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We propose a power-efficient method for transmitting multiple frequency-division multiplexed (FDM) orthogonal frequency-division multiplexing (OFDM) signals in intensity-modulation direct-detection (IM-DD) optical systems. This method is based on quadratic soft clipping in combination with odd-only channel mapping. We show, both analytically and experimentally, that the proposed approach is capable of improving the power efficiency by about 3 dB as compared to conventional FDM OFDM signals under practical bias conditions, making it a viable solution in applications such as optical fiber-wireless integrated systems where both IM-DD optical transmission and OFDM signaling are important.
© 2015 Optical Society of America
1. Introduction
The transmission of complex signals using optics is well known in the industry. The largest example is in hybrid fiber coax networks, where analog optical transmission is used to transmit about 1 GHz bandwidth of video modulated RF subcarriers. More modern systems explore the use of orthogonal frequency division multiplexing (OFDM) to create a modulated signal with useful properties such as easy equalization and optimal bit-loading to fit the channel. In either case, the complex signals have a noise-like characteristic: the amplitude distribution is nearly Gaussian, and the frequency spectrum is mainly flat within the band. This is another way of saying that these signals have a poor peak to average power ratio (PAPR). One of the classic problems of this sort of system is the use of intensity modulation (IM). While IM enables direct detection (DD), it also gives us two constraints on the analog signal being transmitted. The intensity of the light can only be real and positive. The real constraint is most commonly handled by creating spectra with Hermitian symmetry. This comes at the price of spectral efficiency; however, optical links usually have an excess of bandwidth, so this is an easy trade-off to make. The positive constraint is different, because what optics possess in bandwidth they lack in dynamic range and power budget. An ideal IM transmitter will have a linear characteristic for positive inputs, but is clamped at zero for negative inputs (in other words, it is a perfect half-wave rectifier). This results in clipping of the composite modulated signal. The most common way to make the signal positive is to add a large DC offset (so-called DCO-OFDM) before clipping, such that the offset signal, YDC is given by
where XM is the DC bias, which is typically set to be 2~4 standard deviations of the composite modulated signal to make the rectification events rare enough to result in an effective noise floor low enough for the application. The nominal range of YDC is then 0 to 2XM. An example of the YDC transfer characteristic for the case where XM = 3 is shown in Fig. 1.
Fig. 1 Examples of the transfer functions considered in this paper, YDC, YAC, and YSC, where XM = 3, as well as the Probability Density Function (PDF) of a Gaussian distributed signal (μ = 0, σ = 1).
As a practical matter, it must be emphasized that real optical transmitters are not so ideal. For example, directly modulated lasers are not so linear when they approach their threshold current. What is worse, they exhibit a large amount of wavelength chirp near threshold, and this unwanted optical frequency modulation will interact with the dispersive optical channel to produce distortion. Thus, in practical implementations of IM-DD, it is important to constrain the signal waveform during its (mathematical) synthesis, before it is presented to the optical transmitter. This way, the transmitter is kept within its well-behaved domain of operation, and bad performance is avoided.
This paper is organized as follows. In Section 2, we first discuss the working principle of asymmetric clipping [1–3] in improving power efficiency and its limitation in frequency division multiplexed (FDM) systems. In Section 3, we introduce our proposed quadratic “soft” clipping scheme and its applicability in FDM systems. In Section 4, we present the application of the soft clipping scheme to an efficient mobile front-haul system. In Section 5, we conduct numerical simulations to validate the proposed approach. In Section 6, we provide experimental results to further validate the benefit of the soft-clipping technique. Finally, concluding remarks are given in Section 7.
2. Asymmetrical clipping
The inspiration of this work comes from the work of Armstrong et al., who have studied the concept of asymmetrically clipping (AC) of an OFDM signal, termed AC-OFDM [1–3]. The clipped signal YAC as a function of the input signal X is given by
This allows the transmitted signal to have a linear gain (2) given the same range (0 to 2XM), as is shown in the example of the YAC transfer function in Fig. 1 (again for XM = 3). Of course, this extreme nonlinearity produces a large amount of distortion. The clipping function can be decomposed into a linear term and a series of even powered terms. This can be illustrated by considering the Fourier series of the half-wave rectified sine, given in Eq. (2a). One can see the DC and fundamental term, plus a series of the even harmonics of the fundamental. This distortion can be managed using a spectrum plan where only the odd harmonically related carriers are used for transmission. In the limit of narrow modulation bandwidths, the distortion products will be found at the even carriers, and so these do not interfere much with the signals on the odd carriers.A more careful consideration of the effect of modulation bandwidth reveals more. The modulation bandwidth should be maximized, so as to yield the highest data throughput. However, this bandwidth also drives the bandwidth of the distortions products. In the ideal OFDM modulated case, we could consider each carrier to be modulated by a series of symbols, the symbol envelopes being perfect Rect(t/T0) functions. This results in each carrier having a spectrum proportional to the Sinc(f/f0) function. The n’th order distortion products then have a spectrum that is the n’th order convolution of the Sinc function. An interesting property of the Sinc function is that any self convolution of the Sinc is the Sinc once again. That is, the bandwidth of the distortions is the same as the signal. This is what makes the odd channel plan work so well in the OFDM case. The distortion products remain in the even channel spaces, and cause little interference.
Alternatively, in the ideal FDM case, each carrier is modulated with a band-limited modulation signal, with a spectrum proportional to Rect(f/(k⋅f0)), where k is the spectral efficiency factor (k = 1 means full bandwidth). The spectrum of the second order distortion is then the Triangle(f/(k⋅f0)), which has a full width of 2kf0. The fourth order spectrum is a piecewise composition of third order polynomials, and has a full width of 4kf0. The operative fact here is that the Rect() spectrum results in distortion products that get wider and wider, and these cannot be avoided so simply by using the odd carriers only. Our work aims to solve this problem: How to apply the Armstrong-style clipping method to FDM systems?
3. Quadratic “soft” clipping
The first thing we can observe is that the hard clipping function has many higher order terms. These terms have distortion spectral bandwidths that are proportional to the order, and so they quickly contaminate the entire spectrum, rendering it noisier. We want to produce a new clipping function that has fewer higher order terms. The simplest one is a quadratic, given by
If we assume the domain of x is -XM to XM, then YSC has a range of 0 to 2XM, just like the asymmetric clipping function. However, unlike the hard clipping function, this new “soft clipping” function does not have a discontinuity in its slope, and by its very construction it only has the second order distortion term. It also has a DC offset, however, this offset is half of the DCO function with the same output range. The similar example of the YSC transfer characteristic is shown in Fig. 1. Looking at the three transfer functions together, one can see that the soft-clipping is an intermediate between the DC offset and AC clipping ones. As a consequence, the soft-clipping produces intermediate signal gains, but with less overall distortion; in other words, it is a trade-off.The soft clipping function produces an improvement in signal gain for the same average optical power. This is not quite as high as the hard clipping gain, but it is still about 3 dB of improvement. We can analyze our three transmitted signals: YDC, YAC, and YSC. We first assume that X is a Gaussian distributed random signal of mean zero and variance one,symbolized as N0,1(x) (the probability density function of this Gaussian is also shown in Fig. 1, for reference). The distributions of the three transformed output signals are shown in Fig. 2. YDC has simply the same normal distribution, but shifted by the bias level. YAC has one side of the normal distribution, stretched to occupy the same range as the DC case. YSC has a distorted distribution that is skewed towards the left. All of these distributions also have a delta function of probability at zero reflecting the part of the signal that is clipped. From these distributions, one can see how the AC and SC functions work to reduce the average optical power while maintaining the same gain.
Fig. 2 The probability density functions of the three transformed signals. Note that all signals have different magnitude delta functions at zero capturing probability of clipping the signal.
The linear gain is the weighted average of the derivative of the linear term of the transfer function. These are given by
The average transmitted signal can be shown to beA graph of some of these functions and the relative benefit of using the AC and SC schemes is shown in Fig. 3, as a function of the DC offset (XM). Note that we consider the ratio of the gain over the average level as proportional to the signal to noise ratio (SNR) (with constant additive white noise, this is the case). Hence, the ratio of these SNR’s will give the relative benefit of one scheme over the other. We also consider the DCO scheme to be the reference scheme, so its benefit is one. The AC scheme has constant gain and average level, as it does not have an offset. Both the DC and SC schemes have a nearly constant gain. The DC scheme’s average level increases linearly with offset, while the SC scheme increases at a slower rate. As a result, the SC scheme has a benefit of around 2.5 dB at a practical offset of 3, and converges to a benefit of 3 dB in the limit of large offset. The AC scheme has a much larger and ever growing benefit, but at the practical offsets (or practical bias conditions) its benefit is about 5 dB.Fig. 3 Linear gains, power averages, and power-efficiency gains (benefits) of various OFDM formats. DCO: DC-offset OFDM; AC: asymmetrically-clipped OFDM [1–3]; SC: soft-clipped OFDM proposed here.
4. Application to efficient mobile front-haul
Collaborative radio access network (C-RAN) architecture is attractive for next-generation wireless networks [4]. It improves network performance via coordinated multi-point. Mobile fronthaul is a key C-RAN segment between centralized baseband units (BBUs) and remote radio units (RRUs) [5,6], and is primarily based on the common public radio interface (CPRI) that transmits digitized baseband signals via on-off-keying (OOK) [7]. For a typical system configuration using 2 × 2 multiple-input and multiple-output (MIMO) and 3 directional sector antennas, CPRI would require a fronthaul data rate of about 36.86 Gb/s. To support this high data rate using OOK, expensive high-speed optical transmitter and receiver are required. For a typical 40-km fronthaul with standard single-mode fiber (SSMF), optical dispersion compensation will be also required for 36.86-Gb/s OOK transmission, which further increases the fronthaul complexity and cost. To achieve efficient mobile fronthaul, transmission of multiple FDM OFDM signals in a single wavelength channel has recently been demonstrated [8, 9]. Dispersion-penalty-free transmission of six 100-MHz-bandwidth LTE-A-like signals with 36.86-Gb/s CPRI-equivalent data rate over a 40-km SSMF fronthaul was demonstrated by using a single 1550-nm directly-modulated laser (DML) with a modulation bandwidth of only 2 GHz [8]. The dispersion penalty due to the interplay between the fiber chromatic dispersion and the chirp of the DML [10, 11] is avoided by a novel dispersion-penalty mitigation technique based on odd-channel-only mapping in frequency-division multiplexing [8].
Figure 4 shows the architecture of the efficient mobile fronthaul as reported in [8, 9]. At the remote site, multiple RRUs interface with multiple antennas. For downlink transmission, the baseband signals generated in a C-RAN central office are inputted to a DSP-based channel aggregator for channel aggregation. The aggregated digital signal is then converted to an analog signal by a digital-to-analog converter (DAC) to drive an optical transmitter to generate a wavelength channel. The wavelength channel can be further multiplexed with other optical signals by a wavelength-division multiplexer before being transmitted over an optical fiber link. Near the intended RRU site, the wavelength channel is dropped by a wavelength-division de-multiplexer before being received by an optical receiver. The received signal is then digitized by an ADC and de-aggregated by a DSP-based channel de-aggregator to produce the original baseband signals, which are then provided to their corresponding RRUs for radio transmission via the antennas. For uplink transmission, the reverse of the above processes is applied.
Fig. 4 Schematic of the bandwidth-efficient mobile fronthaul architecture with DSP-based channel aggregation and de-aggregation as reported in [8, 9]. Inset: measured optical spectra of the aggregated signals under the odd-channel-only mapping. BBU: baseband unit; RRU: remote radio unit; DAC: digital-to-analog converter; ADC: analog-to-digital converter; TX: optical transmitter; RX: optical receiver; WDM: wavelength division multiplexer.
It is desirable to improve the power efficiency of FDM OFDM signals, so that the efficient mobile fronthaul scheme [8,9] can be readily supported by conventional passive optical networks [12]. Although AC provides high power efficiency of single-channel OFDM, as discussed in the previous sections, it is not viable in FDM-OFDM with multiple channels, even when odd-channel-only mapping is applied. This is because the higher even-order harmonics generated by AC spread ever wider in frequency, and serve to contaminate the odd channels with distortion. On the other hand, our proposed quadratic SC function can avoid this.
Returning to the spectrum plan, only the odd channels will be used, and centered at frequencies (2n-1)⋅f0 where n is a positive integer; however, their bandwidth will be constrained to only 2/3 of the full amount, which is typical for wireless channels. That is, the channel at f0 will have a band that runs from 2/3f0 to 4/3f0. The channel at 3f0 has a band that runs from 8/3f0 to 10/3f0. Such an arrangement will yield gaps between adjacent channels of 4/3f0. This will be the spectral width of the 2nd-order distortion term. Indeed, all the 2nd-order distortions fall exactly in these gaps, leaving the spectrum around the channels to be clear of impairment. This result has been tested by both numerical simulation and experiment, which will be presented in the following sections.
5. Simulation results
We first perform numerical simulations to verify the analytical results presented above. Figure 5(a) shows the simulated spectrum of 24 20-MHz LTE signals that are aggregated using the efficient mobile fronthaul approach with the odd-channel-only mapping and under the DCO bias condition. Each LTE signal is an OFDM signal with an IFFT size of 2048 and with 1200 64-QAM subcarriers used to carry payload data. Due to Hermitian symmetry used for IM/DD, there are 24 images of the original 24 signals. In this case, no inter-signal mixing is observed. Figure 5(b) shows the recovered constellation of the 24th LTE signal (the signal with the highest frequency after channel aggregation) in the back-to-back configuration (without fiber transmission). A low error vector magnitude (EVM) of 0.43% is obtained. Figure 6(a) shows the simulated spectrum of the same aggregated LTE signals under the AC bias condition in the back-to-back configuration. Large inter-signal mixing is observed, due to even-order distortions that are higher than the 2nd order, e.g. the 4th-order distortion, as discussed in the previous section. The 4th-order distortions spread more than the spectral gaps between signals, and will cause inter-signal interference that degrades signal quality. Figure 6(b) shows the recovered constellation of the 24th LTE signal. Indeed, the signal EVM is much degraded to 7.7%, confirming that the asymmetric clipping [1–3] is not suitable for FDM OFDM. Figure 7(a) shows the simulated spectrum of the same aggregated LTE signals under the proposed SC bias condition in the back-to-back configuration. Evidently, inter-signal mixing only causes distortions in the spectral gaps between signals, avoiding performance degradation to actual signals. Figure 7(b) shows the recovered constellation of the 24th LTE signal. A low EVM of 0.5% is obtained, confirming that the soft clipping is viable for FDM OFDM.
Fig. 5 (a) Simulated optical spectrum of 24 20-MHz LTE signals (and their images due to Hermitian symmetry) that are aggregated using the odd-channel-only mapping under the DCO bias condition in the back-to-back configuration; (b) Recovered constellation of the highest-frequency (24th) signal.
Fig. 6 (a) Simulated optical spectrum of 24 20-MHz LTE signals (and their images due to Hermitian symmetry) that are aggregated using the odd-channel-only mapping under the AC bias condition in the back-to-back configuration; (b) Recovered constellation of the highest-frequency (24th) signal.
Fig. 7 (a) Simulated optical spectrum of 24 20-MHz LTE signals (and their images due to Hermitian symmetry) that are aggregated using the odd-channel-only mapping under the proposed SC bias condition in the back-to-back configuration; (b) Recovered constellation of the highest-frequency (24th) signal.
6. Experimental results
We then perform experiments to further verify the analytical results. Figure 8 shows the experimental setup, which is similar to that used in [8, 9]. At the transmitter, we use offline DSP to generate 24 20-MHz LTE OFDM signals. The modulation format is OFDM with 64-QAM subcarrier modulation, which is the highest level modulation specified in LTE. The time-domain signal waveform is stored in an arbitrary waveform generator and outputted by a 5-GSa/s 8-bit DAC. This analog signal is then amplified before driving a 1550-nm DML with a modulation bandwidth of about 2 GHz. Odd-channel-only mapping is used. The center frequencies of the signals after aggregation are (2n-1) × 30.72MHz, where n = 1,2,3,..24, as shown in the measured spectrum in Fig. 9(a). We compare two bias conditions, the conventional DCO bias condition and the proposed SC bias condition. The optical signal is launched into a 20-km standard single-mode fiber (SSMF). After fiber transmission, a variable optical attenuator (VOA) is used to vary the optical power (P) received by an avalanche photodiode (APD). The detected signal is digitized by a 10-GSa/s 8-bit ADC in a real-time sampling scope. The digitized samples are stored in the scope, and later processed by offline DSP for down-sampling, channel de-aggregation, OFDM demodulation, and evaluation of signal error vector magnitude and bit error ratio (BER).
Fig. 9 (a) Experimentally measured spectrum of the aggregated signals after 20-km SSMF transmission with a received power of −22 dBm and (b) Measured EVM as a function of received power, all under the DCO condition.
Figure 9 shows the experimentally measured spectrum of the aggregated signals after 20-km SSMF transmission with P = −22 dBm (a), and the measured EVM as a function of receiver power (b), all under the DCO condition. The spectral power in Fig. 9(a) is normalized to the power at the center frequency. For EVM to be less than 5% (a reasonably low value [7]), the received power needs to be higher than about −19 dBm. Figure 10 shows the experimentally measured spectrum of the aggregated signals after 20-km SSMF transmission with P = −22 dBm (a), and the measured EVM as a function of receiver power (b), all under the proposed SC condition. For EVM to be less than 5%, the received power needs to be higher than about −22 dBm. This means that the SC method offers a power budget improvement of about 3 dB as compared to the conventional DCO method. It is in good agreement with the analytical results presented in Section III.
Fig. 10 (a) Experimentally measured spectrum of the aggregated signals after 20-km SSMF transmission with a received power of −22 dBm and (b) Measured EVM as a function of receiver power, all under the proposed SC condition.
7. Conclusion
We have proposed a power-efficient method for both single OFDM signal and multiple FDM OFDM signals based on quadratic soft-clipping. We have also conducted analytical, numerical, and experimental studies to show that the proposed approach is capable of improving the power efficiency by about 3 dB as compared to conventional FDM OFDM signals under practical bias conditions. With its performance advantage and implementation simplicity, this novel soft-clipping technique is deemed to be viable in applications such as optical fiber-wireless integrated systems where both IM-DD optical transmission and OFDM signaling are important.
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