The multispan effects of Kerr nonlinearity and amplifier noises onShannon channel capacity of a dispersion-free nonlinear optical fiber

We present a general treatment of multispan effects on Shannon channel capacity for dispersion-free nonlinear optical fiber transmission. We have derived a closed-form formula that is exact and applicable for arbitrary input signal power and noise power. The nonlinear interference among the accumulated Kerr nonlinear noises and the amplifier noises along each span will be investigated. The dependence of the channel capacity on various operating renditions and system parameters will be studied. We have derived a simple scaling law for the Shannon capacity as given by log, [1+CN_s^-4/3 N_c^-2/3γ^-2/3P_W ^-2/3] for Nb spans and N_c channels, where γ is the Kerr nonlinearity coefficient and P_W is the average noise power density for each amplifier     

The capacity of downlink fading channels with variable rate andpower

We obtain the Shannon capacity region of the down-link (broadcast) channel in fading and additive white Gaussian noise (AWGN) for time-division, frequency-division, and code-division. For all of these techniques, the maximum capacity is achieved when the transmitter varies the data rate sent to each user as their channels vary. This optimal scheme requires channel estimates at the transmitter; dynamic allocation of timeslots, bandwidth, or codes; and variable-rate and power transmission. For both AWGN and fading channels, nonorthogonal code-division with successive decoding has the largest capacity region, while time-division, frequency-division, and orthogonal code-division have the same smaller region. However, when all users have the same average received power, the capacity region for all these techniques is the same. In addition, the optimal nonorthogonal code is a multiresolution code which does not increase the signal bandwidth. Spread-spectrum code-division with successive interference cancellation has a similar rate region as this optimal technique, however, the region is reduced due to bandwidth expansion. We also examine the capacity region of nonorthogonal code-division without interference cancellation and of orthogonal code-division when multipath corrupts the code orthogonality. Our results can be used to bound the spectral efficiency of the downlink channel using time-division, frequency-division, and code-division, both with and without multiuser detection     

The capacity region of broadcast channels with intersymbolinterference and colored Gaussian noise

We derive the capacity region for a broadcast channel with intersymbol interference (ISI) and colored Gaussian noise under an input power constraint. The region is obtained by first defining a similar channel model, the circular broadcast channel, which can be decomposed into a set of parallel degraded broadcast channels. The capacity region for parallel degraded broadcast channels is known. We then show that the capacity region of the original broadcast channel equals that of the circular broadcast channel in the limit of infinite block length, and we obtain an explicit formula for the resulting capacity region. The coding strategy used to achieve each point on the convex hull of the capacity region uses superposition coding on some or all of the parallel channels and dedicated transmission on the others. The optimal power allocation for any point in the capacity region is obtained via a multilevel water-filling. We derive this optimal power allocation and the resulting capacity region for several broadcast channel models     

Space–Time Power Schedule for Distributed MIMO Links Without Instantaneous Channel State Information at the Transmitting Nodes

A space-time optimal power schedule for multiple distributed multiple-input multiple-output (MIMO) links without the knowledge of the instantaneous channel state information (CSI) at the transmitting nodes is proposed. A readily computable expression for the ergodic sum capacity of the MIMO links is derived. Based on this expression, which is a non-convex function of power allocation vectors, a projected gradient algorithm is developed to optimize the power allocation. For a symmetric set of MIMO links with independent identically distributed channels, it is observed that the space-time optimal power schedule reduces to a uniform isotropic power schedule when nominal interference is low, or to an orthogonal isotropic power schedule when nominal interference is high. Furthermore, the transition region between the latter two schedules is seen to be very sharp in terms of nominal interference-to-noise ratio (INR). For MIMO links with correlated channels, the corresponding space-time optimal power schedule is developed based on the knowledge of the channel correlation matrices. It is shown that the channel correlation has a great impact on the ergodic capacity and the optimality of different power scheduling approaches.     

WDM systems with unequally spaced channels

Crosstalk due to four-wave mixing (FWM) is the dominant nonlinear effect in long-haul multichannel optical communication systems employing dispersion-shifted fiber. A method is discussed to find non-uniform channel separations for which no four-wave mixing product is superimposed on any of the transmitted channels, therefore suppressing FWM crosstalk. The residual crosstalk, due to channel power depletion only, is analytically evaluated for intensity-modulated repeaterless wavelength-division-multiplexed (WDM) systems and compared to experimental results. The theory includes the effect of the channel depletion on the amplitude of each phase-matched FWM wave. The probability of error is evaluated including the statistics of the pattern dependent channel depletion. The BER curve computed for an 8-channel WDM system is found to be in good agreement with experimental results. In the experiment, repeaterless transmission of eight 10 Gb/s WDM channels over 137 km (11 Tb/s-km) of dispersion-shifted fiber was demonstrated and error-free operation was achieved over a wide range of input powers using unequally spaced channels. The same system with equally spaced channels could not achieve a probability of error lower than 10^-6. The use of unequal channel spacing allowed fiber input power to be increased by as much as 7 dB, which could be translated into a fivefold increase of the bit rate per channel (and therefore of the system capacity), or to an increase in the system length of about 30 km     

一种正交频分复用系统的非线性补偿技术

针对正交频分复用(OFDM)系统在功率放大器(PA)非线性较强时的性能问题,基于一种无线设备非线性与无线信道的联合估计技术,提出了一种基于训练序列的OFDM非线性信道估计与补偿技术.首先基于最小二乘(LS)算法进行发射机非线性与无线信道单位脉冲响应的联合估计,然后依次进行无线信道与发射机非线性的补偿.仿真结果显示,提出方法可逼近不考虑PA非线性时OFDM无线通信系统的完美均衡理论解析值.     

A Nash game approach for OSNR optimization with capacity constraint in optical links

This paper develops a Nash game towards optimizing optical signal-to-noise ratio (OSNR) in the presence of link capacity constraint. In optical wavelength-division multiplexing (WDM) networks, all wavelength-multiplexed channels in a link share the optical fiber. In order to limit nonlinear effects, the total power launched into a fiber has to be below a nonlinearity threshold. This can be regarded as the optical link capacity constraint. We formulate an extended OSNR Nash game that incorporates this underlying system constraint. The status of an optical link is considered directly in the cost function. Sufficient conditions for existence and uniqueness of the Nash equilibrium (NE) solution are given. Two iterative algorithms for channel power control are proposed to compute the NE solution: a parallel update algorithm (PUA) and a relaxed PUA (r-PUA). Their convergence is studied under different conditions, both theoretically and numerically.     

A Hammerstein-type equalizer for concatenated fiber-wireless uplink

In optical fiber-based wireless access schemes, the radio signal is transmitted through fiber without frequency conversion radio-over-fiber (ROF). Although the fiber has adequate bandwidth, nonlinear distortion due to electrical to optical (E/O) conversion is a concern. In the uplink, the dynamic multipath wireless channel is followed by this static memoryless ROF link; this forms a Wiener system. In this paper, we propose a Hammerstein type decision feedback equalizer (HDFE) for the fiber-wireless uplink to combat the nonlinear distortion and the wireless channel dispersion. The proposed equalizer is less complex because it handles static and dynamic distortions separately. The nonlinear distortion is compensated first, reducing the power of cross modulation products significantly. Analytical results show that the lower bound of the mean squared error depends on the optical and wireless channel noise. The bit error rate (BER) performance of the HDFE for the nonlinear channel approaches the performance of a decision feedback equalizer (DFE) in a linear channel when the nonlinearity is adequately compensated.     

A New Outer Bound and the Noisy-Interference Sum–Rate Capacity for Gaussian Interference Channels

A new outer bound on the capacity region of Gaussian interference channels is developed. The bound combines and improves existing genie-aided methods and is shown to give the sum–rate capacity for noisy interference as defined in this paper. Specifically, it is shown that if the channel crosstalk coefficient magnitudes lie below thresholds defined by the power constraints then single-user detection at each receiver is sum–rate optimal, i.e., treating the interference as noise incurs no loss in performance. This is the first capacity result for the Gaussian interference channel with weak to moderate interference. Furthermore, for certain mixed (weak and strong) interference scenarios, the new outer bounds give a corner point of the capacity region.      

The Capacity Region of Frequency-Selective Gaussian Interference Channels Under Strong Interference

This paper presents the capacity region of frequency-selective Gaussian interference channels under the condition of strong interference, assuming an average power constraint per user. First, a frequency-selective Gaussian interference channel is modeled as a set of independent parallel memoryless Gaussian interference channels. Using nonfrequency selective results, the capacity region of frequency-selective Gaussian interference channels under strong interference is expressed mathematically. Exploiting structures inherent in the problem, a dual problem is constructed for each independent memoryless channel, in which both mathematical and numerical analysis are performed. Furthermore, three suboptimal methods are compared to the capacity-achieving coding and power allocation scheme. Iterative waterfilling, a suboptimal scheme, provides close-to-optimum performance and has a distributed coding and power allocation scheme, which are attractive in practice.     

Capacity Regions and Bounds for a Class of Z-Interference Channels

We define a class of Z-interference channels for which we obtain a new upper bound on the capacity region. The bound exploits a technique first introduced by Korner and Marton. A channel in this class has the property that, for the transmitter-receiver pair that suffers from interference, the conditional output entropy at the receiver is invariant with respect to the transmitted codewords. We compare the new capacity region upper bound with the Han/Kobayashi achievable rate region for interference channels. This comparison shows that our bound is tight in some cases, thereby yielding specific points on the capacity region as well as sum capacity for certain Z-interference channels. In particular, this result can be used as an alternate method to obtain sum capacity of Gaussian Z-interference channels. We then apply an additional restriction on our channel class: the transmitter-receiver pair that suffers from interference achieves its maximum output entropy with a single input distribution irrespective of the interference distribution. For these channels, we show that our new capacity region upper bound coincides with the Han/Kobayashi achievable rate region, which is therefore capacity-achieving. In particular, for these channels superposition encoding with partial decoding is shown to be optimal and a single-letter characterization for the capacity region is obtained.     

A comparison study of the Shannon channel capacity of various nonlinear optical fibers

A comparative study of the Shannon channel capacity is presented for a dispersion-free fiber, a fiber with constant dispersion, and a fiber with variable dispersion. Improvement of the capacity by optical phase conjugation (OPC) is also investigated. Simple scaling laws are prescribed for the dependence of the optimal capacity on various system settings such as number of spans, number of channels, noise power, channel width, strength of chromatic dispersion, bandwidth of an OPC device, etc.     

Dependence of cross-phase modulation on channel number in fiber WDMsystems

The phase term appearing in the expression for cross-phase modulation due to the optical Kerr effect depends on the sum of the powers carried by each wavelength channel. For this reason, one might expect that the amount of cross-phase modulation would increase with increasing channel number, causing increased interference among channels and hence limiting the total number of channels that a WDM system can support. However, computer simulations of multichannel systems have shown no change in signal distortion as the number of wavelength channels is increased from four to eight. In a simulated three-channel system, the signal distortion of the central channel approaches that of a single-channel system as the wavelength separation is increased to approximately 2 nm. Thus, even a moderate amount of dispersion tends to cancel out the influence of cross-phase modulation, so that beyond a certain wavelength spacing, additional channels do not interfere with the channel under consideration. From these observations, we conclude that cross-phase modulation does not limit the number of wavelength channels that a single optical fiber can support. However, self- and cross-phase modulation are not the only nonlinear effects influencing fiber lightwave systems. Stimulated Raman scattering tends to transfer optical power from short-wavelength channels to channels operating at longer wavelength, degrading their signal-to-noise ratio. The efficiency of this process increases with increasing wavelength spacing. Clearly, a compromise needs to be reached between the conflicting requirements imposed by the optical Kerr effect and by stimulated Raman scattering     

一种基于干扰温度限制的信道与功率联合分配新算法

随着无线通信业务的不断增长,频谱资源越来越紧缺,然而另一方面大量授权的无线频谱却被闲置或者利用率极低,于是认知无线电技术应运而生,已成为无线通信领域的研究热点。认知无线电的基本思想是次用户（认知用户）利用主用户（授权用户）未占用的空闲频谱进行通信,其可用无线资源是根据授权用户的频谱使用情况而动态变化的。因此,能否实现对系统可用无线资源的合理有效管理,对整个认知无线电系统性能的优劣起着决定性作用。本文提出了一种在干扰温度限制下基于公平的功率与信道联合分配算法,该算法在主用户干扰温度及次用户发射功率的双重限制下,以最大化系统容量为基本目标,实现信道与功率的联合分配,并且引入贫困线来保证各个用户信道分配的公平性。论文建立了该问题的非线性规划数学模型,给出了模型的求解方法,并进一步设计了具体分配算法及其步骤。论文对干扰门限分别为-90dBm、-95dBm、-100dBm、-105dBm、-110dBm时的系统归一化容量累积分布函数进行了仿真比较,发现当干扰门限越低时,本文算法的优势越明显。这是因为在干扰门限较低时,干扰温度限制是功率分配的主要制约因素,而本文的算法正是基于干扰门限进行分配的。因此基于干扰温度限制的公平的功率与信道联合分配算法具有良好的性能,在保证了系统的公平性效益的同时,提高了系统的归一化容量。      

无色散光纤信道的非线性演化

在单模光纤中由于非线性效应和拉曼增益效应的共同作用,导致光子在各向同性介质中传输时满足非线性薛定谔方程。利用随机微分方程研究了长距离光纤通信中噪声对光纤信道的影响,给出了光纤信道的动力学机理模型。首先在非线性薛定谔方程的基础上引入噪声项,然后利用It公式将其整理成极坐标系下标准的随机微分方程组,最后利用福克尔-普朗克(Fokker-Planck)方程得到了光脉冲在光纤信道中的概率密度函数,精细地研究了光纤信道的非线性演化规律。即在加入噪声项的情况下,分析了光纤通信的传输性能指标,得到了概率密度函数。     

The Shannon channel capacity of dispersion-free nonlinear opticalfiber transmission

We extend the Shannon information theory to a nonlinear system and present a model for calculating the channel capacity of an optical-fiber transmission system using dispersion-free fiber. For this particular fiber, a closed-form solution for the nonlinear Schroedinger equation exists. This allows us to derive an analytical result for the channel capacity that is exact and valid for arbitrary input power. We will study the single-span case and examine the dependence of the capacity on operating input power, the number of channels (N_c), the noise power (P_W), etc. The maximum capacity is shown to follow a simple scaling law with log₂(1+CN_c^-2/3P _W^-2/3) dependence, multiplexing (WDM) systems     

Performance analysis of deliberately clipped OFDM signals

We analyze the performance of the clipped orthogonal frequency division multiplexing (OFDM) system in terms of peak power reduction capability and degradation of channel capacity. The clipping is performed on the baseband OFDM signals with and without oversampling, followed by the ideal low-pass filter. First, the effect of the envelope clipping on the peak-to-average power ratio (PAPR) and the instantaneous power of the band-limited OFDM signal is studied. We then discuss the channel capacity of the oversampled and clipped OFDM signals over the additive white Gaussian noise and ideally interleaved Rayleigh fading channels. The capacity is calculated based on the assumption that the distortion terms caused by the clipping are Gaussian. It is shown that the SNR penalty due to the clipping can be considerably alleviated by using optimal coding and reducing the information data rate. The results are justified by the simulation results using near optimal turbo codes     

Capacity and optimal resource allocation for fading broadcastchannels .II. Outage capacity

For pt.I see ibid., vol.47, no.3, p.1083-1102 (2002). We study three capacity regions for fading broadcast channels and obtain their corresponding optimal resource allocation strategies: the ergodic (Shannon) capacity region, the zero-outage capacity region, and the capacity region with outage. In this paper, we derive the outage capacity regions of fading broadcast channels, assuming that both the transmitter and the receivers have perfect channel side information. These capacity regions and the associate optimal resource allocation policies are obtained for code division (CD) with and without successive decoding, for time division (TD), and for frequency division (FD). We show that in an M-user broadcast system, the outage capacity region is implicitly obtained by deriving the outage probability region for a given rate vector. Given the required rate of each user, we find a strategy which bounds the outage probability region for different spectrum-sharing techniques. The corresponding optimal power allocation scheme is a multiuser generalization of the threshold-decision rule for a single-user fading channel. Also discussed is a simpler minimum common outage probability problem under the assumption that the broadcast channel is either not used at all when fading is severe or used simultaneously for all users. Numerical results for the different outage capacity regions are obtained for the Nakagami-m (1960) fading model     

Spectrum Efficiency Evaluation with Diversity Combining for Fading and Branch Correlation Impairments

In this paper, closed-form expressions for capacities per unit bandwidth for fading channels with impairments due to Branch Correlation are derived for optimal power and rate adaptation, constant transmit power, channel inversion with fixed rate, and truncated channel inversion policies for maximal ratio combining diversity reception case. Closed-form expressions for system spectrum efficiency when employing different adaptation policies are derived. Analytical results show accurately that optimal power and rate adaptation policy provides the highest capacity over other adaptation policies. In the case of errors due to branch correlation, optimal power and rate adaptation policy provides the best results. All adaptation policies suffer no improvement in channel capacity as the branch correlation is increased. This fact is verified using various plots for different policies. With increase in branch correlation, capacity gains are significantly larger for optimal power and rate adaptation policy as compared to the other policies. The outage probability for branch correlation is also derived and analyzed using plots for the same.     

A generalized suboptimum unequally spaced channel allocationtechnique. I. In IM/DD WDM systems

Four-wave mixing (FWM) is the most serious fiber nonlinearity associated with low-input optical power levels in long-haul multichannel optical systems employing dispersion-shifted fiber. To reduce the crosstalk due to FWM, a generalized suboptimum unequally spaced channel allocation (S-USCA) technique is proposed and investigated. Even though the developed technique is useful in combating FWM crosstalk in wavelength division multiplexing (WDM) lightwave systems with up to 12 channels, its main virtue is in designing multichannel WDM lightwave systems with more than 12 channels. Comparisons of power penalty due to FWM between equal channel spacing (ECS) systems and the S-USCA systems are presented. It is shown that for an intensity modulation/direct detection (IM/DD) transmission system operating in an optical bandwidth of 16 nm with 0 dBm (1 mW) peak optical input power per channel, while a conventional ECS WDM system with 0.84-nm channel spacing cannot even achieve a bit-error rate (BER)=10^-9, the suboptimum technique developed in this paper, for the same minimum channel spacing, can achieve a BER=10^-9 with an FWM crosstalk power of less than 1 dB at the worst channel in a 20-channel WDM system