Optimized Differential Bluetooth DemodulatorBo Yu, Liuqing Yang

Department of ECE, University of Florida, Gainesville, FL 32611and

Chia-Chin ChongDOCOMO USA Labs, 3240 Hillview Avenue, Palo Alto, CA 94304

Abstract—Bluetooth is a short-range communication standardwhich employs Gaussian frequency shift keying (GFSK). Thedesign of low-complexity and high-performance receivers forBluetooth systems is a challenging task. In this paper, we developan optimized differential GFSK demodulator and investigatethe phase wrapping issue in its implementation. Simulationresults show bit-error-rate (BER) performance improvementin comparison with conventional differential demodulators. Wealso compare our proposed demodulator with other existingalternatives in terms of BER performance and complexity.

I. INTRODUCTION

IEEE 802.15.1 Bluetooth Standard is an increasingly pop-ular and widely deployed standard for short-range wirelessapplications operating in the 2.4 GHz unlicensed industrial,scientific and medical (ISM) band. The modulation for Blue-tooth is Gaussian frequency shift keying (GFSK). To facilitatelow complexity receivers with reliable link quality, the demod-ulator needs to be both simple and robust.

The optimum GFSK demodulator is the trellis-based Viterbidecoder [1], [2], [3]. These designs always assume a cer-tain nominal value for the modulation index h. However,the modulation index in Bluetooth is allowed to vary in arelatively large interval, leading to a varying trellis structurefor sequence detection with possibly tremendous number ofstates. In addition, Bluetooth’s frequency hopping makes itdifficult to establish a phase reference which is required forcoherent sequence detection. All these render this optimumreceiver impractical. Hence noncoherent suboptimal receiversare typically used to demodulate GFSK signals.

A simple limiter-discriminator followed by integrate-and-dump post-detection filtering [1], [4] is often employed inBluetooth devices. The resulting receiver is called limiter-discriminator with integrator (LDI) receiver. The LDI receiveris low-cost and easy to implement, but it suffers from arelatively poor power efficiency compared to more sophisti-cated receivers. Another simple noncoherent receiver is thephase differential detector [4], [5]. These receivers are alsoapplicable to a relatively large range of modulation indices.

In [6], Lee proposed a zero-intermediate frequency zero-crossing demodulator (ZIFZCD). The performance of this de-modulator is not as good as the LDI detector. Later, Scholandand Jung proposed another Bluetooth receiver [7] based on thezero-crossing detection of the a real-valued received signalat an appropriately chosen intermediate frequency (IF) incombination with a decorrelating matched filter. The receiver

performance has some improvements compared to the con-ventional baseband zero-crossing detectors and has similarperformance with the baseband differential detector and theLDI detector.

More recently, Lampe et.al. proposed a noncoherent se-quence detector based on Rimoldi’s decomposition [8] of theGFSK transmit signal. This receiver can achieve significantperformance gain of more than 4dB over the discriminator-based detector. The main disadvantage is the complexity thatis required for a two-state trellis search. In their subsequentpaper [9], Ibrahim et.al. considered sequence detection basedon Laurent’s decomposition of the GFSK transmit signal. Theresult shows a performance close to the limit of maximum-likelihood sequence detection. However, this receiver stillrequires the modulation index to be known at the receiver.When the modulation index is not available, it has to gothrough an adaptation period to search for the modulationindex.

In this paper, we develop a low-complexity demodulatorfor Bluetooth receivers, which simply averages the phasebased on the signal-to-noise ratio (SNR) maximizing criterion,and does not require knowledge of the exact modulationindex. Compared to demodulators with similar complexity,such as the LDI, our proposed receiver can achieve superiorperformance. The paper is organized as follows. In Section II,we will briefly introduce the Bluetooth transmission model.The phase noise distribution is derived in Section III. Based onthe phase noise distribution, we will then propose an optimizeddifferential demodulator in Section IV and investigate thephase wrapping problem in Section V. The performance ofthis receiver is studied in Section VI, followed by summarizingremarks in Section VII.

II. BLUETOOTH TRANSMISSION MODEL

In this section, we will briefly describe the Bluetooth radiolink transmission model.

A. Bluetooth Transmitter

The block diagram of the Bluetooth transmitter is shown inFig. 1. The transmitter uses GFSK modulation. A passbandtransmitted GFSK signal can be represented as [5]

s(t, α) =

√2Eb

Tcos[2πf0t + ϕ(t, α) + ϕ0], (1)

where Eb is the energy per bit, T is the symbol period, f0 is thecarrier frequency, ϕ0 is an arbitrary constant phase shift. The

1030978-1-4244-5827-1/09/$26.00 ©2009 IEEE Asilomar 2009

Input bitsNRZ

signal

Gaussian filter

BT=0.5

Voltage

Controlled

Oscillator

Fig. 1. Block diagram of the Bluetooth transmitter

output phase deviation ϕ(t, α) is determined by the input datasequence α = ..., α−2, α−1, α0, α1, α2, ... with αi ∈ {±1}:

ϕ(t, α) = 2πh

n∑i=n−L+1

αiq(t− iT ) + πh

n−L∑i=−∞

αi , (2)

where h is the modulation index, q(t) =∫ t

−∞g(τ)dτ with

g(t) = Q

(γ · BT

(t

T− 1

2

))−Q

(γ · BT

(t

T+

1

2

))

being the frequency pulse with constant γ = 2π/√

ln 2 andQ(·) is the Gaussian Q-function. In the Bluetooth standard[10], the modulation index h can vary between 0.28 and 0.35and the 3dB bandwidth-time product is specified as BT = 0.5with T = 10−6s, giving rise to a g(t) with effective durationof 2T .

A phase trajectory ϕ(t, α) generated by the sequence(1,−1, 1,−1, 1, 1, 1, 1) for a partial response Gaussian pulseof length 2T is illustrated in Fig. 2. As we can see, the phasetrellis is monotonic within every symbol duration, and thetrellis segment per symbol has four different shapes whichvary according to four different bit stream patterns; that is,(000 or 111), (001 or 011), (100 or 110) and (101 or 010).

B. Channel Model

The model for the equivalent complex-baseband receivedsignal is

r̃(t) = sl(t) + η(t), (3)

where sl(t) is the complex envelope of the Bluetooth trans-mitted signal, which has the form

sl(t) = r exp[j(ϕ(t, α) + ϕ0)], (4)

and η(t) = ηr(t) + jηi(t) is a complex white Gaussian noisewhich has zero mean and variance 2σ2. In addition, ηr andηi are independent and each has a distribution of N (0, σ2).Hence the joint distribution of ηr and ηi is

fηrηi(x, y) =1

2πσ2e−

x2+y2

2σ2 . (5)

For phase differential demodulation, one needs to firstextract the phase of the received signal

φ(t) = �r̃(t) = ϕ(t) + φη(t) (6)

where ϕ(t) is determined by the transmitted phase and φη(t) isa random variable due to the additive white Gaussian noise. Tofacilitate the design of an optimum differential demodulator,we will next study the phase noise distribution.

0 1 2 3 4 5 6 7 8−1

−0.5

0

0.5

1

t(μs)

Am

plitu

de

0 1 2 3 4 5 6 7 80

1

2

3

4

5

t(μs)

φ(t,α

)

Fig. 2. GFSK phase trellis.

III. PHASE NOISE DISTRIBUTION

For notational simplicity, let ϕ(t) = 0 without loss ofgenerality. It then follows that φ(t) = φη(t). Denote theprobability density function (PDF) and the cumulative densityfunction (CDF) of the random variable φ(t) as fφ(ϕ) andFφ(ϕ), respectively. We first consider 0 ≤ φ(t) ≤ π/2, onwhich

Fφ(ϕ) = Pr(0 ≤ φ ≤ ϕ)

=

∫ ∞

−r

∫ (r+x) tan ϕ

0

fηrηi(x, y) dy dx.(7)

Then by Leibnitz’s rule, the PDF is derived as

fφ(ϕ) =d

dϕFφ(ϕ)

=1

2πσ2

∫ ∞

−r

e−x2+(r+x)2 tan2 ϕ

2σ2d

dϕ[(r + x) tan ϕ] dx

=1

2πe−

r2

2σ2 +r cosϕ√

2πσe−

r2 sin2 ϕ

2σ2 Q(−r cosϕ

σ

).

(8)It can be readily shown that the formula holds for the entirerange of 0 ≤ φ(t) < 2π as well.

At high SNR (large r), this complicated noise distributioncan be approximated as:

fφ(ϕ) ≈ 1√2πσ/r

· e− 12 ·

ϕ2

(σ/r)2 , (9)

which is simply the Gaussian distribution with zero mean andvariance (σ/r)2. Such a Gaussian approximation of the phasenoise distribution can be very useful if the approximation isaccurate. This is because our demodulator design is intendedto optimize the ultimate bit-error-rate (BER) performance ofthe system. In the case of Gaussian noise, minimizing BERis equivalent to maximizing the SNR, leading to a readily-achievable SNR-maximizing system optimization criterion.This is generally not true for non-Gaussian noise [11].

To assess the accuracy of such an approximation, weperform two sets of simulations. In the first simulation, wecompare the true and approximate PDF curves at variousSNR values. As shown in Fig. 3, we observe that thesecurves are already very close at r2/(2σ2) = 2dB, and are

1031

−4 −3 −2 −1 0 1 2 3 40

0.2

0.4

0.6

0.8

(a)

Exact distributionGaussian distribution

−4 −3 −2 −1 0 1 2 3 40

0.5

1

1.5

(b)

Exact distributionGaussian distribution

Fig. 3. PDF curves for (a) r2/(2σ2) = 2dB; (b) r2/(2σ2) = 8dB.

−6 −4 −2 0 2 4 6 8 1010

−4

10−3

10−2

10−1

SNR (dB)

BE

R

exact phase noiseGaussian phase noise

Fig. 4. BER performance for the exact phase noise and theapproximated Gaussian phase noise.

nearly identical at r2/(2σ2) = 8dB. We also simulate andcompare the BER performance of a simple binary systemwhere the additive noise follows either the true noise PDF orthe Gaussian approximation. The result in Fig. 4 shows thatthe BER performances of the two are very similar. You maynote that the BER gap at higher SNR seems to be larger thanthe gap at lower SNR. However, since the scale is differentin the BER axis, the approximation is actually much better athigher SNR, which conforms to our previous analysis.

So far, we have verified that the noise in the extractedphase can be well approximated as Gaussian distributed. In thefollowing section, we will develop our optimized differentialGFSK demodulator.

IV. OPTIMIZED DIFFERENTIAL GFSK DEMODULATOR

As shown in Fig. 2, the phase trellis is piece-wise monotonicwithin each symbol duration. The direction of the monotonicchange is determined by the binary symbol value. Hence,a differential demodulator can be employed. Essentially, theconventional differential demodulator involves sampling φ(t)in (6) at symbol rate to obtain φ(nT ) and then taking the

T - To T To0

A

t

q(t)

Fig. 5. Linear phase curve model.

difference of the neighboring samples

Δφ(nT ) = φ(nT )− φ(nT − T ).

A decision can then be made based on the sign of Δφ(nT ).However, due to the randomness of the phase noise, the deci-sion based on a single phase sample per symbol lacks sufficientreliability so that conventional differential demodulator suffersfrom degraded performance.

Consider again the phase trellis in Fig. 2. Instead of a singlesample, one can average a portion of every symbol-long trellissegment at each end before taking the difference. Next wewill determine the optimum portion to be averaged so that thedemodulation performance is maximized. As we mentionedbefore, each symbol-long phase trellis segment can take oneof the four shapes depending on the specific symbol sequence.For example, the phase trellis during [0, T ) is linear in the firsthalf and nonlinear in the second half (which we term as typeD); the phase trellis during [T, 2T ) is nonlinear throughout theentire duration (which we term as type C); the phase trellisduring [4T, 5T ) is nonlinear in the first half and linear in thesecond half (which we term as type B); and the phase trellisduring [5T, 6T ) is linear throughout (which we term as typeA). Without loss of generality, we will start from the linearphase segment and then generalize to the nonlinear and partlynonlinear ones.

A linear phase curve segment has a function q(t) = A · t/T ,t ∈ [0, T ). Note that the scalar A depends linearly on themodulation index h. However, since its value does not affectthe optimization result, we will normalize it to 1 for notationalsimplicity (the same for the nonlinear phase curve cases). LetTo be the portion at each end of this segment over which weaverage before taking the difference, as illustrated in Fig. 5.Then the resultant SNR is:

SNRA(To) =

(∫ T

T−To

tT dt− ∫ To

0tT dt

)2

2Toσ2. (10)

As we have shown in the preceding section, the phase noisecan be well approximated as Gaussian distributed even atfairly low SNR. Hence, the SNR-maximizing To is essentiallyalso BER-minimizing. Maximizing SNRA for a given symbolduration T and A2/σ2, we obtain

max0≤To≤T/2

SNRA(To) = max0≤To≤T/2

To(T − To)2

2T 2σ2, (11)

which results in To = T/3. This result indicates that oneshould average the first and last 33% of each symbol-long

1032

Take the

phase

Phase

unwrapping

Take the difference

between the

integrals

(or averages )

Make a

decision

Integrate (or average)

the first 35% of the

phase curve (samples)

for one symbol

Integrate (or average)

the last 35% of the

phase curve (samples)

for one symbol

Fig. 6. Block diagram of our optimized differential demodulator.

phase segment and then take their difference on which adecision can be made.

Next we consider the type-C phase segment during [T, 2T ).The exact function for this segment does not have a closedform (containing the integral of Gaussian Q function). Hencewe use the second-order polynomial approximation; that isx1(t) = −0.9202(t/T )2 − 0.1797(t/T ) + 0.8158, t ∈[0, 0.5T ), and x2(t) = 0.9202(t/T )2− 1.9911(t/T ) + 1.273,t ∈ [0.5T, T ). Accordingly, the resultant SNR after averagingand difference taking is:

SNRC(To) =

(∫ T

T−Tox2(t)dt− ∫ To

0 x1(t)dt)2

2Toσ2. (12)

As a result, the optimal portion should be chosen as To =0.3675T .

Similar results can be obtained for type-B and type-D phasesegments. Due to the space limit, these are omitted here. Sincethe four different shapes occur with equal probability, one canalso find the overall optimum To by solving

max0≤To≤T/2

[SNRA(To) + SNRB(To)

+ SNRC(To) + SNRD(To)],(13)

which results in To = 0.35T .The continuous-time analysis can be readily extended to

the discrete-time sampled phase segments. Let 2K denotethe number of samples per symbol. The problem now is todetermine the optimum M , which is the number of samplesto be averaged at each end of the 2K samples per sym-bol. Denote the second group of M samples as S1, ...., SM ,and the first group of M samples as S′1, ...., S

′M . It then

follows that Si = (K − 1 + i)/(2K − 1) + ηi and S′i =(i− 1)/(2K − 1) + η′i, i = 1, . . . , M . Consider again thetype-A linear phase segment. With the Gaussian approximationof the phase noise, the difference between the two averages is∑M

i=1(Si − S′i)/M , leading to an SNR of

SNR(d)A (M) =

M(2K −M)2

2(2K − 1)2σ2, (14)

where the (d) indicates discrete-time. Clearly, SNR(d)A (M)

bears a form very similar to SNRA(To) in (10). Not surpris-ingly, maximizing SNR(d)

A (M) results in M = 2K/3, whichagrees perfectly with the continuous-time result. Using thesame methodology, we can obtain the optimum M for thegeneral case as M = 0.7K . Thus, by choosing the optimal M ,the SNR is maximized and the error probability minimized.

The block diagram of our optimized differential demodu-lator is shown in Fig. 6. We can see that the only difference

IF filter Delay TPhase shift

LPF

Sample

and hard

decision

Fig. 7. Conventional implementation of differential demodulator [5].

between this new demodulator and the conventional differen-tial one is the averaging part. To evaluate the performancegain by using the optimized differential demodulator, wenotice that the conventional differential demodulator can beconsidered as a special case with M = 1 for any K values.Considering again the type-A phase segment, and evaluating(14) at M = 2K/3 and M = 1 and taking their relativeratio, we obtain the SNR gain over conventional differentialdemodulator as:

SNR(d)A (2K/3)

SNR(d)A (1)

=32

27

K3

(2K − 1)2.

This gain is a function of K because the optimum numberof averaged samples M is dependent on the total number ofavailable samples 2K .

V. PHASE WRAPPING PROBLEM

When realizing these phase differential demodulation algo-rithms, however, there is an implementation problem. Recallthat our differential operations are performed on the phasefunction φ(t) in (6). Unlike the original phase trellis ϕ(t, α)in (2), φ(t) is not only noisy, but also suffers from the phasewrapping problem since it assumes a finite range of 2π.

To solve this problem, conventional differential demodulatoroften takes the form as shown in Fig. 7. In this structure, r(t)is multiplied by a T -delayed and π/2 phase-shifted version ofitself and then sampled at the symbol rate to give the decisionstatistic. These operations will essentially give sin[Δφ(nT )]without explicitly extracting φ(t), thus avoiding the phasewrapping problem. Since the decision only relies on the signof Δφ(nT ), the operations in Fig. 7 leading to sin[Δφ(nT )]preserves this sign information.

However, for large modulation index h ≥ 1, the phasechange Δφ(nT ) over one symbol duration may exceed π. Inthis case, the sin operator cannot preserve the sign of Δφ(nT )anymore.

Hence, in both the conventional and optimized differentialdemodulators, we directly deal with the phase extracted fromthe received signal. Specifically, the phase is unwrapped bysimply adding ±2π when the absolute change between theconsecutive phase samples are greater than the jump toleranceπ. Both methods are tested and compared by simulations inthe next section.

VI. PERFORMANCE EVALUATION AND DISCUSSION

In this section, we will evaluate the performance of ourproposed demodulator by simulations in terms of BER versusEb/N0, where Eb is the signal energy per information bit andN0 is the power spectral density of the additive noise.

We first set the modulation index h as 0.33, which fallswithin the [0.28, 0.35] range specified in the Bluetooth stan-dard. The BER performance for our optimized differential

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0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

Eb/N

0 (dB)

BE

R

LDIoptimized differentialconventional in [5]conventional with phase unwrapping

Fig. 8. BER performance for h = 0.33.

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

Eb/N

0 (dB)

BE

R

LDIoptimized differentialconventional in [5]conventional with phase unwrapping

Fig. 9. BER performance for h = 0.2.

demodulator is shown together with that of the conventionalone and the LDI demodulator in Fig. 8. In particular, forthe conventional differential demodulator, we simulated boththe system in [5] which avoids phase wrapping as well asour direct unwrapping approach. We observe that, to achievea BER of 10−3 as required by the Bluetooth standard, ouroptimized demodulator requires about 18 dB while both theconventional one and the LDI need about 20 dB. Also theoptimized demodulator shows about 1∼2 dB improvementover the conventional one and the LDI especially at high SNR.Note that the implementation of LDI also needs over sampling,so the complexity is similar to our optimized demodulator.

Figs. 9 and 10 show the performance of the three demod-ulators with h equals to 0.2 and 1, respectively. For h = 0.2,the performance of all demodulators degrade compared to h= 0.33. And the performance of the conventional differentialdemodulator with direct phase unwrapping is similar to theapproach in [5]. For h = 1, the performance of all demodula-tors except the conventional differential system in [5] improvesignificantly compared to small h cases. The performance lossof conventional differential demodulator in [5] results fromthe fact that the phase change during one symbol interval may

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

Eb/N

0 (dB)

BE

R

LDIoptimized differentialconventional in [5]conventional with phase unwrapping

Fig. 10. BER performance for h = 1.

exceed π for large h such that the sin operator cannot preservethe sign of Δφ(nT ) anymore. In addition, the Eb/N0 gain ofthe optimized demodulator over the conventional one for largeh is not as significant as the Eb/N0 gain for small h. This isdue to the phase unwrapping operation.

VII. CONCLUSIONS

In this paper, we have developed an optimized phase dif-ferential demodulator for GFSK modulated systems includingBluetooth. Performance of this demodulator has been studiedby theoretical analysis and simulations. It has been shown thatthe optimized demodulator can achieve evident improvementsover the conventional demodulator and provide a favorableperformance with a low implementation complexity, which isdesirable for practical wireless systems including Bluetooth.

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[5] J. B. Anderson, T. Aulin, and C.-E. Sundberg, Digital Phase Modulation,1st ed. Springer, 1986.

[6] E. K. B. Lee, “Zero-crossing baseband demodulator,” in IEEE Interna-tional Symposium on Personal, Indoor and Mobile Radio Communica-tions, vol. 2, September 1995, pp. 466–470 vol.2.

[7] T. Scholand and P. Jung, “Novel receiver structure for Bluetooth basedon modified zero-crossing demodulation,” in IEEE Global Telecommu-nications Conference, vol. 2, December 2003, pp. 729–733 Vol.2.

[8] L. Lampe, R. Schober, and M. Jain, “Noncoherent sequence detectionreceiver for Bluetooth systems,” IEEE Journal on Selected Areas inCommunications, vol. 23, no. 9, pp. 1718–1727, September 2005.

[9] N. Ibrahim, L. Lampe, and R. Schober, “Bluetooth receiver design basedon Laurent’s decomposition,” IEEE Trans. on Vehicular Tech., vol. 56,no. 4, pp. 1856–1862, July 2007.

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[11] B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation PatternRecognition. Cambridge University Press, 2005.

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