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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 28 — Dec. 31, 2012
  • pp: 29404–29412
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Bandpass sampling in heterodyne receivers for coherent optical access networks

Paraskevas Bakopoulos, Stefanos Dris, Bernhard Schrenk, Ioannis Lazarou, and Hercules Avramopoulos  »View Author Affiliations


Optics Express, Vol. 20, Issue 28, pp. 29404-29412 (2012)
http://dx.doi.org/10.1364/OE.20.029404


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Abstract

A novel digital receiver architecture for coherent heterodyne-detected optical signals is presented. It demonstrates the application of bandpass sampling in an optical communications context, to overcome the high sampling rate requirement of conventional receivers (more than twice the signal bandwidth). The concept is targeted for WDM coherent optical access networks, where applying heterodyne detection constitutes a promising approach to reducing optical hardware complexity. The validity of the concept is experimentally assessed in a 76 km WDM-PON scenario, where the developed DSP achieves a 50% ADC rate reduction with penalty-free operation.

© 2012 OSA

1. Introduction

The idea of using coherent technology in future optical access networks has gained traction amongst researchers in recent years, fueled by telecom operators’ requirements of providing high data-rates per user, long reach (50-100 km), and high splitting ratios of up to 1000. These challenges can be addressed in a cost-effective manner by employing coherent heterodyne detection in a pure Wavelength Division Multiplexing (WDM) scheme such as the one advocated in [1

1. H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent Optical Access Networks,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper OTuB1.

]; this provides the necessary receiver sensitivity to mitigate large splitting losses and attenuation due to long transmission length, as well as higher spectral efficiency through the use of advanced modulation formats. Moreover, heterodyne detection in particular is attractive due to requiring simpler optical hardware at the Optical Network Unit (ONU) compared to intradyne reception: Only a single coupler is needed instead of a 90° hybrid, and the number of photoreceivers is halved. In addition, the same ONU laser can be used both as the local oscillator (LO) as well as the upstream transmitter. Besides the obvious benefit of cost reduction, the implication is that the LO can be sufficiently detuned from the downstream signal to avoid the adverse effects of Rayleigh backscattering (RB) from the upstream, as illustrated in Fig. 1
Fig. 1 Illustrating how the downstream signal is allocated to a wavelength further away from the upstream, in order to avoid the effects of Rayleigh backscattering.
. In contrast, an intradyne receiver that requires a LO set to the downstream center frequency must employ a second laser at a different wavelength from the ONU transmitter, to avoid RB.

In a heterodyne coherent optical receiver, the data signal is first downconverted from the optical carrier to an electrical intermediate frequency (IF) determined by the wavelength difference between LO and data transmitter lasers. The modulated electrical carrier is then sampled by an ADC, with further downcoversion to the baseband and subsequent demodulation operations carried out by digital signal processing (DSP). There are two main drawbacks associated with the presence of an IF in such a scheme: (a) A large optoelectronic bandwidth needs to be accommodated. For a signal bandwidth B and IF centered on fc, the highest frequency component will be fc+B/2, which sets the minimum analog bandwidth requirement for both the photoreceiver and ADC. (b) To satisfy the Shannon-Nyquist theorem [2

2. C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng. 37, 10–21 (1949).

], a high sampling rate (fs) is required at the ADC when conventional low-pass sampling is employed. This must be larger than twice the highest frequency component: 2(fc+B/2).

While (a) cannot be easily avoided, it is possible to employ advanced sampling theory concepts to bring the sampling rate, fs, arbitrarily close to the theoretical minimum (2B) needed to accurately reconstruct the modulating baseband signal (a process termed bandpass sampling [3

3. J. Proakis and D. Manolakis, Digital Signal Processing, (Pearson Prentice Hall 2007).

]). However, this requires careful selection of the carrier and sampling oscillator frequencies fc and fs, which in turn must be precise enough to avoid detrimental aliasing effects. This poses the question of whether such a scheme can be used in an optical coherent communication system, where fast (and relatively large) frequency fluctuations occur due to normalized laser linewidths that can typically be as high as Δv∙Ts ≈ 10−3 (where Δv is the linewidth and Ts is the symbol period).

In [4

4. S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-Complexity DSP Using Undersampling for Heterodyne Receivers in Coherent Passive Optical Access Networks,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper We.3.A.4.

] we first presented the bandpass sampling digital heterodyne receiver. In this work, we extend the discussion on the theoretical basis of the concept, and provide additional experimental results to illustrate the functionality at each stage of the DSP. We provide experimental proof that bandpass sampling can be effectively employed in coherent optical receivers to significantly relax the rate and memory requirements on the front-end digitizer, despite the phase noise issues associated with commercial-off-the-shelf lasers. In adhering to the low-complexity nature of ONU hardware, complex digital downconversion (DDC) to the baseband is achieved with a Hilbert transform filter. Thus a single Finite Impulse Response (FIR) filter replaces the two low-pass filters required in a conventional quadrature mixing DDC, roughly halving the number of multiply-accumulate operations needed at this stage. The rate reduction achieved with bandpass sampling also has a positive impact on the filter(s) of the DDC; for a given frequency response, the number of FIR taps required drops proportionally with the sampling rate reduction, thus offering further computational complexity benefits. Finally, no dispersion compensation or adaptive equalization is included in our scheme.

We validate the performance of the proposed digital receiver in a realistic application scenario. Using the developed DSP at the ONU, a 3 GBaud (6 GHz bandwidth) heterodyne-detected QPSK signal residing on a 9.3 GHz IF is sampled at a rate of only 12.5 GSa/s, with negligible performance penalty in a 76 km coherent WDM-PON architecture. This is very close to the theoretical minimum of 2B = 12 GSa/s, and is half the rate required when performing conventional low-pass sampling on the same signal at the Nyquist rate (2×(9.3+3) = 24.6 GSa/s). In addition, it represents a ~2.9-fold rate reduction (normalized to the symbol rate) compared to the recent implementation of Smolorz et al. [5

5. S. Smolorz, E. Gottwald, H. Rohde, D. Smith, and A. Poustie, “Demonstration of a Coherent UDWDM-PON with Real-Time Processing,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper PDP4.

], in the case of the 0.311 Gbaud heterodyne signals that was sampled at 3.7 GSa/s.

2. Principle of bandpass sampling

Bandpass sampling can be used to digitize a continuous bandlimited signal centered on a frequency other than 0 Hz. When sampled by an ADC, spectral replicas of the original signal appear in the digital frequency domain at regular intervals determined by the sampling rate. The process is illustrated in Fig. 2
Fig. 2 Illustrating spectral replications in the digital frequency domain arising from bandpass sampling of an analog signal.
, which shows the positions of the copies of the positive and negative image bands, after sampling at a rate fs < 2(fc+B/2). As long as these resulting replicas do not overlap, aliasing can be avoided, even if the Nyquist criterion is not adhered to. Moreover, a copy of the signal appears near the baseband; bandpass sampling can therefore be thought of as simultaneous digitization and frequency translation. This technique not only offers the potential for relaxing the sampling speed requirement of the ADC below that necessary with traditional low-pass Nyquist sampling; it can also reduce the amount of memory needed to capture a given time interval of a continuous signal, thus decreasing overall computational complexity. It should be noted, however, that even with bandpass sampling, the ADC’s analog bandwidth must still be high enough to accommodate the original signal (>fc+B/2).

It is straightforward to show that aliasing is avoided as long as fs lies within the interval [6

6. R. Lyons, Understanding DSP, (Prentice Hall PTR 2001).

],

2fcBmfs2fc+Bm+1
(1)

Where m can be any positive integer that results in fs > 2B, and determines the number of replications after sampling; in the illustrative example of Fig. 2, m = 4. Clearly the numerators in Eq. (1) are twice the lower (fL) and higher (fU) frequency limits of the original bandlimited signal:

fL=2fcB=2(fcB/2)fU=2fc+B=2(fc+B/2)
(2)

We can rewrite Eq. (1) to relate the bandwidth-normalized sampling rate, fs/B, to the normalized highest frequency component, fU/B. Dividing by B and substituting Eq. (2) into Eq. (1), we can write:

2fLmfUfUBfsB2(m+1)fUB
(3)

This gives rise to Fig. 3
Fig. 3 Regions of acceptable sampling rates (white), normalized to the signal bandwidth, from Eq. (3).
, which is a useful way of visualizing the ‘allowed’ sampling rates for any set of signal parameters. It can be explained as follows: The right-hand side of Eq. (3) gives the lower limit of the normalized sampling rate fs/B for a given value of m. Plotting this for values of m = 1 to 3, we obtain the solid lines of Fig. 3 whose gradients are 2/(m+1). Similarly, the upper limits for a range of values of m are determined by the left-hand side of Eq. (3), and these are shown as dashed lines in the same plot. The resulting white ‘wedges’ correspond to the acceptable operating points where no aliasing occurs. The dotted line is the lower limit of conventional sampling (2fU), as determined by the Shannon-Nyquist theorem. Also evident from the graph is that rates below 2fs/B are not allowed, as this is the absolute minimum given by the Shannon-Nyquist theorem.

With careful selection of our parameters, we could, in principle, design a system that samples a data-modulated electrical IF resulting from optical heterodyne detection, at a sampling rate as low as the theoretical minimum (the tip of one of the wedges in Fig. 3). In practice, however, we are dealing with real-world lasers whose frequency fluctuations would cause the selected operating point to shift horizontally on Fig. 3 and move into the aliasing zone. The problem is exacerbated with increasing m. The wedges of allowed sampling rates become much narrower, placing more stringent limits on the range of frequency uncertainty that can be tolerated without any significant aliasing occurring. The implication of all this is that the operating point must be chosen to be slightly above the theoretical minimum. How much above depends on the laser linewidth, as well as its long-term wavelength stability. It should be noted that similarly, the tolerance of the ADC clock, as well as its jitter would cause vertical movement of the operating point. However, modern ADCs have jitter specifications that are a tiny fraction of the baudrate of a typical optical communication system, so this uncertainty is negligible in practice.

3. Undersampling heterodyne receiver

The proposed digital receiver architecture is shown in Fig. 4
Fig. 4 Proposed DSP for the coherent optical heterodyne receiver using bandpass sampling.
. Before digitization can take place, the heterodyne-detected QPSK-modulated IF which is centered at fc is first passed through an analog bandpass filter (BPF) to suppress out-of-band noise. This is a crucial step to avoid SNR degradation in the signal at the input of the ADC; the bandpass sampling process aggregates noise outside the band of interest and ‘folds’ it into the baseband, thus corrupting the resulting discrete signal. Even an ideal BPF, however, cannot mitigate the thermal noise from the subsequent sampling device; this noise power increases roughly by a factor corresponding to the amount of undersampling (m+1) [7

7. R. G. Vaughan, N. L. Scott, and D. Rod White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39(9), 1973–1984 (1991). [CrossRef]

]. Thus, SNR degradation is negligible as long as overall system noise is dominated by that prior to the ADC, after ideal bandpass filtering.

To better illustrate the functionality of the DSP, we consider signals obtained experimentally. The dotted blue line in Fig. 5(a)
Fig. 5 (a) The received signal in the time domain, oversampled at 50 GSa/s (blue dotted line) and bandpass sampled at 12.5 GSa/s (red circles). Corresponding spectra of the oversampled (b) and bandpass sampled (c) signal.
shows a 3 GBaud QPSK data signal (r(t)) residing on an IF carrier of ~9.3GHz, and oversampled at 50 GSa/s. With no pulse shaping, the bandwidth is approximately twice the baudrate (6 GHz). Figure 5(b) is the corresponding single-sided electrical spectrum. The red circles in Fig. 5(a) are the result of undersampling r(t) at a rate of just fs = 1/Ts = 12.5 GSa/s (derived from Eq. (1) with m = 1), to yield the discrete-time signal r(nTs), n. In the corresponding spectrum of Fig. 5(c), the QPSK signal is now clearly visible near the baseband, with a new center frequency fc2 = fc - mfs = 9.3-12.5 = −3.2GHz (the negative sign simply means that the positive and negative image bands have been reversed, which is the case for odd values of m).

Downconversion to the complex baseband is achieved with a Hilbert-based DDC. r(nTs) is delayed by the filter delay Δ, and along with the output of the Hilbert filter forms the complex signal (Fig. 6(a)
Fig. 6 Evolution of the received signal in the receiver DSP, shown in IQ histogram plots: (a) After the Hilbert DDC. (b) Sampling point selection and decimation to the symbol rate. (c) Removal of the exponential term in Eq. (4). (d) Final constellation after frequency offset and carrier phase estimation.
),

[I(nTs)+jQ(nTs)]exp(jωc2nTs)
(4)

where I and Q are the tributaries of the transmitted signal. The exponential term is essentially a complex rotation with the known carrier frequency (ωc2 = 2πfc2) that results from the frequency translation property of the bandpass sampling process. Rather than removing it at this stage, we first perform symbol clock recovery with the square timing algorithm. This reduces the sample rate from 1/Ts down to the symbol rate 1/Tsym, therefore saving computational resources in subsequent operations. The result is shown in the IQ plot of Fig. 6(b), where the rotating symbols can be seen as a ‘donut’ shape. The rotation due to the exponential term in Eq. (4) is then removed by complex multiplication, and this is evident in Fig. 6(c) where the only rotations left are due to the frequency mismatch between transmitter and LO lasers, as well as their combined linewidths. The final constellation of Fig. 6(d) is obtained after employing standard algorithms for frequency offset estimation [8

8. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007). [CrossRef]

] and carrier phase recovery [9

9. E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” J. Lightwave Technol. 25,2675–2692 (2007).

].

4. Application in a coherent PON

The developed undersampling heterodyne receiver was experimentally evaluated in the coherent PON testbed shown in Fig. 7
Fig. 7 Long-reach coherent PON layout for the evaluation of sub-Nyquistbandpass sampling.
. A long-reach WDM PON architecture was emulated employing a 76 km dual-feeder section followed by a WDM demultiplexer and a 625 m drop section. Standard single-mode fiber was used for both the feeder and drop segments, and the total loss of the Optical Distribution Network (ODN) was 22.5 dB. Due to limited availability of fiber, the dual-feeder section was replaced by a single fiber spool in the experiment.

The Optical Line Terminal (OLT) transmitter generated a 3 Gbaud QPSK downstream (DS) and was realized by cascading a 100 kHz linewidth tunable laser (λDS) with a 20 GHz LiNbO3 IQ modulator, driven by two decorrelated 211-1 PRBS data streams. After propagation in the ODN, this signal was heterodyne-detected at the Optical Network Unit (ONU). The ONU receiver consisted of a tunable LO laser with 1 MHz linewidth (λUS), which was heterodyne-mixed in a 90:10 coupler and detected with an 11.5 GHz photoreceiver. To save the cost of an additional laser at the ONU for upstream transmission [1

1. H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent Optical Access Networks,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper OTuB1.

], the upstream (US) signal was provided by the same LO laser after external modulation in an Electro-Absorption Modulator (EAM) at 3 Gb/s. Simple OOK modulation was preferred over more spectrally-efficient schemes for the US to avoid the use of expensive, specialized components at the ONU. The US was transmitted through the ODN and direct-detected at the OLT receiver. The WDM PON architecture allowed sustainable per-user capacity which in our implementation was 6 Gb/s for the DS and 3 Gb/s for the US, whereas the asymmetry was due to the different modulation formats used for each transmission direction.

The DS performance of the implemented PON was studied for different configurations of the ONU heterodyne receiver. In order to generate a reference measurement the DS was first evaluated without simultaneous US transmission, thus avoiding stimulation of RB. In this scenario the LO was detuned from the downstream to produce an IF at fc = 4.5 GHz, which was adequate for accommodating the signal bandwidth (±3 GHz for the main lobe). The Nyquist rate for low-pass sampling this signal was 15 GSa/s, corresponding to twice the highest frequency component in the received heterodyne signal: 2(fc+B/2). To evaluate this signal, it was digitized in a 16 GHz real-time oscilloscope at 50 GSa/s (oversampled) and was demodulated offline using the standard, Nyquist approach. The BER performance was estimated over ~1.2∙106 bits (Fig. 8(a)
Fig. 8 (a) BER and (b) - (e) eye diagrams of the received DS after 76 km transmission for different wavelength detuning of the US and different sampling rates at the ONU receiver. (b) no US, (c) 4.5 GHz US, (d) 9.3 GHz US oversampled, (e) 9.3 GHz undersampled.
).

Simultaneous transmission of the US signal in the ODN incurred a 3.5 dB BER penalty as a result of back-scattered US arriving at the ONU receiver, overlapping spectrally with the DS signal, as shown in Fig. 9(a)
Fig. 9 Received DS and RB at the ONU for (a) 4.5 GHz and (b) 9.3 GHz detuning of the US.
. The BER measurement for this case (Fig. 8(a)) was performed with the same sampling rate and DSP algorithms as in the reference measurement.

Mitigation of RB degradation can be achieved effectively by reducing the spectral overlap of the received DS with the back-scattered US. This was implemented experimentally by further detuning the upstream and downstream wavelengths to 9.3 GHz (Fig. 9(b)). The new wavelength arrangement improved the BER penalty by 2.5 dB (Fig. 8(a)), yet at the expense of higher requirements in terms of bandwidth and sampling rate. Specifically, increasing the IF resulted in an increase of the frequency components that had to be accommodated up to 12.3 GHz, thus requiring at least 24.6 GSa/s according to the Shannon-Nyquist theorem. In the experimental evaluation, this signal was again oversampled at 50 GSa/s and demodulated offline using the same DSP algorithms.

Although successful for the mitigation of RB penalties, detuning the LO from the DS has direct implications on the necessary hardware for implementing the ONU receiver. Even though 10 GHz electro-optic components are widely available, the high sampling rate requirement can inhibit practical application in a cost-sensitive access network as it not only implies faster digitizers, but also higher processing power and memory availability at the receiver front-end, in order to cope with the increased data traffic. A drastic reduction in sampling rate can be achieved by bandpass sampling of the received signal. Using this scheme and as explained in Section 2, digitization results in an additional downconversion of the DS signal near the baseband (Fig. 10
Fig. 10 Acquired heterodyne spectra for 9.3 GHz detuning (IF) using 50 GSa/s (a) and 12.5 GSa/s (b) sampling rate. Undersampling causes downconversion of the heterodyne signal.
) and aliasing can be avoided as long as the sampling rate lies within the range indicated by Eq. (1) and depicted with the white areas of Fig. 3. Operation at different areas of Fig. 3 corresponds to different factors of reduction over the Nyquist rate for low-pass sampling the electrical IF signal. A theoretical limit for selecting the sampling rate within the operating areas of Fig. 3 is obviously the data signal bandwidth B, which sets a lower limit to the acceptable sampling rate fs > 2B. In our implementation the signal bandwidth was 6 GHz necessitating a sampling rate of at least 12 GHz. Given the 9.3 GHz IF, this indicated that the only acceptable operating area from Fig. 3 was the first wedge corresponding to m = 1. From Eq. (1), the allowed sampling rate range is 12.3-12.6 GSa/s, thus allowing plenty of room for laser linewidth-induced frequency fluctuations (Fig. 11
Fig. 11 Close-up of Fig. 3 (m = 1), showing the chosen operating point.
). A sampling rate of 12.5 GSa/s was chosen due to limitations of the available digitizer. Long-term wavelength stability was adequate for testing under laboratory conditions, whereas in a real ONU implementation a simple feedback loop would be required to control the LO wavelength within the allowed range.

The performance of the developed undersampling DSP receiver was evaluated for 9.3 GHz detuning using the same data sets as for the oversampling case. In our experiment, digitization was performed at 50 GSa/s, with undersampling being emulated in the offline DSP by retaining only 1 out of every 4 acquired samples. The reason for this was to allow emulation of the electrical bandpass filter (BPF) shown in Fig. 4 that is required prior to undersampling, which is a vital function for the operation of the scheme and is assumed to be implemented in hardware in a real system. Figure 8(a) illustrates the BER performance with undersampling, showing no distinguishable penalty compared to oversampling. This result proves the effectiveness of the demonstrated scheme, achieving 50% reduction in the sampling rate with practically no penalty in performance. Moreover this reduction in sampling rate translates to a significant simplification in the filters used at the receiver front-end. Thus in the undersampling case just 10 taps were sufficient to obtain the required response from the Hilbert FIR filter.

5. Conclusion

A simple heterodyne demodulator using bandpass sampling has been implemented. Operation was experimentally verified in a 76 km PON with no implementation penalty, reducing ADC rate requirements by ~50%. This scheme can significantly relax the sampling, processing and memory requirements in the DSP and is therefore particularly attractive for a realistic implementation of coherent PONs, where minimizing hardware complexity and the associated cost are of primary importance.

Acknowledgments

This work was supported by the European Commission through FP7 project GALACTICO (contract number 258407). The authors gratefully acknowledge Yenista Optics for providing the narrow-linewidth tunable laser.

References and links

1.

H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent Optical Access Networks,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper OTuB1.

2.

C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng. 37, 10–21 (1949).

3.

J. Proakis and D. Manolakis, Digital Signal Processing, (Pearson Prentice Hall 2007).

4.

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-Complexity DSP Using Undersampling for Heterodyne Receivers in Coherent Passive Optical Access Networks,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper We.3.A.4.

5.

S. Smolorz, E. Gottwald, H. Rohde, D. Smith, and A. Poustie, “Demonstration of a Coherent UDWDM-PON with Real-Time Processing,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper PDP4.

6.

R. Lyons, Understanding DSP, (Prentice Hall PTR 2001).

7.

R. G. Vaughan, N. L. Scott, and D. Rod White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39(9), 1973–1984 (1991). [CrossRef]

8.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007). [CrossRef]

9.

E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” J. Lightwave Technol. 25,2675–2692 (2007).

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4250) Fiber optics and optical communications : Networks
(060.2840) Fiber optics and optical communications : Heterodyne

ToC Category:
Subsystems for Optical Networks

History
Original Manuscript: October 3, 2012
Revised Manuscript: November 16, 2012
Manuscript Accepted: November 21, 2012
Published: December 19, 2012

Virtual Issues
European Conference on Optical Communication 2012 (2012) Optics Express

Citation
Paraskevas Bakopoulos, Stefanos Dris, Bernhard Schrenk, Ioannis Lazarou, and Hercules Avramopoulos, "Bandpass sampling in heterodyne receivers for coherent optical access networks," Opt. Express 20, 29404-29412 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29404


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References

  1. H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent Optical Access Networks,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper OTuB1.
  2. C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng.37, 10–21 (1949).
  3. J. Proakis and D. Manolakis, Digital Signal Processing, (Pearson Prentice Hall 2007).
  4. S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-Complexity DSP Using Undersampling for Heterodyne Receivers in Coherent Passive Optical Access Networks,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper We.3.A.4.
  5. S. Smolorz, E. Gottwald, H. Rohde, D. Smith, and A. Poustie, “Demonstration of a Coherent UDWDM-PON with Real-Time Processing,” in proceedings of Optical Fiber Communication Conference (OFC), OSA Technical Digest (Optical Society of America, 2011), paper PDP4.
  6. R. Lyons, Understanding DSP, (Prentice Hall PTR 2001).
  7. R. G. Vaughan, N. L. Scott, and D. Rod White, “The theory of bandpass sampling,” IEEE Trans. Signal Process.39(9), 1973–1984 (1991). [CrossRef]
  8. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett.19(6), 366–368 (2007). [CrossRef]
  9. E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” J. Lightwave Technol.25,2675–2692 (2007).

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