Computationally Efficient Maximum Likelihood Sequence Estimation ...

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Computationally Efficient Maximum Likelihood Sequence Estimation and Activity Detection for M -PSK Signals in Unknown Flat Fading Channels S. Gazor, M. Derakhtian and A.A. Tadaion

Abstract—In this paper, we develop a computationally efficient algorithm for the Maximum Likelihood (ML) sequences estimation (MLSE) of an M-ary Phase Shift keying (M –PSK) signal transmitted over a frequency non-selective slow fading channel with an unknown complex amplitude and an unknown variance additive white Gaussian noise. The proposed algorithm also provides the ML estimates of the complex amplitude and the noise variance that are critical in signal activity detection and demodulation in the modern cognitive communication receivers. We prove the optimality of the proposed algorithm and compare its performance via simulation with a recent suboptimal algorithm.

I. I NTRODUCTION Signal activity detection and unknown parameter estimation are of much interest in recent advanced intelligent communication systems. For instance, in cognitive radio, accurate and timely detection of the presence of a signal is a challenging and major problem in spectrum sensing. In the absence of the pilot training signals, the blind estimation of the channel gain is another issue in the demodulation [1]–[5]. We are interested in the implementable methods with low Computational Complexity (CC) for joint Estimation of the Sequence and the unknown parameters as well as the detection of the activity of PSK signals. In many applications, such a joint estimation seems to require very expensive exhaustive search over all possible cases of the unknown symbols [6], [7]. In [6], we derived a signal activity detector for an M ary Phase Shift Keying (M PSK) signal in Additive White Gaussian Noise (AWGN) which involves such a search with an exponential CC of the order of O(M N ), where M is the number of constellation points and N is the number of received symbols. In [6], the authors also proposed a suboptimal algorithm with CC of the order of O(N M ) whose performance is close to the optimal search. Note that the key point in the estimation of the unknown parameters is the estimation of the transmitted symbols. This yields into solving a key discrete optimization problem which can be employed in other application. In this paper, we introduce an optimal algorithm for this maximization problem with the computational complexity of the order of O(N log N ). We use the fact that the phase of the channel can only be identified up to a rotation by any multiple of 2π M . We prove the optimality of this algorithm which shows that the proposed algorithm gives the true value of the Maximum Likelihood (ML) estimates of the symbol sequence.

The remaining of the paper is organized as follows. In Section II, we introduce the problem formulation and assumptions. In Section III, we propose and analytically prove our algorithm. In Section IV we provide our new algorithm for two widely used applications, the activity detection and blind demodulation of Differential PSK (DPSK) signal. Section V concludes the paper. II. P ROBLEM S TATEMENT Consider the activity detection of an AWGN as the following binary hypothesis  H0 : r = n, if PSK signal H1 : r = acM + n, if PSK signal

M PSK signal in problem: is absent, is active,

(1)

where r = [r0 , · · · , rN −1 ]T is the base band representation of the received signal vector, n = [n0 , · · · , nN −1 ]T is the complex zero-mean white Gaussian noise with unknown variance σ 2 = E[|ni |2 ], a = |a|ejθ is the unknown complex amplitude of the received signal, which is assumed constant during the observation period and c = [c0 , · · · , cN −1 ]T is the data vector. The digitally modulated sequence the set of n 2πk cn is selected from o M –PSK alphabets CM = ej M , k = 0, · · · , M − 1 , where M is the number of constellation points, which is assumed to N be known. In this paper, we do not treat the data vector c ∈ CM as random and estimate it via ML estimation. We assume no Inter-Symbol Interference (ISI) and neglect synchronization inaccuracies in the sampling time and in the carrier frequency. In practice, the synchronization may be achieved using several methods, such as transmitting pilot-tones for all users or nondata-aided algorithms [8]. The pdf of observation under each hypothesis is:   1 1 2 f (r; σ 2 , H0 ) = exp − krk , (2a) N σ2 (πσ 2 )   1 1 2 2 f (r; a, c, σ , H1 ) = exp − 2 kr − ack (2b) . σ (πσ 2 )N The ML estimates of the unknown deterministic parameters (a, c, σ 2 ) for the problem in (1) and the Generalized Likelihood Ratio (GLR) detector are derived in [6]. The ML estimate of σ 2 under each hypothesis is derived by maximizing the above pdfs, ( c2 = 1 krk2 − |b H1 : σ a|2 , 1 N (3) 1 c 2 2 H0 : σ0 = N krk ,

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and the ML estimate of a under hypothesis H1 is,
1 |b a| = max Re e−jθ cH r . N θ,cn∈CM

(4)

where (.)H denotes the transpose conjugate. In [6], it is suggested that the GLR test be implemented as a computationally expensive search with the exponential order of complexity. Thus to reduce the computational cost substantially, they introduced a suboptimal alternative for the GLR detector namely Simplified GLR (S-GLR). In this paper, we propose a computationally efficient algorithm for the implementation of the GLR and the ML estimates of the unknown parameters. An interesting phenomenon we employ here is that the phase can only be identified up to a rotation by a multiple of 2π M. Using this fact and reduction of the search space, we decrease the complexity order of the search from the exponential to the linear. The main mathematical problem is obtained by substituting the estimated values of a and σ 2 in the pdf (2) and maximizing over remaining unknown parameters result in N the following optimization over c ∈ CM and θ ∈ [0, 2π),
ˆ = arg max Re e−jθ cH r . (ˆc, θ) (5) θ,cn ∈CM

Note that such a discrete optimization problem can have other applications, e.g., in the estimation of the received signal amplitude, the SNR estimation and in signal demodulation.

III. P ROPOSED A LGORITHM In this section, we develop a fast solution for the maximization in (5). Lemma 1: The optimum value in (5) remains invariant under the transformation rn′ = ejφn rn for all ejφn ∈ CM . Proof: From (5), we can easily see that ! N −1 X −jθ jφn ∗ jφn ˆ (ˆ c, θ) = arg max Re e (e cn ) (e rn ) . θ,cn ∈CM

n=0

Since (e cn ) is also in CM for all ejφn ∈ CM , defining ∆ c′n = ejφn cn we have ! N −1 X ˆ = arg max Re e−jθ (cˆ′ , θ) c′∗ (ejφn rn ) , jφn

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θ,c′n ∈CM

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ˆ are related by cn = e−jφn c′n . Interestingly, this where cˆ′ and c ˆ gives the optimal solution of (5) for simply means that (cˆ′ , θ) jφn the transformed data e rn . The invariance of (5) becomes ˆ and r′ in (5). trivial by substituting (cˆ′ , θ) We use the above lemma as a first step of the proposed algorithm and transform the input data {r0 , r1 , · · · , rn } by applying discrete rotations rn′ = ejφn rn (note that ejφn is a constellation point) such that all rotated data rn′ fall inside the π π arc [− M ,M ). These N rotations can be efficiently performed using a COordinate Rotation DIgital Computer (CORDIC) algorithm [9]. It is easy to
see that the rotated data is unique. ∂ Setting ∂θ Re e−jθ cH r = 0 and using Lemma 1, we easily could show that the optimization problems in (4) and (5) are also equivalent with   θ = ∠c′ H r′ , ′H ′ (6) r c .  c′ = arg max ′ cn ∈CM

From (5), we conclude that ifθ is known, c′n can be simply determined by maximizing Re e−jθ c′ H r′ over all possible c′n ∈ CM . This means to take c′n from CM which has the nearest phase to e−jθ rn′ . However since θ is unknown, we should determine c′n for each value of θ ∈ [0, 2π), in this way; H among them, c′ which maximizes c′ r′ is the solution. We must note that the possible values of θ that should be verified, is not finite. In the following lemma, we prove that it is enough to perform a search only over specified discrete values of θ. Since the optimization criterion is invariant under the group of permutation, we sort the data vector r′ based on the phases of its elements in an ascending order and denote the resulting sorted vector by z = [z0 , · · · , zN −1 ]T where ∠z0 ≤ ∠z1 ≤ · · · ≤ ∠zN −1 . Such a sort can be performed with a CC of order of N log2 N operations. In the following, we prove that ∆ it suffices to investigate only for the discrete values of θn = π ∠zn + M , for all n = 0, · · · N −1. We also partition our search space [0, 2π M ) into sub intervals I0 = [0, θ0 ), I1 = [θ0 , θ1 ), · · ·, 2π IN −1 = [θN −2 , θN −1 ), IN = [θN −1 , 2π M ), where [0, M ) = N ∪i=0 Ii . Lemma 2: In order to determine θ that maximizes (6), it suffices to perform our search over N values each taken from one of intervals Ii for all i = 0, · · · , N ; i.e., one of these N different values of θ maximizes (6). These values are specified in the proof. Proof: From   ˆ = arg max Re e−jθ (c′ H r′ ) (ˆc, θ) ′ θ,cn ∈CM

2πq it is easy to see that the pairs (cˆ′ ej M , θˆ − 2πq M ) for all q = also optimal solutions; since in (6) we have 0, · · · , M − 1 are2πq ′H ′ ′ j M H ′ ) r . Therefore, it suffices to search on c r = (c e θ over any interval with a length of 2π M , e.g., we search over θ ∈ [0, + 2π ). M As mentioned before, the optimum choice of c′n is the member of CM which has the nearest phase to ∠(e−jθ zn ) = π . Thus for θ ∈ I0 , we have c′n (I0 ) = 1, n = θn − θ − M π 0, · · · , N − 1 since all the phases ∠(e−jθ zn ) = θn − θ − M π π ′ are within the interval [− M , M ). The notation cn (Ik ) is the optimum value of c′n provided that θ ∈ Ik . For all values θ ∈ Ik = [θk−1 , θk ), we have π ∠(e−jθ zn ) + = θn − θ ∈ (θn − θk , θn − θk−1 ]. M Since the sequence, θn is already sorted, for n < k we have π −jθ θn − θk−1 ≤ 0 and θn − θk > − 2π zn )+ M must M ; thus ∠(e 2π be a negative number greater than − M . Similarly for n ≥ k, π it is easy to see that, ∠(e−jθ zn ) + M must be positive and 2π smaller than M . This simply means that we should choose  −j 2π e M for n < k, (7) c′n (Ik ) = 1 otherwise.

Note that we do not need to perform any computation for the last interval θ ∈ IN . Based on the above lemmas, we propose the following algorithm for the determination of dˆn . Step 1: Transform the input data {r0 , r1 , · · · , rn } by applyn ing discrete rotations rn′ = ejφn rn where φn = 2πl M

GAZOR ET. AL.

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for some ln ∈ {0, · · · , M − 1} such that all rotated π π ,M ). Record the data rn′ fall inside the arc [− M ′ vector c ← [exp(jφ0 ), · · · , exp(jφN −1 )]T . Step 2: Sort the new values rn′ , n = 0, · · · , N − 1, increasingly, according to their phase values. We call the new values z, such that zn′ = rq′ n , n = 0, · · · , N − 1. Let q = [q0 , · · · , qN −1 ]T denote the index of the unsorted values, and update c′ by the same permutation, i.e., c′n ← c′qn , n = 0, · · · , N − 1. Step 3: Denoting   1 1 ··· 1 1 2π  ej M 1 ··· 1 1     .. ..  ′ C = . (8)  .  2π  2π j j  e M e M ··· 1 1 

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compute w = C ′ z and find the element of the result with the maximum absolute value as the desired maximum value. Let r denotes the index of the element with maximum value, i.e., r = arg max |wk |. In

Fig. 1. Performance comparison of the proposed GLR detector and S-GLR detector in [6] for BPSK signals, in terms of Pd versus SNR, assuming that N = 7 and Pfa = 0.01.

that case the optimal value for c′ is obtained by the multiplication of the k th row of C ′ by the stored value in c′ from previous steps. Step 3: Let p = [p0 , · · · , pN −1 ]T denote the inverse of the permutation q obtained in the sort operation in Step 2, and apply this inverse-sort to c′ in order to find the estimated c as follows:

0.6 dB in SNR, respectively for symbol lengths of 10. In Figure 4, we compare the same results as in the previous figure for 8PSK signals.

k

b c = [c∗p0 , · · · , c∗pN −1 ]T .

The above matrix product may be implemented employing a fast algorithm with a CC of order of 2N . Thus the CC of this algorithm is of order of N log2 N due to the sort algorithm. We must note that the symbol sequence can be determined correspondingly using the maximization algorithm. IV. S IMULATION R ESULTS We employ our algorithm in two well-known applications; signal activity detection and data extraction. The GLR test for the presence detection of an M -PSK signal, while the complex amplitude of the signal and the noise variance are unknown, involves the exhaustive search over symbol sequence. We implement this detector using our algorithm, with substantially lower CC. Figures 1 and 2 illustrate the performance of our implementable GLR detector compared with the suboptimal S-GLR detector in [6]. We observe that in Figure 1, for a BPSK signal with symbol length of N = 7 the proposed GLR outperforms about 0.085 dB in SNR and in Figure 2, for an 8PSK signal with symbol length of N = 8 the proposed GLR outperforms about 0.06 dB better in SNR. Symbol sequence estimation in the frequency non-selective slow fading channel with unknown complex amplitude of the received signal is another important application of the proposed optimization algorithm. Figure 3 shows that the symbol error rate of our proposed ML symbol sequence estimator compared with the well-known DBPSK symbol-bysymbol estimator is improved significantly about 0.5 dB and

V. C ONCLUSION In this paper we proposed an algorithm with low computational complexity for activity detection and the estimation of the transmitted symbols in M -PSK signals in slow flat fading environment. The estimation is developed assuming that the complex amplitude of the received signal and the noise variance are unknown. This ML estimates not only are used in the signal activity detection problem, but also in the estimation of the signal amplitude in fading channels. Simulation examples illustrate and compare the performance of the proposed detector/estimator with the ones obtained using the suboptimal algorithms we derived in our previous research. R EFERENCES [1] J.F. Kuehls and E. Geraniotis, “Presence Detection of Binary-PhaseKeyed and Direct-Sequence Spread-Spectrum Signals Using a PrefilterDelay-and-Multiply Device,” IEEE Journal on Selected Areas in Communications, vol. , vol.8, No.5, pp.915-933, Jun. 1990. [2] A. Polydoros and K. Kim, “On the Detection and Classification of Quadrature digital modulations in band-limited noise,”, IEEE Transactions on Communications, vol. , vol.38, No.8, pp.1199-1211, Aug. 1990. [3] W.A. Gardner, C.M. Spooner, “Signal Interception: Performance Advantages of Cyclic Feature Detectors,” IEEE Trans. on Communication, pp.149-159, Jan. 1992. [4] S.M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall, 1998. [5] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, 1998. [6] A.A. Tadaion, M. Derakhtian, S. Gazor, M.M. Nayebi, M.R. Aref, “Signal Activity Detection of Phase-Shift Keying Signals,” IEEE Transactions on Communications, vol. , vol. 54, no. 8, pp. 1439-1445, Aug. 2006. [7] M. Derakhtian, A. A. Tadaion, S. Gazor and M. M. Nayebi, “Invariant Activity Detection of a Constant Magnitude Signal with Unknown Parameters in White Gaussian Noise,” IET Communications, May. 2009.

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[8] B.D. Hart, “Maximum Likelihood Sequence Detection Using a Pilot Tone,” IEEE Trans. Veh. Technology, Vol.9, No.42, pp.550-560, Mar. 2000. [9] Duprat, J., Muller, J.-M., “The CORDIC algorithm: new results for fast VLSI implementation,” IEEE Transactions on Computers, Vol.42, No.2, pp.168-178, Feb. 1993.

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SNR (dB) Fig. 4. Symbol Error Rate versus SNR for the proposed ML sequence estimator (N = 10) and the traditional Symbol by Symbol estimator of D8PSK signal.