Low-Complexity Hybrid Adaptive Blind Equalization Algorithm for High-Order QAM Signals

Abstract

It is well known that the constant modulus algorithm (CMA) presents a large steady-state mean-square error (MSE) for high-order quadrature amplitude modulation (QAM) signals. In this paper, we propose a low-complexity hybrid adaptive blind equalization algorithm, which augments the CMA error function with a novel constellation matched error (CME) term. The most attractive advantage of the proposed algorithm is that it is computationally simpler than concurrent CMA and soft decision-directed (SDD) scheme (CMA+SDD), and modified CMA (MCMA), while the approximation of steady-state MSE of the proposed algorithm is same with CMA+SDD, and lower than MCMA. Extensive simulations demonstrate the performance of the proposed algorithm.

1. Introduction

In the past, single-carrier communication was the modulation format of choice. In this context, inter-symbol interference (ISI) becomes the main factor affecting system performance. Recently, the orthogonal frequency division multiplexing (OFDM) protocol is being used in the transmission scheme for the majority of new communications systems. One of the most important advantages of OFDM systems is their robustness to multipath channels, thanks to the introduction of a cyclic prefix (CP), which efficiently combats ISI in time-dispersive channels. However, the insertion of a CP is pure redundancy, which decreases the spectral efficiency [1]. Additionally, when the CP length is shorter than the channel length, the effect of frequency-selective fading cannot be completely eliminated, and the intercarrier interference and ISI will be introduced. Blind equalization techniques without regular training/pilot symbols can be utilized to mitigate the problems [2,3].

The constant modulus algorithm (CMA), pioneered by Godard [2], is by far the most popular blind equalization algorithm, since it can be easily implemented with good convergence properties [4]. For constant modulus signals, such as 4-QAM, a CMA-based fractionally spaced equalizer (FSE) can achieve a zero steady-state mean-square error (MSE) in noiseless channels. For non-constant modulus signals, such as high-order QAM signals which have been used for a long time in LTE mobile networks [5], it suffers a relatively large misadjustment resulting in a large steady-state MSE [4,6].

A possible solution is to enable CMA until the eye diagram opens, and then switch to another adaptive equalization to minimize the residual error and compensate the phase offset. However, ensuring such a transfer is challenging, and sensitive to signal constellation, channel characteristics, and signal-to-noise (SNR) [7]. To this end, several soft-switching methods have been proposed. Weerackody et al. [8] provided a dual-mode type algorithm. In such an algorithm, the blind equalization mode works at higher error levels, while the mode similar to decision-direct (DD) works at lower error levels. De Castro et al. [7] suggested a typical low-complexity concurrent CMA and DD scheme (CMA+DD). Rather than switching to DD adaptation if CMA converges, they proposed enabling a DD equalizer concurrently with a CMA equalizer. The weight adaptation of DD equalizer follows that of CMA. To avoid error propagation due to incorrect decisions, the DD adjustment only works if the CMA adaptation is judged to be successful. At a cost of complexity, CMA+DD equalizer can obtain a dramatic performance improvement over CMA. To ease computational requirements in CMA+DD, Chen [9] proposed a concurrent CMA and soft DD scheme (CMA+SDD), wherein the soft DD part is an exponential weighted function of the distance between the equalizer soft output and the tentative decision. This soft decision enables a simultaneous update of both CMA and SDD weight vectors without error propagation. Compared with CMA+DD, CMA+SDD is reported to have simpler computational requirements, faster convergence rate and identical steady-state equalization performance. As a tradeoff, Xie et al. [10] proposed a concurrent dithered signed-error CMA (DSE-CMA) and SDD algorithm which can compensate the phase shift and provide better convergence and steady-state behavior than DSE-CMA while with less computation than CMA+SDD. Silva et al. [11] proposed a soft-switching blind equalization based on a convex combination of a blind equalization algorithm and a DD algorithm. The combination is adapted in a blind manner, and can automatically switch between the component filters, avoiding MSE level settings a priori. Like CMA+SDD, this combination algorithm provides faster convergence rate and lower steady-state MSE than CMA+DD. Although its computational complexity was not clearly discussed, it still suffers high computation complexity because of the update of mixing parameters.

Recently, some hybrid blind equalization algorithms have attracted attention due to their relative simplicity and good performance. This hybrid approach, named constellation matched error (CME), combines CMA criterion with a penalty term, and with zero values at constellation points coordinates [12]. The most common CME functions encountered in the literature are polynomials of high order and powers of cosine functions [13,14]. Afterwards, Thaiupathumpa et al. [15] added a CME to the cost function of generalized Sato algorithm [16]. He [17] added a CME to the cost function of multimodulus algorithm [18]. Sheikh and Fan [19] added a CME to the cost function of sliced multimodulus algorithm [20]. Labed et al. [21] introduced a CME, which is the product of l1-norm of the deviations of equalizer output from the constellation points [22].

In this paper, we propose a low-complexity hybrid adaptive blind equalization algorithm in which the hybrid cost function is a weighted sum of two separate well-defined error terms; one is identical to the CMA case, and the other is a new CME term based on a simple coordinate transformation. The approximation of the steady-state MSE of the proposed algorithm is derived, which is the same as that of the CMA+SDD [9] and lower than that of the modified CMA (MCMA) [14]. The most attractive advantage of the proposed algorithm is that, it is computationally simpler than CMA+SDD and MCMA. Extensive simulations demonstrate the performance of the proposed algorithm.

In Section 2, we review the blind equalization based on conventional CMA, CMA+SDD, and MCMA, respectively. In Section 3, a novel hybrid blind equalization algorithm is proposed. Section 4 addresses the computational complexity per weight update and the steady-state MSE analysis of the proposed algorithm. Section 5 presents simulation results in different situations. Finally, Section 6 provides the main conclusions of the paper.

 

2. Blind Equalization Using CMA, CMA+SDD, and MCMA

Equalization algorithms can be implemented in symbol-spaced form or fractionally spaced equalizer form (FSE). Here, we concentrate on fractionally spaced implementations due to their inherent advantages [23]. The schematic diagram of the T/2 FSE without channel noise is shown in Fig. 1. Given a fractionally spaced channel of finite length 2M, the even and odd sets of channel coefficients can be collected into vectors as ce = [c(0), c(2),⋯,c(2M-2)]T and co = [c(1), c(3),⋯,c(2M-1)]T, respectively. Similarly, given a fractionally spaced equalizer of finite length 2N, the even and odd sets of equalizer coefficients can be collected into vectors as fe = [f(0), f(2),⋯,f(2N-2)]T and fo = [f(1), f(3),⋯,f(2N-1)]T, respectively.

Fig. 1.Structure of T/2 FSE

A pair of (M + N −1) × N convolution matrices Co and Ce are given by

We define the transmitted symbol vector as ak = [a(k),a(k−1),⋯,a(k −M − N + 2)] , and the compound matrix and vector quantities as

Then, the output of the equalizer can be expressed as

In Fig. 1, xo(k) and xe(k) are the outputs of 2 sub-channels co and ce, respectively. Define xo,k = [xo(k), xo(k−1),⋯, xo(k − N +1)]T and xe,k = [xe(k), xe(k−1),⋯, xe(k − N +1)]T, the input of the equalizer is thus given by

Then

2.1 CMA

In the T/2 FSE, the most popular adaptive blind equalization algorithm CMA2-2 [1] (CMA for simplicity) can be used to minimize a cost function defined by the CM criterion

where R is a fixed value of the transmitted sequence {a(⋅)} and defined as

CMA seeks to minimize the cost function (6) using stochastic gradient; hence, the weight vector of the equalizer is updated by

with a step-size μ. Here, the symbol * denotes complex conjugate transposition, and eo(k) denotes an instantaneous error.

Obviously, the CM criterion penalizes deviations in the modulus (i.e. magnitude) of the equalized signal |z(k)| away from the fixed value . The CM criterion can successfully equalize signals characterized by source alphabets not possessing a constant modulus (e.g., 16-QAM), as well as those possessing a constant modulus (e.g., 4-QAM) (see Fig. 2). Unfortunately, toward non-constant modulus and constant modulus signals, CMA shows different performance.

Fig. 2.(a) Nonconstant modulus source constellation (16-QAM) versus (b) constant modulus source constellation (4-QAM).

For a constant modulus signal, a CMA-based FSE can achieve perfect equalization (i.e. MSE = 0 ) in a noiseless environment. For example, for a 4-QAMconstant modulus signal as shown in Fig. 2(b), constellation points are {±1± j} and if z(k) = a(k) in the steady state, then eo(k) = 0 which terminates the updating of the weight vector, i.e., MSE = 0 .

For a non-constant modulus signal, a CMA-based FSE cannot achieve perfect equalization. For example, for a 16-QAM nonconstant modulus signal as shown in Fig. 2(a), constellation points are {±1± j,±1± 3j,±3 ± j,±3 ± 3j} and if z(k) = a(k) then eo(k) ≠ 0 which results in the weight vector keeps updating with a non-vanishing term and vibrates around the mean solution. Thus, z(k) ≠ a(k), i.e. MSE ≠ 0 .

The steady-stateMSE of CMA for the complex-valued cases can be approximated by [6]

where E{||xk||2} is the received signal norm.

2.2 CMA+SDD

Chen [9] proposed a blind equalization scheme consisting of a CMA equalizer and a SDD equalizer. The output of CMA+SDD is

where fc and fd are the weight vectors with the same length 2N of the CMA equalizer and the SDD equalizer, respectively.

The CMA equalizer adjusts the weight vector fc as (8)

where μ is the step-size of the CMA equalizer.

The weight vector fd of the SDD equalizer is updated by [9]

where and U is an integer, μd is the step-size of the SDD equalizer, and apq are four symbol points in a decision region which the equalizer output z(k) belongs to. Typically, ρ is chosen to be < 1 .

Compared with CMA+DD, CMA+SDD blind equalizer has simpler computational requirements, faster convergence rate and identical steady-state equalization performance. For obtaining the approximate steady-state MSE of CMA+SDD, CMA+SDD is equivalent to the blind equalizer described by a hybrid cost function [24]. The two time evolution equations (11) and (12) are changed to be one time evolution with respect to one tap weight vector

where f = fc + fd and α = μd / μ.

Now the steady-state MSE of CMA+SDD for the complex-valued cases can be approximated by [24]

2.3 MCMA

The Modified CMA (MCMA) [14] is the state-of-the-art hybrid method for the CMA, which augments the CMA error function with a constellation matched error (CME) term. The CME term provides MCMA with knowledge of the constellation allowing for faster convergence rate and greater reduction of MSE. The general form of the cost function for the MCMA is given by

where the subscripts r and i correspond to the real and imaginary components, respectively, and β is the CME weighting factor. The weight vector of MCMA is updated by

where ηk is the CME term defined as [14]

where g(γ) is the CME function. The following CME function for QAM constellations is zero at symbol points [14]

where n is an integer number and 2d is the minimum distance between symbols. In this paper, d is set to be 1.

For n =1, the CME function is a cosine square, i.e. g(γ) = cos2 (γπ / 2), as shown in Fig. 3(a). In this case, it has the lowest complexity, and eo(k) in (16) is expressed as

Fig. 3.The CME functions of MCMA (a) and proposed algorithm (b) for the 16-QAM signal.

The difference between MCMA and CMA+SDD is that MCMA is based on one filter, while CMA+SDD has two filters. For CMA+SDD, its output is the sum of the output of the two filters, which operates under two cost functions. The MCMA equalizer, on the other hand, has one filter, whose weight vector is updated according to one cost function, which includes both CMA and CME terms. While in [24], the CMA+SDD blind equalizer can be equivalent to the blind equalizer described by a hybrid cost function, whose SDD part can be set to be a CME term, as shown in (13).

For n =1 and d = 1, the steady-state MSE of MCMA can be approximated by [14]

where π2β / 2 is usually larger than 2R.

 

3. Proposed Algorithm

3.1 The CME term of the proposed algorithm

In our algorithm, construction of the CME term according to the transmitted symbol characteristics is the key to improve algorithm performance. We assume a transmitted symbol a has coordinates (ar,ai) , i.e. a = ar + jai, in a constellation diagram. For M-QAM, we have

where g , h , and L are integers, and the minimum distance between symbol points is set to be 2 (as shown in Fig. 2) for simplicity. For high-order QAM signals, L ≥ 2 .

To form the CME term of the proposed algorithm, we firstly introduce a coordinate transformation criterion

where sgn(⋅) is a sign function, n is an integer between 1 and L −1, and χL−1 is the new coordinate transformed from the original coordinate χ0 by using (22).

In (22), let χ0,r = ar and χ0,i = ai, then χL−1,r and χL−1,i can be obtained as sr and si, respectively. Then the process of change for different high-order M-QAM signals can be described as follows.

1) For a 16-QAM signal ( L = 2 ) with ar = ai = {±1,±3} ,

2) For a 64-QAM signal ( L = 3 ) with ar = ai = {±1,±3,±5,±7} ,

3) For a 256-QAM signal ( L = 4 ) with ar = ai = {±1,±3,±5,±7,±9,±11,±13,±15},

It is clear that the following functions take zero values at every symbol point of high-order M-QAM signals by using the coordinate transformation criterion

For the equalizer output signal ( z = zr + jzi ), let χ0,r = zr and χ0,i = zi in (22), then χL−1,r and χL−1,i are obtained as yr and yi respectively,

when the output signals are the same as the transmitted signals.

So the CME function of the proposed algorithm is given by

Take a 16-QAM signal for example, from (23) and (28) the CME function of the proposed algorithm, as shown in Fig. 3(b), is

From Fig. 3(b), the CME function (29) of the proposed algorithm satisfies the following properties.

1) The function is symmetric with respect to each alphabet when −4 ≤ χ ≤ 4 .

2) The minimum values are zeros and only occur at the constellation points ±1 and ±3 . If the equalizer output is near the constellation points, it has a local maximum 0.5 which is reached at the center points between two consecutive alphabets. If the equalizer output is far away from the constellation points, i.e., |χ|> 4 , the function will place the higher penalty.

With property 2), the proposed algorithm can provide better performance than MCMA, since the minimum values of MCMA’s CME function occur not only at the constellation points but also at the unexpected position, as shown in Fig. 3(a). For instance, for a 16-QAM signal with constellation points ±1 and ±3 , if the received signal is far away from the constellation points, then this signal is equalized to ±5 (even ±7) by the CME function of the MCMA, since at these points the values of the CME function are also zeros. So the CME function of the proposed algorithm has lower misadjustment than that of the MCMA, which means the proposed algorithm can obtain better convergence performance.

From (17) and (28), we get the CME term of the proposed algorithm

where yr(k) and yi(k) are derived from zr(k) and zi(k) by using (22), respectively.

3.2 Proposed algorithm

The general form of the cost function of the proposed algorithm is given by

The weight vector of the proposed algorithm is updated by

Like the MCMA, the weighting factor λ in (32) trades off amplitude and constellation-matched errors. Large λ increases convergence time, whereas small λ diminishes the advantage of CME in reducing the steady-state residual errors. Accordingly, λ is chosen to satisfy [14]

The CME function of the proposed algorithm only has a local maximum 0.5, so in (33) the local maximum is used. It cannot affect the performance of the proposed algorithm because when the equalizer output is far away from the constellation points, the CME function places the right higher penalty, i.e. enforces it to be close to the constellation points. The above equation translates to λ < 57.6 for 16-QAM, λ < 1120.2 for 64-QAM, and λ < 12770 for 256-QAM, respectively.

 

4. Performance Analysis

In this section, we show the complexity of the proposed algorithm. Without loss of generality, we derive an analytical expression for the steady-state MSE of the proposed algorithm for 16-QAM in a noiseless environment.

4.1 Computational complexity

The computational requirements of the proposed algorithm (for 16-QAM), CMA, CMA+SDD, and MCMA are summarized in Table 1. And the computational requirements of the proposed algorithm for different high-order M-QAM signals are shown in Table 2. From Table 1, CMA+SDD requires 12×2N + 29 multiplications, 14×2N + 21 additions and 4 exponential operations for each iteration, which indicates a higher complexity than that of CMA. Therefore, CMA+SDD obtains a dramatic performance improvement over CMA at a cost of complexity. The proposed algorithm requires 8×2N +10 multiplications and 8×2N + 6 additions and 4 sign operations for each iteration. It has the similar computational complexity to CMA. So, compared with the CMA+SDD, a considerable computational saving is obtained by the proposed algorithm. Moreover, the MCMA equalizer for calculating the sine function may be balanced for constraints such as speed, accuracy, portability, or range of accepted input values. Hence the computational complexity of the proposed algorithm is also simpler than that of the MCMA.

Table 1.Comparison of computational complexity per weight update

Table 2.The computational requirements of the proposed algorithm per weight update for different high-order M-QAM signals

4.2 Steady-state MSE Performance

Without loss of generality, for the 16-QAM signal we derive the steady-state MSE performance of the proposed algorithm without noise using the method proposed in [6]. The key formula for evaluating the steady-state MSE of the adaptive algorithm is [6]

This implies that the terms T1 and T2 are identical. From this equality, we can obtain an approximate expression for the steady-state MSE E{|ea(k)|2}. And ea(k) is defined as

The eo(k) of the proposed algorithm is in (32)

where yr(k) = zr(k) − 2sgn[zr(k)] and yi(k) = zi(k) − 2sgn[zi(k)] for the 16-QAM signal.

For the 16-QAM signal, sr(k) = ar(k) − 2sgn[ar(k)] = {±1} and si(k) = ai(k) − 2sgn[ai(k)] = {±1} . And in the steady state, sgn[yr(k)] = sgn[sr(k)] = sr(k), sgn[yi(k)] = sgn[si(k)] = si(k), sgn[zr(k)] = sgn[ar(k)], and sgn[zi(k)] = sgn[ai(k)]. So in steady state the eo(k) can be rewritten as

The analysis that follows for the proposed algorithm is based on the assumptions made in [6] regarding the independence of the transmitted symbol a(k) and ea(k), and the independence of μ 2||x(k)||2 and the equalizer output z(k). Furthermore, we can conclude the independence of s(k) and ea(k), and the independence of μ2||x(k)||2 and y(k). In addition, we drop the time index k for simplicity and assume the step-size μ is sufficiently small and the value |ea(k)|2 is reasonably small in steady state.

For T1, we obtain

The term A can be approximated by 2μE(2|a|2-R)E{|ea|2} as shown in [6]. Then

For T2, we obtain

The term B can be approximated by μ2E{|a|6 −2R|a|4+R2|a|2}E{||x||2} as shown in [3]. The term D can be neglected for small μ and small |ea|. By substituting z by a − ea, we get

By expanding F and neglecting the terms containing μ2|ea|2 for some μ and small |ea| and using the independence of a and ea, we obtain the approximation F ≈ 0 . Now we have

From the equality T1 = T2, the steady-state MSE of proposed algorithm can be approximated by

Similarly, the steady-state MSE expressions for 64-QAM and 256-QAM can be calculated, which are the same as the equation (43).

It is clear that when the weighting factor λ is zero, equation (43) is identical with the steady-state MSE of the conventional stochastic gradient descent CMA as shown in (9). The steady-state MSE of the proposed algorithm is linearly proportional to the step-size μ and the received signal norm E||x||2. In steady state, the weighting factor λ only exits in the denominators of (43), and the MSE will decrease with the increase of λ.

It is important to note that the MSE expressions of CMA (9), CMA+SDD (14), MCMA (20) and proposed algorithm (43) have the same numerator. Therefore, the relationships between them depend on the denominators. Specifically,

1) MSECMA+SDD < MSECMA, since α = μd/μ >0.

2) MSEProposed algorithm < MSECMA , since λ > 0 .

3) MSEProposed algorithm ≈ MSECMA+SDD, when λ = α.

4) MSEProposed algorithm ≈ MSECMA+SDD > MSEMCMA > MSECMA, when λ =α =π2β/ 2 .

 

5. Simulation Results

Firstly, we provide simulation results that compare the experimental performance with the one predicted by the analysis in this section. In this simulation, the transmitted data symbols are 16-QAM, 64-QAM and 256-QAM, respectively. The channel considered in this simulation is given by c1 = [0.1,0.3,1,−0.1,0.5,0.2]T [6]. A 10-tap FIR filter is used as a T/2-FSE. Under different step-size μ (from 3×10−5 to 5×10−5) with fixed weighting factor λ = 20 and λ = 50 , respectively, Fig. 4(a) shows two groups of curves for the experimental results of the steady-state MSE and the theoretical results of (43) for 16-QAM. Under different weighting factor λ (from 0 to 50) with fixed step-size μ = 3×10−5 and μ = 4×10−5, respectively, Fig. 4(b) shows two groups of curves for the experimental results of the steady-state MSE and theoretical results of (43) for 16-QAM. From these two figures, we can observe that two steady-state MSEs are matched reasonably well, and linearly proportional to the step-size μ, and increase with decreasing λ. The similar simulation results for 64-QAM and 256-QAM are shown in Fig. 5 and Fig. 6, respectively.

Fig. 4.Theoretical and experimental steady-state MSE curves against step-size μ (a) and weighting factor λ (b) in a noise free environment for 16-QAM signal and channel c1.

Fig. 5.Theoretical and experimental steady-state MSE curves against step-size μ (a) and weighting factor λ (b) in noise free environment for 64-QAM signal and channel c1.

Fig. 6.Theoretical and experimental steady-state MSE curves against step-size μ (a) and weighting factor λ (b) in noise free environment for 256-QAM signal and channel c1.

Secondly, the performance of discussed blind equalization algorithms for 64-QAM are compared. Additionally, to illustrate the advantage of the proposed algorithm we consider the constant norm algorithm (CNA) proposed in [25], which has proven to be particularly adapted to square constellations. Since we pay more attention to the CMA+SDD [9], the same simulation parameters described in [9] are considered. The channel c2 in [9, Table 2] is used and the parameters of the algorithms are set as follows: N= 11 and μ = 5×10−7 for every algorithm, λ = 400 for the proposed algorithm, α = λ and ρ = 0.6 for the CMA+SDD, and β = 2λ/π2 for the MCMA. Fig. 7 shows the simulation results for four different SNRs. The MSE traces of these algorithms shown in Fig. 7 verify that the proposed algorithm and the CMA+SDD have the similar steady-state MSE and convergence rate which are better than those of CMA, CNA and MCMA. Although in the case of lower SNR the steady-state MSE of the proposed algorithm is not much lower than that of the MCMA, the convergence rate of the proposed algorithm is much faster than that of the MCMA. It is interesting to note that the advantages of the proposed algorithm are more significant in higher SNR.

Fig. 7.Performance comparison of CMA, CNA, MCMA, CMA+SDD, and proposed algorithm for 64-QAM and channel c2 with different SNR values. (a) SNR=20dB. (b) SNR=30 dB. (c) SNR= 40dB. (d) SNR=50 dB.

Finally, we compare the performance of discussed blind equalization algorithms through two different channels for 256-QAM. From n=0 to n=40000, we consider the cable channel c3 from the signal processing information base (SPIB, located at http://spib.rice.edu/) and at n=40000, it is changed abruptly to channel c4 = [0.36 0.86 0.36]T [26]. In both cases, the remaining parameters of the algorithms are set as follows: N= 21 and μ =10−8 for every algorithm, λ =1800 for the proposed algorithm, α = λ and ρ = 0.7 for the CMA+SDD, and β = 2λ/π2 for MCMA. Additionally, we do not consider CNA in this simulation since the CNA is very suitable for high-order QAM constellation but not including 256-QAM [25]. In practice, CMA achieves better performance than CNA for 256-QAM. The learning curves of these algorithms are depicted in Fig. 8. We can observe that the proposed algorithm has the lowest steady-state MSE, and the fast convergence rate as CMA+SDD even in the variable channels.

Fig. 8.MSE traces for CMA, MCMA, CMA+SDD, and proposed algorithm for 256-QAM through channels c3 and c4.

 

6. Conclusions

In this paper, we studied a novel low-complexity hybrid blind equalization algorithm that minimizes a cost function made up of CMA and coordinate transformation-based CME terms. The steady-state performance analysis of the proposed technique was developed. The proposed algorithm has a significantly lower computational complexity than the CMA+SDD, and has the same steady-state performance as the CMA+SDD. When compared with the MCMA, the proposed algorithm has lower computational complexity, lower steady-state MSE and faster convergence rate. Simulation results for 16-QAM, 64-QAM, and 256-QAM confirm the effectiveness of the new blind algorithm over existing CMA, CAN, CMA+SDD, and MCMA.

In future work, we will consider applying the proposed algorithm instead of CNA in [27] to improve the performance of blind equation used into OFDM/OQAM. In addition, we will find a way to combine the linear programming technique with the proposed blind equalization to further improve the performance, since the recent research of linear programming applied to blind equalization has proved a significant advance [28].