PERFORMANCE IMPROVEMENT OF CONSTANT ...

c ICIC International 2013 ISSN 1881-803X

ICIC Express Letters Volume 7, Number 4, April 2013

pp. 1377–1383

PERFORMANCE IMPROVEMENT OF CONSTANT MODULUS ALGORITHM BLIND EQUALIZER FOR 16 QAM MODULATION Faramarz Asharif1 , Shiro Tamaki1 , Mohammad Reza Alsharif1 Mahdi Khosravy1 and Heung Gyoon Ryu2 1

Graduate School of Science and Engineering University of the Ryukyus No. 1, Senbaru, Nishihara, Okinawa, Japan { faramarz asharif; mahdykhosravy }@yahoo.com; { shiro; asharif }@ie.u-ryukyu.ac.jp 2

Department of Electronic Engineering Chungbuk National University 52, Naesudong-ro, Heungdeok-gu, Cheongju, Chungbuk 361-763, Korea [email protected]

Received June 2012; accepted August 2012 Abstract. In this paper, we aim to blindly equalize the channel. The deployed modulation is 16QAM. Therefore, Blind Equalizer is suggested. Blind equalizers do not require any known training sequence for the startup period, but can rather perform at any time equalization directly on the data stream. Moreover, the pilot signal can be cut off due to utilization of Blind Equalizer so that the amount of transmitted data is reduced. Thus faster computational process is expected. In this paper, employment of 16QAM modulation has been proposed for Constant Modulus Equalizer (CME) and Modified Constant Modulus Equalizer (MCME) in time domain process. However, it is more difficult for a blind equalizer to perform over 16QAM, and this modulation has been realized. In order to evaluate the each scheme, we have analyzed with the Normalized Mean Square Error (NMSE). Consequently Modified Constant Modulus Equalizer enhances the performance. Keywords: Blind equalizer, Adaptive filter, Constant modulus algorithm, Frequency domain constant modulus algorithm

1. Introduction. In communication systems through transmitting the signal to receiver side, the transmitted signal will be distracted by channel. The effect of channel includes attenuation, distortion and phase offsets. Therefore, channel estimation is required in order to avoid these effects. However, there is no information about the reference signal. In modern digital communication systems many schemes of data correction and recovering are known. The error is dependent on the channel distortion and additive noise. Therefore, there are many schemes for error correction such as CRC (Cyclic Redundancy Check), Adler-32. However, utilization of these data error correcting schemes is limited due to existence of channel complexity and an insufficiency of signal processing. Other scheme is to implement the pilot in OFDM signal in order to measure the channel impulse response. However, this method is independent from channel complexity and additive noise. Therefore, in order to overcome the signal distortion and attenuation, channel estimation is required. By estimating of channel signal distortion or attenuation, channel effects can be reduced and receive the desirable signal in the receiver side. However, generally speaking, in communication system the desired transmitted signal is unknown and moreover the channel which is an entirely unknown multipath impulse response is quite hard to estimate. Therefore, basic concept of this study is to equalize the received signal without having any training sequence. To estimate the unknown channel without any training is known as blind equalization. The difference of blind equalization and conventional method equalization is that trained equalization or non-blind adaptive 1377

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F. ASHARIF, S. TAMAKI, M. R. ALSHARIF, M. KHOSRAVY AND H. G. RYU

filter algorithms are used for equalization by using a training signal to update the weight. In the recent systems well known methods are based on training sequences, where a part of signal is known and repeated and the equalizer is based on matching with its output to the reference signal by adapting its parameters to minimize Mean Square Error regarding reference signal. Unfortunately, the training sequence consumes a considerable part of the overall message. For this reason, recently much research effort has been devoted to blind equalization algorithms. Blind equalization or self-recovering algorithms have no training sequence. Therefore, they do not require an extra bandwidth; moreover, the bandwidth efficiency potential is increased, and hence the bit rate can be improved, but the main weaknesses of these approaches are their high computational process complexity and slow adaptation compared with conventional non-blind equalizer. The most known and popular blind algorithm is the Constant Modulus Algorithm (CMA) [1,2]. Among several classes of methods, the Fractionally Spaced Constant Modulus Algorithm has proven to be a successful algorithm due to its simplicity and robustness to channel effects and performance which is the convergence of learning curve. In next chapters, we will discuss about basic concept of equalization in detail for blind equalization, CMA and Modified CMA, respectively. 2. Blind Equalization. In general case of communication systems, transmitter side transmits the desired signal which is unknown to receiver side and receiver side will receive the transmitted signal. The problem is that between transmitter and receiver, there is an unknown multipath channel too. Because of these, the received signal from transmitter side will be attenuated and distorted by the multipath channel. In addition there is an additive noise in receiver side. Therefore, the received signal entirely becomes different from desire signal. In order to avoid these effects, we have to design a filter regarding to multipath channel and at least suppress the additive noise. By designing the filter which is similar to multipath channel, the attenuation and distortion can be reduced. Thus in order to recover the received signal, blind equalization is one of the important processes in the receiver side. Figure 1 indicates the effects of multi-path channel and additive noise which may cause the attenuation, distortion and interferences.

Figure 1. The effects of channel and additive noise Here “Tx” and “Rx” stand for Transmission side and receiver side, respectively. The equivalent figure of Figure 1 is shown as follows.

Figure 2. The equivalent block diagram of Figure 1 From Figure 2, the received signal can be expressed as follows: y(n) = h(n) ∗ x(n) + n(n)

(1)

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where, ∗ indicates the convolution between transmitted signal and n(n) is additive noise. Also the model of multipath channel h(n) is shown as follows: X h(n) = αi δ(n − τi ) (2) i

where, αi and τi are attenuation factor and time-delay, respectively. As it is clear in Equation (1), the received signal will contain totally different information compared with reference signal which is transmitted signal. Therefore, estimation of unknown multipath channel is required. The structure of blind equalization is shown in Figure 3.

Figure 3. The structure of blind equalization Through Figure 3, it is obvious that the estimated multipath channel ˆh(n) represents the FIR (Finite Impulse Response) Adaptive Filter. So, by making the inverse system, the multipath channel can be canceled. In other words in order to obtain the desired signal, the received signal is deconvoluted with estimated multipath channel which is blindly equalized. If the estimated channel is equivalent to unknown multipath channel then we obtain the desired signal. In following equations, the derivation of equalized signal ye (n) has been shown. From Figure 3, we have ye (n) = yr (n) − (ˆ y (n) − ye (n)) (3) ˆ yˆ(n) = h(n) ∗ ye (n) and by substitution, ˆ ye (n) = yr (n) − (h(n) ∗ ye (n) − ye (n)) ˆ → yr (n) = h(n) ∗ ye (n)

(4)

Therefore, the equalized signal can be obtained by taking the deconvolution ˆ if h(n) = h(n) ˆ ye (n) = deconv(yr (n), h(n)) deconv(x(n) ∗ h(n), h(n)) ˆ → = deconv(x(n) ∗ h(n) + n, h(n)) +deconv(n(n), h(n)) ˆ ˆ = deconv(x(n) ∗ h(n), h(n)) + deconv(n(n), h(n)) = x(n) + deconv(n(n), h(n)). Though the additive noise is remained, however the desired signal is obtained by equalizing or deconvoluting the multipath channel. The method of Steepest Descent is the most ˆ ˆ known scheme to obtain h(n) which it is finding h(n) that minimize the cost function. ˆ Below, briefly shows the derivation of h(n). Error function of equalizer is e(n) = yr (n) − yˆ(n)

(5)

and cost function is Mean Square Error where h.i indicates the expectation.

JBE (n) = e2 (n)

(6)

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ˆ And the weighted vector or h(n) is updated according to Steepest Descent ˆ + 1) = h(n) ˆ h(n + µ(−∇ˆ JBE )

(7)

h

where, ∇hˆ = ∂∂hˆ and µ is computing step size. The above equation can be rewritten as follows. ∂ ˆ + 1) = h(n) ˆ h(yr (n) − yˆ(n))2 i h(n − µ ∂ h(n) ˆ ˆ yˆ(n) = h(n) ∗ ye (n) ∂ ˆ = h(n) − µ ∂ h(n) h(yr2(n) − 2yr (n)ˆ y (n) + yˆ2 (n)i → ˆ H ˆ = h (n)ye (n) ∂ ˆ = h(n) − µ ∂ h(n) (h(yr2 (n)i − 2 h(yr (n)ˆ y (n)i + h(ˆ y 2(n)i) ˆ By substituting the estimated channel signal we have: D E D E ∂ 2 H H 2 ˆ ˆ ˆ ˆ h(n + 1) = h(n) − µ ∂ h(n) hyr (n)i − 2 h (n)ye (n)ˆ y (n) + (h (n)ye (n)) ˆ D E D E ∂ ∂ ∂ 2 H H ˆ ˆ ˆ ˆ hyr (n)i − 2 ∂ h(n) h(n)ye (n)ˆ y (n) − ∂ h(n) ye (n)h(n)h (n)ye (n) = h(n) − µ ∂ h(n) ˆ ˆ ˆ ˆ = h(n) + µ(2 hyr (n)ye (n)i − 2 hye (n)ˆ y (n)i) Here ·H stands for transposition and conjugation of a vector. The instantaneous value of weight vector is ˆ + 1) = h(n) ˆ h(n − 2µye (n)(ˆ y (n) − yr (n)). Eventually the update weight vector becomes as follows: ˆ + 1) = h(n) ˆ h(n + 2µe(n)ye (8) 3. Constant Modulus Algorithm. In previous chapter, we introduced the basic concept of blind equalizer. Constant Modulus Algorithm (CMA) is a carrier-phase independent blind equalizer that is based on the signal modulus. Therefore, it is different from conventional method blind equalizer. Following indicates the cost function and error function of CMA.

JCM A (n) = e2CM A (n) (9) eCM A (n) = |ˆ y (n)|2 − R2 where R2 is a constant modulus which is defined by

4 |x(n)| R2 =

|x(n)|2

The weight vector is updated by the following. ˆ CM A (n + 1) = h ˆ CM A (n) − µ∇JCM A h

ˆ CM A (n) − µ2 (|ˆ =h y(n)|2 − R2 )∇|ˆ y (n)|2

(10)

(11)

(12)

Here we have, ˆ CM A (n) ∗ ye (n) |ˆ y (n)|2 = yˆ(n)ˆ y ∗(n) yˆ(n) = h and . H ˆ∗ ˆ ˆ H (n)ye (n) =h =h CM A (n)ye (n)ye (n)hCM A (n) CM A By substituting the estimated channel signal we have: D E ˆ CM A (n + 1) = h ˆ CM A (n) − µ2 (|ˆ ˆ H (n)ye (n) × y H (n)h ˆ H (n) . h y(n)|2 − R2 )∇h CM A e CM A The instantaneous value of updated wheight vector is ˆ CM A (n + 1) = h ˆ CM A (n) − 2µeCM A (n)ˆ h y (n)ye (n)

(13)

The CMA is widely used in practice for its robustness. However, towards the high order modulation signals, the performance of learning curve will be degraded. The Reason of this is that the error function of CMA attempts to drive the equalizer output to lie on a


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circle radius “R”. Therefore, for overcoming the disadvantage of CMA, the cost function and error function can be defined for Modified CMA (MCMA) as follows:

JM CM A = e2M (n) eM (n) = |Re[ˆ y (n)] − 2sgn[Re[x(n)]] + j(Im[ˆ y (n)] − 2sgn[Im[x(n)]])|2 − R′2

|Re[ˆ y (n)] − 2sgn[Re[x(n)]] + j(Im[ˆ y (n)] − 2sgn[Im[x(n)]])|4 ′2 R =

|Re[ˆ y (n)] − 2sgn[Re[x(n)]] + j(Im[ˆ y (n)] − 2sgn[Im[x(n)]])|2

(14)

Re[·], Im[·] and sgn[·] indicates the real part, imaginary part and signs, respectively. Eventually the update vector becomes as follows: ˆ + 1) = h(n) ˆ h(n − µˆ y ∗(n)ˆ y (n)eM (n)(ye (n) − 2sign[Re[ye (n)]] − j2sign[Im[ye (k)]]) (15) Through such coordinates transforming, all signals locate on the same unit circle. 4. Simulation and Results. In order to evaluate the several algorithms, we have simulated a multipath channel with additive noise for 16QAM modulation. The performance evaluation of the algorithm calculated by normalized mean square error as expressed below

NMSE = 10 log10

n P

|yr (i) − yˆ(i)|2

i=1 n P

(16) 2

|yr (i)|

i=1

Figures 4 and 5 show the channel specification in frequency and time domain response, respectively. Figure 6 shows the transmitted signal with 16 QAM modulation, convolution with channel, CMA procedure in receiver side and finally MCMA are scattered. Moreover, Figure 6 indicates the inequalized signal is entirely distorted by multi-path channel and additive noise. After equalization with CMA, received signal is recovered though it contains phase shift. Subsequently, phase offset has overcome by Modified CMA. Eventually the received signal is scattered almost equivalently to the 16 QAM modulation. Figure 7 shows the learning cure of both methods. In this figure, it can be confirmed that Modified CMA enhanced the performance compared with CMA.

Figure 4. Frequency response of unknown channel

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Figure 5. Impulse response of unknown channel

Figure 6. Constellation of 16QAM modulation and CMA and MCMA in time-domain process 5. Conclusion. In this paper, we applied two different schemes of blind equalizer in time domain over 16QAM modulation. Eventually by simulation results, the performance is evaluated. Modified CMA enhanced the performance compared with CMA. For realizing the adaptive modulation, more fast convergence learning curve is required. Thus, as a future work in order to improve the performance frequency domain process is suggested due to reduction of computational complexity. Acknowledgment. This work was supported by Japan Society for Promotion of Science KAKENHI 22560837 and Marubun Research Promotion Foundation. We wish to deeply express our gratitude to their supports.


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Figure 7. NMSE of CMA and MCMA in time-domain process REFERENCES [1] W. Rao, W. Tan, D. Li, G. Dai, F. Xia, L. Fan, J. Liu and H. Xu, Concurrent blind equalization suitable for 16-QAM signal, International Conference on Wireless Communications & Signal Processing, 2009. [2] A. Dr. A. M. Nassar and E. W. E. Nahal, New blind equalization technique for constant modulus algorithm (CMA), IEEE International Workshop Technical Committee on Communications Quality and Reliability, 2010. [3] H. H. Dam, S. Nordholm and H.-J. Zepernick, Frequency domain blind equalization for MIMO systems, IEEE the 16th International Symposium Personal, Indoor and Mobile Radio Communications, 2005. [4] K.-A. Lee, W.-S. and S. M. Kuo, Subband Adaptive Filtering Theory and Implementation, John Wiley & Sons, Ltd, 2009.