Carrier Phase Recovery For Constant ModulusQAM Signals Under Flat Fading

Kunio Takaya and David E. DoddsDepartment of Electrical Engineering, University of Saskatchewan

57 Campus Drive, Saskatoon, Sask. S7N 5A9 , CanadaPhone: (306) 966-5392, Fax: (306) 966-5407

Telecommunications Research Laboratories (TRLabs)108-15 Innovation Blvd., Saskatoon, Sask. S7N 2X8 , Canada

Phone: (306) 668-9300, Fax: (306), 668-1944email: kunio_takaya@engr.usask.ca

email: david_e_dodds@engr.usask.ca

Abstract

When π/4 offset DQPSK signals are subjected to timevarying multi-path reception, both amplitude and phaseof the signal experience severe distortion. This effectis called flat (in frequency) fading, which causes sud-den signal weakening often observed in a moving vehicle.AGC (Automatic Gain Control) can compensate ampli-tude fluctuation since the modulus of π/4 offset DQPSKis constant. On the other hand, phase fluctuation mustbe corrected based on the knowledge of amplitude fluctu-ation. It is known that instantaneous phase differentialcan be estimated from the amplitude, provided if signal isof minimum (or maximum) phase. This paper discusseshow to estimate the phase when flat fading obscures thephase of constant modulus QAM carrier signal by usingthe Hilbert transform and the amplitude information.

I. Introduction

The emerging PCS standard IS-54 has adopted π/4 offsetDQPSK digital modulation scheme for wireless cellularsystems. The π/4 offset DQPSK is phase modulation of aconstant modulus. Mobile RF channels experience shortterm fading due to the standing wave established by thesuperposition of waves travelling over multiple paths ofdifferent lengths. The nulls of the standing wave occurevery half wavelength of the RF wave. It has been re-ported that a vehicle travelling a velocity of 100km/hwill encounter a null at every 6.3ms if RF frequency isat 850MHz. [2] The π/4 offset DQPSK operating at asub-carrier frequency is affected by this flat fading of RFpropagation channel. Flat fading implies that all fre-quencies receives the same amount of fading at any time.A constant modulus QAM signal of the Nth order canbe represented by

s0(t) = A0 ej[ωt+ 2π

N n(t)+θ]. (1)

s0(t) is a complex wave to be usually observed in the I-Qphase plane. Where, n(t) is one of the integer numbern = 0, 1, 2, or N − 1 representing a symbol of N possiblesymbols for the constant modulus QAM. θ is an offsetangle. In the case of π/4 offset DQPSK, N = 4 andθ = π/4. When this signal is transmitted through a flat

fading channel, it receives amplitude and phase distor-tion. The received and demodulated signal in the baseband is, therefore,

s(t) = A(t) ejφ(t). (2)

Where, A(t) is amplitude fluctuation and φ(t) includesthe phase shift by encoding and phase fluctuation. In theproblem of estimating the carrier phase, the phase fluc-tuation due to multi-path propagation and the Dopplereffect needs to be estimated. If the carrier phase fluc-tuation is estimated entirely from the magnitude, andonly the special case of the constant modulus QAM isconsidered, the phase shift introduced by QAM modu-lation can be ignored, since the QAM modulation doesnot affect the amplitude A(t). The phase fluctuation,now represented by φ(t) severely affects the detection ofthe modulated phase of 2π

N n(t). Automatic gain control(AGC) can keep A(t) reasonably constant so that the keyto the problem is to estimate φ(t). Practical RF chan-nels are not flat over the frequency range to fs/2 (half asampling frequency) or a raised cosine frequency responsewhich warrants the channel to be free from inter-symbolinterference (ISI). However, it is assumed here that thechannel itself is ideal.

In order to separate the phase, we now consider ln s(t)instead of s(t) and differentiate it with respect to time t.

d

dtln s(t) =

1

A(t)

d

dtA(t) + j

d

dtφ(t). (3)

Thus, the problem of estimating the phase φ(t) now be-comes a problem of finding the imaginary part j d

dtφ(t)

from the real part 1A(t)

ddtA(t). There is a feasible so-

lution to the problem of finding the real part from theimaginary part or vice versa, presented in the articles[1][5]. A similar work applied to the specific case of π/4offset DQPSK problem estimating the phase differentialfrom the amplitude is reported in the article [2].

dφ(t)

dt= H

[

1

A(t)

dA(t)

dt

]

, (4)

where H is the Hilbert transform. A more detailed inter-pretation for this solution is given in the next section. It

is known that this solution requires signals to satisfy theminimum or maximum phase condition.

II. Recovery of Real or Imaginary Part of aComplex Signal

Suppose we have a complex signal s(n) = x(n) + j y(n),for which s(n) is defined for the range of n = 0, 1, · · · , N−1. Define its mirror image sequence xm and ym(n) suchthat

xm(n) = x(N − n− 1), ym(n) = y(N − n− 1). (5)

We first augment the signal x+jy with xm(n) and ym(n)to double the length to make it twice as long.

(x) + j(y) =⇒ s1 = (x, xm) + j(y,−ym) (6)

The resulting vector (x, xm) is apparently symmetric and(y,−ym) is asymmetric. Define the operator ASYMthat transforms a symmetric vector to an asymmetricvector to make the subsequent discussion simple. Forexample, ASYM(x, xm) = (x,−xm). Also, DFT andIDFT are the discrete Fourier transform and the in-verse discrete Fourier transform operator, respectively.Consider the following operations:

IDFT {ASYM{DFT (x, xm)}}= (0, 0) + j(x, xm) (7)

IDFT {ASYM{DFT (y, ym)}}= (y,−ym) + j(0, 0) (8)

Since (x, xm) is symmetric, its DFT results in a real sym-metric vector. Converting the symmetric DFT into anasymmetric vector, the last operator of IDFT producesan symmetric imaginary vector. Similarly, the DFT ap-plied to the asymmetric (y,−ym) results in an imaginaryasymmetric vector. Forcing this to become a symmetricvector by the asymmetric operator, the last operator ofIDFT produces an real asymmetric vector. Since the re-sult of Eq. 7 is imaginary and the result of Eq. 8 is real,these are now combined to make a new vector,

s2 = (y,−ym) + j(x, xm) (9)

We now seek for a condition to reconstruct a zero paddedoriginal signal s having the size of 2N from the two sig-nals s1 and s2. That is,

(x, 0) + j(y, 0)

=1

2[ s1 + s2 ]

=1

2[(x, xm) + j(y,−ym)] +

1

2[(y,−ym) + j(x, xm)]

=1

2[(x+ y, xm − ym) + j(y + x,−ym + xm))] (10)

Therefore, we find the necessary conditions, xm = ymand xm = ym. This implies that x = y and x = y,because xm and ym are the mirror images of x andy. Recalling the fact that the three stage operations ofIDFT {ASYM{DFT (., .)}} followed by swapping thereal part and the imaginary part is equivalent to the dis-crete Hilbert transform H{.}, we have x = H{x} and

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.5

1

1.5

2

2.5

Amplitude Envelope

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−70

−60

−50

−40

−30

−20

−10

0

Phase Differential

Figure 1: Amplitude and phase differential changes overa course of time. (Phase in logarithmic scale)

y = H{y}. Thus, we can conclude with the final result,

x = y = H{y} (11)y = x = H{x}. (12)

The real part and the imaginary part of the signals(n) = x(n) + jy(n) are related by the Hilbert trans-form. However, so called minimum phase condition mustbe met for the above relationship to hold. If a zero ofthe signal sequence s(n), n = 0, 1, · · · , N − 1 is locatedat Rejθ, the contribution of its reciprocal zero R−1e−jθ

is equal in magnitude, but different in phase. There-fore, it is known that phase cannot be uniquely deter-mined from magnitude, unless the minimum phase con-dition is imposed to the signal sequence s(n). Findingthe phase from the magnitude in the polar coordinates,or the imaginary part from the real part in the cartesiancoordinates is possible only when all zeros of the signalsequence is confined within the unit circle, i.e. minimumphase. This condition causes problems when the methodis implemented in a practical problem.

III. A Model of Flat Fading Channels

A model of the constant modulus signal encoded withN code words, received in a moving vehicle, affected bymulti-path propagation, can be represented by the fol-lowing equation.

s(t) =∑

i

ci ej{ωt+ 2π

N n(t)+αi

√βit2+γi+ζi} (13)

The summation over the subscript i means to accountall paths of propagation. ci represents the gain in pathi. αi, βi, γi and ζi are the constants that depend onthe the locations of the base station, reflection surfacesand the location and velocity of the vehicle. n(t) is aninteger representing a symbol being transmitted. A casein which reflections come from three near-by surfaces isshown in Fig. 1. The amplitude shows dramatic changesthat accompany changes in the phase differential (firsttime derivative of the phase). This information of phasedifferential can only be calculated by the signal modelgiven in Eq. 13 but not available for practical receivers.

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−2

0

2

CW Real

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−2

0

2

CW Imag

0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

2

3

Magnitude

Figure 2: A QPSK waveform affected by flat fading

Large spikes can be observed in the graph of phase defer-ential when the amplitude encounters the maxima or theminima. It is also apparent that the time derivative ofthe amplitude is reflected to the phase differential, sup-porting the basic relationship given by Eq. 4. Receivedconstant modulus QAM signals affected by multi-pathpropagation is compensated in terms of amplitude, per-haps by means of automatic gain control (AGC). How-ever, the phase fluctuation due to the multi-path effectcan only be compensated with the help of Eq. 4. Further-more, the calculated phase differential must be integratedto obtain the phase.

IV. Simulation Results

Using a QPSK signal generated according to the signalmodel represented by Eq. 13, the performance of thephase estimation method given by Eq. 4 was studied.The real and the imaginary part of the complex waves(t) and its amplitude are shown in Fig. 2. In Fig. 3,The phase fluctuation without the phase modulation isshown in the top panel. The second panel shows thephase part of the modulated signal s(t). The third panelfrom the top shows the phase after removing the phasefluctuation shown in the top panel from the second panel.The fourth panel is the same phase as the third panel,but plotted in angle measured within the range of −π to+π, which corresponds to the angle of the randomly gen-erated 4 possible QPSK code words. The bottom panelis the amplitude. The graphs in Fig. 3 demonstrates thateven if the QPSK signal is severely affected by flat fad-ing, the demodulated phase signal properly representsthe transmitted code words as long as we keep track ofthe phase fluctuation, exactly.

Following Eq. 4, the magnitude differential dA(t)dt was

calculated as shown in the top panel of Fig. 4. Dividingthis by the magnitude shown in the second panel, thethird panel showing the phase differential divided by themagnitude was obtained. Then, the Hilbert transformwas applied to the graph in the third panel, by using thealgorithm discussed in Section II. The graph in the fourthpanel is the phase differential signal necessary to calcu-late the accumulated total phase change. By integratingthe phase differential, the total phase accumulated fromthe onset of the signal was obtained as shown in the bot-

200 400 600 800 1000 1200 1400 1600 1800 20000

5

10

phas

e un

mod

.

200 400 600 800 1000 1200 1400 1600 1800 2000

−5

0

5

phas

e m

od.

200 400 600 800 1000 1200 1400 1600 1800 2000

−10

−5

0

5

diff.

200 400 600 800 1000 1200 1400 1600 1800 2000−2

0

2

sign

al

200 400 600 800 1000 1200 1400 1600 1800 2000

123

mag

nitu

de

Figure 3: Demodulation process of the QPSK signal un-der flat fading using the known phase.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.04−0.02

00.020.04

Mag

diff

w

0 200 400 600 800 1000 1200 1400 1600 1800 2000

123

Mag

w

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−1−0.5

00.5

Mag

Diff

/Mag

0 200 400 600 800 1000 1200 1400 1600 1800 2000−1.5

−1−0.5

0

Hilb

ert T

f

0 200 400 600 800 1000 1200 1400 1600 1800 20000

5

10

est P

hase

Figure 4: Estimation of the phase under flat fading bymeans of the Hilbert transform.

tom panel of Fig. 4. The known phase fluctuation andthat of the calculated are overlaid in this panel.

The graph that starts out higher, then crosses over tocome below the other line is the calculated total phase, inthe bottom panel of Fig. 4. The other line which showsa jump around 1900 in the horizontal scale is the knownphase. Comparing these two graphs, one can notice asimilarity between the two graphs. However, they do notquite agree each other. In this simulation, the minimumphase condition was not checked. The minimum phasecondition is, generally speaking, satisfied for a long signalsequence such as this case. So, it is not difficult obtain areasonable similarity of the calculated phase to the truephase. However, when the signal is segmented in muchsmaller signal sequences, those sequences that occur atnulls or near zero in the magnitude tends to be non-minimum phase. The reason for this disagreement is dueto those hidden non-minimum phase signal segments.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 5: The constellation of QPSK code sequence.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 6: The constellation of demodulated QPSK codesequence after magnitude and phase are compensated.

The constellation of the QPSK code sequence used inthe signal is shown in Fig. 5. The constellation obtainedafter demodulation is shown in Fig. 6, where both of themagnitude and phase are compensated. Because of thediscrepancy observed in Fig. 4, the stars in Fig. 6 aredistributed to form four clusters.

V. Conclusion

The method to recover carrier phase for constant modu-lus QAM signals affected by flat fading was discussed inthis paper. The basic principle given by Eqns. 3 and 4is a feasible approach to the problem. Since this prob-lem is of the non-linear channel equalization problem,other methods using higher order statistics such as thirdor fifth order cummulants are applicable. However, theproposed method is far simpler and requires much lesscomputations. The simulation shown demonstrated thatthe method works reasonably well even in the severe fad-ing due to the Doppler effect of a moving vehicle. How-ever, the weakness in recovering the imaginary part fromthe real part by the Hilbert transform is the minimumphase condition. Once the minimum phase condition isviolated by a given signal sequence, the result is incorrect.This happens more likely when the magnitude becomesnear zero. However, calculated is fortunately the phasedifferential, which is to be integrated to yield the phase.Ignoring the periods where the magnitude suddenly dropsdown to near zero merely introduces an phase offset fromthat time and afterward. It is quite possible to estimatethis offset from the rotational shift of clusters in the con-stellation. The study of adaptive carrier phase recoverybased on this notion is currently under way. This paperreported a comprehensive simulation study of the basicprinciple.

References

[1] Kunio Takaya, ”Recovery of Phase Using PowerSpectrum and Bispectrum,” Journal of Circuits,Systems and Computers, Vol. 7, No. 4 pp. 261-272,World Scientific Publishing Company, 1997.

[2] M. Fattouche, and H. Zaghloul, ”Estimation ofphase differential of signals transmitted over fad-ing channels,” Electronic Letters, vol. 27, no.20, pp.1823- 1824, September 26, 1991.

[3] David E. Dodds, Kunio Takaya and Qingyi Zhang,”Frame Synchronization for Pilot Symbol AssistedModulation”, 1999 IEEE Canadian Conference onElectrical and Computer Engineering, Edmonton,Alberta, May 9-12, 1999.

[4] Kunio Takaya and David E. Dodds, ”BispectrumPhase Estimation For π/4 DQPSK Signals UnderFlat Fading”, ISPACS99 1999 IEEE InternationalSymposium on Intelligent Signal Processing Pro-ceedings, pp. 419-422, Phuket, Thailand, December8-10, 1999.

[5] Herbert B. Voelcker, ”Toward a Unified Theory ofModulation, Part-I: Phase-Envelope Relationship”,IEEE Proceedings 54-3, pp. 340-353, 1966.