New coherent laser communication detection scheme based on channel-switching method Fuchuan Liu, Jianfeng Sun,* Xiaoping Ma, Peipei Hou, Guangyu Cai, Zhiwei Sun, Zhiyong Lu, and Liren Liu Key Laboratory of Space Laser Communication and Detection Technology, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China *Corresponding author: [email protected] Received 12 December 2014; revised 1 February 2015; accepted 14 February 2015; posted 17 February 2015 (Doc. ID 229409); published 25 March 2015
A new coherent laser communication detection scheme based on the channel-switching method is proposed. The detection front end of this scheme comprises a 90° optical hybrid and two balanced photodetectors which outputs the in-phase (I) channel and quadrature-phase (Q) channel signal current, respectively. With this method, the ultrahigh speed analog/digital transform of the signal of the I or Q channel is not required. The phase error between the signal and local lasers is obtained by simple analog circuit. Using the phase error signal, the signals of the I∕Q channel are switched alternately. The principle of this detection scheme is presented. Moreover, the comparison of the sensitivity of this scheme with that of homodyne detection with an optical phase-locked loop is discussed. An experimental setup was constructed to verify the proposed detection scheme. The offline processing procedure and results are presented. This scheme could be realized through simple structure and has potential applications in cost-effective high-speed laser communication. © 2015 Optical Society of America OCIS codes: (060.1660) Coherent communications; (060.5060) Phase modulation; (060.2920) Homodyning. http://dx.doi.org/10.1364/AO.54.002738
1. Introduction
Compared with intensity modulation and direct detection scheme, phase modulation and coherent detection with a local laser attracts an increasing amount of attention because of its improved sensitivity and ability to achieve a high data rate. The binary phase shift keying (BPSK) modulation format with a relative simple modulator is still widely used in coherent laser communication. In particular, it has potential applications in cost-effective, light-weighting situation in either free-space or optical fiber communication. BPSK modulation and homodyne detection with an optical phase-locked loop (OPLL) is the communication scheme with highest sensitivity [1]. However, a very narrow linewidth laser and short loop 1559-128X/15/102738-09$15.00/0 © 2015 Optical Society of America 2738
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delay are required to achieve a stable OPLL [2–5]. Given that the loop delay is large, PLL bandwidth is usually limited below 1 MHz, and system stability is difficult to maintain when semiconductor lasers have large phase noise and frequency drift [6]. Digital coherent receivers are used in coherent laser communication [7,8]. These coherent receivers comprised a 90° optical hybrid, two balanced photodetectors, two high-speed analog-digital converters (ADCs), and a parallel digital signal processor (DSP), and the phase estimation method is utilized as the demodulation algorithm [9,10]. However, the overall receiver exhibits very high cost and the requirement of the sufficient speed for ADC and DSP in high data rate is a disadvantage. In the present paper, a new coherent laser communication detection scheme based on the channelswitching method, where neither the OPLL nor high-speed ADC and DSP are needed, is proposed
for the application of the BPSK modulation format. The detection front end also consists of a 90° optical hybrid, two balanced photodetectors which output inphase (I) channel signal current and quadraturephase (Q) channel signal current, respectively [11,12]. The phase error between the signal and local lasers is obtained by simple analog circuit. Using the phase error signal, the signals of the I∕Q channel are switched alternately. This technique is the basic idea of the channel-switching method. The principle of the proposed detection scheme is presented and the sensitivity of this detection scheme, at the shot noise limit, is discussed. Compared with the sensitivity of homodyne detection with OPLL, the sensitivity of the proposed scheme indicates that a 3 dB deterioration in sensitivity results from the fact that the incoming signal beam is split into two separate beams with equal optical power. However, no more than 6 dB deterioration exists in sensitivity, because the signal optical power in homodyne detection is no less than a quarter of that with the channel-switching method when the signal-to-noise ratio (SNR) is equal. An experimental setup was constructed to verify this detection scheme. The offline processing procedure and the bit error rate (BER) test results are also presented. This method can be realized through basic electric modules, such as mixer, comparator, flip-flop, and multiplexer. Thus, the proposed scheme has potential applications in cost-effective high-speed laser communication. Section 2 presents the basic principle of the homodyne detection scheme and the digital coherent receiver and its limitations. Section 3 presents the detection scheme based on the channel-switching method in detail. The laboratory experimental demonstrations are shown in Section 4. The conclusion is presented in Section 5, and the Appendix A is given afterwards. 2. Basic Principle of BPSK Modulation and Coherent Detection Schemes
The complex field of the incoming signal beam can be represented by ES
p PS expfjωS t Δφt φS tg;
(1)
where PS denotes the signal power, ωS is the signal angular frequency, and φS t is the random phase noise, and the phase modulation Δφt mt × π;
p PLO expfjωLO t φLO tg;
A. Homodyne Detection Scheme with OPLL
For ideal homodyne detection, the phase locking of the local laser is satisfied to that of the signal, that is ωs − ωLO t φs t − φLO t 0. The modulation sequence can be recovered from the alternate current after blocking the direct current. The alternate current is p iac t 2R PS PLO cosΔφt;
(3)
(4)
where R is the responsivity of the photodiode. If I 1 represents bit “1”, I 0 represents bit “0”, then p I 1 −I 0 2R Ps PLO :
(5)
Modulation data can be recovered by simply setting the threshold. Moreover, two main kinds of Gaussian random noises exist during the detection period: thermal noise and shot noise. For the shot noise limited system, which is the ideal situation for the coherent detection scheme, the mean square noise currents of bits “1” and “0” are equal to the value of shot noise. When the dark current of the photodiode is ignored, and PLO ≫ PS is satisfied, the mean square noise current of the shot noise can be represented by [13] σ 2s 2qRPLO Δf ;
(6)
where q is the electron charge and Δf is the bandwidth of the photodetector. The SNRs of bits “1” and “0” are also equal, represented by SNR
I 21 2RPS : qΔf σ 2s
(7)
The ideal BER [14] for PSK coherent detection is represented as a function of SNR by 1 BER erfc 2
r SNR ; 2
(8)
where erfc is complementary error function. Moreover, according to Eqs. (7) and (8), the ideal sensitivity of the homodyne detection scheme could be represented by
(2)
where mt is the modulation digital 0/1 sequence at a specific data rate. This 0∕π phase modulation on signal beam is called BPSK. A local beam must be mixed with the signal beam to recover the initial modulation mt sequence. The local beam can be represented by ELO
where PLO is the local power, ωLO is the local angular frequency, and φLO t is the random phase noise.
PS
erfc−1 2 BER2 qΔf : R
(9)
The real sensitivity is not as good as that described in Eq. (9), because to achieve phase locking a signal power penalty is caused by the loop feedback in the OPLL. For example, the structure of a costas PLL is depicted in Fig. 1 and a signal power penalty is caused by the existence of the Q channel. However the data are recovered from the I channel only, most 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
2739
p Ps PLO cosfθerr Δφtg; p iQ t R Ps PLO sinfθerr Δφtg; iI t R
Fig. 1. Basic structure of an optical costas phase-locked loop.
of the signal power is transmitted into the I branch. For instance, the optical signal power is divided 7:1 to the I branch and Q branch in [15]. Ideally, the signal power of the Q-channel branch could be approximately omitted. In addition, a very narrow linewidth laser and short loop delay are required to obtain an OPLL with absolute stability. PLL bandwidth is usually limited below 1 MHz because of large loop delay, and system stability is difficult to maintain when semiconductor lasers have large phase noise and frequency drift. This technical difficulty inherent in OPLL has not been solved perfectly. Some reports used photonic integrated circuit (PIC) and electronic integrated circuit (EIC) to decrease the loop delay, and greater than 1 GHz loop bandwidth could be achieved [16]. This method maybe a promising solution for OPLL. However it exhibits very complicated structure, and the requirements for PIC and EIC are still strict. B.
Digital Coherent Receiver
Digital coherent receivers could be used in coherent laser communication. These coherent receivers, the basic structure of which is shown in Fig. 2, are comprised of a 90° optical hybrid, two balanced photodetectors, two high-speed ADCs and a parallel DSP, and the phase estimation method is utilized as demodulation algorithm. In view of BPSK as an example, two high-speed ADCs simultaneously sample the I and Q channel signal currents detected by two photodetectors. The signal currents of I and Q are represented by
(10a) (10b)
where Δφt is the phase modulation ( 0; π), and θerr ωs − ωLO t φs t − φLO t is the phase error. The sampled signal of I and Q becomes the real part and imaginary part of Yt, that is, Yt iI t jiQ t. The square cancels the phase modulation Δφt, given that Y 2 t ∝ expj2θerr . Phase error θerr is estimated by using N samples, and the complex amplitudes Y 2 t of N samples are summed. Thus, the phase estimate θe is represented as [10] " # N X 1 2 Y k : θe arg 2 k1
(11)
This phase estimate for BPSK is the same for QPSK in [10], with the product factor in Eq. (11) equal to 1/2 instead of 1/4. A modulation symbol ( 0; 1) is obtained by discriminating the phase modulation Δφt fargYk − θe g. However, the data should be differentially precoded to solve phase ambiguity. Moreover, a frequency control feedback loop is needed to maintain the intermediate frequency [7]. For a long-haul optical fiber communication system, the ADC-DSP-based digital coherent receiver is powerful and the impairments stemming from narrowband filtering, chromatic dispersion could be compensated through DSP algorithms. However, in a short distance optical fiber communication system or free-space laser communication system, where dispersion effects are not severe, the application of DSP with high cost may be wasteful. Moreover, with the data rate increasing, the speed of ADC and DSP is a disadvantage. Although advanced modulation formats can be an alternative, the structures of the modulator for these formats are sophisticated. 3. Coherent Detection Scheme Based on Channel-Switching Method
This coherent laser communication detection scheme based on the channel-switching method is applied for the BPSK modulation format. In this scheme, neither the OPLL nor the high-speed ADC and DSP are needed. A. Basic Principle of This Coherent Detection Scheme
The detection front end of this detection scheme is the same as that of the digital coherent receivers. This detection front end comprises a six-port 90° optical hybrid and two balanced photodetectors. Both signal and local optical power are divided 1:1 to the I branch and Q branch. If the signal beam and local beam are represented by the same equations as Eq. (1) and Eq. (3), then iI and iQ can be represented by Fig. 2. Basic structure of a digital coherent receiver. 2740
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iI t R
p Ps PLO cos θerr cosΔφt;
(12)
iQ t R
p Ps PLO sin θerr cosΔφt;
(13)
where θerr ωs − ωLO t φs t − φLO t is the phase error, and Δφt mt × π is the phase modulation. To simplify the analysis, the θerr can also be assumed to be θerr 2nπ θ;
(14)
where n is an integer and θ ∈ −π∕4; 7π∕4, thus, n θerr − θ∕2π. First, define idm as the demodulation source signal and according to Eqs. (12) and (13), the channelswitching method can be described in detail as follows: when when when when
−π∕4 < θ < π∕4, then jiI j > jiQ j, idm iI ; π∕4 < θ < 3π∕4, then jiQ j > jiI j, idm iQ ; 3π∕4 < θ < 5π∕4, then jiI j > jiQ j, idm −iI ; 5π∕4 < θ < 7π∕4, then jiQ j > jiI j, idm −iQ .
Thus, the demodulation source signal idm can be represented by θ − 2nπ idm θerr ; t rect err π∕2 θerr − 2n 1π iI − rect π∕2 θerr − 2n 1∕2π rect π∕2 θerr − 2n 3∕2π iQ ; − rect π∕2
(15a)
where the rect function is defined as rectx
1 0
jxj < 0.5 : jxj > 0.5
threshold I D 0: rt
1; idm < I D : 0; idm > I D
(17)
According to the Eqs. (16) and (17), the modulation data mt is recovered, mt rt:
(18)
Generation of the switch trigger: The multiplied signal is represented by 1 imul θerr iI iQ R2 Ps PLO sin2 θerr ; (19) 2 when θerr π∕4 2nπ or 5π∕4 2nπ, then imul R2 Ps PLO ∕2 at its peak; when θerr −π∕4 2nπ or 3π∕4 2nπ, then imul −R2 Ps PLO ∕2 at its valley. Thus, the moment that the peak and the valley of the multiplied signal imul arise could be the switch moment. It is important to note that the peak of imul arises both when θerr π∕4 2nπ and θerr 5π∕4 2nπ, and the data should be differentially precoded to solve the phase ambiguity. To facilitate understanding, the general view of iI , iQ , imul and idm are shown in Fig. 3, wherein the horizontal axis represents data bits, the vertical axis represents amplitude (peak amplitude of iI , iQ is assumed to be “1”). The switch trigger signal source is generated through a mixer, the differential circuit module, and a zero-crossing comparator. The differential circuit module is used to convert the determined peak and valley of the multiplied signal imul to the test of zero-crossing moment of the differential signal of imul through the comparator. The differential signal of imul is represented by
(15b)
When θerr π∕4 2nπ or 3π∕4 2nπ, the I∕Q channels should be switched. A mixer circuit could be used to cancel the phase modulation and obtain the multiplied signal of the iI and iQ , imul θerr iI iQ 1∕2 R2 Ps PLO sin2 θerr , which incorporates the phase error θerr. The generation of the switch trigger using the multiplied signal is the key of this method and is presented later. Substituting Eqs. (12) and (13) into Eq. (15), the idm can be represented by (detailed mathematical derivation is presented in Appendix A)
dimul R2 Ps PLO ωIF cos2 θerr ; dt
(20)
where ωIF ωs − ωLO denotes the intermediate frequency. The switch trigger signal source output from
θerr − 2n C∕2π rect idm θerr ; t π∕2 C0 ! 3 X
× cosθerr − Cπ∕2 p R Ps PLO cosmt π:
16
The demodulation data rt could be obtained from idm by simply thresholding process with a decision
Fig. 3. General view of (a) iI , (b) iQ , (c) imul , and (d) idm . 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
2741
the comparator can be represented as a digital signal by dimul itrig θerr sgn dt θerr − 2nπ θerr − 2n 1∕2π − rect rect π∕2 π∕2 θerr − 2n 1π rect π∕2 θ − 2n 3∕2π ; (21a) − rect err π∕2 where the sgn function is defined as 8 x>0
A new coherent laser communication detection scheme based on the channel-switching method is proposed. The detection front end of this scheme comprises a 90° optical hybrid and two balanced photodetectors which outputs the in-phase (I) channel and quadrature-phase (Q) channel signal current, respectively. With this method, the ultrahigh speed analog/digital transform of the signal of the I or Q channel is not required. The phase error between the signal and local lasers is obtained by simple analog circuit. Using the phase error signal, the signals of the I∕Q channel are switched alternately. The principle of this detection scheme is presented. Moreover, the comparison of the sensitivity of this scheme with that of homodyne detection with an optical phase-locked loop is discussed. An experimental setup was constructed to verify the proposed detection scheme. The offline processing procedure and results are presented. This scheme could be realized through simple structure and has potential applications in cost-effective high-speed laser communication. © 2015 Optical Society of America OCIS codes: (060.1660) Coherent communications; (060.5060) Phase modulation; (060.2920) Homodyning. http://dx.doi.org/10.1364/AO.54.002738
1. Introduction
Compared with intensity modulation and direct detection scheme, phase modulation and coherent detection with a local laser attracts an increasing amount of attention because of its improved sensitivity and ability to achieve a high data rate. The binary phase shift keying (BPSK) modulation format with a relative simple modulator is still widely used in coherent laser communication. In particular, it has potential applications in cost-effective, light-weighting situation in either free-space or optical fiber communication. BPSK modulation and homodyne detection with an optical phase-locked loop (OPLL) is the communication scheme with highest sensitivity [1]. However, a very narrow linewidth laser and short loop 1559-128X/15/102738-09$15.00/0 © 2015 Optical Society of America 2738
APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015
delay are required to achieve a stable OPLL [2–5]. Given that the loop delay is large, PLL bandwidth is usually limited below 1 MHz, and system stability is difficult to maintain when semiconductor lasers have large phase noise and frequency drift [6]. Digital coherent receivers are used in coherent laser communication [7,8]. These coherent receivers comprised a 90° optical hybrid, two balanced photodetectors, two high-speed analog-digital converters (ADCs), and a parallel digital signal processor (DSP), and the phase estimation method is utilized as the demodulation algorithm [9,10]. However, the overall receiver exhibits very high cost and the requirement of the sufficient speed for ADC and DSP in high data rate is a disadvantage. In the present paper, a new coherent laser communication detection scheme based on the channelswitching method, where neither the OPLL nor high-speed ADC and DSP are needed, is proposed
for the application of the BPSK modulation format. The detection front end also consists of a 90° optical hybrid, two balanced photodetectors which output inphase (I) channel signal current and quadraturephase (Q) channel signal current, respectively [11,12]. The phase error between the signal and local lasers is obtained by simple analog circuit. Using the phase error signal, the signals of the I∕Q channel are switched alternately. This technique is the basic idea of the channel-switching method. The principle of the proposed detection scheme is presented and the sensitivity of this detection scheme, at the shot noise limit, is discussed. Compared with the sensitivity of homodyne detection with OPLL, the sensitivity of the proposed scheme indicates that a 3 dB deterioration in sensitivity results from the fact that the incoming signal beam is split into two separate beams with equal optical power. However, no more than 6 dB deterioration exists in sensitivity, because the signal optical power in homodyne detection is no less than a quarter of that with the channel-switching method when the signal-to-noise ratio (SNR) is equal. An experimental setup was constructed to verify this detection scheme. The offline processing procedure and the bit error rate (BER) test results are also presented. This method can be realized through basic electric modules, such as mixer, comparator, flip-flop, and multiplexer. Thus, the proposed scheme has potential applications in cost-effective high-speed laser communication. Section 2 presents the basic principle of the homodyne detection scheme and the digital coherent receiver and its limitations. Section 3 presents the detection scheme based on the channel-switching method in detail. The laboratory experimental demonstrations are shown in Section 4. The conclusion is presented in Section 5, and the Appendix A is given afterwards. 2. Basic Principle of BPSK Modulation and Coherent Detection Schemes
The complex field of the incoming signal beam can be represented by ES
p PS expfjωS t Δφt φS tg;
(1)
where PS denotes the signal power, ωS is the signal angular frequency, and φS t is the random phase noise, and the phase modulation Δφt mt × π;
p PLO expfjωLO t φLO tg;
A. Homodyne Detection Scheme with OPLL
For ideal homodyne detection, the phase locking of the local laser is satisfied to that of the signal, that is ωs − ωLO t φs t − φLO t 0. The modulation sequence can be recovered from the alternate current after blocking the direct current. The alternate current is p iac t 2R PS PLO cosΔφt;
(3)
(4)
where R is the responsivity of the photodiode. If I 1 represents bit “1”, I 0 represents bit “0”, then p I 1 −I 0 2R Ps PLO :
(5)
Modulation data can be recovered by simply setting the threshold. Moreover, two main kinds of Gaussian random noises exist during the detection period: thermal noise and shot noise. For the shot noise limited system, which is the ideal situation for the coherent detection scheme, the mean square noise currents of bits “1” and “0” are equal to the value of shot noise. When the dark current of the photodiode is ignored, and PLO ≫ PS is satisfied, the mean square noise current of the shot noise can be represented by [13] σ 2s 2qRPLO Δf ;
(6)
where q is the electron charge and Δf is the bandwidth of the photodetector. The SNRs of bits “1” and “0” are also equal, represented by SNR
I 21 2RPS : qΔf σ 2s
(7)
The ideal BER [14] for PSK coherent detection is represented as a function of SNR by 1 BER erfc 2
r SNR ; 2
(8)
where erfc is complementary error function. Moreover, according to Eqs. (7) and (8), the ideal sensitivity of the homodyne detection scheme could be represented by
(2)
where mt is the modulation digital 0/1 sequence at a specific data rate. This 0∕π phase modulation on signal beam is called BPSK. A local beam must be mixed with the signal beam to recover the initial modulation mt sequence. The local beam can be represented by ELO
where PLO is the local power, ωLO is the local angular frequency, and φLO t is the random phase noise.
PS
erfc−1 2 BER2 qΔf : R
(9)
The real sensitivity is not as good as that described in Eq. (9), because to achieve phase locking a signal power penalty is caused by the loop feedback in the OPLL. For example, the structure of a costas PLL is depicted in Fig. 1 and a signal power penalty is caused by the existence of the Q channel. However the data are recovered from the I channel only, most 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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p Ps PLO cosfθerr Δφtg; p iQ t R Ps PLO sinfθerr Δφtg; iI t R
Fig. 1. Basic structure of an optical costas phase-locked loop.
of the signal power is transmitted into the I branch. For instance, the optical signal power is divided 7:1 to the I branch and Q branch in [15]. Ideally, the signal power of the Q-channel branch could be approximately omitted. In addition, a very narrow linewidth laser and short loop delay are required to obtain an OPLL with absolute stability. PLL bandwidth is usually limited below 1 MHz because of large loop delay, and system stability is difficult to maintain when semiconductor lasers have large phase noise and frequency drift. This technical difficulty inherent in OPLL has not been solved perfectly. Some reports used photonic integrated circuit (PIC) and electronic integrated circuit (EIC) to decrease the loop delay, and greater than 1 GHz loop bandwidth could be achieved [16]. This method maybe a promising solution for OPLL. However it exhibits very complicated structure, and the requirements for PIC and EIC are still strict. B.
Digital Coherent Receiver
Digital coherent receivers could be used in coherent laser communication. These coherent receivers, the basic structure of which is shown in Fig. 2, are comprised of a 90° optical hybrid, two balanced photodetectors, two high-speed ADCs and a parallel DSP, and the phase estimation method is utilized as demodulation algorithm. In view of BPSK as an example, two high-speed ADCs simultaneously sample the I and Q channel signal currents detected by two photodetectors. The signal currents of I and Q are represented by
(10a) (10b)
where Δφt is the phase modulation ( 0; π), and θerr ωs − ωLO t φs t − φLO t is the phase error. The sampled signal of I and Q becomes the real part and imaginary part of Yt, that is, Yt iI t jiQ t. The square cancels the phase modulation Δφt, given that Y 2 t ∝ expj2θerr . Phase error θerr is estimated by using N samples, and the complex amplitudes Y 2 t of N samples are summed. Thus, the phase estimate θe is represented as [10] " # N X 1 2 Y k : θe arg 2 k1
(11)
This phase estimate for BPSK is the same for QPSK in [10], with the product factor in Eq. (11) equal to 1/2 instead of 1/4. A modulation symbol ( 0; 1) is obtained by discriminating the phase modulation Δφt fargYk − θe g. However, the data should be differentially precoded to solve phase ambiguity. Moreover, a frequency control feedback loop is needed to maintain the intermediate frequency [7]. For a long-haul optical fiber communication system, the ADC-DSP-based digital coherent receiver is powerful and the impairments stemming from narrowband filtering, chromatic dispersion could be compensated through DSP algorithms. However, in a short distance optical fiber communication system or free-space laser communication system, where dispersion effects are not severe, the application of DSP with high cost may be wasteful. Moreover, with the data rate increasing, the speed of ADC and DSP is a disadvantage. Although advanced modulation formats can be an alternative, the structures of the modulator for these formats are sophisticated. 3. Coherent Detection Scheme Based on Channel-Switching Method
This coherent laser communication detection scheme based on the channel-switching method is applied for the BPSK modulation format. In this scheme, neither the OPLL nor the high-speed ADC and DSP are needed. A. Basic Principle of This Coherent Detection Scheme
The detection front end of this detection scheme is the same as that of the digital coherent receivers. This detection front end comprises a six-port 90° optical hybrid and two balanced photodetectors. Both signal and local optical power are divided 1:1 to the I branch and Q branch. If the signal beam and local beam are represented by the same equations as Eq. (1) and Eq. (3), then iI and iQ can be represented by Fig. 2. Basic structure of a digital coherent receiver. 2740
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iI t R
p Ps PLO cos θerr cosΔφt;
(12)
iQ t R
p Ps PLO sin θerr cosΔφt;
(13)
where θerr ωs − ωLO t φs t − φLO t is the phase error, and Δφt mt × π is the phase modulation. To simplify the analysis, the θerr can also be assumed to be θerr 2nπ θ;
(14)
where n is an integer and θ ∈ −π∕4; 7π∕4, thus, n θerr − θ∕2π. First, define idm as the demodulation source signal and according to Eqs. (12) and (13), the channelswitching method can be described in detail as follows: when when when when
−π∕4 < θ < π∕4, then jiI j > jiQ j, idm iI ; π∕4 < θ < 3π∕4, then jiQ j > jiI j, idm iQ ; 3π∕4 < θ < 5π∕4, then jiI j > jiQ j, idm −iI ; 5π∕4 < θ < 7π∕4, then jiQ j > jiI j, idm −iQ .
Thus, the demodulation source signal idm can be represented by θ − 2nπ idm θerr ; t rect err π∕2 θerr − 2n 1π iI − rect π∕2 θerr − 2n 1∕2π rect π∕2 θerr − 2n 3∕2π iQ ; − rect π∕2
(15a)
where the rect function is defined as rectx
1 0
jxj < 0.5 : jxj > 0.5
threshold I D 0: rt
1; idm < I D : 0; idm > I D
(17)
According to the Eqs. (16) and (17), the modulation data mt is recovered, mt rt:
(18)
Generation of the switch trigger: The multiplied signal is represented by 1 imul θerr iI iQ R2 Ps PLO sin2 θerr ; (19) 2 when θerr π∕4 2nπ or 5π∕4 2nπ, then imul R2 Ps PLO ∕2 at its peak; when θerr −π∕4 2nπ or 3π∕4 2nπ, then imul −R2 Ps PLO ∕2 at its valley. Thus, the moment that the peak and the valley of the multiplied signal imul arise could be the switch moment. It is important to note that the peak of imul arises both when θerr π∕4 2nπ and θerr 5π∕4 2nπ, and the data should be differentially precoded to solve the phase ambiguity. To facilitate understanding, the general view of iI , iQ , imul and idm are shown in Fig. 3, wherein the horizontal axis represents data bits, the vertical axis represents amplitude (peak amplitude of iI , iQ is assumed to be “1”). The switch trigger signal source is generated through a mixer, the differential circuit module, and a zero-crossing comparator. The differential circuit module is used to convert the determined peak and valley of the multiplied signal imul to the test of zero-crossing moment of the differential signal of imul through the comparator. The differential signal of imul is represented by
(15b)
When θerr π∕4 2nπ or 3π∕4 2nπ, the I∕Q channels should be switched. A mixer circuit could be used to cancel the phase modulation and obtain the multiplied signal of the iI and iQ , imul θerr iI iQ 1∕2 R2 Ps PLO sin2 θerr , which incorporates the phase error θerr. The generation of the switch trigger using the multiplied signal is the key of this method and is presented later. Substituting Eqs. (12) and (13) into Eq. (15), the idm can be represented by (detailed mathematical derivation is presented in Appendix A)
dimul R2 Ps PLO ωIF cos2 θerr ; dt
(20)
where ωIF ωs − ωLO denotes the intermediate frequency. The switch trigger signal source output from
θerr − 2n C∕2π rect idm θerr ; t π∕2 C0 ! 3 X
× cosθerr − Cπ∕2 p R Ps PLO cosmt π:
16
The demodulation data rt could be obtained from idm by simply thresholding process with a decision
Fig. 3. General view of (a) iI , (b) iQ , (c) imul , and (d) idm . 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
2741
the comparator can be represented as a digital signal by dimul itrig θerr sgn dt θerr − 2nπ θerr − 2n 1∕2π − rect rect π∕2 π∕2 θerr − 2n 1π rect π∕2 θ − 2n 3∕2π ; (21a) − rect err π∕2 where the sgn function is defined as 8 x>0
