## Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM |

Optics Express, Vol. 20, Issue 1, pp. 317-337 (2012)

http://dx.doi.org/10.1364/OE.20.000317

Acrobat PDF (2735 KB)

### Abstract

Nyquist sinc-pulse shaping provides spectral efficiencies close to the theoretical limit. In this paper we discuss the analogy to optical orthogonal frequency division multiplexing and compare both techniques with respect to spectral efficiency and peak to average power ratio. We then show that using appropriate algorithms, Nyquist pulse shaped modulation formats can be encoded on a single wavelength at speeds beyond 100 Gbit/s in real-time. Finally we discuss the proper reception of Nyquist pulses.

© 2011 OSA

## 1. Introduction

1. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express **16**(2), 841–859 (2008). [CrossRef] [PubMed]

3. X. Liu, S. Chandrasekhar, B. Zhu, P. Winzer, A. Gnauck, and D. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000 km of ultra-large-area fiber and five 80-GHz-grid ROADMs,” J. Lightwave Technol. **29**(4), 483–490 (2011). [CrossRef]

4. R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, B. Nebendahl, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse modulation transmitter generating rectangular shaped spectra of 112 Gbit/s 16QAM signals,” in *Signal Processing in Photonic Communications*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper SPMA5. http://www.opticsinfobase.org/abstract.cfm?URI=SPPCom-2011-SPMA5

## 2. Advanced filtering in optical WDM networks

5. M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in *Optical Fiber Communications Conference*, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMJ5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OMJ5

6. R. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. **28**(4), 662–701 (2010). [CrossRef]

7. A. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. **28**(4), 423–433 (2010). [CrossRef]

*M*-ary QAM signals, the spectral occupancy does not alter significantly when changing the number of bits

*b*transmitted per symbol. Thus increasing the number of constellation points

*M =*2

*leads directly to an increase in spectral efficiency. However, transmitting an additional bit per symbol implies doubling the number of constellation points, so that for a constant average power the required signal-to-noise ratio (SNR) increases significantly. This is also true if the spectral efficiency is increased by polarization division multiplexing (PDM) or polarization switching [8*

^{b}8. M. Sjödin, P. Johannisson, H. Wymeersch, P. A. Andrekson, and M. Karlsson, “Comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express **19**(8), 7839–7846 (2011). [CrossRef] [PubMed]

*F*[9]. If the optimization of spectral efficiency is the ultimate target, all frequency components outside the Nyquist band must be removed by filters. As a consequence, the time domain signal changes from pulses that are clearly separated in time (e. g., non-return-to-zero format, NRZ) to pulses that overlap their neighbors.

_{s}*M*-ary QAM NRZ signal for three different WDM channels centered at optical frequencies

*f*

_{0},

*f*

_{1}, and

*f*

_{2}. The spectra are significantly wider than

*F*, but can be reduced to the Nyquist bandwidth without loosing any signal information. However, appropriate filtering is required to achieve the best possible transmission quality. The sinc-shaped spectrum of an NRZ signal should be filtered such that the resulting spectrum is of rectangular shape under the assumption that the frequency response of the channel is flat in the region of interest. Therefore the side lobes must be removed, and the spectrum within

_{s}*F*must be flattened. If there are slopes in the channel’s frequency response, or the noise accumulated in the system is not constant over frequency, a pre- and de-emphasis filtering scheme should be applied. In a properly filtered WDM spectrum comprising the same three carrier wavelengths as in Fig. 1(a), the channels can now be placed next to each other located on a frequency grid the minimum spacing of which is dictated by the symbol rate

_{s}*F*(Fig. 1(b), Nyquist-WDM [10

_{s}10. G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the Performance of Nyquist-WDM Terabit Superchannels Based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM Subcarriers,” J. Lightwave Technol. **29**(1), 53–61 (2011). [CrossRef]

11. D. Hillerkuss, R. Schmogrow, M. Meyer, S. Wolf, M. Jordan, P. Kleinow, N. Lindenmann, P. Schindler, A. Melikyan, X. Yang, S. Ben-Ezra, B. Nebendahl, M. Dreschmann, J. Meyer, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, L. Altenhain, T. Ellermeyer, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “32.5 Tbit/s 16QAM Nyquist – WDM transmission, Opt. Express (submitted).

*S*

_{21}(

*f*) as in Fig. 2(a) could be used. The difficulty is to build optical filters with frequency responses that drop significantly inside just a few MHz. Optical filters based on liquid crystals may offer an opportunity to perform such filtering [12

12. Z. Dong, J. Yu, H. C. Chien, N. Chi, L. Chen, and G. K. Chang, “Ultra-dense WDM-PON delivering carrier-centralized Nyquist-WDM uplink with digital coherent detection,” Opt. Express **19**(12), 11100–11105 (2011). [CrossRef] [PubMed]

13. X. Zhou, L. Nelson, P. Magill, B. Zhu, and D. Peckham, “8x450-Gb/s, 50-GHz-spaced, PDM-32QAM transmission over 400km and one 50GHz-grid ROADM,” in *Optical Fiber Communications Conference*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-PDPB3

14. R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. **22**(21), 1601–1603 (2010). [CrossRef]

*F*of the digital filter based transmitter is achieved without changing the hardware. Analog filters are generally fixed with respect to their frequency responses and cannot easily be altered.

_{s}## 3. Nyquist pulse modulation and OFDM: A comparison

1. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express **16**(2), 841–859 (2008). [CrossRef] [PubMed]

*z*) is 1 for |

*z*| < 1/2 and zero otherwise, see Eq. (12) in the Appendix. By Fourier transforming Eq. (1) we obtain the frequency domain representation of the

*i-*th temporal OFDM symbol, i. e., a set of

*N*spectral sinc-functions centered at frequencies

*f*,

_{k}*i*th spectral Nyquist symbol, i. e., a set of infinitely many temporal sinc-functions centered at times

*t*,

_{k}*c*

_{i}_{1}=

*c*

_{i}_{2}=

*c*

_{i}_{3}= 1 form a specific OFDM symbol with width

*T*and positioned at

_{s}*t*= 0. The OFDM spectrum is a superposition of three spectral sinc-functions located at frequencies

*f*

_{k}_{-1},

*f*, and

_{k}*f*

_{k}_{+1}, which are separated by

*F*. In Fig. 3(b), the superposition of three temporal sinc-functions is seen which are located at times

_{s}*t*

_{k}_{-1},

*t*, and

_{k}*t*

_{k}_{+1}and separated by

*T*. These Nyquist pulses are modulated with

_{s}*c*

_{i}_{1}=

*c*

_{i}_{2}=

*c*

_{i}_{3}= 1 and form a specific spectral Nyquist symbol with width

*F*and position at

_{s}*f*= 0. It consists of three spectrally sinusoidal Nyquist “subcarriers”. The graphs in Fig. 3(a) and (b) represent Eq. (1) - (4) for

*i*= 0, i. e., for an OFDM and a Nyquist symbol positioned at

*t*= 0 and

*f*= 0, respectively.

*k*= 0, then each of the three OFDM or Nyquist symbols shown here consists of only one temporal zero-frequency (

*f*

_{0}= 0) or spectral zero-time (

*t*

_{0}= 0) “sinusoidal”, respectively. For OFDM, the three temporal symbols are positioned at times (

*i*−1)

*T*, i

_{s}*T*and (

_{s}*i*+ 1)

*T*

_{s}, Fig. 3(c). The resulting spectrum is located within a sinc-shaped envelope having its first zeros at –

*F*and +

_{s}*F*. Due to the different positions of the temporal symbols we see three spectral sinusoidals within the (green) spectral envelope. For Nyquist pulses, the three temporal sinusoidals inside the (green) sinc-shaped pulse envelope with zeros at –

_{s}*T*and +

_{s}*T*correspond to three spectral symbols positioned at frequencies (

_{s}*i*−1)

*F*,

_{s}*iF*, (

_{s}*i*+ 1)

*F*, Fig. 3(d).

_{s}*k*or

*i*of Eq. (1) - (4) to zero while varying the other one, and we present the signal generation in both frequency and time domain.

_{k}_{= 0}(

*f*) centered at

*f*= 0 is shifted by a finite number of equidistant frequency steps

*kF*,

_{s}*k*= 0…

*N*−1. The resulting sub-spectra are modulated by complex coefficients

*c*. The total OFDM spectrum

_{ik}*X*

^{(0)}(

*f*) for

*i*= 0 is formed by superimposing all

*N*subcarrier spectra (Σ stands for summation), resulting in an OFDM symbol located at

*t*= 0 only.

*y*

_{k}_{= 0}(

*t*) centered at

*t*= 0 is shifted by an infinite number of equidistant time steps

*kT*,

_{s}*k*= −∞… + ∞. The impulses are modulated by complex coefficients

*c*. The total Nyquist pulse

_{ik}*y*

^{(0)}(

*t*) for

*i*= 0 is formed by superimposing all “subcarrier” pulses, resulting in a Nyquist pulse sequence at one carrier “frequency”

*f*= 0 only.

*x*

_{k}_{= 0}(

*t*) comprising only one carrier “frequency”

*f*= 0 is shifted by an infinite number of equidistant time steps

*iT*,

_{s}*i*= −∞… + ∞. These sub-pulses are modulated by complex coefficients

*c*. The total OFDM time signal

_{ik}*x*(

*t*) for

*f*= 0 is formed by superimposing infinitely many temporal sub-pulses.

*Y*

_{k}_{= 0}(

*f*) comprising only one Nyquist pulse (“carrier”) at

*t*= 0 is shifted by a finite number of equidistant frequency steps

*iF*,

_{s}*i*= 0…

*N*−1. The resulting sub-spectra are modulated by complex coefficients

*c*. The total Nyquist symbol

_{ik}*Y*(

*f*) at

*t*= 0 is formed by superimposing all

*N*sub-spectra.

*receivers*can be built similar to the transmitter scheme depicted in Fig. 4. To this end, the received signal would enter from the right, the symbol Σ would represent a splitter, and local oscillators with complex conjugate time dependency (OFDM signal) or complex conjugate Nyquist pulses (Nyquist signal) mix with the incoming signals to recover the modulation coefficients

*c*having integrated over the symbol period

_{ik}*T*(for OFDM signal) or over all times (for Nyquist signals). Forming the complex conjugate means reverting the signs of frequency steps

_{s}*F*and time steps

_{s}*T*, respectively.

_{s}*time*division multiplexing (OTDM) technique.

## 4. Oversampled Nyquist pulses with finite-length

15. L. Lo Presti and M. Mondin, “Design of optimal FIR raised-cosine filters,” Electron. Lett. **25**(7), 467–468 (1989). [CrossRef]

*q*(typically

*q*= 1.2, 2, …) is needed. In this paper we have chosen

*q*= 2. This way we will subsequently save FPGA resources since sampling points of adjacent symbols fall onto the same time slot. However, smaller oversampling factors such as

*q*= 1.2 suffice if adequate anti-aliasing filters are available. This would allow us to reduce the required processing speed and DAC sampling rate but comes at the cost of an increased processing complexity.

*R*can be constructed by a sequence of

*R*delay elements

*T*/

_{s}*q*with

*T*= 1 /

_{s}*F*, and

_{s}*R*+ 1 taps in-between. The tapped signals are weighed by a number of

*R*so-called filter coefficients

*h*and summed up to form the filter output, Fig. 5 . A “one-tap” filter with order

_{r}*R*= 0 reproduces the filter input.

*R*is shown in Fig. 6 . The left column shows the impulse response of each filter. The effective windowing is indicated by a green rectangle. The linearly scaled corresponding transfer functions are seen in the middle column. The right column displays these same transfer functions on a logarithmic scale. The spectra of the single pulses (white lines) are plotted together with simulated data (colored). A two-fold oversampling

*q*= 2 is used in this context.

^{15}− 1 serves as origin for simulated complex data. As a reference, these complex data

*c*modulate NRZ pulses, one of which is displayed in Fig. 6(a), left column. The linearly scaled sinc-shaped power spectrum of this elementary impulse is seen in Fig. 6(a), middle column. The logarithm of the same power spectrum is shown as a white line in Fig. 6(a), right column, together with the ensemble-averaged power spectrum for the simulated data. For all power spectra a possibly existent discrete carrier line is omitted.

_{ik}*R*with

*R*+ 1 taps corresponds to the rectangular time window within which the function is defined (left column, green). The convolution of the rectangular spectrum of an infinitely extended temporal sinc-pulse with the sinc-shaped spectrum of the rectangular time window leads to the power spectra depicted in Fig. 6(b)-(d), middle and right column. As the filter order

*R*increases from

*R*= 16 to

*R*= 1024, the spectrum evolves towards an ideal rectangle rect(

*f*/

*F*) with a spectral width equal to the Nyquist bandwidth

_{s}*F*for complex data. Already for

_{s}*R*= 32 a significant increase of the spectral efficiency is to be seen in comparison to NRZ modulation. For

*R*= 1024 the ideal rectangular spectrum is approximated even more closely. However, due to Gibbs’ phenomenon, strong ringing at the steep spectral slopes is to be observed. Non-rectangular window functions like Hann or Hamming windows lead to smoothened spectra and a stronger suppression of the side lobes. However, this advantage comes at the price of a widened spectrum and thus a reduced spectral efficiency.

## 5. Spectral efficiency and peak-to-average power ratio

*R*and the oversampling factor

*q*, see Fig. 6. For convenience and without loss of generality we choose the spectral symbol

*i*= 0 which lies symmetrical to

*f*= 0. The spectrum then reads

*z*) denotes the sine integral [17], see text before Eq. (38) in the Appendix. Power spectra computed from Eq. (5) closely match the graphs of Fig. 6 which are obtained by simulations. To determine the bandwidth

*B*=

*B*

^{Nyq}, we find the first spectral zeros to the right and to the left of the main band by a numerically exact evaluation of Eq. (5). From these results we extract a simple empirical relation to estimate the SE of digitally generated Nyquist signals:

*N*, or in other words by the size of the inverse fast Fourier transform (IFFT) used for signal generation. For our discussion we disregard more advanced OFDM techniques such as a cyclic prefix, guard bands or the introduction of pilot tones that would decrease the SE. The resulting SE then is [1

1. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express **16**(2), 841–859 (2008). [CrossRef] [PubMed]

*x*(

*t*), we assume without loss of generality that the

*N*subcarriers are modulated with a random sequence of real coefficients

*c*= ± 1. In this case the maximum amplitude is seen if all

_{ik}*N*maxima of the temporal sinusoidals happen to add constructively at one point in time, see Fig. 3(a) at

*t*= 0 and Eq. (33) in the Appendix. The average power of such a random OFDM signal is the sum of the average powers of the

*N*orthogonal subcarriers. For arbitrary modulation coefficients

*c*, the average power for

_{ik}*c*= ±1 has to be divided by a format dependent factor

_{ik}*k*

^{2}[18

18. R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” Photon. Technol. Lett., DOI: *10.1109/LPT.2011.2172405*. [CrossRef]

*Q*=

*N*. We thus approximate the PAPR

_{OFDM}by

*k*

^{2}= 1 is seen in Fig. 7(b), red line. With increasing number

*N*of subcarriers the value for PAPR

_{OFDM}increases linearly. However, the probability that an OFDM signal actually has this peak amplitude decreases with the complexity of the

*M*-ary QAM modulation and with the number

*N*of the subcarriers.

*c*= ±1. Although the local extrema of a single sinc-impulse are not located at times

_{ik}*t*/

*T*= −0.5, 0.5, 1.5…, i. e., not in the center of the interval between zeros, it can be shown that the extrema of superimposed Nyquist pulses are located at exactly these times, Eq. (24) in the Appendix. For a worst-case consideration all contributions sum up constructively, so in order to obtain the maximum compound signal we sum up the absolute values of sinc-pulses at

_{s}*t*/

*T*= 1 / 2. If the compound Nyquist signal was constructed with infinitely extended sinc-functions, the maximum signal power would not converge when the number of Nyquist pulses increases. Nevertheless, sinc-functions located far away from the time of summation only contribute little to the sum. For a finite approximation of a sinc-impulse as described in Section 4, only

_{s}*R*/

*q*pulses can contribute. Here the filter order

*R*denotes the number of time intervals

*T*/

_{s}*q*for

*q*-fold oversampling, i. e.,

*R*stands for the length of the impulse response. We find the maximum power, see Eq. (25) in the Appendix with

*Q*=

*R*/

*q*

*R*. Yet, while

*R*, the probability for finding

*R*sinc-pulses interfering constructively decreases as well similarly to the OFDM case.

*y*

^{(0)}(

*t*) encoded with real coefficients

*c*= ±1. According to [19] we find the average power

_{ik}*R*, orthogonality as implied by Eq. (10) is still a good assumption, so that the average power of truncated Nyquist sinc-impulses is close to

*c*, Eq. (10) has to be divided by a format dependent factor

_{ik}*k*

^{2}[18

18. R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” Photon. Technol. Lett., DOI: *10.1109/LPT.2011.2172405*. [CrossRef]

_{Nyquist}then follows from the ratio of maximum power

*P*and average power

_{max}*k*

^{2}= 1 is plotted in Fig. 7(b), blue line. Unlike OFDM signals where the PAPR increases linearly, Eq. (8), the PAPR of Nyquist signals does not, due to the temporal decay of its elementary sinc-impulse. However, neither for OFDM nor for single-carrier Nyquist pulses the PAPR converges with increasing IFFT size

*N*or filter order

*R*, respectively.

*F*might add up constructively as well. However, the Nyquist channel spacing is typically in the order of several GHz [11

_{s}11. D. Hillerkuss, R. Schmogrow, M. Meyer, S. Wolf, M. Jordan, P. Kleinow, N. Lindenmann, P. Schindler, A. Melikyan, X. Yang, S. Ben-Ezra, B. Nebendahl, M. Dreschmann, J. Meyer, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, L. Altenhain, T. Ellermeyer, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “32.5 Tbit/s 16QAM Nyquist – WDM transmission, Opt. Express (submitted).

13. X. Zhou, L. Nelson, P. Magill, B. Zhu, and D. Peckham, “8x450-Gb/s, 50-GHz-spaced, PDM-32QAM transmission over 400km and one 50GHz-grid ROADM,” in *Optical Fiber Communications Conference*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-PDPB3

20. P. Milder, R. Bouziane, R. Koutsoyannis, C. Berger, Y. Benlachtar, R. Killey, M. Glick, and J. Hoe, “Design and Simulation of 25 Gb/s Optical OFDM Transceiver ASICs,” in *37th European Conference and Exposition on Optical Communications*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper We.9.A.5. http://www.opticsinfobase.org/abstract.cfm?URI=ECOC-2011-We.9.A.5

21. R. Schmogrow, M. Winter, D. Hillerkuss, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time OFDM transmitter beyond 100 Gbit/s,” Opt. Express **19**(13), 12740–12749 (2011). [CrossRef] [PubMed]

22. D. Rafique and A. D. Ellis, “Nonlinear penalties in long-haul optical networks employing dynamic transponders,” Opt. Express **19**(10), 9044–9049 (2011). [CrossRef] [PubMed]

## 6. Implementation

### 6.1 FPGA based DSP

*q*= 2), a series of 128 / 2 = 64 modulated sinc-pulses form the output signal, Fig. 8. The FIR filter is realized in the time domain by convolving the complex modulation coefficients

*c*with a sampled sinc-shaped impulse. All possible products of

_{ik}*c*and an elementary sinc-impulse are sampled (dots in Fig. 8(a)), quantized and stored within LUTs. For illustration purposes we choose

_{ik}*c*= ±1 as coefficients for the spectral symbol

_{ik}*i*= 0. The LUT outputs are delayed by a multiple of

*T*, Fig. 8(b), and all samples belonging to the same point in time are added, Fig. 8(c). The resulting output is fed to a clipping module, Fig. 8(d), which then delivers the output Nyquist waveform as seen in Fig. 8(e). Red dots mark the position of the sinc-pulse maxima ± 1. Continuous operation for an infinite number of Nyquist pulses is achieved through cyclic buffering of samples that are used within adjacent computation windows.

_{s}### 6.2 Experimental setup

_{3}Mach-Zehnder modulator (MZM) serving as an I/Q-modulator, and an erbium-doped fiber amplifier (EDFA). We modulate a continuous wave (CW) external cavity laser (ECL) with in-phase (I) and quadrature-phase (Q) data as shown in Fig. 9 . Within the FPGAs, complex Nyquist pulses are calculated from a 2

^{15}− 1 PRBS in real-time as outlined above, and passed on to the DACs. The polarization division multiplexing (PDM) stage then emulates a polarization multiplexed signal [2

2. D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s^{−1} line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics **5**(6), 364–371 (2011). [CrossRef]

*q*= 2 can be reduced if adequate electrical or optical anti-aliasing filters are available.

## 7. Experimental results

## 8. Nyquist pulse reception – required electrical bandwidth and clock phase recovery

*M*-ary QAM signal is similar to the reception of a conventional, unfiltered NRZ signal. The complex-modulated optical field is down-converted to the baseband by a coherent receiver (e. g. 90° hybrids with balanced photo-detectors). The electrical signal is then sampled by analog-to-digital converters (ADC) before being processed in the digital domain. Despite all similarities we identified two differences when receiving Nyquist pulses as are discussed in the following.

### 8.1 Receiver bandwidth impact

### 8.2 Clock phase recovery

12. Z. Dong, J. Yu, H. C. Chien, N. Chi, L. Chen, and G. K. Chang, “Ultra-dense WDM-PON delivering carrier-centralized Nyquist-WDM uplink with digital coherent detection,” Opt. Express **19**(12), 11100–11105 (2011). [CrossRef] [PubMed]

*M*-ary QAM. Since this method neglects the phase of the complex received signal it can be applied prior to carrier phase recovery. Hence standard algorithms for carrier phase recovery can be employed. The influence of a phase error on the signal quality (here represented by EVM) is depicted in Fig. 12(d). Once the optimum clock phase is found, a feedback control minimizing the signal’s EVM is perfectly suited even for real-time systems.

## 9. Conclusions

*F*. Finally, reception of Nyquist signals is explained and demonstrated, pointing out similarities and differences to standard NRZ reception techniques.

_{s}## Appendix

*t*and

*f*stand for time and frequency.

**Orthogonality relations**

**Series expansions**

**Peak power of a sum of sinc-functions**

*s*(

*z*) of equidistantly shifted sinc-functions, Eq. (17). We are interested in a worst-case estimation of the maximum power |

*s*

_{max}|

^{2}. To this end we assume a constant height of all sinc-functions by choosing coefficients |

*ψ*| = 1 with equal magnitude. The signs of the coefficients

_{m}*ψ*are then selected such that a maximum

_{m}*s*

_{max}(

*z*

_{max}) is found at some position

*z*

_{max}. We start by expanding the special function

*z*. This value can be exceeded by choosing the expansion coefficients

*s*(

*z*) is obtained when all sinc-functions have the same sign in the

*z*-interval under consideration. This is true forwhere the coefficients

*m*= (0, + 1); (−1, + 2); (−2, + 3); … have all a positive sign in the interval 0 <

*z*<

*Z*. The resulting function

*s*(

*z*) is monotonic in 0 <

*z*<

*Z*and symmetrical with respect to

*z*=

*Z*/2, so that the superposition of each pair has its maximum at this point, as will be explained in the following.

*f*(

*u*) which is monotonic in an interval −

*U*<

*u*< +

*U*(

*U*> 0). In this interval the sum

*s*(

_{f}*u*) =

*f*(

*u*) +

*f*(−

*u*) has an extremum if

*s*(

_{f}’*u*) =

*f’*(

*u*) −

*f’*(−

*u*) = 0, i. e., for

*u*= 0. This result as applied to Eq. (23) means that the maximum is found at the symmetry point

*z*

_{max}=

*Z*/ 2 of the sum

*s*(

*z*),

*Q*sinc-functions, from which the maximum power

**Average power of an oversampled sinc-function**

*qz*/Z−

*m*) (oversampling factor

*q*), we expand

*ψ*(

*z*) = sinc(

*z*/Z) Eq. (17), but this time in terms of oversampled sinc-functions sinc(

*qz*/Z−

*m*). We find the expansion coefficients

*ψ*= sinc(

_{m}*m*/

*q*) according to Eq. (18) and write

*ψ*(

*z*) in the power relation Eq. (20) and by applying the orthogonality relation Eq. (14) we find the average power

*q*(preferably

*q*= 2) is needed to simplify the filtering of a Nyquist channel. The sinc(

*z*/

*Z*)-function is then represented not by a number of

*Q*base functions as in Eq. (25), but by

*q Q*base-functions, and again orthogonality is lost in the strict sense. Nevertheless we approximate Eq. (26) by

*q Q*is large enough, the average power should be still close to 1,

*q Q*of base functions, but the so far assumed equal modulus for all expansion coefficients must be modified if QAM modulated signals come into play. In this case, the approximated average power Eq. (29) needs to be divided by a format dependent factor

*k*[18

^{2}18. R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” Photon. Technol. Lett., DOI: *10.1109/LPT.2011.2172405*. [CrossRef]

**PAPR for a Nyquist signal**

**Peak power of an OFDM symbol**

*Q*sinusoidal carriers constant within a window of width

*Z*be given by

*α*of the

_{m}*Q*carriers are chosen accordingly and all symbols have maximum values, then all amplitudes add up leading to:

**Average power of an OFDM symbol**

*Q*subcarriers. Similar to the arguments leading to Eq. (30), the average power in a symbol needs to be divided by a format dependent factor

*k*[18

^{2}*10.1109/LPT.2011.2172405*. [CrossRef]

**PAPR of an OFDM symbol**

**Spectrum of a Nyquist signal**

*Y*

^{(0)}(

*f*) of Eq. (3) (representing the spectrum symmetrical to

*f*= 0 of an infinitely extended baseband Nyquist sinc-impulse) with a sinc-shaped spectrum

*W*(

*f*,

*R*) (representing the spectrum of a rectangular time window

*w*(

*t*) = rect[

*t*/ (

*RT*/

_{s}*q*)] which depends on the number of filter taps

*R*and the oversampling factor

*q*),

## Acknowledgments

## References and links

1. | W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express |

2. | D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s |

3. | X. Liu, S. Chandrasekhar, B. Zhu, P. Winzer, A. Gnauck, and D. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000 km of ultra-large-area fiber and five 80-GHz-grid ROADMs,” J. Lightwave Technol. |

4. | R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, B. Nebendahl, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse modulation transmitter generating rectangular shaped spectra of 112 Gbit/s 16QAM signals,” in |

5. | M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in |

6. | R. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

7. | A. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. |

8. | M. Sjödin, P. Johannisson, H. Wymeersch, P. A. Andrekson, and M. Karlsson, “Comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express |

9. | H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. |

10. | G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the Performance of Nyquist-WDM Terabit Superchannels Based on PM-BPSK, PM-QPSK, PM-8QAM or PM-16QAM Subcarriers,” J. Lightwave Technol. |

11. | D. Hillerkuss, R. Schmogrow, M. Meyer, S. Wolf, M. Jordan, P. Kleinow, N. Lindenmann, P. Schindler, A. Melikyan, X. Yang, S. Ben-Ezra, B. Nebendahl, M. Dreschmann, J. Meyer, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, L. Altenhain, T. Ellermeyer, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “32.5 Tbit/s 16QAM Nyquist – WDM transmission, Opt. Express (submitted). |

12. | Z. Dong, J. Yu, H. C. Chien, N. Chi, L. Chen, and G. K. Chang, “Ultra-dense WDM-PON delivering carrier-centralized Nyquist-WDM uplink with digital coherent detection,” Opt. Express |

13. | X. Zhou, L. Nelson, P. Magill, B. Zhu, and D. Peckham, “8x450-Gb/s, 50-GHz-spaced, PDM-32QAM transmission over 400km and one 50GHz-grid ROADM,” in |

14. | R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett. |

15. | L. Lo Presti and M. Mondin, “Design of optimal FIR raised-cosine filters,” Electron. Lett. |

16. | E. Sun, B. Tian, Y. Wang, and K. Yi, “Quasi-orthogonal time division multiplexing and its applications under Rayleigh fading channels,” in |

17. | M. Abramowitz and I. A. Stegun, eds., “Exponential Integral and Related Functions,” in |

18. | R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” Photon. Technol. Lett., DOI: |

19. | I. S. Gradstein and I. M. Ryshik, eds., |

20. | P. Milder, R. Bouziane, R. Koutsoyannis, C. Berger, Y. Benlachtar, R. Killey, M. Glick, and J. Hoe, “Design and Simulation of 25 Gb/s Optical OFDM Transceiver ASICs,” in |

21. | R. Schmogrow, M. Winter, D. Hillerkuss, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time OFDM transmitter beyond 100 Gbit/s,” Opt. Express |

22. | D. Rafique and A. D. Ellis, “Nonlinear penalties in long-haul optical networks employing dynamic transponders,” Opt. Express |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 14, 2011

Manuscript Accepted: December 3, 2011

Published: December 20, 2011

**Citation**

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, "Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM," Opt. Express **20**, 317-337 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-317

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### References

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- D. Rafique and A. D. Ellis, “Nonlinear penalties in long-haul optical networks employing dynamic transponders,” Opt. Express19(10), 9044–9049 (2011). [CrossRef] [PubMed]

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