EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEur. Trans. Telecomms. 2011; 22:93–97

Published online 18 November 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.1452

LETTER: WIRELESS SYSTEMS

Iterative MMSE estimation of SINR for turbo-codedFH-SS systems in partial-band interference andfading channelsAhmad Gomaa1∗ and Ayman Elezabi2

1 Electrical Engineering Department, University of Texas at Dallas, Richardson, TX 75080, USA2 Electronics Engineering Department, American University in Cairo, Cairo, Egypt

ABSTRACT

The minimum mean square error (MMSE) estimate of the signal-to-interference-and-noise ratio (SINR) is derived forturbo-coded slow frequency hopping systems in the presence of partial-band interference and Rayleigh fading. The MMSEestimator operates without training symbols and iteratively uses the turbo decoder extrinsic information to improve theaccuracy of the SINR estimate. Performance comparisons are made with a genie-aided MAP-based estimator and near-optimal performance is obtained for all but impractically small hop sizes. Copyright © 2010 John Wiley & Sons, Ltd.

KEYWORDS

frequency hopping; minimum mean square error (MMSE); partial-band interference; SNR estimation; turbo coding

*Correspondence

Ahmad Gomaa, Electrical Engineering Department, University of Texas at Dallas, Richardson, TX 75080, USA.E-mail: aarg 2010@yahoo.com

Received 15 July 2009; Revised 3 July 2010; Accepted 1 September 2010

1. INTRODUCTION

In this paper, we consider turbo-coded Frequency-HoppingSpread Spectrum (FH-SS) systems operating in partial-bandinterference and fading channels. On the AWGN channel,turbo decoding requires the computation of the so-calledchannel measurement log-likelihood ratio (LLR) which isa function of the received signal-to-noise ratio (SNR) [1].When channel fading and interference are present, the turbodecoder requires an estimate of the instantaneous signal-to-interference-plus-noise ratio (SINR) to be incorporated inthe LLR. In this paper, we consider the problem of SINRestimation in turbo-coded slow frequency hopping (SFH)communication systems using coherent binary phase shiftkeying (BPSK) in fading channels with partial-band noise(PBN) interference. In the absence of training symbols, afew estimation techniques [2--4] have been proposed forobtaining the LLR that differ in their performance, com-plexity, underlying assumptions, and the amount of sideinformation available at the receiver. In Reference [2], amaximum a posteriori (MAP)-based approach is used thatdepends on quantizing the probability density function ofthe fading magnitude. This method gives performance closeto that of perfect SNR knowledge for interference-freechannels. The method, however, is not extensible, or at

least is difficult to extend, to channels with interference.In Reference [3], the Expectation-Maximisation (EM) esti-mate of the signal-to-noise ratio (SNR) in each hop wasdeveloped for noncoherent frequency shift keying (NFSK).However, no interference was assumed there either. In Ref-erence [4], the EM estimate of the SINR was obtained andhad the advantage of not requiring any information on thestatistics of fading or interference. Except for large hopsizes, its performance for BPSK leaves room for improve-ment compared to the case with perfect SINR knowledge.Other related works include References [5--6]. In Refer-ence [5], nonorthogonal continuous-phase FSK modulationtechnique is considered for the turbo-coded SFH systemwhere the modulation index is optimised in the presence ofPBN interference. In Reference [6], an iterative technique isdeveloped for joint carrier phase recovery and interferencedetection for FH systems.

In this work, we derive the minimum mean square error(MMSE) estimate of the SINR for channels with fadingand PBN. Unlike the EM estimate we derived in Reference[4], the MMSE estimate obtained in this paper requires apriori statistics on fading and interference. We again usean iterative technique in which the turbo decoder extrinsicinformation is fed back to the SINR estimator. We comparethe performance of the proposed MMSE-based method with

Copyright © 2010 John Wiley & Sons, Ltd. 93

Iterative MMSE estimation of SINR A. Gomaa and A. Elezabi

a genie-aided version of the MAP-based method of Ref-erence [2] in which we provide perfect knowledge of thenoise variance that varies from hop to hop. This is becausea MAP-based approach that estimates both the channel andinterference variance is not available. Clearly, the perfor-mance of the genie-aided MAP-based method is an upperperformance bound on the MAP-based approach. The restof the paper is organized as follows. The system model isdescribed in Section 2. The MMSE estimate of the SINR isderived in Section 3. Performance comparisons between theMMSE, EM and MAP-based methods are provided usingextensive simulations in Section 4. The paper is concludedin Section 5.

2. SYSTEM MODEL

We consider SFH using Coherent BPSK (CBPSK). The hoprate is such that the fading channel is approximately staticover a hop interval. The thermal background noise is addi-tive white Gaussian noise (AWGN) with two-sided powerspectral density (PSD) of weight No/ 2. The PBN interfer-ence is modeled as additional Gaussian noise that existsover part of the complete hopping band. In fact, this modelis widely used in the literature for the PBN interference (seeReferences [4--6] and references therein). The fraction of thetotal hopping band in which the interference exists is calledthe duty factor of the PBN, ρ, where 0 < ρ ≤ 1. The equiv-alent full-band interference two-sided PSD weight is NI/2.Then, the total interference power is distributed uniformlyover a fraction ρ of the total spread spectrum system band-width with a PSD of weight NI/(2ρ). For CBPSK, the realpart of the phase-compensated complex low pass receivedsymbol is

x = d A√

Ec + w (1)

where A is the Rayleigh-distributed fading magnitude, Ec isthe energy of the code bit, d represents the transmitted datasymbols where d = ± 1. The noise element is representedby w which has zero mean and variance B where B = No/2if the bit is interference-free and B = (No + NI/ρ)/2 if thebit suffers PBN interference. The received vector of thehop is defined as X = [x1 x2 . . . . . . xN ] where N denotes thenumber of bits per hop (bph). Furthermore, it is reasonableto assume that B is constant over one hop.

3. MMSE ESTIMATE OF SINR

For CBPSK, the channel measurement LLR of each codebit is well known to be

LLR = 2A√

Ecx

/B (2)

Hence, we need to estimate the quantities A and B. Inthis paper, we derive the MMSE estimates of both of the

fading magnitude, A, and the noise variance, B, assumingthat we have a priori knowledge of Ec, No, NI and ρ. Theseparameters are usually fixed or slowly varying and may beknown or estimated prior to communication. The MMSEestimates of A and B (denoted by AandB) are obtained by

minimising E

((A−A

)2∣∣∣X, B

)and E

((B−B

)2∣∣∣X, A

),

respectively, where the operators E ( ) and | denote the sta-tistical expectation and conditioning, respectively. Knowingthat the MMSE estimate [7] equals the conditional expec-tation, we write

A =∞∫

A=0

A fA(A |X,B )dA (3)

B = Bo P(B = Bo| X,A) + BI P(B = BI| X,A) (4)

where

Bo = No/2, BI = (No + NI/ρ)/2 (5)

First, we carry out some manipulations of Equation (4).From Bayes’ rule, we get

P(B = Bo/I

∣∣X,A)

= fX

(X

∣∣B = Bo/I, A)P(B = Bo/I

)/fX(X| A) (6)

where

P(B = BI ) = ρ, P(B = Bo) = 1−ρ (7)

Assuming that the channel interleaver is large enough torender the channel memory-less and applying the theoremof total probability, we write

fX

(X

∣∣Bo/I, A)

=N∏

i=1

fXi

(xi

∣∣Bo/I, A)

=N∏

i=1

(P(di = 1)fXi

(xi

∣∣Bo/I, A , di = 1)

+ P(di = −1)fXi

(xi

∣∣Bo/I, A , di = −1))

=N∏

i=1

(P(di = 1)√

2πBo/I

exp

(−

(xi−A

√Ec

)2/

2Bo/I

)

+ P(di = −1)√2πBo/I

exp

(−

(xi + A

√Ec

)2/

2Bo/I

))

(8)

where P(di = 1) and P(di = −1) represent the probabilitiesthat di = +1 and di = −1, respectively, and are obtainedfrom the extrinsic turbo decoder outputs on each iteration.

94 Eur. Trans. Telecomms. 2011; 22:93–97 © 2010 John Wiley & Sons, Ltd.DOI: 10.1002/ett

A. Gomaa and A. Elezabi Iterative MMSE estimation of SINR

From the theorem of total probability, fX(X |A ) in Equation(6) can be written as follows

fX(X |A ) = P(B = Bo

)fX

(X

∣∣B = Bo, A)

+ P(B = BI

)fX

(X

∣∣B = BI, A)

(9)

Back to Equation (3) and using Bayes’ rule, we write

fA(A |X, B ) = fX(X |B, A )fA(A)/fX(X| B) (10)

where fX(X |B, A ) is obtained in Equation (8), andfX(X |B ) can be written as follows

fX(X |B ) =∞∫

A=0

fX(X |B , A)fA(A)dA. (11)

It is clear from Equation (8) that fX(X |B, A ) can bedecomposed into 2N terms, and the result of the integral inEquation (3) would contain 2N terms. However, the inte-gral complexity will grow exponentially with the hop size.Hence, we follow the method in Reference [2] and useLloyd-Max quantiser [8] to quantise the fading amplitudesinto M regions, R1, R2, . . . , RM with M output levels (cen-troids), A1, A2, . . . , AM . Then, we approximate the integralin Equation (3) by a summation as follows:

A =M∑

k=1

AkP(A∈Rk|X,B). (12)

Using Bayes’ rule, we write

P(A∈Rk|X,B) = fX(X |A∈Rk, B )P(A∈Rk)

fX(X| B)(13)

We calculate fX(X |A∈Rk, B )in the same way we calcu-lated fX(X |B, A ) in Equation (8) as follows

fX(X |A∈Rk, B )

=N∏

i=1

(P(di = 1)√

2πBexp

(−

(xi−Ak

√Ec

)2/

2B

)

+ P(di = −1)√2πB

exp

(−

(xi + Ak

√Ec

)2/

2B

))(14)

Furthermore, fX(X |B ), the denominator of Equation(13), is obtained from the total probability theorem as fol-lows

fX(X |B ) =M∑

k=1

(fX(X |B , A∈Rk)P(A∈Rk)) (15)

According to Equations (4) and (12), it appears that theestimate of the fading magnitude depends on the variance

and vice versa. This inter-dependence may be manipulatediteratively as follows:

A(l+1) =M∑

k=1

AkP(A∈Rk|X,B(l)

)(16)

B(l+1) = Bo × P(B = Bo| X,A(l+1)

)+BI × P

(B = BI | X,A(l+1)

)(17)

where l = 0,1, . . . denotes the iteration number. The initialvalue of the variance estimate is set as follows:

B(0) = 1

2

1

N

N∑i=1

y2i︸ ︷︷ ︸

MeasurementsPart

+ ρBI + (1−ρ)Bo︸ ︷︷ ︸AprioristatisticsPart

(18)

where yi is the imaginary part of the phase-compensatedcomplex low pass received ith symbol. The term yi is a purenoise element which has the same statistical characteristicsof w. Observe that we use both receiver measurements andapriori statistics to compute the initial variance estimate.

4. PERFORMANCE COMPARISONS

Throughout our simulations we use the IS2000 rate halfturbo encoder with a block size of 1024 information bitsper frame. The number of iterations used is 10. The channelinterleaver is a simple matrix interleaver whose size is madeequal to the coded turbo frame size as in Reference [4]. Forcomparisons with the MAP-based method of Reference [2],the number of regions M in the Lloyd-Max quantiser is set to8, as in Reference [2]. We denote the information bit energyby Eb where Eb = 2Ec since a rate-half coding is used.

In Figure 1, the required Eb/NI to achieve a bit-error-rate(BER) of 2 × 10−4 is plotted against ρ for various SINRestimation techniques; namely our proposed MMSE estima-tion technique, the EM estimation technique in Reference

Figure 1. Required Eb/NI to achieve BER = 2 × 10−4 as functionof � for various estimation techniques with Eb/No = 10 dB and

N = 64 bph.

Eur. Trans. Telecomms. 2011; 22:93–97 © 2010 John Wiley & Sons, Ltd. 95DOI: 10.1002/ett

Iterative MMSE estimation of SINR A. Gomaa and A. Elezabi

Figure 2. BER performance for various hop sizes (N = 64 (dash-dotted), 32 (dashed) and 8 (solid)) with � = 0.5 and Eb/No = 5 dB.

[4], and the MAP estimation technique of A in Reference[2] with B known perfectly (genie-aided) in each hop. Forreference, we also include the performance in the case ofperfect knowledge of both A and B. We provide the MAPtechnique with the perfect knowledge of B in each hop sinceit is not straightforward to compute the MAP estimate of thevariance. It can be seen from Figure 1 that the performanceof the MMSE technique is virtually the same as that of theMAP-based technique which is given the true value of thetime-varying B. Furthermore, the performance of our pro-posed MMSE technique exceeds that of the EM technique,albeit by a very small margin for this hop size, because theMMSE estimate exploits the a priori statistics of the fadingchannel and the PBN interference.

In Figure 2, the BER is plotted versus Eb/NI for varioushop sizes. Several observations may be made from Figure 2.First, decreasing N always improves the performance withperfect SINR knowledge because smaller N leads to a largernumber of hops per frame. Hence, the coded turbo framewill undergo a larger number of independent fading andinterference channels which means an increase in diversity.This is based on fixing the overall number of bits in theturbo block size [2--4]. Second, the gap between the perfectSINR knowledge scheme and the estimated SINR schemesincreases with decreasing N which is expected because thenumber of observations becomes insufficient for reliableestimates. Third, the superiority of the proposed MMSE-based technique over the EM-based technique is evidentfor moderate and small hop sizes. Fourth, the MMSE per-formance is close to that of the genie-aided MAP-basedmethod even for small hop sizes. We reiterate that theMAP-based method proposed in Reference [2] assumedinterference-free channels. Finally, we note that the errorfloors in Figure 2 are due to the background noise.

In Figure 3, we plot the BER against the SNR forextremely small hop sizes. We compare the performancesof the practical schemes (MMSE and EM) where we do notassume the knowledge of A or B. We observe that decreas-ing the hop size below 8 bph degrades both performances.The reason is that the diversity advantage (caused by theincreased number of hops per frame) is dominated by theinsufficient number of observations (caused by reducing the

Figure 3. BER versus Eb/No for MMSE and EM approachesunder small hop sizes (N = 1, 4 and 8 bph) with � = 0.5 and

Eb/NI = 10 dB.

hop size) causing the overall performance to degrade. How-ever, we notice that the MMSE performance is still morerobust than that of the EM even for 1 bph thanks to the apriori statistics exploited by the former.

5. CONCLUSION

The MMSE estimate of the SINR was derived for slow FHsystems under fading and interference conditions. Perfor-mance comparisons via simulations were carried out fordifferent SINR, SNR, interference duty factors and hopsizes. The performance of the proposed technique wasshown to be very close to a genie-aided version of the MAP-based estimator of Reference [2], which was proposed forinterference-free channels, where the latter is given the per-fect estimate of the interference power in each hop. Wealso compared the performance of the proposed MMSEmethod with the EM method and demonstrated a signifi-cant performance advantage for the MMSE-based methodfor all but large hop sizes. The MMSE approach requiresthe knowledge of the fading distribution and the second-order statistics of the PBN which can be estimated prior toreception.

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Eur. Trans. Telecomms. 2011; 22:93–97 © 2010 John Wiley & Sons, Ltd. 97DOI: 10.1002/ett