An Algorithm to Mitigate Channel Distortion inBlind Modulation Classification

Gaurav Jyoti PhukanBharat Electronics Limited

BangaloreIndia

Email: g.phukan@iitg.ac.in

P K BoraDepartment of EEE

Indian Institute of Technology, GuwahatiIndia

Email: prabin@iitg.ernet.in

Abstract—This paper presents a method for classificationof digital modulation schemes in an asynchronous receptionscenario. The proposed classification is based on likelihoodprinciples and without prior knowledge of channel response.We propose blind channel estimation in non-cooperative sce-nario using conventional Decision Directed Least Mean Squarealgorithm while the signal is subjected to channel distortion andunknown gain. Convergence characteristics of the LMS algorithmis modulation dependent and after choosing the best equalizer, thefinal decision for Modulation Classification is made by likelihoodprinciples. Experimental results are presented to compare theperformance with the optimal classifier. The scope for furtherimprovement is outlined.

Keywords: Modulation classification (MC), maximum like-lihood (ML), log likelihood function, quadrature amplitudemodulation (QAM), Inter Symbol Interference, fuzzy-c meansalgorithm.

I. INTRODUCTION

Blind Modulation Classification (MC) has become an inte-gral part of communication surveillance for military applica-tions. Due to the recent developments in cognitive radio appli-cations, MC research has found considerable importance [1]in the civilian field. There are two broad approaches to mod-ulation classification (MC), namely the likelihood based (LB)approach and the feature based (FB) approach [2]–[5]. In bothcases, adverse channels conditions provide a major challengeto the classification accuracy [6]. In majority of applications,MC faces the challenge of classifying the modulation schemein the signal which is received with multi-path propagationeffect and channel noise. Issues due to channel noise andunknown gain were addressed in [1], [7]. Frequency selectivefading due to multi-path effect causes phase shift, distortionand random scaling in the received signal. In the likelihoodbased approach, the resulting Inter Symbol Interference (ISI)deteriorates the MC performance due to modal mismatch inthe likelihood function. For severe channel distortion, LBclassification completely fails if mitigation measures are nottaken. In addition to the ISI, the issue of amplitude mismatch[7] needs to be addressed to resolve the modal mismatch. Inblind modulation classification, neither prior information ofthe signal nor channel information is available. Conventionalblind equalization used for communication systems avoids thetraining sequence to conserve band-width [8] and increase

throughput. In this case, the modulation scheme being a knownparameter, a Decision Directed Least Mean Square (DDLMS)approach is commonly employed [8]. Decision making forthe DDLMS algorithm is straight-forward with the numberof levels of the base band symbols a known parameter.For a blind scenario, the performance of DDLMS algorithmdeteriorates due to uncertainty of the symbol levels. As theorder of modulation is changed, the Mean Square Error (MSE)increases [9].

In this paper, we study the effect of channel distortion inmodulation classification. The LB approach is followed dueto the availability of an optimum solution. As a mitigationmeasure for channel distortion and resulting modal mismatchin likelihood based MC, the DDLMS algorithm is adaptedwith amplitude normalization and quantization. A pool ofpossible modulation schemes is generated for using the blindequalization algorithm. Experimental results are presented forcomparing the performance with conventional approach.

II. PERFORMANCE OF LIKELIHOOD BASED

MODULATION CLASSIFICATION WITH CHANNEL

DISTORTION

The received signal model for the signal sequence in fre-quency selective fading conditions can be expressed as follows𝑦(𝑛) =

𝐴∞∑

𝑙=−∞𝑥(𝑙)ℎ(𝑛𝑇 − 𝑙𝑇 + 𝜖𝑇𝑇 ) exp(𝑗2𝜋𝑓𝑜𝑛𝑇 + 𝑗𝜃𝑛) + 𝑔(𝑛),

(1)where {𝑥(𝑙)} is the original signal sequence, 𝐴 is an

unknown amplitude factor, ℎ(.) accounts for the residualbaseband channel effect, 𝑇 is the symbol period, 𝜖𝑇 is thetiming error, 𝑓𝑜 is the frequency offset, 𝜃𝑛 is the randomphase jitter and 𝑔(𝑛) is the complex additive white Gaussiannoise. Here we assume that 𝑇 has been estimated without anysignificant error and effects of 𝜖𝑇 , 𝑓𝑜 and 𝜃𝑛 is negligible. Thereceived signal model now can be approximated as

𝑦(𝑛) = 𝐴

∞∑𝑙=−∞

𝑥(𝑙)ℎ(𝑛𝑇 − 𝑙𝑇 ) + 𝑔(𝑛), (2)

The most popular and widely discussed LB classifier isbased on the Bayes rule and it is termed as the optimal978-1-4673-5952-8/13/$31.00 c⃝ 2013 IEEE

modulation classifier. The initial work in this area was pro-posed in [2], [3], [10].The approach to digital modulationclassification based on the maximum likelihood classificationis to first derive the log likelihood function of the receivedsignal based on the assumed signal model. Next, the valueof the likelihood function is determined for all the candidatemodulation schemes, using their constellation characteristics.

Suppose there are 𝐶 modulation schemes in an 𝑀 -arysystem. Hence there will be 𝐶 types of constellations givenby,

𝐼𝑙 = 𝑥𝑙1, 𝑥𝑙2, ....., 𝑥𝑙𝑀 , 𝑙 = 1, 2, ..., 𝐶 (3)

The modulation classification now becomes the problem oftesting 𝐶 hypotheses: 𝐻𝑙 is the hypothesis that the constella-tion is 𝐼𝑙,

Assume 𝑁 data samples are received. The received symbolsare given by

yk = [𝑦𝐼𝑘, 𝑦𝑄𝑘]𝑇, 𝑘 = 1, 2, ..., 𝑁 ; (4)

where 𝑦𝐼𝑘 is the in-phase component and 𝑦𝑄𝑘 is the quadraturephase component of 𝑘𝑡ℎ sample. Maximizing the a posterioriprobability is equivalent to maximizing the maximum likeli-hood function. Therefore, the optimal hypothesis 𝐻𝑙

∗ is givenby

𝐻𝑙∗ =argmax

𝐻𝑙

P(𝐻𝑙∣y1,y2, ...,yk)

= argmax𝐻𝑙

𝐿(y1,y2, ...,yk∣𝐻𝑙)(5)

As the channel is considered to be AWGN, the likelihoodfunction can be expressed as [11]

𝑁∏𝑘=1

p(yk∣𝐻𝑙) =

𝑁∏𝑘=1

𝑀𝑙∑𝑖=1

1

𝑀𝑙

1√2𝜋𝜎

exp

( −1

2𝜎2∣∣yk − xli∣∣2

)(6)

where 𝑥𝑙𝑖 is the 𝑖𝑡ℎ constellation point of hypothesis 𝐻𝑙 and𝜎2 is the variance of the AWGN.

Now, the log likelihood function is given by

ln{𝐿(y1,y2, ...,yk∣𝐻𝑙)}

= ln

{𝑁∏

𝑘=1

𝑀𝑙∑𝑖=1

1

𝑀𝑙

1√2𝜋𝜎

exp

( −1

2𝜎2∣∣yk − xli∣∣2

)}

=

𝑁∑𝑘=1

ln

{𝑀𝑙∑𝑖=1

1

𝑀𝑙

1√2𝜋𝜎

exp

( −1

2𝜎2∣∣yk − xli∣∣2

)}(7)

The above equation can be simplified by omitting theconstant terms:

𝑙(y1,y2, ...,yk∣𝐻𝑙) =𝑁∑

𝑘=1

ln

{𝑀𝑙∑𝑖=1

1

exp

( −1

2𝜎2∣∣yk − xli∣∣2

)}(8)

A pool of modulation containing BPSK, QPSK, 8QAMand 16QAM is considered to calculate the average probabilityof correct classification (𝑃𝐶𝐶). The performance of the LBclassifier in channel with random gain is shown in Fig. 1.

Fig. 2 shows the performance of the LB classifier in a fadingchannel. As a result of fading, the received signal constellationsuffers from phase shift, distortion and random scaling asshown in Fig. 3

−15 −10 −5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR in dB

Pcc

Channel Gain 1Channel Gain 0.9Channel Gain 0.8Channel Gain 0.7Channel Gain 1.5Channel Gain 3Channel Gain 8

Fig. 1. Performance of LB classifier with randon scaling

−15 −10 −5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR in dB

Pcc

h=(1,0,0)h=(0.9,0.1,0)h=(0.8,0.2,0)h=(0.7,0.2,0.1)

Fig. 2. Performance of LB classifier in fading channel

Fig. 3. Distortion in received signal. (a)Received signal with random gain(b)Multi-path fading, h=(0.2, 0.3, 1)

III. BLIND CHANNEL ESTIMATION AND AMPLITUDE

NORMALIZATION

As a measure to mitigate multi-path fading effect, a blindequalization and amplitude normalization method is adopted.The proposed scheme is shown in Fig. 4.

Fig. 4. Blind DDLMS scheme in Likelihood Based MC

There are two main approaches to blind adaptive equaliza-tion: based on Stochastic Gradient Descent (SGD) and HigherOrder Statistics (HOS). We consider the first method here andadapt the same for blind MC. The SGD approach iterativelyminimizes a chosen cost function over the choices of equalizercoefficients [12]. In the MC scheme of Fig. 4, we assume alinear FIR filter W(𝑧) with weight vector 𝑤 to compensate forthe channel response 𝐻(𝑧). The equalizer coefficients need tobe adjusted such that the output of the equalizer 𝑦𝑒𝑞(𝑛) canbe quantized to yield a reliable estimate of the channel input𝑥(𝑛). The decision device is a quantizer, which forces thevalue of the output symbol level to the nearest symbol value.The output of the decision device is represented by

�̂�(𝑛) = 𝑄(𝑦𝑒𝑞(𝑛)) = 𝑥(𝑛− 𝛿), (9)

where 𝛿 is a constant integer delay. This delay being a constantfactor, does not effect the performance of the MC scheme. Theequalization process is to identify the coefficients of the thefading channel, defined as

𝐻(𝑧) =

∞∑𝑘=0

ℎ(𝑘)𝑧−𝑘 (10)

The channel output is given by

𝑦(𝑛) = ℎ(𝑛) ∗ 𝑥(𝑛) + 𝑔(𝑛), (11)

where ∗ denotes the convolution operator, g(n) is the whiteGaussian noise with constant power spectral density 𝑁0. Allthe ISI in a typical Zero Forcing (ZF) equalizer is removed if

𝐻(𝑧)𝑊 (𝑧) = 𝑎𝑧−𝛿, 𝑎 ∕= 0 (12)

Here the equalizer output becomes

𝑦𝑒𝑞(𝑛) = 𝐴𝑥(𝑛− 𝛿), (13)

where 𝐴 is a scaling factor. Theoretically the generated equal-izer takes the form of a Infinite Impulse Response (IIR) filter.However, for practical convenience, the adaptive equalizeris implemented by FIR filters. Let the coefficients of the

equalizer be

w = [𝑤0 𝑤1 . . . 𝑤𝑚]𝑇 ,𝑚 < ∞ (14)

The received signal vector is

y(𝑛) = [𝑦(𝑛) 𝑦(𝑛− 1) . . . 𝑦(𝑛−𝑚)]𝑇 (15)

Thus the output of the equalizer is

𝑦𝑒𝑞(𝑛) = w𝑇 y(𝑛), (16)

where, the equalizer transfer function is

W(𝑧) =

𝑚∑𝑖=0

𝑤𝑖𝑧−𝑖 (17)

The conventional Decision Directed LMS (DDLMS) algorithm[13] is employed to minimize the error function using iterativesteps. However, in a blind scenario, the training sequence andthe number of levels of symbol values are unknown. Theiterative expression in such scenario can be given by

w(𝑛+ 1) = w(𝑛) + 𝜇[𝑄(𝑦𝑒𝑞(𝑛))− 𝑦(𝑛)]y*(𝑛) (18)

where 𝜇 is a small positive stepsize and Q(.) is the decisionfunction to estimate the incoming signal sequence. The blindequalizer iteratively minimizes the Mean Square Error (MSE)cost function

𝐸{∣𝑒𝑛∣2} = 𝐸∣𝑄(𝑦𝑒𝑞(𝑛)− 𝑦𝑒𝑞(𝑛)∣2 (19)

The iterations are continued till the MSE becomes very small,so that the equalizer output 𝑦𝑒𝑞(n) is a close estimate of theoriginal channel input 𝑥(𝑛). Hence, by selection of the Q(.)function, the blind equalizer will be able to track moderatevariations in the channel dynamics. The most common deci-sion function employed for blind equalization in communica-tion systems is a hard or soft [13] decision quantizer.

Selection of Decision function for blind MC scenario:Unlike conventional communication system, where the IQconstellation pattern is known, in blind MC there is no priorknowledge of the constellation pattern. In this scenario, thedecision function is proposed to be selected based on the oneyielding best convergence of the DDLMS algorithm. Sincethe received signal is distorted with various factors includingunknown channel gain, fixing up of the absolute quantizationlevels for a certain modulation scheme is not possible. Tomitigate this issue, normalization of the received constellationis carried out before subjecting it to decision device.

Normalization requires to estimate the apparent constella-tion points using a clustering algorithm [7]. The fuzzy C-means (FCM) clustering is a method of clustering whichallows a sample of data to belong to two or more clusters.This method is based on the minimization of the followingobjective function:

𝐽𝑚 =

𝑁∑𝑖=1

𝐶∑𝑗=1

𝑢𝑚𝑖𝑗 ∣∣𝑦𝑖 − 𝑐𝑗 ∣∣2, 1 ≤ 𝑚 < ∞ (20)

where 𝑚 is any real number greater than 1, 𝑢𝑖𝑗 is the degree

of membership of 𝑦𝑖 in the cluster j, 𝑦𝑖 is the 𝑖𝑡ℎ of d-dimensional measured data, 𝑐𝑗 is the d-dimension center of thecluster, and ∣∣.∣∣ is any norm expressing the similarity betweenany measured data and the center. The Fuzzy partitioning iscarried out through an iterative optimization of the objectivefunction shown above, with the update of membership 𝑢𝑖𝑗 andthe cluster centers 𝑐𝑗 by

𝑢𝑖𝑗 =1∑𝐶

𝑘=1

( ∣∣𝑦𝑖−𝑐𝑗 ∣∣∣∣𝑦𝑖−𝑐𝑘∣∣

) 2𝑚−1

, 𝑐𝑗 =

∑𝑁𝑖=1 𝑢𝑖𝑗 .𝑦𝑖∑𝑁𝑖=1 𝑢

𝑚𝑖𝑗

(21)

The iterative process will terminate when max𝑖𝑗{∣𝑢(𝑘+1)𝑖𝑗 −

𝑢𝑘𝑖𝑗 ∣} < 𝜀, where 𝜀 is a termination criterion, 0 ≤ 𝜀 ≤ 1 and

𝑘 is iteration step.The number of valid cluster centers in a received signal is

an unknown parameter. We assume the value of 𝑀 to be thenumber of constellation points of the highest order modulationin the pool of candidate modulations.

The final cluster centers are expressed as

𝐶(𝑘)𝑓𝑖𝑛𝑎𝑙 = [𝑐𝑗 , 𝑗 = 1, 2, ...,𝑀 ]; (22)

The maximum signal amplitude in a constellation is the outer-most constellation point or the amplitude of the outermost ringof constellation points. Power normalization is carried out byconsidering the detected cluster centers as signal constellationpoints. The average power of the cluster centers is given by

𝑃𝑎𝑣 =1

𝑀

[𝑀∑𝑖=1

(𝑐2𝐼𝑖 + 𝑐2𝑄𝑖)

], 𝑐𝑖 =

√𝑐2𝐼𝑖 + 𝑐2𝑄𝑖, (23)

where, 𝑐𝐼𝑖 and 𝑐𝑄𝑖 are in-phase and quadrature-phase com-ponents of 𝑖𝑡ℎ cluster center 𝑐𝑖. To ensure that the averageenergy over all cluster centers are unity, scale factor of 1√

𝑃𝑎𝑣

is used to approximately normalize the received constellation.Assume the there are 𝐶 modulation schemes in a pool

of possible modulation schemes. There will be 𝐶 types ofconstellations. The decision function is designed such thatthe distance of the normalized equalizer output from anyof the symbol values of the assumed constellation modelis minimum. The Euclidian distance of power normalized𝑦𝑛 from the model constellation points for any of the 𝐶modulation schemes is expressed as

𝑑(𝑙) = ∣∣𝑦𝑛 − 𝑠(𝑙)∣∣, 𝑙 = 1, 2, . . . 𝑁𝑞 (24)

where 𝑦𝑛 is the normalized output of the blind equalizer,𝑠(𝑙) is the 𝑙𝑡ℎ constellation point and 𝑁𝑞 is the number ofconstellation points in the assumed modulation scheme. Thedecision function 𝑄(.) can be expressed as

𝑄 (𝑦𝑛) = 𝑠

(argmin

𝑙(𝑑(𝑙)

)(25)

Out of the pool of assumed modulation schemes, blind equal-ization is carried out for each of the 𝐶 constellation models.The appropriate constellation model fitting into the received

signal is decided based on the convergence characteristics ofthe recursive DDLMS algorithm. The criterion for convergenceis examined based on Mean Square Error (MSE) after a fixednumber of iterations [9]. The Mean Square Error (MSE) ofthe DDLMS algorithm can be expressed as

∣𝑒𝑙(𝑁)∣2 = ∣𝑄(𝑦𝑛(𝑁)− 𝑦𝑛(𝑁)∣2, 𝑙 = 1, 2, . . . , 𝐶 (26)

where 𝑁 is the number of fixed iteration steps for examiningthe convergence. The constellation model 𝑙𝑒 is selected as themodel which best fits into the incoming signal, such that

𝑙𝑒 = argmin𝑙(∣𝑒𝑙(𝑁)∣2) (27)

For decision making of the received signal modulation, twolikelihood functions can be generated. The first one with powernormalized signal 𝑦𝑘 from the equalizer output. The secondlikelihood function is generated using quantized output ofthe decision device. The first log-likelihood function with thenormailized values is

𝑙(𝑦1, 𝑦2, ..., 𝑦𝑘∣𝐻𝑙) =

𝑁∑𝑘=1

ln

{𝑀𝑙∑𝑖=1

1

𝑀𝑙

1

exp

( −1

2𝜎2∣∣𝑦𝑘 − 𝑥𝑙𝑖∣∣2

)}(28)

where, 𝑦𝑘 is the power normalized amplitude of the equalizerand 𝑥𝑙𝑖 is the power normalized amplitude of the 𝑖𝑡ℎ constel-lation point in the I-Q plane.

The final likelihood function is defined as

𝐿𝑐 = 𝑙(𝑦1, 𝑦2, ..., 𝑦𝑘∣𝐻𝑙); (29)

The decision rule is defined as

𝐻𝑙∗ = argmax

𝐻𝑙

𝐿𝑐(𝑦1, 𝑦2, ..., 𝑦𝑘∣𝐻𝑙), (30)

The above decision rule is subjected to the selection of theappropriate constellation model in blind equalizer as per (27).

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The performance of the proposed scheme is studied usingthe Monte Carlo simulation method to calculate probabilityof correct classification 𝑃𝐶𝐶 against SNR. The comparison ismade with the optimum classifier having the complete knowl-edge of the channel response. Cases with various channelconditions are experimented.

Fig. 5 show that, blind channel estimation for MC ispossible within a certain limit of channel distortion. Therecursive algorithm does not converge in severe multi-pathscenarios, where the direct signal path is small as comparedto the delayed reflected paths.

Fig. 6 (a) and (b) show the received signal constellationafter subjected to channel distortion by impulse responseℎ = (0.8, 0.2, 0.1) and equalized output after normalizationfor a 16QAM signal at 10 dB SNR. Fig. 6 (b) is the powernormalized constellation of the equalizer output.

Fig. 7 shows the MSE for a range of iteration steps fordifferent constellation models, when the original signal is16QAM. For a mismatch in the constellation model, the MSE

−15 −10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR in dB

Pcc

Optimum classifier with known channel rtesponseProposed scheme with h=(0.9, 0.1, 0) Proposed scheme with h=(0.75, 0.3 0.1)

Fig. 5. Comparative performance of the proposed scheme

Fig. 6. (a) Received constellation, (b) Equalized using the blind scheme

Fig. 7. Constellation model: (a) BPSK, (b) QPSK, (c) 8QAM, (d)16QAM

does not converge. However, in cases of a pool of possiblemodulation schemes like BPSK and QPSK, the convergencecharacteristic is ambiguous to make a decision. In such cases,although the purpose of blind equalization for MC scenario isserved, exact determination of the modulation scheme is doneby the ML principle.

V. CONCLUSION

In this paper we have considered the case of blind chan-nel estimation for Modulation Classification scenario, whereprior knowledge of signal constellation is not available. Thechannel estimation is carried out using conventional DDLMSalgorithm supported by decision making using the convergencecharacteristics for various constellations models in the pool ofpossible modulations considered. The scheme performs wellfor moderately distorted signals. For severe channel distortiondue multi path effect, blind equalization suffers from theproblem of divergence in the MSE. The future work may focuson addressing this issue. Also, the decision error due to outlierpoints within the unstable portion of the MSE plot can beexcluded to increase probability of correct classification.

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