Denoising Higher-order Moments for Blind Digital Modulation Identification in Multiple-antenna Systems
Sofiane Kharbech, Eric Pierre Simon, Akram Belazi, Wei Xiang
TThis article has been accepted for publication in a future issue of the journal IEEE Wireless Communications Letters, but has not been fully edited. Contentmay change prior to final publication. Citation information: DOI 10.1109/LWC.2020.2969157, IEEE Wireless Communications Letters.
Denoising Higher-order Moments for Blind DigitalModulation Identification in Multiple-antennaSystems
Sofiane Kharbech,
Member, IEEE,
Eric Pierre Simon, Akram Belazi, and Wei Xiang,
Senior Member, IEEE
Abstract —The paper proposes a new technique that substan-tially improves blind digital modulation identification (DMI)algorithms that are based on higher-order statistics (HOS). Theproposed technique takes advantage of noise power estimationto make an offset on higher-order moments (HOM), thus gettingan estimate of noise-free HOM. When tested for multiple-antenna systems, the proposed method outperforms other DMIalgorithms, in terms of identification accuracy, that are basedonly on cumulants or do not consider HOM denoising, even fora receiver with impairments. The improvement is achieved withthe same order of complexity of the common HOS-based DMIalgorithms in the same context.
Index Terms —Cognitive radio, modulation identification,higher-order statistics, multiple-antenna systems, denoising fea-tures.
I. I
NTRODUCTION W ITH the continuous and fast development of intelligentcommunication systems, signal detection is alwaysa critical issue to consider. In intelligent transmission suchas cognitive radios, signal detection is no more limited todetecting energy, and it goes beyond, e.g., demodulatingunknown signals. Modulation identification is the step thatsucceeds energy detection and precedes signal demodulation.When both source signals and channel parameters are un-known, we are in a blind context that naturally requires ablind process of modulation recognition. Despite their highidentification accuracy, maximum-likelihood-based techniquesfor modulation identification often suffer from the substantiallyhigh complexity. Feature-based algorithms of modulation iden-tification give an alternative that provides a good performanceand complexity trade-off.As low computational complexity features and widely em-ployed in digital modulation identification (DMI), higher-order statistics (HOS), i.e., higher-order moments (HOM)and higher-order cumulants (HOC), have always exhibited agood identification performance [1]–[8]. Employed HOS in
The source code for this work is available on https://github.com/sofiane-kharbech/Denoising-HOM-for-DMI
S. Kharbech is with the Laboratory IEMN/IRCICA (UMR-CNRS-8520),University of Lille, Lille 59100, France, and with the Laboratory Sys’Com-ENIT (LR-99-ES21), Tunis El Manar University, Tunis 1002, Tunisia (e-mail:sofi[email protected]).E. P. Simon is with the Laboratory IEMN/TELICE (UMR-CNRS-8520),University of Lille, Lille 59100, France (e-mail: [email protected]).A. Belazi is with the Laboratory RISC (LR-16-ES07), Tunis El ManarUniversity, Tunis 1002, Tunisia (e-mail: [email protected]).W. Xiang is with the College of Science and Engineering, James CookUniversity, Cairns, QLD 4870, Australia (e-mail: [email protected]). that context are estimated from noisy observations. EstimatedHOC are insensitive to noise [9], unlike the estimated HOM.As such, most of HOS-based DMI algorithms rely on HOCas features [1], [5], [7], [8]. However, many other HOS-based DMI algorithms attempt to improve the identificationperformance by including a set of HOM [2]–[4], [6].Since blind estimation of the noise power is widely ad-dressed in the literature, this motivated us to consider de-noising the estimated HOM. The main contribution of thispaper is to further improve the performance of a DMI systemthrough the use of noise-free HOM as part of HOS. Further-more, to enhance the blindness aspect, we make use of aclassifier that does not require prior training. Also, within theframework of this paper, we consider multiple-input–multiple-output (MIMO) systems as an essential part of state-of-the-art wireless systems. Moreover, multi-antenna systems areamply involved in the subject of DMI [2]–[6], [8], [10]–[13].As far as we know, there is no yet attempt on offsettingnoise in HOM in the blind DMI context. In more detail, thecontributions of the paper are three-fold: (i) The derivationof the noise-free HOM formulas for the baseband digitally-modulated signals that in turn require the derivation of theHOM formula for the complex-valued Gaussian noise; (ii) Thedenoising approach is integrated into the MIMO system. Itallows noiseless HOM for each receive antenna, taking intoaccount the effect of source separation processing; and (iii)The denoising technique improves the identification accuracyunder the influence of various receiver impairments whilemaintaining the same complexity order of the DMI system.The rest of the paper is organized as follows. Section IIdescribes the signal model as well as the identification process.In Section III, we give analytical formulas for the denoisedmoments. Section IV includes a discussion of the presentedresults. Finally, Section V concludes the paper.II. S
YSTEM M ODEL
In this section we formulate the mathematical model of thereceived signals and we present a description of each block ofthe identification process. For better readability, Table I definesthe notation used in the paper.
A. Signal Model
We consider a frequency-flat block-fading MIMO systemwith N t transmit and N r receive antennas ( N t < N r ). The a r X i v : . [ ee ss . SP ] F e b his article has been accepted for publication in a future issue of the journal IEEE Wireless Communications Letters, but has not been fully edited. Contentmay change prior to final publication. Citation information: DOI 10.1109/LWC.2020.2969157, IEEE Wireless Communications Letters. TABLE I: Notation d = Equality in distribution C Set of complex numbers N + Set of natural numbers, 0 is excluded Imaginary unit a , a , A Scalar, vector, matrix I d Identity matrix of size d × d ( . ) T Transpose operator ( . ) ∗ Complex conjugate ( . ) H Hermitian transpose | . | Modulus of a complex number ∠ . Argument of a complex number ˆ x Estimator of x N Real-valued normal (or Gaussian) distribution CN Complex-valued normal distribution U Uniform distribution E { . } Expected value of a random variable !! Double factorialdiag − ( . ) Main diagonal vector of a matrixi.i.d. independent and identically distributed n th received baseband signal at antenna i is expressed as y i ( n ) = N t (cid:88) j =1 h ij x j ( n ) + w i ( n ) , (1)where y i ( n ) is the i th element of the received MIMO symbol y ( n ) ∈ C N r × , x i ( n ) is the j th element of the transmittedMIMO symbol x ( n ) ∈ C N t × (source signals are i.i.d.), h ij isthe element ( i, j ) of the spatially-uncorrelated MIMO channelmatrix H ∈ C N r × N t , and w i ∼ CN (cid:0) , σ w (cid:1) is a circularlysymmetrical complex Gaussian noise at the receive antenna i . B. Identification Process
The overall process relies on a process that is widely usedfor the blind DMI issue in MIMO systems [2]–[6] whiletaking advantages of noise power estimation (e.g., [2], [5]) fordenoising moments. This is in addition to the use of a blindclassifier instead of a classifier that requires prior training.Fig. 1 depicts the implemented detection process on onereceive antenna. The proposed scheme is composed of threemain stages, namely: (i) the blind source separation (BSS) stepto blindly recover the source signal in conjunction with a noisepower estimator; (ii) the denoising-based feature extractionstage allows a better characterization of the modulation scheme(features are denoted by µ for HOM and by κ for HOC);and (iii) the modulation scheme at each antenna is estimatedvia minimum distance (MD) classification. The estimatedmodulations ˆ θ j are gathered for the final decision. The mostfrequent modulation scheme is regarded as the final decision.For BSS, we make use of the simplified constant modulusalgorithm (SCMA) [14], which is a simplified version ofthe well-known constant modulus algorithm. SCMA aims atfinding a matrix G ∈ C N r × N t termed the separator so that therecovered MIMO symbol ˆ x ( n ) is estimated as ˆ x ( n ) = G T y ( n ) = G T Hx ( n ) + ˜ w ( n ) , (2)where ˜ w ( n ) = G T w ( n ) is the filtered noise. Assuming perfectBSS, i.e., G T H = I N t , we have ˆ x ( n ) = x ( n ) + ˜ w ( n ) . (3) The common eigenvalue-based technique [15] is an obviouschoice to estimate the noise power and the number of transmit-ters at once. This technique can be applied as it is in a blindcontext [2]. The features extraction process with denoising,as the main contribution of the paper, is discussed in detailin the next section. For ensuring blind DMI, we make useof a blind classifier, i.e., a classifier that does not need tobe trained on test signals with known modulation schemesand particular values of the signal-to-noise ratio (SNR). TheMD classifier is the simplest for that goal as it calculates theEuclidian distance of a feature vector with all the theoreticalones, and then selects the closest. Featuresextractionand denoising
MDclassificationBSSNoise powerestimation . . .
Fig. 1: Denoising-based blind modulation identification scheme for areceive antenna.
III. H
IGHER - ORDER S TATISTICS FOR
DMI
AND D ENOISING M OMENTS
Table II shows how a digital modulation scheme can becharacterized by a set of HOS. Since it is already proved thatHOC are noise-insensitive [9], we focus on the derivation ofthe HOM in this section. For a given signal x and integers p and q , ≤ q ≤ p , the HOM of order p is expressed as µ pq ( x ) = E (cid:8) x p − q x ∗ q (cid:9) . (4) TABLE II: Theoretical values of the deployed set of higher-orderstatistics for the simulated pool of modulation schemes [1], [16].These values are obtained using noiseless signals of zero mean andunit variance. B - PS K Q - PS K - PS K - A S K - A S K - QA M µ .
64 1 . − . µ .
64 1 .
77 0 µ .
64 1 .
77 1 . µ .
92 3 .
62 0 µ − .
92 3 . − . µ .
92 3 .
62 0 µ .
92 3 .
62 1 . µ .
25 7 .
92 3 . κ − − . − . − . κ − − . − .
24 0 κ − − − − . − . − . κ
16 0 0 8 .
32 7 .
19 0 κ − .
32 7 .
19 2 . κ
16 0 0 8 .
32 7 .
19 0 κ
16 4 4 8 .
32 7 .
19 2 . In fact, not all moments need to be denoised. One canprove that µ p is noise-insensitive (cf. Appendix). To derivethe noise-free moments for q > , we have to derive theirformulas in terms of the noise power σ w . Towards this end, we will consider y = x + w as a given mix of a digital modulatedsignal x in the baseband and a circularly symmetrical complexGaussian noise w ∼ CN (cid:0) , σ w (cid:1) independent of x . Theconsidered moments (Table II, q > ) are derived as follows. µ ( y ) = µ ( x ) + 3 µ ( x ) σ w (5) µ ( y ) = µ ( x ) + 4 µ ( x ) σ w + E (cid:8) w w ∗ (cid:9) (6) µ ( y ) = µ ( x ) + 5 µ ( x ) σ w (7) µ ( y ) = µ ( x ) + 8 µ ( x ) σ w + 6 µ ( x ) E (cid:8) w w ∗ (cid:9) (8) µ ( y ) = µ ( x ) + 9 µ ( x ) σ w + 9 µ ( x ) E (cid:8) w w ∗ (cid:9) + E (cid:8) w w ∗ (cid:9) (9) µ ( y ) = µ ( x ) + 16 µ ( x ) σ w + 36 µ ( x ) E (cid:8) w w ∗ (cid:9) + 16 µ ( x ) E (cid:8) w w ∗ (cid:9) + E (cid:8) w w ∗ (cid:9) (10)In (6), (8)–(10), to have formulas in relation to σ w , we shouldderive µ pq ( w ) (i.e., E { w q w ∗ q } ) for q > . For that purpose,we introduce Theorem 1. Theorem 1.
Let s ∼ CN (cid:0) , σ s (cid:1) , µ pq ( s ) = (cid:16) p (cid:17) ! σ ps , if q = p , , elsewhere . (11) Proof. s = s r + s i , where s r and s i are two independent, real-valued normal random variables, i.e., s r d = s i ∼ N (cid:16) , σ s (cid:17) .For q = p/ , we have µ pq ( s ) = E {| s | p } (cf. Appendix) = E (cid:110)(cid:0) s r + s i (cid:1) p/ (cid:111) = p/ (cid:88) k =0 (cid:18) p/ k (cid:19) E (cid:8) s p − kr s ki (cid:9) = p/ (cid:88) k =0 (cid:18) p/ k (cid:19) E (cid:8) s p − kr (cid:9) E (cid:8) s ki (cid:9) . Using the moments of a real-valued normal variable derivedin [17], we arrive at E { s nr } = ( n − (cid:18) σ s (cid:19) n/ , if n is even , , if n is odd . This leads to µ pq ( s ) = p/ (cid:88) k =0 (cid:18) p/ k (cid:19) ( p − k − (cid:18) σ s (cid:19) p/ − k × (2 k − (cid:18) σ s (cid:19) k = (cid:16) p (cid:17) ! (cid:18) σ s (cid:19) p/ p/ (cid:88) k =0 ( p − k − k − k ! ( p/ − k )! (cid:124) (cid:123)(cid:122) (cid:125) = 2 p/ = (cid:16) p (cid:17) ! σ ps . (cid:4) Hence, Theorem 1 results in µ ( w ) = 2 σ w , µ ( w ) =6 σ w , and µ ( w ) = 24 σ w , and the moments are properlyderived in terms of the noise power. Considering our context,the denoised moments at each receive antenna j are givenbelow. µ ( x j ) = µ (ˆ x j ) − µ (ˆ x j )ˆ σ (cid:101) wj µ ( x j ) = µ (ˆ x j ) − µ ( x j )ˆ σ (cid:101) wj − σ (cid:101) wj µ ( x j ) = µ (ˆ x j ) − µ (ˆ x j )ˆ σ (cid:101) wj µ ( x j ) = µ (ˆ x j ) − µ ( x j )ˆ σ (cid:101) wj − µ (ˆ x j )ˆ σ (cid:101) wj µ ( x j ) = µ (ˆ x j ) − µ ( x j )ˆ σ (cid:101) wj − µ ( x j )ˆ σ (cid:101) wj − σ (cid:101) wj µ ( x j ) = µ (ˆ x j ) − µ ( x j )ˆ σ (cid:101) wj − µ ( x j )ˆ σ (cid:101) wj − µ ( x j )ˆ σ (cid:101) wj − σ (cid:101) wj , (12)where µ ( x j ) = µ (ˆ x j ) − ˆ σ (cid:101) wj and ˆ σ (cid:101) wj is the estimatedpower of the filtered noise at the receive antenna j . Thevariance is estimated as (cid:2) ˆ σ (cid:101) w , · · · , ˆ σ (cid:101) wN t (cid:3) T = diag − (cid:16) E (cid:110)(cid:101) w (cid:101) w H (cid:111)(cid:17) = diag − (cid:16) E (cid:110) G T ww H G ∗ (cid:111)(cid:17) = diag − (cid:16) G T E (cid:8) ww H (cid:9) G ∗ (cid:17) = ˆ σ w diag − (cid:16) G T G ∗ (cid:17) , (13)where ˆ σ w is the estimated noise power of the channel. Fur-thermore, it is worth noting that, to offset the scale factor thatcan be introduced by BSS non-ideality, all the employed HOSare self-normalized, i.e., divided by µ p/ .IV. N UMERICAL R ESULTS
In this section, we evaluate the performance of our proposedscheme as characterized by the probability of correct identifi-cation, P ci . Computer simulations are based on the modulationpool of Table II and different MIMO antenna configurations.Fig. 2 shows the performance of the identification system,with perfect estimation of σ w for the following scenarios:(i) a set of HOS is used without denoising HOM; (ii) onlya set of HOC is used; and (iii) a set of HOS is usedwith denoising HOM. Distinctly, the third scenario, whichrepresents our proposal, exhibits a better performance thanthe other comparative ones. For example, considering theMIMO antenna configuration × within the simulated SNRrange, DMI in scenario (iii) attains an average gain of about15% and 7% compared to a DMI in scenarios (i) and (ii),respectively. Regarding scenarios (i) and (iii), the performancegain is further higher at lower SNRs. It is noted in the samefigure that for all the investigated scenarios, the identificationperformance decreases when ∆ = N r − N t decreases. Infact, this phenomenon is due to the proportionality betweenthe effectiveness of the BSS and ∆ [2]. In respect of theperformance drop due to a lower ∆ , DMI endures an averageperformance loss of 10% for scenario (iii) compared to about17% and 15% for scenarios (i) and (ii), respectively. Thismeans that the performance gain in scenario (iii) is more -2 0 2 4 6 8 10 12 14 SNR (dB) P c i Fig. 2: Probability of correct identification in terms of SNR. Thesimulated scenarios are as follows. With moments denoising [ ∗ ],without moments denoising [ ◦ ], and cumulants only [ (cid:79) ]. MIMOantenna configurations are × [ ] and × [ ]. -2 0 2 4 6 8 10 12 14 SNR (dB) P c i Fig. 3: Probability of correct identification in terms of SNR underdifferent standard deviations (std) of ε . MIMO antenna configurationis × . The plots in gray are copied from Fig. 2 for comparisonwith the HOC-only and non-denoised HOM senarios. significant for the MIMO antenna configuration × . There-fore, the proposed method is more resistant against the BSSimpairments when ∆ becomes smaller.Robustness to imperfect estimation of the channel noisepower (i.e., ˆ σ w = σ w + ε , where ε is the estimation error) isinvestigated in Fig. 3. Obviously, the performance undergoesdegradation as the variance of the channel’s noise powerestimator becomes higher. Nevertheless, the denoising remainsrelevant, in particular at lower SNRs, and still outperforms theHOC-only approach.Fig. 4 assess the reliability of the denoising-based processagainst impairments of the baseband receiver (local oscillator).To generate the phase noise, we employ one of the most com-monly used procedures [18]. Globally, due to the robustnessof the utilized BSS algorithm over the phase noise, the three scenarios have not endured a significant performance loss.However, the denoising-based approach has almost maintainedthe same performance. This is unlike the effect of the carrierfrequency offset (CFO), where a drop in performance is clearlyobserved for all approaches. Indeed, this is due to deficientBSS since the CFO induces a time-variation effect, and theSCMA algorithm is designed for time-invariant channels [19].Yet, the DMI algorithm with the denoising approach remainsthe best compared to the other ones.V. C ONCLUSION
In summary, after deriving the noise-free moments for DMI,simulation results proved that modulation detection throughdenoising is more efficient that the deployment of classicalprocesses for detection like the HOC-based ones and otherschemes that do not proceed to denoise moments. Moreover,as it did not assume particular values of the SNR, the proposedapproach of denoising is also applicable to hierarchical clas-sification. With regards to complexity, additional operationsrelated to denoising terms have a constant time complexity.Besides, since the evaluated identification scheme is basedon the ones like [2], [4] which follow the same complexityorder (both have a polynomial running time), the outstandingperformance of our proposal is achieved more blindly and withthe same order of computational complexity of the HOS-basedDMI systems in MIMO channels. More generally, employingdenoised HOM for any HOS-based DMI algorithm, whichassumes a priori knowledge of the noise power or considersits estimation (e.g., [2], [4], [12]), improves the identificationaccuracy without an increase in the computational complexityor at least its order. A
PPENDIX D ERIVING µ p ( x + w ) AND µ pq ( s ) a) Deriving µ p ( x + w ) : Let the mix x + w of a basebanddigitally-modulated signal x and a noise w ∼ CN (cid:0) , σ w (cid:1) independent of x , µ p ( x + w ) = E { ( x + w ) p } = p (cid:88) k =0 (cid:18) pk (cid:19) E (cid:8) x p − k w k (cid:9) = p (cid:88) k =0 (cid:18) pk (cid:19) E (cid:8) x p − k (cid:9) E (cid:8) w k (cid:9) . (14)However, w is a circularly symmetrical Gaussian process, i.e., E (cid:8) w k (cid:9) = 0 ∀ k ∈ N + , thus, µ p ( x + w ) = E { x p } = µ p ( x ) . (15) b) Deriving µ pq ( s ) : Let s = re θ ∼ CN (cid:0) , σ s (cid:1) , where r = | s | and θ = ∠ s are independent, µ pq ( s ) = E (cid:8) s p − q s ∗ q (cid:9) = E (cid:110) r p − q e ( p − q ) θ r q e − qθ (cid:111) = E { r p } E (cid:110) e ( p − q ) θ (cid:111) , (16)however, θ ∼ U ( − π, π ) , consequently, E (cid:110) e ( p − q ) θ (cid:111) = , if q = p , , elsewhere . (17) -2 0 2 4 6 8 10 12 14 SNR (dB) P c i -4 -3 -2 -1 Normalized frequency -30-20-100 N o r m a li z ed m agn i t ude ( d B ) -1 0 1-1-0.500.51 (a) Phase noise effect. -2 0 2 4 6 8 10 12 14 SNR (dB) P c i (b) CFO effect with normalized frequency offset of the order − . Fig. 4: Probability of correct identification in terms of SNR in consideration of local oscillator imperfections. The simulated scenarios include:with moments denoising [ ∗ ], without moments denoising [ ◦ ], and cumulants only [ (cid:79) ]. The MIMO antenna configuration is × . The plotsin gray are copied from Fig. 2 for comparison. The plots inside (a) show the frequency response of the filter used to generate the phasenoise (characteristics of the power spectrum density mask are . − and − dBc/Hz for the phase noise normalized-bandwidth and level,respectively), and the related effect observed on noise-free 16-QAM constellation as an example. Thus, µ pq ( s ) = E {| s | p } , if q = p , , elsewhere . (18)A CKNOWLEDGMENT
The authors would like to thank Dr. Octavia A. Dobre fromMemorial University of Newfoundland (NL, Canada) and Dr.Chad M. Spooner from NorthWest Research Associates (CA,USA) for the useful discussions on higher-order statistics.R
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