In this paper we develop a blind modulation-index estimator for\ud
a combined CPM/OFDMReceiver. The performance of the estimator\ud
in an AWGN channel is assessed by simulation and analysis\ud
and its suitability for our receiver is established

Hoeksema, F.W.

Schiphorst, R.

Slump, C.H.

English

In this paper we develop a blind modulation-index estimator for a combined CPM/OFDM Receiver. The performance of the estimator in an AWGN channel is assessed by simulation and analysis and its suitability for our receiver is established

Schiphorst, Roelof

Slump, Cornelis H.

NARCIS 

Modulation-Index Estimation in a Combined CPM/OFDM Receiver

University of Twente Research Information

MODULATION-INDEX ESTIMATON IN A COMBINED CPM/OFDM RECEIVER1FokkeHoeksema, 2Roel Schiphorst and 3KeesSlump1f.w.hoeksema@utwente.nl1, 2 ,3 University of Twente, EEMCS department, Signals and Systems Group,P.O. Box 217, 7500 AE Enschede, The NetherlandsABSTRACTIn this paper we develop a blind modulation-index estimator fora combined CPM/OFDM Receiver. The performance of the esti-mator in an AWGN channel is assessed by simulation and anal-ysis and its suitability for our receiver is established.1. INTRODUCTIONIn our SoftwareDefined Radio (SDR) project [1] we com-bined two different types of standards, Bluetooth [2] andHiperLAN/2 [3] on one flexible platform. The imple-mented part consists of the radio front-end of a receiver, sofrom antenna (RF signal) till and including demodulation(raw bits). For this purpose, a CMOS wide band analogfront-end (0.4 GHz - 6 GHz), a data-converting PCI boardand demodulation software on a Pentium 4 processor wereimplemented. So, we built a multi-band (2.4 GHz and5.6 GHz) and multi modulation-scheme (Continuous PhaseModulation (CPM) and Orthogonal Frequency DivisionModulation (OFDM)) receiver of which we believe it canbe used for other standards as well [4].We decided to re-use the HiperLAN/2 OFDM functionsfor Bluetooth CPM demodulation purposes. This couldbe done by using the Laurent-decomposition technique forCPM signals [5], [6]. This not only allowed us to re-useQAM functionality already present in the OFDM archi-tecture, but also lowered the required SNR for Bluetoothreception from 17 dB for a discriminator-based design toapproximately 11 dB for the Laurent design [7].There is a catch though: the Bluetooth standard was de-signed for low-cost analog design (discriminator-based)and allows devices to have a modulation index in the range0.28 ≤ h ≤ 0.35. Unfortunately it appeared that the Lau-rent design is very sensitive to a mismatch between themodulation index of the receiver hRX and of the transmit-ter hT X .This sensitivity was confirmed by an experiment using theCPM blocks present in MATLAB Simulink’s Communi-cations Blockset. In figure 1 for three transmitter modu-lation indices, hT X = 0.28, 0.29, 0.30 the number of biterrors out of 10000 transmitted bits is shown as a functionof the modulation index of the receiver hRX . It can beobserved that only in case |hRX −hT X| ≤ 0.002 no bit er-rors occurred due to modulation-index mismatch. So, a re-quired accuracy of (at least) less than 1% is necessary. We					 	 	 	 		     	 		  Figure 1: Sensitivity to modulation-index mismatch.therefore concluded that some form of modulation-indexestimation had to be present in our architecture, moreoverthis estimator is required to operate satisfactorily at 11 dBSNR. The questions are: can we find a modulation-indexestimator that satisfies these requirements? Can it be ablind estimator or is some well-known data needed? Canwe incorporate the estimator in the protocol proceduresof Bluetooth? E.g. in the Inquiry(Scan) and Page(Scan) protocol procedures, in which frames (packets)of 68 bits are used [2]?In this paper we present a blind modulation-index estima-tor and test it against the requirements specified above.In section 2, CPM and the Bluetooth version thereof arebriefly explained. Subsequently the estimator is given (sec-tion 3). Then, simulation results are shown for an AWGNchannel (section 4) and the no-noise case is analyzed insection 5. The paper ends with a discussion and someconclusions.2. CONTINUOUS PHASE MODULATIONBluetooth uses Gaussian Frequency Shift Keying (GFSK)as modulation scheme [2]. GFSK is a particular form ofContinuous Phase Modulation and can be described usingthe CPM formalism [8].The complex envelope s̃(t) of the transmitted signal isgiven bys̃(t) = A ej φ(t,α) (1)Proceedings of SPS 2004 (the 2004 IEEE Benelux Signal Processing Symposium)171in which A is the signal amplitude, φ(t, α) the signal phaseand α the sequence of data symbols αi ∈ {−1, 1}. The re-lation between the signal phase and the sequence of sym-bols α is given byφ(t, α) = 2 π h∑iαi . q(t − i T) (2)in which T is the bit period, h is the modulation index andq(t) the phase response for which we assume thatq(t) =0 if t ≤ 01/4 if t = L/2 . T1/2 if t ≥ L . T(3)holds, with L the correlation length.Assume that αk = 0 for k < 0. Symbol αk for k ≥ 0 istransmitted during the k-th interval Ik = {t | k T ≤ t <(k + 1) T}. The phase function can be written as a trellisencoder. During the k-th interval:φ(t, α) = η(t, C k, αk) + 8k (4)holds, withη(t, C k, αk) , 2 π hk∑i=k−L+1αi . q(t − i T) (5)and correlative state C k = [ αk−L+1, ..., αk−1 ]. The phasestate is8k ,{0 if k < Lπ h∑k−Li=0 αi mod 2 π if k ≥ L .(6)The form of the phase function we use below (k ≥ L)is given by φ(t, α k) , φ(t, α) for t ∈ Ik and α k =[ α0, α1, ..., αk ] the vector of transmitted symbols untiland into the k-th interval Ik:φ(t, α k) = 2 π hk∑i=k−L+1αi . q(t − i T) + π hk−L∑i=0αi .(7)We forget about the modulus operation in the phase stateexpression (6). In our simulator we use a phase-unwrapperthat avoids any modulo-2 π problems.For GFSK the correlation length L = 2 and the phase-response q(t) is determined by the Gaussian pre-modulationfilter [6], [8]. Below, the analysis is continued for the(Bluetooth) GFSK case only - generalization to generalCPM is believed to be straightforward.3. MODULATION-INDEX ESTIMATORIn the (k-1)-th interval t ∈ Ik−1 (with k ≥ 3) we haveφ(t, α k−1) = 2 π h (αk−2 . q(t − (k − 2)T) ++ αk−1 . q(t − (k − 1)T) + π h∑k−3i=0 αi .(8)In the k-th interval t ∈ Ik (with k ≥ 2) we haveφ(t, α k) = 2 π h (αk−1 . q(t − (k − 1)T) ++ αk . q(t − kT) + π h(∑k−3i=0 αi + αk−2).(9)The phase metric for k ≥ 3 is defined as1φ[k] , 1/π (φ(kT, α k) − φ((k − 1)T, α k−1)). (10)Observe that k counts the bit periods and that we assumethat timing synchronization has been established. Rewrit-ing (10) using (8) and (9) gives1φ[k] = h ({1 − 2q(T)} . αk−2 + 2q(T) . αk−1) =, h (Q2 . αk−2 + Q1 . αk−1).(11)in which Q2 = 1 − 2q(T) and Q1 = 2q(T) can be deter-mined by applying L = 2 in (3), so Q1 = Q2 = 1/2. Thesquared phase metric is1φ[k]2 = h2 (Q22 . αk−22 + Q12 . αk−12 ++2 Q2 . Q1 . αk−2 . αk−1) == h2 (Q22 + Q12 + 2 Q2 . Q1 . αk−2 . αk−1)(12)as αi 2 = 1. Define the sum of the squared phase metricover a number of N bit periods asS1φ2 [N] ,k0+N−1∑k=k01φ[k]2 (13)(for some k0, say k0 ≥ 3). Then, with (12),S1φ2 [N] = h2 (∑k0+N−1k=k0 Q22 + Q12 ++ 2 Q2 . Q1 αk−2 . αk−1) =, h2(N .∑Q2 +2 Q2 . Q1∑k0+N−1k=k 0αk−2 . αk−1)(14)in which we defined∑Q2 = Q22 + Q12. Assume thatthe sequence of symbols α is an i.i.d. zero mean randomprocess. In this case, the last term in (14) results in anincoherent addition of the summand αk−2 . αk−1. For largeN the term may assumed to be zero.This observation leads to the definition of both the estima-tor ĥ2 for the squared modulation index and the modulation-index estimator ĥ itself:ĥ2 ,1N .∑Q2. S1φ2 [N], andĥ ,√ĥ2.(15)In section 5 an analysis of the modulation-index estimatoris presented. However, we first demonstrate its perfor-mance by a simulation for an AWGN channel.4. SIMULATIONThe channel is modelled as a complex baseband AWGNchannel. The relation between transmitted complex enve-lope s̃(t) and received one r̃ (t) is given by r̃ (t) = s̃(t) +ñ(t), in which ñ(t) is a complex zero-mean white Gaus-sian process. As the power of the received complex enve-lope is given by σr̃ 2 , E[|r̃ (t)|2] = E[|s̃(t)|2]+E[|ñ(t)|2] =A2 + σñ 2, we define the SNR asProceedings of SPS 2004 (the 2004 IEEE Benelux Signal Processing Symposium)172SN R ,A2σñ2 . (16)A trace of the phase metric 1φ[k], generated using a MAT-LAB Simulink model, is shown in figure 2. In the left partof the figure, the no-noise case (110 dB SNR) is shown,in the right part an SNR of 11 dB was used. The sen-0 500 1000 1500 2000 2500 3000 3500 4000-1-0.8-0.6-0.4-0.200.20.40.60.81phase metric                             (a) SNR= 110 dB.0 500 1000 1500 2000 2500 3000 3500 4000-1-0.8-0.6-0.4-0.200.20.40.60.81phase metric                             (b) SNR= 11 dB.Figure 2: Phase metric (10) vs. elapsed time since thebeginning of thetransmission in [bit period] for h = 0.28.sitivity to noise of the modulation-index estimate ĥ canbe seen in figure 3. The lowest curve depicts the resultin the no-noise case and corresponds to the trace in fig-ure 2(a); the top curve of figure 3, for an SNR of 11 dBcorresponds to the phase metric in figure 2(b). A curvein between the top and bottom curve has half the noisepower of the curve above it. As can be seen, all curvesreach some form of steady state after approximately 2250bit periods, although at a level (bias) depending on theSNR. This bias appears to be approximately linear in thenoise power: doubling of the noise power approximatelydoubles the bias. In figure 4 this observation is verified.Estimated Modulation Index (h_TX = 0.28)0.2740.2780.2820.2860.290.2940.2980.3020.3060.310.3140 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000Time [bit period]Estimated hSNR = 11 dB 14 dB 17 dB 20 dB 23 dB 110 dBFigure 3: Estimated modulation-index ĥ vs. elapsed timein [bit period] for h = 0.28.The once-per-bit-period estimated modulation-index ĥ isaveraged over the last 1024 bit periods of the trace toproduce ĥAv . We see the relative error in the estimatedmodulation-index ĥAv−hh (in dB) as a function of the SNRin dB (SN RdB) that was used to produce the simulationtrace (the nominal SNR). The figure can be interpreted byusing10.10 log (ĥAvh− 1) = −α . SN RdB + β. (17)Relative estimation error and SNRy = -1.0563x + 1.8262R2 = 0.9995-24-21-18-15-12-99 12 15 18 21 24SNR  [dB]10*10log(h_Av / h - 1)  [dB]Linear (10 * 10log(h_Av/h - 1))Figure 4: Relation between relative estimation error andnominal SNR (curveof measured ĥAv and curve fit).This givesĥAvh= 1 + 10β10 . 10−α .SN RdB10 (18)with (see figure 4) α = 1.0563 and β = 1.8262. The es-timated modulation-index ĥ for any particular bit periodcan now be corrected by a correction factor that compen-sates (18):ĥcor r.h= 11 + 10β10 . 10−α .SN RdB10.ĥh(19)In case ĥ is converged, ĥcor r. ≈ h. The result of apply-ing the correction factor is given in figure 5. As can beseen, after approximately 2250 bit periods the estimatedand SNR-corrected modulation-index falls within the re-quired modulation index range.Estimated Modulation Index, with SNR correction  (h_TX = 0.28)0.27400.27800.28200.28600.29000.29400.29800.30200.30600.31000.31400 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000Time [bit period]Estimated h (SNR corrected)SNR = 11 dB 14 dB 17 dB 20 dB 23 dB 110 dBFigure 5: Noise-corrected estimated modulation-indexĥcor r. vs. elapsed time in [bit period] for h = 0.28.Although, in this paper, simulation based curvefitting hasbeen applied to relate an a priori known SNR (the nomi-nal SNR) to the correction factor, we believe that by mea-surement of the SNR during signal reception a correctionfactor can be computed. The thing to do is to arrive ata formal (mathematical) relationship between the (mea-sured) SNR and a correction factor. The authors observedthat this can be done for the AWGN case.5. ANALYSISIn this section we assess the observed behaviour of theestimator in the no-noise case. The interest is in establish-Proceedings of SPS 2004 (the 2004 IEEE Benelux Signal Processing Symposium)173ing the relation between expectation and variance of theestimators in (15), the number of transmitted bits N andthe modulation index h. Under the assumption of zeromean i.i.d. input symbols, the expectation and variance ofS1φ2 [N] can be derived from (14):E[S1φ2 [N]]= h2 . N .∑Q2 andE[ (S1φ2 [N])2 ] = h4 .(N2 . (∑Q2)2 + 4Q22 Q12 . N)so σ 2S1φ2 [N]= 4Q22 Q12h4 . N.(20)To determine the moments of the estimator ĥ2 of the squaredmodulation index, (15) is needed. One arrives atµĥ2 =1N .∑Q2. E[S1φ2 [N]]= h2σ 2ĥ2= 1N2 . (∑Q2)2 . σ2S1φ2 [N]= 4Q22 Q12h4(∑Q2)2 .1N.(21)For GFSK Q2 = 1/2, Q1 = 1/2 and∑Q2 = 1/2 (seejust below (11)), soσ 2ĥ2= h4N. (22)The relation between the expectation and variance of ĥ2and those of ĥ can be found by an approximation that re-lates the moments of a function y = g(x) to the momentsof the ordinate x [9, p.152 - (5-62),(5-63)]. This resultsfor g(x) = √x inµ√x ≈√µx −1µx√µx.σ 2x8σ 2√x≈14 µx. σ 2x .(23)With x ≡ ĥ2 and √x ≡ ĥ we find (using both (21) and(22)):µĥ ≈√µĥ2 −1µĥ2√µĥ2.σ 2ĥ28= h . (1 − 18N)σ 2ĥ≈14 µĥ2. σ 2ĥ2= 14h2.h4N= h24N(24)The agreement of analysis and simulation can be estab-lished by considering the confidence interval I ĥ , [µĥ −α σĥ , µĥ + α σĥ] of the estimate ĥ. With high proba-bility an estimate ĥ will lie in this interval, provided thatα is chosen high enough (say 2 to 3). By substitution ofthe expectation and standard deviation in (24) into I ĥ, andobserving the behaviour for increasing N, one can showthat for approximately N ≥ 2300 a 1% relative error canbe reached. There is close agreement between simulationand analysis.6. DISCUSSION AND CONCLUSIONSWe presented a blind modulation-index estimator and as-sessed its performance. It was observed that the estimatedmodulation-index depended on the the SNR of the AWGNchannel. It was shown that by measuring the SNR and byapplying a correction factor, the steady state estimate iswithin 1% of the true modulation index, even in the 11 dBSNR case. In order for this correction mechanism to bepractical applicable however, a formal (mathematical) re-lationship between measured SNR and correction factorneeds to be established.As can be seen from the convergence speed in the traces,approximately 2250 bit periods are needed for adequateestimation (both in the no-noise and in the noise case inwhich SNR correction is applied). This slow convergencecame as a surprise to the authors. However, by interpret-ing the confidence interval I ĥ of the estimate ĥ, it couldbe shown that the results of analysis and simulation com-pletely agreed with one another.The convergence speed seems far too slow for the 68-bitlength of the packets involved in the Bluetooth Inquiry(Scan) and Page(Scan) protocol procedures. Althoughwe did not investigate other blind methods for suitabil-ity in our system, we believe that exploring some form ofdata-aided modulation-index estimation (using an 68 bitpacket) may be worthwhile.7. REFERENCES[1] R. Schiphorst, F.W. Hoeksema, and V.J. Arkesteijn, “Thebluetooth-HiperLAN/2 SDR receiver project web site,”http://www.sas.el.utwente.nl/home/SDR/,2002, 2003, 2004.[2] Bluetooth SIG, “Specification of the bluetooth system -core,” Technical Specification Version 1.1, Bluetooth SIG,February 2001.[3] ETSI, “Broadband radio access networks (BRAN); HIPER-LAN type 2; system overview,” Technical Report ETSI TR101 683 V1.1.1 (2000-02), ETSI, February 2000.[4] V.J. Arkesteijn, R. Schiphorst, F.W. Hoeksema, E.A.M.Klumperink, B. Nauta, and C.H. Slump, “A combinedreceiver front-end for bluetooth and hiperlan/2,” 4thPROGRESSworkshop on Embedded Systems and Software,Nieuwegein, October 2003.[5] P.A. Laurent, “Exact and approximate construction of digi-tal phase modulations by superposition of amplitude modu-lated pulses,” IEEE Transactions on Communications, vol.34, pp. 150–160, 1986.[6] G. Kawas Kaleh, “Simple coherent receivers for partial re-sponse continuous phase modulation,” IEEE Journal on Se-lected Areas in Communications, vol. 7, no. 9, pp. 1427–1436, December 1989.[7] R. Schiphorst, F.W Hoeksema, and C.H. Slump, “A (sim-plified) bluetooth maximum a posteriori probability (map)receiver,” Proceedings of IEEE SPAWC2003, Rome, June2003.[8] J.B. Anderson, T. Aulin, and C.E. Sundberg, Digital PhaseModulation, Plenum Press, 1986.[9] A. Papoulis, Probability, RandomVariables, and StochasticProcesses, McGraw-Hill Book Company, 1981.Proceedings of SPS 2004 (the 2004 IEEE Benelux Signal Processing Symposium)174

A (simplied) bluetooth maximum a posteriori probability (map) receiver,

A combined receiver front-end for bluetooth and

Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses,

Probability, Random Variables, and Stochastic Processes,

radio access networks (BRAN); HIPERLAN type 2; system overview,

Simple coherent receivers for partial response continuous phase modulation,

Specication of the bluetooth system -core,

The bluetooth-HiperLAN/2 SDR receiver project web site, http://www.sas.el.utwente.nl/home/SDR/,

https://ris.utwente.nl/ws/files/5474510/sps2004a.pdf

Modulation-Index Estimation in a Combined CPM/OFDM Receiver

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University of Twente Research Information