adaptation algorithm or when the stepsize is small enough [29].
When the DFE structure is considered, we can let wk=
h fTk bTk
iT
, xk=£
rTk ˆaTk−1¤T
(3.7) where f and b are the coefficient of the FFF and FBF relatively, and ˆa is the tentative decision.
Thus, the above LMS algorithm can be directly applied in the DFE coefficient adaptor.