308 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 16, NO. 1, JANUARY 2004
The Effect of Interferometer Phase Error on
Direct-Detection DPSK and DQPSK Signals
Keang-Po Ho, Member, IEEE
Abstract—Based on the well-known Marcum’s function,
closed-form formulae are derived to evaluate the error probability of direct-detected differential phase-shift keying (DPSK) and differential quadrature phase-shift keying (DQPSK) signals with interferometer phase error. For a signal-to-noise ratio penalty less than 1 dB, the phase error must be less than 16 and 6 for DPSK and DQPSK signals, respectively.
Index Terms—Differential phase-shift keying (DPSK), differ-ential quadrature phase-shift keying (DQPSK), direct detection, phase error.
I. INTRODUCTION
D
IFFERENTIAL phase-shift keying (DPSK) [1]–[5] and differential quadrature phase-shift keying (DQPSK) [6]–[9] modulation have received renewed attention recently for long-haul transmission or spectrally efficiency systems. When DPSK and DQPSK signals are directly detected using a balanced receiver following an interferometer, one of the major receiver imperfections is the phase error at the asymmetric Mach–Zehnder interferometer [10]–[14]. Although the phase error can be eliminated by active delay-line stabilization [10], the degradation due to phase error needs to be calculated accu-rately, as an application, to specify the accuracy requirement for the stabilization algorithm.Previous works evalulate the sensitivity penalty due to phase error by experiment [12], simulation [11], [14], or semianalyt-ical technique [13]. As shown in the Appendix , for a matched filter-based receiver, the formula from [15] can be used to cal-culate the symbol or bit-error rate (BER) with phase error. How-ever, the series summation from [15] has large terms oscillating between positive and negative values, giving some numerical difficulties. The closed-form expression of [15] is not used in [11], [12], [14].
In this letter, analytical expressions based on the well-known Marcum’s function [16, pp. 43–44] are used to evaluate the error probability for DPSK and DQPSK signals with phase error. Originally for noncoherent detection of correlated binary signals, the formulae from [16, Sec. 5.4.4] are adapted for DPSK and DQPSK signals with phase error. The correlation coefficient, an essential parameter, as a function of phase error is given for the first time in this letter for DPSK or DQPSK signals with phase error.
Manuscript received April 25, 2003; revised July 28, 2003. This work was supported in part by the National Science Council of R.O.C. under Grant NSC-92-2218-E-002-034.
The author is with the Institute of Communication Engineering and Depart-ment of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: kpho@cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/LPT.2003.819359
II. DPSK SIGNALS
If the path difference in the asymmetric interferometer does not match to the frequency of the signal, the phase error is equal to , where is the frequency mismatch and is the symbol period. At the output of the balanced receiver, ig-noring the constant factor of coupler loss, photodetector respon-sivity, and receiver gain, the signal is
(1) As the signal of (1) is the same as that of the decision vari-able for envelope detection of correlated binary signals [16, Sec. 5.4.4], the equivalent correlation coefficient is [16, eq. (5.4-54)], denoted as
(2) Using [16, eq. (5.4-54)], the BER is
(3) where is the well-known first-order Marcum’s func-tion, and is the signal-to-noise ratio (SNR). The error proba-bility of (3) assumes a matched filter [16, Sec. 5.1.2], the typical cases of [13], [14]. Within the matched filter, only the amplifier noise in the same polarization as the signal is considered, the typical cases of [17]–[19]. The results are applicable to both re-turn-to-zero or nonrere-turn-to-zero line codes.
The BER of (3) used an SNR twice that in [16, eq. (5.4-54)] as the correlated binary signal is , uses two time slots, and doubles the energy per symbol. While difficult to prove analytically, from numerical results, the BER of (3) and (5) from the Appendix are the same.
III. DQPSK SIGNALS
DQPSK signals have four constellation points, corresponding to four correlated signals with noncoherent envelope detection. Without phase error, the BER with Gray code is that of (3) with for a correlated angle of [16, eq. (5.2-71)] .
Without going into details, with phase error, the BER is
1041-1135/04$20.00 © 2004 IEEE
HO: EFFECT OF INTERFEROMETER PHASE ERROR ON DIRECT-DETECTION DPSK AND DQPSK SIGNALS 309
Fig. 1. BER as a function of SNR for (a) DPSK and (b) DQPSK signals with interferometer phase error.
(4) The BER of (4) is numerically the same as (6) from the Ap-pendix . From the two terms of (4), due to the phase error, the two adjacent points of the signal constellation of the DQPSK signal have different correlation coefficients. The correlated an-gles to the adjacent points are increased (corresponding to and ) or decreased (corresponding to and ) by the phase error of from .
IV. NUMERICALRESULTS
From (3) and (4), the BER is independent of the sign of the phase error. In later parts of this letter, only the results of positive phase error are shown.
Fig. 1 shows the BER as a function of SNR . Fig. 1(a) is plotted for a DPSK signal with phase errors of 10 , 20 , 30 , and 40 . Fig. 1(a) also shows the error probability with no phase error of [16, eq. (5.2-69)]. Fig. 1(b) is plotted for a DQPSK signal with phase errors of 5 , 10 , 15 , and 20 . In Fig. 1(b), the BER is plotted versus the SNR instead of SNR per bit in [16, Fig. 5.2-13]. Fig. 1(b) also shows the error probability with no phase error [16, eq. (5.2-71)].
Fig. 2 shows the SNR penalty as a function of phase error for both DPSK and DQPSK signals. The SNR penalty is calculated for a BER of , corresponding to a required SNR of 13 and 18 dB for DPSK and DQPSK signals, respectively.
The curve of SNR penalty of a DPSK signal in Fig. 2 has insignificant difference with the corresponding curves in [12, Fig. 3], [13, Fig. 2], and [14, Fig. 5] (required the adjusting of axis). The phase error for an SNR penalty of 1 dB is about 16 (or 4.5% of 360 ) for a DPSK signal. In all of [12]–[14], the phase error of 1-dB SNR penalty is about 4% to 5% from simulation or analysis. With narrow bandwidth and from [13], the SNR penalty due to phase error is more or less independent of the optical filter before the interferometer. For SNR penalty less than 2 dB and from [14], the SNR penalty due to phase error is more or less independent of the electrical filtering after the balanced receiver. Just as with the theoretical results from
0 10 20 30 40 0 0.5 1 1.5 2 2.5 3 3.5 4
Phase Error (deg)
SNR Penalty (dB)
DPSK DQPSK
Fig. 2. SNR penalty as a function of interferometer phase error for DPSK and DQPSK signals.
[12]–[14], the SNR penalty of Fig. 2 is smaller than that from measurement [12], [14]. As explained in [14], this discrepancy is probably due to the nonideal signal source used in the exper-iment. The 10% (or 36 ) mismatch of [11] gives a penalty of about 3.5 dB.
For the same SNR penalty, the DQPSK signal is about 2.7 times more sensitive to the phase error than is the DPSK signal, more or less the same ratio as the experimental and simulated results in [12]. The phase error for an SNR penalty of 1 dB is about 6 (or 1.7% of 360 ) for a DQPSK signal, the same as a ratio of mismatched frequency to the symbol rate of [12, Fig. 3] by simulation. The measurement of [12] shows a larger penalty than that of Fig. 2 due to nonideal signal source [14].
V. CONCLUSION
Based on the well-known Marcum’s function, two ana-lytical formulae are derived to evaluate the BER of DPSK and DQPSK signals with interferometer phase error at the receiver. The error probability is numerically verified as the same as that from the series summation of [15] and the Appendix . For an SNR penalty less than 1 dB, the phase error of the interferometer must be less than 16 and 6 for DPSK and DQPSK signals, re-spectively, the same as the simulation or analytical results from [12]–[14].
APPENDIX
From [15], the error probability for a DPSK signal with phase error is
(5) where is the th order modified Bessel function of the first kind.
From the factor of , the terms of (5) oscillate between positive and negative values. Although the summation of (5) converges, the calculation is numerically challenging for small error probability. Note that the multiplication factor of the sum-mation is a small value of , and the summation has a value of about for small error probability. For large SNR , the summation has very large terms although the error
310 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 16, NO. 1, JANUARY 2004
probability is small. The error probability is the difference be-tween 1/2 and a value a little bit smaller than 1/2. As a compar-ison, the Marcum’s function is the summation of all positive small terms [16, pp. 43–44].
For DQPSK signals, following the derivation of [15], we get the BER as
(6) Similar to (5), the summation in (6) is difficult to calculate for small BER. Although the BER of (6) has never been derived before, it comes almost directly from [15].
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