Peer ReviewedPostprint (published version

Castells, J.

Castro, F. J.

Sánchez, J. J.

Vázquez Grau, Gregorio

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•m .e 1' f of SIGNAL PROCESSING V: Theories and Applications L. Torres, E. Masgrau, and M.A Lagunas (eds.) © Elsev/er Science Publishers B. V., 1990 1843 LINEAR PHASE ADAPTIVE LINE ENHANCER FOR IMPROVING THE PERFORMANCE OF PHASE SYNCHRONIZERS F.Castro. J.Castells. J.Sanchez. G. Vazquez. Department of Signal Theory and Communications E.T.S.E. Telecomunicaci6.Apto 30002, (08080) Barcelona (Spain) ·Abstract: This paper deals with the problem of symbol recovery under low SNR conditions and nonlinear channel effects. In order to improve the performance of phase synchronizers the use of some linear ALE methods is proposed: classic ALE, CMA, and ML filtering. Comparisons among them are made from different signal processing aspects such as capacity of noise removal and speed properties. A linear regression is shown to be a good quick form for phase and frequency adquisition. Finally two types of ALE structures are suggested. I • INTRODUCTION The main goal of this paper is the inclusion of advanced DSP methods in classical communication structures. In our study we specially consider open loop diagrams to achieve good solutions for burst PSK modulation. As it is well known, in communication systems good estimation of amplitude, phase and synchronize are required. An easy method for symbol recovery without demodulation process can be implemented with a bank of matched filters to each possible waveform. Only a good symbol synchronization is required. Unfortunately, this structure is is not flexible and cannot adapt itself to the variant characteristics of the incoming signal. Therefore a Doppler deviation, for instance, forces the system to work in .a different frequency which is not the nominal one, so the performance becomes worse. Thus, this paper suggests the use of a linear regression method as frequency and phase estimator. In order to improve it, we deal with an ALE, CMA and ML filtering as a previous process. 11 • THE LINEAR REGRESSION METHOD. A linear regression is an easy and quick form for obtaining the phase and frequency of an unmodulated carry signals from its phase samples. The next picture shows the block diagram of a frecuency and phase estimator based on linear regression. x(n) Figure 1 If an arctangent phase detector is used, (fig 2), the phase unwrapping reconstructs a segment of the phase straight line. ·-----------. ' Phase detector 1 1\l( n) ' ' ' ATN .I!...I~D: l~ :-{uPhase J J •' 1 l£ 1o , nwra 1."' ' ,_ ' ' ' ' ! -------- ----' Figure 2 A Then, a linear regression from the noisy samples cl> ( n) is performed : A If cl> ( n) represents the least square linear, then A 2 E'(l\>(n)-1\>(n)) ~min (1) A A A where cl> ( n) = ro on T + 9 o Minimizing the cost function (1) we achieve the optimal 1\ 1\ solution for ro 0 and 9 0 , that is "E{lc~>EnFJEEnFEEc!>EnF ro=~JDJDJiiJJJDJ.;.._JDJJ0 Et1 2)- E(n) E(n) " EEc~>EnF Etl 2 )- E{NFEEnc~>EnFF 9 = 2 0 E(n ) - E(n) E(n) 1\ 1\ Graphically ro 0 and a 0 can be seen as the slope and the origin ordinate of the straight line that matches the sample set <j>(n). The sample window length becomes a tradeoff between the ability of the system to adap itself to frecuency shifts and the noise sensitivity. This work was granted by the CICYT National Research Plan on Information and Communication Technologies of Spain. -- -----~JJJJJ1844 The structure presents a thereshold level in noise and a maximum estimated frequency since a nonlinear operation (the phase unwrapping) is employed; a high level of noise masks the 27t crossings of the sawtooth phase signal. We can improve the properties of the linear estimator placing a previous equalizer that removes both noise and modulation from the signal. An adaptive equalizer can adjust its behavior to the statistics of the incoming signal. Ill • ADAPTIVE SYSTEMS . Introduction In the next sections we'll suppose that symbol synchronization is known. Furthermore only digital angular modulation is employed, due to its immunity in front of noise, channel nonlinearities, etc. For this reason the use of linear phase adaptive systems is desired (information "travels" on the phase). In'put __.,Output Collateral Information Figwe 3 An adaptive system obtains from the incoming signal a specific information (output signal), thanks to an additional or collateral external information. We can consider two types of collateral information: - Independent information from the input and output signals. For example, the modulus of the output signal (CMA algorithm), the nominal carry frequency (ML filtering), etc. - Dependent information. For example, a reference signal (ALE). The .type of collateral information and its use determine the different methods of adaptive filtering. • The Adaptive Line Enhancer (ALE) The ALE [1], which basic scheme can be seen in figure 4, was initially designed as a degenerate form of a noise canceller. x(n) Figure 4. Classic ALE structure The ALE uses the LMS adaptive algorithm to recursivelly adjust the weights of the filter. So W(n) = W(n-1) + 2 fl e(n) X(n) with e(n) = x(n) - y(n) The collateral information employed is an appropiate delayed replica of the incoming signal. The delay .6 is chosen to obtain a high level of correlation at the input of the comparator. Therefore if with x(n)= A sin (r.o0nT + e0) + No(n) r.oo=27t/T o then .6 is forced to be .6= m T0 with m= 1,2, ... Thus the error signal e(n) has several incorrelated components, basically noise, and it's used to properly adjust the frecuency response of the filter. ALE reachs a high level of noise cancelling, even in very low SNR cases, as shown below. A 8 c Figure 5. (A) Ideal signal (B) Noisy signal (C) Ouptut signal of the filter SNR= OdB For a fast convergence a weight vector W(0)=1/LF (1, cos(r.o0), ... cos(r.o0(LF-1 ))) is used, so initially the algorithm performs a OFT. The fl parameter controls the stability, convergence speed and filter capacity of the ALE. In tt no is imp1 out~ Whe pert< prod algo1 . c The rest1 aver The cons The Thus This cor re If then has r In thE lly :le )n ld ·ly ry In the classic ALE scheme it can be shown the reference is noisy because it's obtained from the incoming signal. An improved scheme extracts the reference from the filter output. x(n) i W t-1 ...... ---...... -tiii.,Y(n) I w+-+. eEnF~ Figure 6. Improved ALE scheme. When the input signal is a modulate signal the performance of the ALE becomes worse because no ~ produces a high level of correlation. Then the CMA algorithm must be used. - Constant Modulus Algorithm (CMA) The objective of this adaptive filtering (2] (3] is to restore the incoming signal to a form wich, on the average, has a constant instantaneous modulus. The collateral information used is the knowledge about the constant modulus of the output signal. The error function to minimize is t= E «~~~tJNF O F Thus, using a gradient search algorithm W(n+1) = W(n)- 11 <IIY<n>li-1) y(n) x·(n) This CMA was initially introduced as a method of correcting the multipath effects: If x(n)= A exp(j(ro 0 nT + 90 + 9(n)) + No(n) then y(n) defined as y(n)=x(n) + a x(n-M has not constant modulus. In the next figure the correcting process is shown A B c Figure 7. (A) Ideal BPSK signal (B) Multipath corrupted signal (C) Output signal of CMA. 1845 With unmodulated signals the CMA performance becomes worse than ALE, since ALE works with a reference signal, which supplies speed and noise removal capability. - Maximum Likehood (ML) Filtering The ML filter (3] uses the knowledge of the nominal frequency of the carry signal as collateral information. The filter is designed under the assumption on minimizing the output power. min ( Py )= wH R w under the constrain equation sHw. 1 where S = (1, exp(-j ro 0 T) ,.exp(-j ro 0 (N-1) T)) T and R = Autocorrelation matrix of X data. The function to be minimized is t= Py- A (SH W -1) being A. a Lagrange multiplier. Taking the gradient V t=O= R Wopt +AS Since SH Wopt =1 then wopt= R'1 s (SH R·1sr1 If, as usually, the R matrix is unknown an adaptive algorithm can be used. Thus, if then where W(n+1) = W(n)- 11 V t W(n+ 1) = P (W(n) - 11 y(n) X(n)) + F P= I-S (SH Sf1 sH F= S (SH Sf1 (Proyection vector) (Shift vector) The filter works respecting a single tone steered at ro0 and removing the rest components as possible. In the particular case of considering white noise then R-1 = constant I Then Wopt = R-1 S (S H R-1 sr1 = S/L where L = OFT length This is equivalent to perform a OFT of the input signal steered at the frecuency ro0. OFT(x(n)) = L x(n) exp(-j ro0 n m T)) with m=1 The OFT filtering provides information about the amplitude and phase of the signal. 1846 Let's assume a MPSK transmission. Then x(n)= A exp(j ro0 n T +90 + e(n)) + No(n) We can estimate the phase of each incoming symbol taking only one sample per symbol of the OFT argument output. The modulation must be removed from the set of OFT samples. To do it, the phase samples are multiplied by M. Thus the modulation phase term become 27t m jumps and they are removed by the phase unwrapping system. Symbol synchronization can be achieved making a L length OFT over the whole data string (being L = number of samples per symbol). When the OFT window is cantered just on a symbol the modulus is maximum. In other cases it clearly decreases. Therefore adjusting the sampling symbol instants (L samples separated) to the modulus of the OFT output the symbol synchronization can be achieved. The OFT results in an attractive usefull method in burst transmission mode thanks to its speed and simplicity. IV • CONCLUSIONS The analysis of the previous methods suggests the implementation of two different structures: - Structure based on ALE filtering x(n) Figure 8 The adaptive characteristics adjusts the behavior of the filtering according to the variations of signal parameters (noise, frequency). No information about nominal frequency is required, so a wide frequency range can be tracked. The convergence time makes it not fast enough for burst transmission mode cases. However an initial OFT vector supplies a high speed improvement. Under modulated signals ALE only provides a first approximation for frequency estimation. - Structure based on ML filtering (OFT) x(n) Figure 9 The quick processing time is very usefull in burst mode thanks to the open loop process employed. Furthermore the structure can be used both in modulated and unmodulated cases. This system requires the knowledge of the nominal carry frequency, so only Doppler deviations can be tracked. REFERENCES [1) John R. Treichler, "Transient and Convergent Behavior Of the Adaptive Line Enhancer" IEEE Trans Acoust, Speech, Signal Processing, vol. ASSP-27, N0.1, FEBRUARY 1979. [2) John R. Treichler and Brian G. Agee,"A New Approach to Multipath Correction of Constant Modulus Signals". IEEE Trans Acoust, Speech, Signal Processing, vol. ASSP-31 .N0.2, APR 1983. [3) John R. Treichler and Michael G. Larimore, "New Processing Techniques Based on the Constant Modulus Adaptive Algorithm". IEEE Trans Acoust, Speech, Signal Processing, vol ASSP.33, N0.2, APR.1985. [4) Otis Lamont Frost, "An Algorithm for Linearly Constrained Adaptive Array Processing". IEEE Proceedings, vol 60, NO 8, August 1972. l SI L. I p n ., • p r • t u ( e a t r I L p b r c a 0 c L " t I L r t I I V c • 

Linear phase adaptive line enhancer for improving the performance of phase synchronizers

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