1

Adaptive Non-Data-Aided Compensation for I/Q Mismatchand Frequency Offset in Low-IF Receivers

Shafayat Abrar1, Azzedine Zerguine2 and Asoke Nandi3

1Department of Electrical Engineering, COMSATS Institute of Information Technology, Islamabad 440002Department of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261

3Department of Electronic and Computer Engineering, BrunelUniversity, Uxbridge UB8 3PH

Abstract—In this work, we present an adaptive non-data-aidedcompensator for the in-phase/quadrature-phase (I/Q) mismatch inlow-intermediate frequency (heterodyne) receivers. In particular,the adaptive I/Q mismatch algorithm is derived by exploiting theuncorrelatedness between the desired and the image signals, andis used to compensate for gain imbalance, phase-offset error andfrequency-offset error.

Index Terms—I/Q imbalance, adaptive filter

I. I NTRODUCTION

Due to components mismatches in analogue electronics andresulting in-phase/quadrature-phase (I/Q) imbalances, the perfor-mance of a heterodyne receiver may degrade significantly [1].To compensate these imbalances and remove image signal fromthe desired band, statistical independence based adaptivemethodswere introduced (refer to [2] and references therein). Here, inthis work, we discuss a simple algorithm for non-data-aidedI/Q compensation which is derived by exploiting the correlationproperties of desired and image signals.

The paper is organized as follows. In Section II, a generalsignal model for an imbalanced analog front-end is presented.In Section III, a cost function sensitive to imbalance is proposedand optimized to yield into an adaptive algorithm for imbalancecompensation. In Section IV, an iterative method is derivedfor theestimation of frequency-offset. Simulation results are presented inSection V and conclusion are drawn in Section VI.

II. SYSTEM MODEL

The system model is shown in Fig. 1. In the absence of additivenoise, the received RF signal is given by

rRF(t) = 2ℜs(t)ei2πfCt+ 2ℜq(t)ei2πfIt, (1)

wherefC is the carrier frequency of desired band,fI = 2fLO−fCis the central frequency of the image band, ands(t) and q(t)are the baseband desired and image signals, respectively. Thequadrature mixture is assumed to suffer with amplitude mismatchε, frequency mismatch∆f = fC − fLO − fIF and phasemismatchθ. The impairmentsε andθ are assumed to be frequencyindependent. The digital IF signalrn, sampled at the rate of1/T ,is expressed as:

rn = (βsn + αq∗n) e+i2πn(fIFT+Ω)

+ (βqn + αs∗n) e−i2πn(fIFT+Ω),

(2)

where

α = 0.5(1− (1 + ε)eiθ

), (3a)

β = 0.5(1 + (1 + ε)e−iθ

), (3b)

LPF A/D

LPF A/D

rRF(t)

cos(2 fLO t)

(1+ ) sin(2 fLO t+ )i

LPF

LPF

exp( i2 fIFnT)

exp(+i2 fIFnT)

rn

xn

zn

Analogue

Processing

Digital

Processing

fLO is local oscillator freq.fIF is desired IF freq. T is symbol period.xn

zn

Adaptive

gain/phase

mismatch

removal

sn

qn

Iterative

frequency

mismatch

removal

sn

qn~

~ ˆ

ˆ

Fig. 1. System model of a heterodyne receiver and mismatch compen-sators.

Ω = T∆f , sn = s(t)|t=nT and qn = q(t)|t=nT . After down-conversion and low-pass filtering, we obtain the baseband signalsxn andzn:

xn = (βsn + αq∗n) e+i2πnΩ, (4a)

zn = (βqn + αs∗n) e−i2πnΩ. (4b)

The digital processor at receiver uses the baseband signalsxnand zn to estimate the impairments,θ, ǫ and Ω. Assumingthat impairments are perfectly known, then the desired and imagesignals are expressed as

sn = sne−i2πnΩ =

(1+γ∗

1−|γ|2

)(xn − γ z∗n) e

−i2πnΩ, (5a)

qn = qne+i2πnΩ =

(1+γ∗

1−|γ|2

)(zn − γ x∗

n) e+i2πnΩ, (5b)

whereγ = α/β∗. The impairmentsε andθ are related toγ, wecan show:

θ = angle

1− γ

1 + γ

, and ε =

∣∣∣∣1− γ

1 + γ

∣∣∣∣− 1. (6)

III. E STIMATION OF GAIN IMBALANCE γ

Exploiting the fact that the desired and image signalssn andqn are mutually uncorrelated, optimum estimates were obtainedin [3] as follows:

γ(1)opt =

B −√B2 − 4|A|22A∗

, (7a)

γ(2)opt =

B +√B2 − 4|A|22A∗

, (7b)

2

whereA := Exnzn, andB := E(|xn|2 + |zn|2

). For vanishing

imbalance, i.e.,A → 0, we haveγ(1)opt → 0 and γ(2)

opt → ∞.Note that authors in [3] preferred to use the root with smallermagnitude, i.e.,γ(1)

opt.In this work, we propose to obtain the value ofγ adaptively by

minimizing a cost which is measure of the correlation betweenthe estimated signals,sn and qn, mathematically it is expressedas

γ† = argminγ

∣∣∣∣E (xn − γz∗n) (zn − γx∗n)

∣∣∣∣2

, (8)

Note that this cost is insensitive to frequency offset error, Ω, whichfacilitates separate estimation ofΩ. To obtain a gradient-basedadaptive algorithm forγ, we use

γn = γn−1 − µ(∇γ |C|2

)∗, (9)

for γ = γn−1 andC := E(xn−γn−1z∗n)(γn−1x

∗n−zn). Note that

the auxiliary variableC can be expressed asC = A−B γn−1 +A∗ γ2

n−1, whereA andB are as specified in (7); next, we find

∂|C|2∂γn−1

=∂|C|2∂C

∂C

∂γn−1= C∗ (2A∗ γn−1 −B) , (10)

Replacing the statisticsA, B and C with their respective esti-mates, we get the following gradient-based algorithm:

An = λ gAn−1 + (1− λ g)xnzn,

Bn = λ gBn−1 + (1− λ g)(|xn|2 + |zn|2

),

Cn = An −Bn γn−1 + A∗n γ

2n−1,

γn = γn−1 + µ g Cn(Bn − 2An γ∗n−1) ,

(11)

whereµ g is a positive step-size and0 < λ g < 1 is a forgetting-factor. SubstitutingAn andBn in (7), we obtainγ(1)

n−1 andγ(2)n−1

as the estimates ofγ(1)opt and γ(2)

opt, respectively. Usingγ(1)n−1 and

γ(2)n−1, we can express (11) as follows:

γn = γn−1 − η(γn−1 − γ

(1)n−1

)(γn−1 − γ

(2)n−1

)

×(γn−1 −

γ(1)n−1 + γ

(2)n−1

2

)∗

,(12)

where η = 2µ|A|2 and 0.5(γ(1)n−1 + γ

(2)n−1) is the estimate of

saddle point (see Fig. 2(a)). This implies that, depending oninitialization, the update may converge either toγ(1)

opt and γ(2)opt.

Under no imbalance condition, however, as one of the roots isrequired to be zero, the update has a natural tendency to convergeto the root with smaller magnitude provided that the step-size islarge enough to help escape the other root (see Fig. 2(b)-(c)).

IV. ESTIMATION OF FREQUENCYOFFSETΩ

A. For PSK Signals:

The presence of frequency-offset error contaminates the esti-mated signalsn by the factore+i2πnΩ. SupposeΩn−1 is theavailable estimate ofΩ, then sn is expressed as

sn = sne−i2πnΩn−1

=1 + γ∗

n−1

1− |γn−1|2(xn − γn−1 z

∗n) e

−i2πnΩn−1 ,(13)

If sn is an m-PSK, then the maximum likelihood approachestimatesΩ, as given by,

Ω =angle

∑N

k=0

(sn−k s

∗n−1−k

)m

2πm, (14)

whereN denotes number of symbols [4]. Note that this estimatorassumes that the signal has constant modulus; in the presence ofgain imbalance, however, we would need gain normalization toensure this property. Denoting∆n := Ωn−1 − Ω, and assumingno additive noise, note that

(s ∗n−1sn

|sn−1| · |sn|

)m=

(s ∗n−1e

i2π(n−1)Ωn−2 · sne−i2πnΩn−1

|sn−1| · |sn|

)m

≈(s∗n−1e

−i2π(n−1)∆n−1 · snei2πn∆n

)m

= ei2πm(∆n−1+n(∆n−∆n−1)),(15)

Further assuming∆n ≈ ∆n−1, we obtain

∆n ≈ 1

2πmangle

(s ∗n−1sn

|sn−1| · |sn|

)m, (16)

With the aid of (16), an iterative estimate ofΩn is obtained as

∆n = λ d∆n−1 + (1− λ d)

(s ∗n−1sn

|sn−1sn|

)m,

Ωn = λ oΩn−1 + (1− λ o)angle

∆n

2πm, (17)

whereλ d andλ o are positive forgetting factors.

B. For QAM Signals:

The estimator (17) is not useful for frequency-offset estimationin quadrature amplitude modulation due to its multi-modulusconstellation. Assuming that the gain imbalance has been com-pensated and denotingΘ = 2πnΩ, we have

sn = sne−iΘ = sne

i(Θ−Θ) (18)

Denoting Θe := Θ − Θ, we can show that the fourth-orderstatistics ofsn contains the information of unknownΘe

E(s4n,I + s4n,Q

)=

1

4E(s4n,I + s4n,Q − 6s2n,Is

2n,Q

)cos(4Θe)

+constant(19)

Note thatcos(4Θe) is maximum (that is equal to+1) whenΘe =0 and it is minimum (that is equal to−1) when Θe = ±π/4.So the unknown phase is compensated if it is between−π/4 and+π/4. For phase ambiguity due to the multiples of90 degree maybe compensated using differential encoding. Further note that, forQAM signals,E

(s4n,I + s4n,Q − 6s2n,Is

2n,Q

)is a negative quantity

which helps us formulate minimization of the following costforthe recovery of unknown phase:

minΘ

E(s 4n,I + s 4

n,Q

)(20)

Notice thatsn,I = ℜ[sn] = sn,I cos Θ + sn,Q sin Θ and sn,Q =ℑ[sn] = −sn,I sin Θ+ sn,Q cos Θ, these relations help us obtainthe following:

∂

∂ΘEs 4

n,I = +4Es 3n,I sn,Q, (21a)

∂

∂ΘEs 4

n,Q = −4Es 3n,Qsn,I , (21b)

3

These statistics may be computed iteratively and lead to thefollowing gradient-based algorithm:

Gn = λ tGn−1 + (1− λ t) s3n,Qsn,I ,

Hn = λ tHn−1 + (1− λ t) s3n,I sn,Q,

Θn = Θn−1 + µ tρn, (ρn := Gn −Hn), (22)

whereµ t is a positive step-size andλ t is a positive forgetting-factor less that one. Note that the algorithm (22) does not (explic-ity) exploit the fact thatΘ = 2πnΩ . Exploiting this information,we modify the problem (20) as follows:

J := minΘ,Ω

E(s 4n,I + s 4

n,Q

), s.t. Θ = 2πnΩ (23)

The optimization of (23) may be realized as separate mini-mizations with respect toΘ and Ω; however, the resulting twoupdates must also satisfy the constraint in (23). To realizesuch anoptimization, we introduce an auxiliary (or intermediate)variableΣn, minimize the cost w.r.t. it, and obtain acoarse(but gradient-based adaptive) estimate ofΩ. Since the relationΘ = 2πnΩ canequivalently be expressed asΘn = Θn−1+2πΩ, whereΘn is thetrue value ofΘ at timen. With these considerations, we suggestto solve

Θn = Θn−1 + 2πΣn−1 −µ t

4

∂J∂Θn

,

Σn = Σn−1 − µ s

4

∂J∂Σn

,(24)

Once Θ is known, afine estimate ofΩn is Ωn = Θn/(2πn);however, in practice, theΩn is not explicitly required to becomputed as the knowledge ofΘn is equivalently sufficient forthe purpose. Note that∂J /∂Θn = −ρn, where the statisticalerror quantityρn is as specified in (27). The derivative of costJw.r.t. Σn requires attention; note that

∂J∂Σ

=∂Θ

∂Σ

∂J∂Θ

, (25)

The constraint in (23) allows us to expressΘn ≈ Θn−1 +2πΣn−1, which gives

∂Θn∂Σn−1

≈ ∂Θn−1

∂Σn−1+ 2π ≈ ∂Θ0

∂Σn−1+ 2πn ≈ 2πn, (26)

Note that the gradient∂Θ/∂Σ is growing linearly in time whichis analytically correct but its use in the update expressionmaycause divergence. One possible way to handle this situationisto use a diminishing step-size to overcome the linear growthof∂Θ/∂Σ. However, a diminishing step size usually leads to slowconvergence and requires exhaustive experimentation to determinehow rapidly the step-size must decrease in order to preventscenarios in which it (the step-size) becomes too small whentheiterates are far from the required estimate. The other solution isto simply drop this gradient factor as it is always positive andhas no role in determining the direction of the update. We preferto adopt the latter proposal while using a fixed but very smallstep-sizeµ s for Σn to ensure the stability and low jitter.

Θn = Θn−1 + 2πΣn−1 + µ tρn,

Σn = Σn−1 + µ sρn, (27)

Taking thez-transform of (27), we get

Θ(z) = Θ(z)z−1 + 2πΣ(z)z−1 + µ tρ(z),

Σ(z) = Σ(z)z−1 + µ sρ(z),(28)

Combining the two expressions in (28), we obtain

Θ(z) = Θ(z)z−1 + µ tρ(z) +2πµ sρ(z)z

−1

1− z−1(29)

Denotingoρ (z) := ρ(z)z−1/(1 − z−1), we obtain an alternate

form of (28) as follows:

oρn=

oρn−1 + ρn−1,

Θn = Θn−1 + µ tρn + 2πµ s

oρn, (30)

Experimentally, we have found that a suitable value ofµ s is close

to the square ofµ t, i.e., µ s ≈ (µ t)2 .

V. SIMULATION RESULTS

We carry out simulations to evaluate the performance of theproposed estimators. The baseband signals in the desired andimage bands are expressed assn = an +wn andqn = bn + vn,respectively, wherean and bn are transmitted quadraturephase-shift keying (QPSK) symbols, andwn andvn denoteadditive white Gaussian noise. The signal-to-noise ratios(SNRs)of the received signalssn and qn are taken as 30 dB. Theforgetting factors were selected asλ g = λ d = λ o = 0.998and the step-sizeµ = 6× 10−4. At time zero, adaptive/iterativeparameters were initialized asA0 = 1, B0 = 2, ∆0 = 1, γ0 = 0,and Ω0 = 0. The frequency offsetΩ = 1× 10−4, the amplitudemismatchε = 0.8, and the phase mismatchθ = 10 (this givesα = −0.3863 − i0.1563, β = 1.3863 − i0.1563 resulting inγ(1)opt = −0.2877 − i0.0803 andγ(2)

opt = −3.2245 − i0.8999).

Experiment 1: In this experiment, we study convergencebehaviour of update (11) for small and relatively large step-sizes(for QPSK signal).

Refer to Fig. 2(a) for the contour plot of the cost where theglobal minima,γ(1)

opt and γ(2)opt, and the saddle point0.5(γ(1)

opt +

γ(2)opt) are labeled. Next in Fig. 2(a) and (b), we provide traces of

convergence for small and relatively large step sizes, respectively.It can be noticed that for small step-size (i.e.,µ g = 5 × 10−5),when γn is initialized nearγ(2)

opt, it converged toγ(2)opt; however,

for relatively large step-size (i.e.,µ g = 1× 10−4), regardless ofthe initialization,γn is found to be always converging to the rootwith smaller magnitude, i.e.,γ(1)

opt.Further, withµ g = 1 × 10−4, refer to Fig. 3(a)-(d) and Fig.

3(e)-(f) for scatter plots and convergence traces, respectively;both estimators can be noticed to be converging steadily totrue values. Refer to Fig. 3(g) for the traces of empiricallyobtained mean square errorE|sn − sn|2 and squared absolutecorrelation|Esnqn|2. Both indices are decreasing along iterationand attaining a stable floor in steady-state; this means that, as aresult of successful convergence, estimated signalsn is gettingclose to desired signalsn and imageqn is rejected fromsn.Note that 1000 symbol points are used in each scatter plot (for asingle realization) and traces (in Fig. 3) were averaged over 500independent realizations.

Experiment 2:

4

VI. CONCLUSIONS

In this work, an adaptive non-data aided in-phase / quadrature-phase imbalance compensator for heterodyne receiver was de-veloped. Simulation results showed that the proposed adaptivescheme can successfully compensate for frequency-independentimbalances.

+

+

o

o and + indicate saddle point and minima, resp.

ℜ[γ]

ℑ[γ

]

γ(1)optγ

(2)opt

0.5(γ(1)opt + γ

(2)opt)

−3 −2 −1 0−2

−1

0

1

−4 −3 −2 −1 0

−2

0

2

µ = 5x10−5 and Iterations = 30000

ℜ[γ]

ℑ[γ

]

−4 −3 −2 −1 0

−2

0

2

µ = 1x10−4 and Iterations = 8000

ℜ[γ]

ℑ[γ

]

Fig. 2. (a) Contour plot of cost function forε = 0.8 and θ = 10,and (b)-(c) convergence trajectories ofγn for small and relatively largestep-sizes.

REFERENCES

[1] S. Mirabbasi and K. Martin, “Classical and modern receiver architec-tures,” IEEE Commun. Mag., vol. 38, pp. 132–139, Nov. 2000.

[2] M. Valkama, M. Renfors, and V. Koivunen, “Advanced methods forI/Q imbalance compensation in communication receivers,”IEEE Trans.Signal Process., vol. 49, no. 10, pp. 2335–2344, Oct. 2001.

[3] G.-T. Gil, Y.-D. Kim and Y.H. Lee, “Non-data-aided approach toI/Q mismatch compensation in low-IF receivers,”IEEE Trans. SignalProcess., vol.55, no.7, pp.3360–3365, July 2007.

[4] U. Mengali, Synchronization Techniques for Digital Receivers,Springer 1997.

VII. A CQUISITION ABILITY OF DMD SYNCHRONIZER

A. Steady-state mean square deviation

A good measure for the performance of synchronizer is thesteady-state error deviation (or variance). We definemean square

−1 0 1−1

0

1

(a) sn

−2 0 2−2

0

2(b) xn

−1 0 1

−1

0

1

(c) sn

−1 0 1−1

0

1

(d) sn

0 2000 4000 60000

0.1

0.2

0.3

(e) |γn|

0 2000 4000 60000

0.5

1

x 10−4 (f) Ωn

SimulatedTrue value

SimulatedTrue value

0 1000 2000 3000 4000 5000 6000

−30

−20

−10

0

10

Iterations

[dB

](g) MSE and SC traces

|Esnqn|2

E|sn − sn|2

Fig. 3. Scatter plots and convergence traces for QPSK.

−1 0 1−1

0

1

(a) sn

−2 0 2−2

0

2(b) xn

−1 0 1

−1

0

1

(c) sn

−1 0 1−1

0

1

(d) sn

Fig. 4. Scatter plots for 8PSK.

deviationasMSD = σ2ψ = E[ψ2

∞] (rad2) and compute it for thethree algorithms. Letψk = θ−φk be the parameter error at timeinstantk. Using a generalized form of adaptive phase estimator,

5

−4 −2 0 2 4−4

−2

0

2

4(a) sn

−5 0 5

−5

0

5

(b) xn

−5 0 5−5

0

5(c) sn

−4 −2 0 2 4−4

−2

0

2

4(d) sn

Fig. 5. Scatter plots for 16QAM.

0 1000 2000 3000 4000 5000−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

Iterations

Fre

quen

cy−

offs

et e

stim

ates

True Ω = 10−4

Ωn for λ t = 0.996Ωn for λ t = 0.999

Fig. 6. Convergence traces:Ωn = Θn/(2nπ) for 16QAM; Θn isestimated as specified in (22). Smaller values ofλ t can ensure shortersettling time while causing relatively large over/undershoot.

0 2000 4000 6000 8000 10000−50

−40

−30

−20

−10

0

10

Iterations

NM

SE

:E

( Ωn/Ω−

1) 2

Adaptive Frequency Offset Recovery

16QAM : ǫ = 0.8, θ = 10,Ω = 1 × 10−4, SNR = 30 dB,λ t = 0.98, λ g = 0.998, µ g =1×10−4

1-step

One-StepSolutionµ t =2×10−4

..

2-step

Two-StepSolutionµ t = 1.5×10−4

µ s = µ2t

Fig. 7. Frequency-offset recovery: normalized MSE traces for 16QAM.

we can obtain

ψk+1 = θ − φk+1 = ψk − µφEz (31)

whereEz is a nonlinear function ofzk. Squaring and averagingthe update (31), we get

E[ψ2k+1

]= E

[ψ2k

]+ µ2

φE[E2z

]− 2µφE [ψkEz] (32)

whereEz = (z2k,I − z2k,R)zk,Izk,R. Using the following approx-imations for some angleϕ ≪ 1: sin(ϕ) ≈ ϕ, and cos(ϕ) ≈1− 0.5ϕ2, and some simple algebra, it is possible to show that

E[E2z |ψk

]≈ c2ψ

2k + c3,

E [Ez|ψk] ≈c12ψk,

(33)

which make it further possible to rewrite (32) as

E[ψ2k+1

]≈ AE

[ψ2k

]+ B,

where A .= 1− c1µφ + c2µ

2φ,

and B .= c3µ

2φ,

(34)

while parametersc1, c2 andc3 are obtained as follows:

c1.= 4E

[3x2

Ix2R − x4

R

], (35a)

c2.= 2E

[x8R − 28x6

Rx2I + 35x4

Rx4I

], (35b)

c3.= 2E

[x6Rx

2I − x4

Rx4I

], (35c)

Upon successful convergence, limk→∞ E[ψ2k+1] =

limk→∞ E[ψ2k] is true, which yields

MSD = σ2ψ =

B1−A =

c3µφc1 − c2µφ

(rad2) (36)

B. Loop gain for givenMSD

By rearranging Equation (36), we can obtain the value of loop-gain for the requiredMSD and given signal statistics:

µφ =c1 MSD

c3 + c2 MSD(37)

Equation (37) is very useful for simulation study that we don’tneed to make trial-and-error guesses to obtain proper values ofloop-gains for fair comparison.

C. InstantaneousMSD and Phase-Error Trajectories

We denoteE[ψ2k] as the(instantaneous) mean square deviation

at time indexk. From the expression (34), it is easily verified thatwe have,

E[ψ2k

]= MSD+ (ρ−MSD) e−kτσ (38)

whereMSD is obtained from (36), whileρ and τσ are obtainedasρ = (θ − φ0)

2 and τσ = − ln (A). The ρ is the initial valueof MSD andτ determines the convergence speed of the adaptivesynchronizer (in the mean square error sense). Similarly, we canobtain

E [ψk] = (θ − φ0)e−kτµ (39)

whereτµ = − ln(1− µφc1

2

)and it helps us to readily obtain

E [φk] = θ(1− e−kτµ

)+ φ0e

−kτµ (40)

6

D. Convergence time for given target

We define theconvergence timeas the number of iterations,K,that is needed for itsE[ψ2

k] to reach(1+ ε) times its steady-statevalue MSD = E[ψ2

∞], for some givenε > 0. That is, it is thetime k = K at which we have

E[ψ2K ] = (1 + ε)MSD = (1 + ε)

B1−A (41)

We assume that the estimator is initialized withφ0, whichprovidesE[ψ2

0 ] = E[(φ0 − θ)2] = (φ0 − θ)2. Next, we rewriteexpression (34) more conveniently as

E[ψ2k

]−MSD = AE

[ψ2k−1

]+ B −MSD

= A(E[ψ2k−1

]−MSD

)

= Ak(E[ψ2

0

]−MSD

).

(42)

At k = K, solving (41) and (42) together we get

K =1

ln(A)ln

(εB

(1−A)(φ0 − θ)2 −B

)(43)

Using the definitions ofρ andτσ , we can rewrite (43) in a muchsimpler form as follows:

K = ln

(ε ·MSD

ρ−MSD

)− 1

τσ

(iter.) (44)

Substituting the values ofA andB from (34), (??), (??) and (??)into (43) yields the convergence time for Chung-, Mathis- andproposed (??) algorithm.

E. Simulation Evidences of Acquisition Ability

1) Acquisition ability and steady-state behavior:In Fig. 8,we compute the steady-state mean deviationσψ =

√MSD

versus the loop gainµφ (as specified in (36)) for a 32-QAMsource (γ = 18.7), in a noise free environment, and comparethem with those obtained from simulation; the values ofθ weretaken to be±5,±15 and ±30. The simulation results areobtained by averaging 1000 Monte-Carlo (MC) experiments eachof 50,000 iterations. Observe that the analytical and simulationresults are conforming with each other, and the values ofMSDare independent of the target valueθ.

Fig. 8. Analysis vs simulation: values ofσψ for 32-QAM

2) Convergence behavior at sameMSD floor: In Fig.9, we compare the theory and simulation convergence-timefor a 32-QAM source with unknown targetθ = 10

in a noise free environment. The step-sizes (µφ) are se-lected such that the steady-state phase deviation(σψ) is 2

(MSD = 20 log10 (2π/180) ≈ −29dB). Observe that1) the con-vergence behavior of Mathis algorithm is slightly better than thatof Chung algorithm,2) the proposed update has a convergencespeed significantly faster than Chung and Mathis algorithms,and 3) the theoretical values ofK are in good agreement withsimulation values for all three adaptive estimators.

Fig. 9. The convergence behavior of Chung, Mathis and proposedalgorithms, acquiringθ = 5.

3) Steady-state behavior at same convergence speed:In Fig.10, we compare the steady-stateMSD while keeping the sameconvergence-time in a noisy condition. Notice that theMSD floorof the proposed update is significantly lower than those of Chungand Mathis algorithms.

Fig. 10. The steady-state behavior of Chung, Mathis and proposedalgorithms, acquiringθ = 10.

Fig. 11. The convergence behavior of the proposed algorithms, acquiringθ = 20 with different initialization.

4) Initialization strategy:

VIII. T RACKING ABILITY OF THE DMD SYNCHRONIZER

A. Steady-State Tracking Error

Due to the presence of frequency offset of the carrier loop, theresulting phase offset drifts over time. Assume the phase offsetis drifting linearly at the (normalized) rateΩ, i.e., the true phaseoffset at thekth update is given by1 φk = θ+ kΩ. The deviationof the estimated parameterφk from the true phase offsetθ is thusgiven by:

ψk = θ + kΩ− φk (45)

Using (45), we modify (31) to obtain

ψk+1 = ψk + Ω− µφEz (46)

Taking the ensemble average, the steady-state system satisfieslimk→∞ E[ψk+1] = limk→∞ E[ψk]. Using the approximations(??), we obtain thetracking error (TE) as given by

TE.= E[ψ∞] ≈ 2Ω

µφc1(rad) (47)

B. Total MSD and Optimal Loop Gain

In the presence of frequency offset, the total mean squaredeviation (TMSD) can be obtained by combining the distortioncontributed by acquisition and tracking phenomenon, whichgives

TMSD.= MSD+ TE

2 =c3µφ

c1 − c2µφ+

4Ω2

µ2φc

21

(rad2) (48)

Notice that the first and the second term in (48) increases anddecreases monotonically with the loop-gainµφ, respectively. Theoptimal loop gain,µopt

φ is obtained by seeking the minimum of(48). We need to solve∂TMSD/∂µφ = 0, which gives

c31c3µ3 − 8c22Ω

2µ2 + 16c1c2Ω2µ = 8c21Ω

2. (49)

Solving for the positive (real-valued) root of (49), we get thefollowing optimum value ofµφ:

µopt

φ =(4c4)

1

3

3c3c31− Ω2c24

5

3

(3c3c

41 − 4Ω2c32

)

3c3c31c1

3

4

+8Ω2c223c3c31

(50)

1The normalized frequency offsetΩ is defined asΩ = 2πf∆

Rs, where

Rs is the symbol-rate andf∆ is the difference between transmitter andreceiver local-oscillator frequencies.

7

wherec4 is obtained as

c4 = −144Ω4c32c3c41 + 27c23c

81Ω

2 + 128Ω6c62

+ 33

2 Ω2c23c61

√27c3c41 − 32Ω2c32

c3

(51)

Notice that the optimum loop-gain depends only on the statisticsof the QAM signal and the frequency offsetΩ; moreover, itincreases with theΩ.

C. Simulation Evidences for Tracking Performance

1) Validating (48) and (50):Due to the space limitation, wevalidate Expressions (48) and (50) for the proposed algorithmonly. Refer to Figure 12, where theTMSD is obtained bysimulation and compared with our analytical result (48) for32-QAM with three different values ofΩ. Since (48) has been derivedfor an ideal noise-free condition, we have not used additivenoisein the simulation as well. It is pretty clear that Expression(48)coincide with simulation results for all three values ofΩ. Itis found that theTMSD seeks its minimum atµφ equals to2.965 × 10−14, 4.672× 10−14 and7.331× 10−14 for Ω equalsto 5 × 10−5, 1 × 10−4 and 2 × 10−4, respectively – which areexactly the sameµφ as obtained from Expression (50).

Fig. 12. Plots of analytical and simulationTMSD with three differentΩ for proposed algorithm.

To find out the analytical performance of the proposed algo-rithm in relation to Chung and Mathis algorithms, we have plottedtheTMSD versusΩ (that is Expression (48)) in Figure 13 for 32-QAM using optimal loop-gain (50). Notice that, for all values of

Fig. 13. TMSD of Chung, Mathis and proposed algorithms versus(normalized) frequency offset with optimal loop-gain for 32-QAM.

Ω, the TMSD of the proposed algorithm is substantially smallerthan those of others. For example, atΩ = 10−5, the TMSD ofChung and Mathis algorithms are approximately10−3; while theTMSD of the proposed algorithm reaches this level (that is10−3)at Ω ≈ 5 × 10−4. It clearly indicates that the frequency-offsettolerance of the proposed algorithm is about fifty (50) timesmorethan those of Chung or Mathis algorithms for 32-QAM (in anideal noise-free condition).

2) Tracking the frequency offset with phase wrapping:Herewe present an experiment to demonstrate the tracking behaviorof the three algorithms in a more realistic scenario. We havesetthe SNR level of 30 dB, a constant (initial) carrier phase errorθ0 = 0.5 radian, and a considerably high frequency offset,Ω =1 × 10−3. The actual and estimated phase are assumed to bewrapped before and after the synchronizer, respectively, such that|θk|, |φk| ≤ π/2 radian. Figures 14-16 show the carrier trackingfor 32-, 128- and 512-QAM signals, respectively. In these figures,the dashed line and the full line indicate, respectively, the originaland the estimated carrier phase. From these results, it appears thatclearly, our proposed method performs better than the classicaltechniques, which show, in these cases, very large trackingerrors.

Fig. 14. Tracking behavior of Chung, Mathis and proposed algorithmsfor θ0 = 0.5 rad,Ω = 1× 10−3 for 32-QAM.

Fig. 15. Tracking behavior of Chung, Mathis and proposed algorithmsfor θ0 = 0.5 rad,Ω = 1× 10−3 for 128-QAM.

Fig. 16. Tracking behavior of Chung, Mathis and proposed algorithmsfor θ0 = 0.5 rad,Ω = 1× 10−3 for 512-QAM.

3) Tracking a random slow-varying phase:Here we presentan experiment to demonstrate the tracking behavior of the threealgorithms in a (random) slow-varying phase environment for 32-QAM. To generate a slow-varying phase, we used the followingpiece of MATLAB code:

function theta = random_phase(N)B = fir1(1e3,1e-4);Y = filter(B,1,randn(1,N)); Y = Y-mean(Y);theta = Y/max(abs(Y))*pi/6;

For a fair comparison, we first obtained the optimum value ofloop-gain (µφ) for the three algorithms. It was done by calculatingthe TMSD for several different values ofµφ; the value ofµφwhich resulted in the lowestTMSD was selected as an optimum.Refer to Figure 17, whereTMSD has been obtained for Chung-,Mathis- and proposed algorithm. TheTMSD curves have beenobtained by averaging 1000 independent phase initialization foreach value ofµφ, while 30 different values ofµφ have been usedfor each algorithm. To further assist the fairness, in each indepen-dent trial, the same value of phase generated by the MATLABfunction random_phase.m is used for all algorithms. Figure

Fig. 17. The tracking behavior of Chung, Mathis and proposedalgo-rithms, acquiring a random slow-varying phase|θk| ≤ π/6 at SNR= 35dB.

17 also depicts the random (reference) phaseθk, estimated phaseφk, and the phase estimation errorψk = θk − φk. Notice thatthe value ofψk for the proposed algorithm is considerably smallas compared to the others. This experiment clearly indicates thatthe proposed algorithm has a good tracking capability and itcanoutperform other existing adaptive synchronizers.