We demonstrate full flex-grid operation with Nyquist frequency division multiplexing. The technique supports high spectral efficiency, asynchronous operation of channels, variable channel loading with different modulation formats and dynamic bandwidth allocation. Data from different sources with different bit and symbol rates are encoded onto electrical Nyquist pulses with different electrical subcarrier frequencies, and then transmitted optically. We give details on the transceiver design with digital signal processing and investigate the implementation penalty as a function of several design parameters such as limited filter length and effective number of bits. Finally, experiments are performed for receivers with direct detection, intradyne and remote heterodyne reception.
© 2014 Optical Society of America
Asynchronously multiplexing various tributaries on a flexible grid allows for transmission at highest spectral efficiency (SE) even when traffic patterns change. In the past, most efforts in increasing the SE were concentrated on reducing guard bands (GB) in wavelength division multiplexed (WDM) schemes and on applying advanced modulation formats such as M-ary quadrature amplitude modulation (QAM) [1,2]. Pulse-shaping as a form of digital signal processing (DSP) can help increasing the spectral efficiency (SE) further, .As an example, orthogonal frequency division multiplexing (OFDM)  has emerged as a promising candidate for both, long-haul and access networks. In OFDM, a channel consists of a group of sinc-shaped overlapping equidistant spectra with identical width. And indeed, OFDM offers a high SE and dynamic bandwidth allocation (DBA). Yet, the granularity in OFDM is limited by the number of subcarriers (SC) and with this by the processing capabilities. In addition, the data on the frequency-locked SCs need to be transmitted in synchronism, and even small deviations of the SC frequencies from an equidistant spacing cannot be tolerated to avoid severe crosstalk . Tolerance towards synchronicity can for instance be obtained by introducing a cyclic extension of the symbols (cyclic prefix, CP). However, a CP reduces the symbol rate and hence the SE. Recently, Nyquist-WDM was introduced [5,6] using modulated sinc-shaped Nyquist pulses having a rectangular spectrum .
In this paper, we propose electrical Nyquist frequency division multiplexing (NFDM) and demonstrate it with up to 5 independent SCs . All SCs are encoded with Nyquist sinc-shaped pulses, resulting in rectangular non-overlapping spectra each representing one electrical NFDM channel. Like OFDM, NFDM supports variable channel loading, i.e., choosing the channels modulation formats individually. However, NFDM offers truly independent SCs offering flexible symbol rate per channel . As SC bands are near the fundamental Nyquist limit, no guard bands are required. Synchronization of different transmitters is not required thanks to the non-overlapping spectra. In addition, the narrow shape of each SC offers a high chromatic dispersion (CD) and polarization mode dispersion (PMD) tolerance inside each electrical SC band. As an advantage, no CP is needed for dispersion compensation. Due to the versatility of the concept, both constant and flexible electrical subcarrier grids are supported.
The paper is organized as follows. In Section 2, the DSP transceiver design is discussed. The impact of parameters such as filter length and resolution of digital-to-analog converters (DAC) is investigated for 16QAM NFDM signals, which are then transmitted optically. In Section 3 we demonstrate a five-channel 16QAM flex-grid operation. Finally, in Section 4 we measure the error vector magnitude (EVM) and the bit error ratio (BER) as a function of the optical signal-to-noise ratio (OSNR) for signals with electrical quadrature phase shift keying (QPSK) and electrical 16QAM using optical receivers with direct detection, intradyne and remote heterodyne reception.
2. Transceiver design: digital signal processing
We begin with a discussion of the NFDM transceiver and its DSP implementations in time and frequency domain, and comment on the respective advantages of each approach.
The schematic NFDM transceiver design is depicted in Fig. 1. The complex data signals are fed to a number of Nyquist pulse transmitters (Tx) comprising pulse-shapers. The employed impulse response is a sinc-function in time with the symbol duration TsFig. 1. The complex signal either modulates an optical carrier by I/Q-modulation, resulting in a quadrature-amplitude modulated optical field (QAM, upper inset in Fig. 1),Fig. 1. The notation indicates averaging over a few optical periods. We find
In the following, we discuss the Tx and Rx implementation in the digital domain using a time domain and frequency domain approach.
2.1 Time domain DSP
The transmitter schematic for time domain (TD) digital signal processing (DSP) of N NFDM channels is depicted in Fig. 2. A time domain DSP implementation is interesting for applications with short, numerically inexpensive filters with a few taps only .
The data to be transmitted are represented by time-discrete samples, Fig. 2(a). We describe the process for real-valued data coefficients −1 and + 1 only, but an extension to complex data is easily possible. The data are mapped on one out of N electrical subcarriers, the largest frequency of which determines the minimum sampling rate for all SC channels. The data are represented by one sample per symbol (SPS). For generating a guard band between adjacent Nyquist spectra, we up-sample by a factor of q (here: q = 4) by inserting zeroes in-between adjacent samples. The up-sampling factor q determines the spectral width and, for a given sampling rate, also the maximum number of NFDM channels.
For TD pulse shaping, finite-duration impulse response (FIR) filters are employed . First the input signal u(t) is convolved with the filter’s impulse response h(t). The filter output is mixed with each Tx’s subcarrier (SC). For avoiding run-time multiplications, filtering and SC mixing can be combined in one pre-calculated module which is stored in a look-up table, see Appendix.
The TD-NFDM receiver concept is schematically depicted in Fig. 2(b). At the input, matched filters select the channel of interest and simultaneously perform a down-conversion, see Appendix. In a next step, the signal is down-sampled with a clock recovery method as described in , and data are recovered.
2.2 Frequency domain DSP
As an alternative to TD processing, frequency domain (FD) DSP can improve the resource utilization especially for long filter impulse responses .
Performing the convolution in the frequency domain is referred to as fast convolution. First, a mn-point fast Fourier transform (FFT) converts a block of mn TD data (here ± 1) to the frequency domain, see Fig. 3(a). The output is zero-padded by M −□mn. for (M / mn)-fold oversampling. Each channel is then shifted to the position of its respective SC. Addition of all channel spectra yields the aggregate signal spectrum that is transferred to the TD by an M-point IFFT. The output is parallel-serial converted and transmitted.
At the Rx the compound signal is serial-parallel converted and transformed to the frequency domain by an M-point FFT, see Fig. 3(b). The filter Hn(f) selects the mn points corresponding to the chosen channel out of the M inputs. A subsequent mn-point IFFT recovers the data using a clock recovery  to synchronize the IFFT with the FFT.
The resource tradeoff between time domain and frequency domain DSP with respect to different filter lengths has been investigated in  and . However, a fast convolution  is performed block-wise with FFT-IFFT transform pairs using the overlap-and-add or overlap-and-save method. This implies some processing overhead, which has not been mentioned when comparing FD-DSP and TD-DSP.
2.3 Implementation penalties due to limited filter length and DAC resolution
So far, ideal non-causal signals have been assumed, leading to perfectly rectangular channel spectra without any inter-channel interference (ICI). However, in a real-word implementation at least two compromises need to be made.
First, the ideal sinc-shaped filter impulse response h(t) needs to be approximated with a discrete FIR filter having a finite number R of coefficients, thus leading to a finite-length impulse response of length L. The resulting NFDM channel spectra, therefore, deviate from an ideal rectangle .
In order to experimentally investigate the effect of filter order R on the observed ICI, we use the setup depicted in Fig. 4 and compare the outcome to simulations. An Agilent M9180A arbitrary waveform generator (AWG) with 12 bit resolution and a sampling rate of 12 GSa/s stores pre-computed NFDM signals comprising five channels of equal bandwidth. All channel centers are spaced by the Nyquist bandwidth of the channels, so that guard bands do not exist. The electrical outputs of the AWG drive an optical I/Q-modulator that modulates an external cavity laser (ECL). The center wavelength of this ECL is set to 1550 nm, and the linewidth is specified as 100 kHz. An erbium-doped fiber amplifier (EDFA) amplifies the optical signals. The Agilent N4391A optical modulation analyzer (OMA) coherently receives the NFDM signals. The OMA comprises two 90° optical hybrids (one for each polarization) and two real-time oscilloscopes with two 32 GHz channels each. The oscilloscopes sample at a maximum rate of 80 GSa/s.
In order to evaluate the influence of ICI as a function of filter length, we transmit de-correlated 1.5 GBd 16QAM data in five adjacent zero-guard band NFDM channels, which are consecutively numbered from 0 to 4. We record the error vector magnitude (EVM) [13,14] for the middle channel 2. The length L of the filter’s impulse response depends on the filter order R, the oversampling factor q, and the symbol duration ,Fig. 5(a). Simulations (black) and measurements (blue) coincide except for an EVM implementation penalty of 1 percentage point. This stems from the fact that simulations are performed without any added noise to isolate the effect of ICI. Insets of Fig. 5(a) show two constellation diagrams for L = 64 Ts and L = 512 Ts, respectively.
Second, we investigate the influence of the digital-to-analog converter (DAC) resolution on the achievable signal quality. Figure 5(b) depicts the outcome of simulations (left) and measurements (right). We have computed all NFDM waveforms using floating point double precision (52 bit significant, 1 hidden bit, an 11 bit exponent, and a sign bit) before reducing the number of quantization steps to the resolution listed on the horizontal axis of Fig. 5(b). From simulations and measurements we see that the use of long filters tends to require higher DAC resolutions. However, DACs with a physical resolution of at least 6 bit suffice without compromising the signal quality significantly.
3. Flex-grid operation
A distinct feature of NFDM is the support of so-called flex-grid operation, i. e., the NFDM channel position and bandwidth can be chosen at will. In order to demonstrate this capability an aggregate NFDM signal is created consisting of five individual channels. We first assess the back-to-back performance at high OSNR and demonstrate the feasibility of flex-grid operation within an experiment using the setup of Fig. 4. We then simulate the performance and depict the bit error ratio (BER) as a function of the signal-to-noise power ratio (SNR) per bit .
For the experiment and the simulation, the DSP transceivers described in Section 2.1 were implemented using Matlab. Each of the five channels carries de-correlated pseudo-random binary sequences (PRBS) mapped with either QPSK or 16QAM. The measured channel spectra transporting 16QAM signals are depicted in Fig. 6(a) along with their back-to-back constellation diagrams. In order to illustrate the flex-grid feature, the NFDM channels were chosen to have an arbitrary bandwidth and center frequency. The symbol rates are chosen as fraction sof the AWG sampling rate 12 GSa/s: fs / 15 (black), fs / 32 (red), fs / 8 (green), fs / 16 (blue), and fs / 10 (cyan). The length of the FIR filter’s impulse response in used for pulse shaping is L = 128 × Ts. We see that the performance of the channel depends on its width. The finite length of the filter results in a non-ideal out-of-band suppression and thus in an overlap with neighboring channels. The smaller the channel is, the larger the de-correlated inter-channel interference (ICI) becomes. It can be represented by Gaussian noise and causes the significant increase of the BER for the red channel.
In a next step, the BER of the individual channels as a function of the SNR per bit  was determined by simulations. For varying the SNR, additive white Gaussian noise (AWGN) with different power was superimposed to the aggregate NFDM signal. The results for QPSK and 16QAM modulation formats are shown in Fig. 6(b). As expected, the SNR per bit is independent of the channel bandwidth, but depends strongly on the modulation format. The experiment as well as the simulation shows that the properties of all the channels can indeed be chosen independently proving that NFDM offers full flex-grid capability without significant ICI penalties if appropriate FIR filters are used. Still, measurements and simulations differ due to implementation penalties mainly at the receiver in this measurement, causing the increase of BER for the narrow channels.
In this subsection we investigate the performance of an NFDM signal with five equal channels for different receiver techniques. First, we investigate the performance using a cost-efficient setup based on intensity modulation with direct detection (IM/DD). As a comparison, an optical coherent receiver is used in Subsection 4.2 for achieving the optimum performance. Last, a combination of direct and optical coherent detection is used, i. e., remote heterodyne reception. Here the optical local oscillator is supplied together with the signal itself by the transmitter and received by a single photodiode.
In the following, the signal quality is determined by the optical signal-to-noise power ratio (OSNR) as well as by signal-to-noise power ratio per bit (SNR per bit, ). The OSNR is obtained by measuring the power in the signal with an optical spectrum analyzer (OSA) along with the noise that is averaged over two 0.1 nm wide bins 2 nm above and below the signal. The ratio of the two powers then determines the OSNR. The SNR per bit is calculated from the OSNR in proportion to the bandwidth B of the signal and the bit rate Rb and the number p of polarization states used for transmission 
4.1 Direct detection of NFDM signals
For a direct detection scheme, we generate NFDM modulated optical signals by intensity modulation of the external cavity laser (ECL), see experimental setup in Fig. 7(a). To overcome the limited capability of our AWG, we employ a Xilinx XCVFX200T field programmable gate array (FPGA) and a high-speed Micram DAC. The FPGA stores pre-computed waveforms and drives the DAC. We used a maximum sampling rate of 25 GSa/s. Image rejection low-pass filters (red) remove image spectra appearing at the DAC output. Although intensity modulated optical fields can be generated using cost-efficient directly modulated lasers (DML), the high signal bandwidth shown here calls for external modulation. To this end we use a single optical Mach-Zehnder modulator (MZM) as opposed to the previous section where an optical I/Q-modulator has been employed. The modulator is biased at its quadrature point where the transfer function between optical intensity and electrical drive signal is approximately linear. A variable optical attenuator (VOA) is used to change the input power to the EDFA and thus to adjust the optical SNR (OSNR). The optical spectrum analyzer (OSA) measures the OSNR and assesses optical noise within a bandwidth of 0.1 nm. The optical power launched onto the photodiode (PD) is kept constant throughout the measurements. The same Agilent real-time oscilloscope, which was used in the previous section, samples the electrical output of the PD. Further processing is performed offline including EVM and BER measurements.
We generate NFDM signals comprising three channels carrying either QPSK or 16QAM data. Each channel operates at a symbol rate of 3.125 GBd (25 GSa/s / q, q = 8), resulting in a bit rate of 18.75 Gbit/s for QPSK and 37.5 Gbit/s for 16QAM. The aggregate signal spectrum, composed of three channels with their complex conjugates, is shown in Fig. 7(b). It shows the spectrum as transmitted through the fiber at the receiver. The overall bandwidth is 6 × 3.125 GHz = 18.75 GHz leading to a spectral efficiency of 1 bit/s/Hz for QPSK and 2 bit/s/Hz for 16QAM, respectively. Since we want to use a direct detection receiver, polarization multiplexing is not feasible, and only real signals can be transmitted. Therefore, Hermitian symmetry must be obeyed, i. e., channels at negative frequencies (lower optical sidebands) must carry the complex conjugate data of their corresponding channels at positive frequencies (upper optical sidebands). Hence only three out of six spectra in Fig. 7(b) are independent.
We vary the OSNR and measure BER and EVM. The equivalent BER calculated from measured EVM (lines) [13,14] and the directly measured BER (squares) for all three independent channels are plotted in Fig. 8. While the OSNR and SNR per bit are determined for the complete signal, the SNR of each individual channel should be different due to the roll-off of the spectrum. We thus give the mean BER over all channels for comparison. Asexpected, the mean BER corresponds well with the performance of ch1 with an SNR between those of ch0 and ch2. Figure 8(a) shows results for QPSK modulation, while Fig. 8(b) depicts measurements obtained for 16QAM modulated NFDM carriers. The error floor observed in Fig. 8(b) for 16QAM signals at high OSNR stems from the electronic noise of the transmitter and receiver. This noise is below the noise that would normally be picked up from amplification in a long reach system. In any case, one can observe that BER and EVM provide similar results, whereas EVM allows quicker measurements across a larger range.
4.2 Intradyne reception of NFDM signals
A coherent receiver enables the reception of complex data encoded by an optical I/Q modulator on an optical carrier, Fig. 9(a). In this case, upper and lower optical sidebands are uncorrelated and transmit the real (I) and imaginary parts (Q) of complex NFDM data. As before, the pre-computed signals are stored in two synchronized FPGAs each driving a high-speed Micram DAC. Again, the OSNR is adjusted by a VOA in front of the EDFA. The OMA already described in Section 2.3 coherently receives the NFDM waveforms. Intradyne reception is performed with a second, OMA-internal ECL used as local oscillator. Further processing is performed offline including BER and EVM assessment. The measured aggregate signal spectrum is depicted in Fig. 9(b). The signal comprises five independent channels, each having a symbol rate of 3.125 GBd in a 3.125 GHz optical bandwidth. The resulting data rates are 5 × 3.125 GBd × 2 bit = 31.25 Gbit/s for QPSK, and 62.5 Gbit/s for 16QAM with 4 bit/symbol, respectively. This corresponds to spectral efficiencies of 2 bit/s/Hz (QPSK) and 4 bit/s/Hz (16QAM). The spectral efficiencies can be doubled if polarization division multiplexing (PDM) is applied . As before, the roof-shaped spectral roll-off in Fig. 9(b) is due to the combined electrical frequency responses of Tx and Rx. The roll-off can be compensated at the Tx by a pre-emphasis. Here, we have decided to perform equalization at the Rx instead, for not wasting effective resolution of the DACs to the pre-compensation.
The measured BER (squares) and the equivalent BER obtained from EVM measurements (lines)  of QPSK modulated NFDM signals at different OSNR are depicted in Fig. 10(a). Channel ch2 located in the center of the spectrum (Fig. 9(b), green spectrum) performs best as expected, because for this channel the highest SNR is observed since it is not significantly suffering from the roll-off. We virtually find the same performance for corresponding channels (i.e. ch1 and ch3 as well as ch0 and ch4) due to the symmetry of the spectrum resulting in the same SNR. When comparing the results from coherent detection to the directly detected signals, we find that coherent detection allows for much lower OSNR for a targeted BER. For 16QAM modulated NFDM signals the BER vs. OSNR results are shown in Fig. 10(b). The OSNR is once more measured for the total signal, but the BER is determined for each channel individually. Due to the roll-off over the spectrum, the SNR of the center channel is higher than that of the outer channels. Thus we give also the mean BER for comparison. The mean BER is nearly identical to the performance of ch2 and ch3, which is once more due to the roll-off of the spectrum. Again measured BER (squares) and equivalent BER (lines) coincide. An increased error floor is observed as compared to the 16QAM BER results with the IM/DD scheme in the previous section, see Fig. 8(b).
The roll-off could be compensated by applying a pre-emphasis. However, this pre-emphasis would reduce the effective number of bits by at least one bit. Taking into account the effective number of bits (ENOB = 5) provided by the DACs, the signal quality would degrade severely by the pre-emphasis according to Fig. 5.
4.3 Remote heterodyne reception of NFDM signals
As a third option next to direct and coherent detection, a remote heterodyne setup is investigated as to be seen in Fig. 11(a). To this end we generate two optical carriers from an ECL source by driving an MZM with an 18.75 GHz sinusoidal tone. The two resulting optical carriers with a spacing of 2 × 18.75 GHz = 37.5 GHz are separated using a Finisar WaveShaper. The lower frequency carrier is modulated with the complex NFDM waveform comprising five QPSK modulated channels. The upper frequency carrier is used for remote heterodyne reception. The combined spectrum of signal and auxiliary carrier is schematically shown as an inset in the upper right of Fig. 11(a). The total signal bandwidth that has to be covered by the PD in the Rx is 50 GHz, i.e., twice the signal bandwidth. An EDFA and a VOA control the signal power Pin fed to the Rx’s semiconductor optical amplifier (SOA). Here we chose an SOA to demonstrate the typical use of such a detection scheme in low-cost applications such as access networks. Finally, the broadband PD converts the optical waveform to an electrical carrier frequency of 37.5 GHz. Complex electrical down-converters select one of the five NFDM channels to be sampled by a 25 GSa/s, 8 GHz real-time oscilloscope. Due to bandwidth limitations we were only able to down-convert two of the five channels (blue and red) centered at electrical carrier frequencies of 31.25 GHz and 34.345 GHz, respectively.
The outcome in terms of measured BER (squares) and equivalent BER from EVM (lines) for different Rx input powers Pin is illustrated in Fig. 11(b). At low input powers the signal quality degrades due to thermal noise that is added by the broad-band electrical amplifier required for amplification of the output electrical signal of the PD. Whereas the increasing BER for high input powers (Pin > −12 dBm) stems from non-linear processes in the SOA . An access network based on NFDM with remote heterodyne reception is investigated in .
Nyquist frequency division multiplexing offers several advantages over OFDM including the transmission of asynchronous channels with variable bandwidth and channel spacing, such enabling flex-grid operation. We discussed the design of NFDM transceivers with time domain and frequency domain digital signal processing. We further experimentally demonstrated non-coherent and coherent transmission links, and we investigated the performance of channels with QPSK and 16QAM modulated signals.
Equivalency of up-converting a shaped sampled signal and up-converting the shaping sampled impulse response
Here we prove the equivalence of mixing a baseband signal u(t) with an oscillator before applying a filter h(t) on one hand, and of the convolution of the same baseband signal with a frequency shifted filter on the other hand:Eq. (7) and filter the periodic spectrum Eq. (12) with the baseband filter . The result is then shifted to by a mixing process. With a proper design of the baseband filter we pick the baseband signal spectrum , and the infinite sum reduces to one term,Eq. (8) numerically to , and then multiply with the periodic signal spectrum Eq. (12). Again, the infinite sum reduces to one term, if the shifted baseband filter is properly designed to select the signal spectrum at ,Fig. 12. In the first case, Eq. (7) and Fig. 12(a), the filter cuts out the partial spectrum centered at frequency f = 0, and the mixing process shifts this partial spectrum by an integer multiple of the sampling frequency . In the second case, Eq. (8) and Fig. 12(b), the filter is centered at an integer multiple of the sampling frequency , and then cuts out the partial spectrum generated by the sampling process at this very frequency. Obviously, the two cases lead to the same final spectrum.
This work was supported by the EU projects ACCORDANCE, OTONES, and FOX-C, the Agilent University Relations Program, the German BMBF project CONDOR, by the Karlsruhe School of Optics & Photonics (KSOP), and by the Helmholtz International Research School for Teratronics (HIRST). We further acknowledge support from Xilinx (XUP), Micram Microelectronic GmbH, Finisar, the Deutsche Forschungsgemeinschaft (DFG), and financial support from the Open Access Publishing Fund of the Karlsruhe Institute of Technology (KIT).
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