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In wireless communications, knowledge of channel coefficients is required for coherent demodulation. Lloyd-Max iteration is an innovative blind channel estimation method for narrowband fading channels. In this paper, it is proved that blind channel estimation based on single-level Lloyd-Max (SL-LM) iteration is not reliable for nonconstant modulus constellations (NMC). Then, we introduce multilevel Lloyd-Max (ML-LM) iteration to solve this problem. Firstly, by dividing NMC into subsets, Lloyd-Max iteration is used in multilevel. Then, the estimation information is transmitted from one level to another. By doing this, accurate blind channel estimation for NMC is achieved. Moreover, when the number of received symbols is small, we propose the lacking constellations equalization algorithm to reduce the influence of lacking constellations. Finally, phase ambiguity of ML-LM iteration is also investigated in the paper. ML-LM iteration can be more robust to the phase of fading coefficient by dividing NMC into subsets properly. As the signal-to-noise ratio (SNR) increases, numerical results show that the proposed method’s mean-square error curve converges remarkably to the least squares (LS) bound with a small number of iterations.

In wireless communication systems, channel state information (CSI) is necessary for coherent demodulation or precoding, and channel estimation is required at the receiver. Data-aid (DA) estimation methods make use of pilot, which is known both at transmitter and at receiver. On the contrary, blind estimation (BE) methods do not use any symbols known priorly at the receiver, thus saving transmitting power and bandwidth.

In [

LS solution of DA estimation was introduced by Crozier et al. [

For this problem, the paper proposes a BE method based on multilevel Lloyd-Max (ML-LM) iteration. By multilevel iteration and by transmitting estimation information from one level to another, the proposed method achieves accurate blind channel estimation for NMC with less received symbols. Moreover, when the number of received symbols is small, we introduce lacking constellations equalization (LCE) algorithm to reduce the influence of lacking constellations (LCs). As the signal-to-noise ratio (SNR) increases, the proposed method’s mean-square error curve converges remarkably to the LS bound with a small number of iterations.

The paper is organized as follows. Section

The notation is defined as follows:

When the coherence time of the channel is large enough, channel coefficients will change very slowly in time domain. Then fading coefficients are invariant in certain intervals. When the bandwidth of the channel is narrow, the channel is frequency-nonselective, namely, flat-fading. Under this condition, the system model is established as

Suppose that

Lloyd-Max iteration [

Suppose that the initial quanta are the MPSK constellation points:

Defining

For every

By repeating steps

Then, the estimator can be deduced as

It can be noted that Lloyd-Max algorithm is based on the principle of DD-LS. In step

Traditional Lloyd-Max iteration, which is called SL-LM iteration, is reliable for CMC. If the offset phase satisfies

On the conditions of SNR 30 dB, QPSK modulation with initial phase

QPSK symbols with different fading coefficients.

However, when the modulus of constellations is nonconstant, even if

On the conditions of SNR 30 dB, square 16QAM constellations, received symbols with different fading coefficients are shown in Figure

Square 16QAM symbols with different fading coefficients (first quadrant).

As we can see in Figure

It is proved that SL-LM iteration is unreliable for NMC. In order to solve this problem, we introduce ML-LM iteration in the following section.

In order to solve the problem above, we propose a BE method based on ML-LM iteration for NMC. For example, if the modulation mode is square 16QAM, the iteration process can be divided into two levels as follows.

The two-level Lloyd-Max iteration consists of L1 and L2. By repeating L1 and L2 until a stopping criterion is met or for a desired number of iterations, the mean value of four L2 estimators is the final estimator.

Supposing that the NMC are

Divide NMC

which satisfy

The mean value of a set

Then, the subsets satisfy

Define initial L1 quanta as

Define L1 regions of received symbols

If

Calculate the L1 estimator of fading coefficient as follows:

Initial L2 quanta are deduced as

Define L2 regions of received symbols

If

Calculate the L2 estimator of fading coefficient as follows:

If a desired number of iterations are met, (

In practice, the number of iteration levels should be set properly. For some high-order modulation modes, such as 256QAM, we must increase the number of levels to guarantee the well performance of the algorithm.

If the number of received symbols is small, it is a high probability event that transmitted constellations of all received symbols have not included all NMC. If a constellation has not been transmitted in the interval of received symbols, we call it lacking constellation (LC). If LCs exist,

LC in square 16QAM constellations (first quadrant).

For this case, we introduce LCE algorithm. For square 16QAM constellations, the L1 quantum can be determined only by 3 constellations in an L1 region

For square 16QAM constellations, LCE algorithm can be concluded as follows.

For every

Define the number of transmitted constellations

Initialize

For every

If a region satisfies

For other high-order modulations, the geometry of constellations is more complex. In an L1 region, how many constellations can determine an L1 quantum is not fixed. In practice, LCE should be modified based on the modulation mode.

Phase ambiguity is a classical problem in blind channel estimation. The reason can be concluded that we cannot determine the transmitted constellation of a received symbol. Some valuable ideas have been given to eliminate it, such as differential modulation and coding [

For square 16QAM constellations, as shown in Figure

Subsets of square 16QAM constellations.

If we divide constellations into subsets according to Figure

Because

In this section, we test the performance of ML-LM iteration through Monte Carlo simulation. The modulation mode is square 16QAM. Suppose that the fading coefficient is

The offset phase

In Figures

NMSE comparisons of SL-LM and ML-LM iteration when

NMSE comparisons of SL-LM and ML-LM iteration when

NMSE comparisons of SL-LM and ML-LM iteration when

NMSE comparisons of SL-LM and ML-LM iteration when

NMSE comparisons of ML-LM iteration for different numbers of iterations.

The

The lower bound in Figures

The number of iterations

When

Because of its high information rate, NMC are widely used in modern communication system. For blind channel estimation based on SL-LM iteration, NMC will result in quantization errors in the first step of iterations. The paper proposes a blind channel estimator based on ML-LM iteration for NMC. By dividing NMC into subsets, Lloyd-Max iteration is used in multilevel. Estimation information is transmitted from one level to another. Then quantization errors are eliminated. Moreover, when

For multipath channels, which produce frequency selectivity, the proposed scheme can be combined with orthogonal frequency division multiplexing (OFDM) scheme to achieve blind channel estimation. For every subchannel in OFDM, the channel is flat-fading and still satisfies the model in (

Phase ambiguity of ML-LM iteration is also analyzed in the paper. The restriction range of no phase ambiguity can be maximum by dividing NMC into subsets properly. How to eliminate the restriction of phase ambiguity will be researched in future work.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful for reviewers for their conscientious reviewing. This work was supported by the National Natural Science Foundation of China with the no. 61372098 and Scientific Research Project of Hunan Education Department with the no. YB2012B004.