Communication for wideband fading channels : on theory and practice

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 163-167).This dissertation investigates some information theoretic aspects of communication over wideband fading channels and their applicability to design of signaling schemes approaching the wideband capacity limit. This work thus leads to enhanced understanding of wideband fading channel communication, and to the proposal of novel efficient signaling schemes, which perform very close to the optimal limit. The potential and limitations of such signaling schemes are studied. First, the structure of the optimal input signals is investigated for two commonly used channel models: the discrete-time memoryless Rician fading channel and the Rayleigh block fading channel. When the input is subject to an average power constraint. it is shown that the capacity-achieving input amplitude distribution for a Rician channel is discrete with a finite number of mass points in the low SNR regime. A similar discrete structure for the optimal amplitude is proven to hold over the entire SNR range for the average power limited Rayleigh block fading channel. Channels with a peak power constraint are also analyzed. When the input is constrained to have limited peak power, we show that if its Kuhn-Tucker condition satisfies a sufficient condition, the optimal input amplitude is discrete with a finite number of values.(cont.) In the low SNR regime, the discrete structure becomes binary. Next, we consider signaling over general fading models. Multi-tone FSK, a signaling scheme which uses low duty cycle frequency-shift keying signals (essentially orthogonal binary signals, is proposed and shown to be capacity achieving in the widceband limit. Transmission of information over wideband fading channels using Multi-tonc FSK is considered by using both theoretic analysis and numerical simulation. With a finite bandwidth and noncoherent detection, the achievable data rate of the Multi-tone FSK scheme is close to the wideband capacity limit. furthermore, a feedback scheme is proposed for Multi-tone FSK to improve the codeword error performance. It is shown that if the receiver can feedback received signal quality to the transimitter. a significant improvement in codeword error probability can be achieved. Experimental results are also obtained to dlenlonstrate features and practicality of Multi-tone FSK.by Cheng Luo.Ph.D

Unified Capacity Limit of Non-coherent Wideband Fading Channels

In non-coherent wideband fading channels where energy rather than spectrum is the limiting resource, peaky and non-peaky signaling schemes have long been considered species apart, as the first approaches asymptotically the capacity of a wideband AWGN channel with the same average SNR, whereas the second reaches a peak rate at some finite critical bandwidth and then falls to zero as bandwidth grows to infinity. In this paper it is shown that this distinction is in fact an artifact of the limited attention paid in the past to the product between the bandwidth and the fraction of time it is in use. This fundamental quantity, called bandwidth occupancy, measures average bandwidth usage over time. For all signaling schemes with the same bandwidth occupancy, achievable rates approach to the wideband AWGN capacity within the same gap as the bandwidth occupancy approaches its critical value, and decrease to zero as the occupancy goes to infinity. This unified analysis produces quantitative closed-form expressions for the ideal bandwidth occupancy, recovers the existing capacity results for (non-)peaky signaling schemes, and unveils a trade-off between the accuracy of approximating capacity with a generalized Taylor polynomial and the accuracy with which the optimal bandwidth occupancy can be bounded.Comment: Accepted for publication in IEEE Transactions on Wireless Communications. Copyright may be transferred without notic

Multiuser non coherent massive MIMO schemes based on DPSK for future communication systems

The explosive usage of rich multimedia content in wireless devices has overloaded the communication networks. Moreover, the fifth generation (5G) of wireless communications involves new requirements in the radio access network (RAN) which require higher network capacities and new capabilities such as ultra-reliable and low-latency communication (URLLC), vehicular communications or augmented reality. All this has encouraged a remarkable spectrum crisis in the RF bands. A need for searching alternative techniques with more spectral efficiency to accommodate the needs of future emerging wireless communications is emerging. In this context, massive MIMO (m-MIMO) systems have been proposed as a promising solution for providing a substantial increase in the network capacity, becoming one of the key enabling technologies for 5G and beyond. m-MIMO provides high spectral- and energy-efficiency thanks to the deployment of a large number of antennas at the BS. However, we have to take into account that the current communication technologies are based on coherent transmission techniques so far, which require the transmission of a huge amount of signaling. This drawback is escalating with the excessive available number of antennas in m-MIMO. Therefore, the differential encoding and non coherent (NC) detection are an alternative solution to circumvent the drawbacks of m-MIMO in coherent systems. This Ph.D. Thesis is focused on signal processing techniques for NC detection in conjunction with m-MIMO, proposing new constellation designs and NC detection algorithms, where the information is transmitted in the signal differential phase. First, we design new constellation schemes for an uplink multiuser NC m-MIMO system in Rayleigh fading channels. These designs allow us to separate the users' signals at the receiver thanks to a one-to-one correspondence between the constellation for each user and the received joint constellation. Two approaches are considered in terms of BER: each user achieves a different performance and, on the other hand, the same performance is provided for all users. We analyze the number of antennas needed for those designs and compare to the required number by other designs in the literature. It is shown that our designs based on DPSK require a lower number of antennas than that required by their counterpart schemes based on energy. In addition, we compare the performance to their coherent counterpart systems, resulting NC-m-MIMO based on DPSK capable of outperforming the coherent systems with the suitable designs. Second, in order to reduce the number of antennas required for a target performance we propose a multi-user bit interleaved coded modulation - iterative decoding (BICM-ID) scheme as channel coding for a NC-m-MIMO system based on DPSK. We propose a novel NC approach for calculating EXIT curves based on the number of antennas. Then using the EXIT chart we find the best channel coding scheme for our NC-m-MIMO proposal. We show that the number of users served by the BS can be increased with a 70% reduction in the number of antennas with respect to the case without channel coding. In particular, we show that with 100 antennas for error protection equal design for all users and a coding rate of 1/2 we achieve the minimum probability of error. Third, we consider that current scenarios such as backhaul wireless systems, rural or suburban environments, and even new device-to-device (D2D) communications or the communications in higher frequencies (millimeter and the emerging ones in terahertz frequencies) can have a predominant line-of-sight (LOS) component, modeled by Rician fading. For all these new possible scenarios in 5G, we analyze the behavior of the NC m-MIMO systems when we have a Rician fading. We present a new constellation design to overcome the problem of the LOS channel component, as well as an associated detection algorithm to separate each user in reception taking into account the characterization of the constellation. In addition, for contemplating a more realistic scenario, we propose grouping users which experience a Rayleigh fading with those with Rician fading, analyzing the SINR and the performance of such combination in a multi-user NC m-MIMO system based on M-DPSK. The adequate user grouping allows unifying the constellation for both groups of users and the detection algorithm, reducing the complexity of the receiver. Also, the number of users that may be multiplexed may be further increased thanks to the improved performance. In the fourth part of this Thesis, we analyse the performance of multi-user NC m- MIMO based on DPSK in real environments and practical channels defined for the current standards such as LTE, the future technologies such as 5G and even for communications in the terahertz band. For this purpose, we use a metric to model the time-varying characteristics of the practical channels. We employ again the EXIT charts tool for analyzing and designing iteratively decoded systems. This analysis allows us to obtain an estimate of the degradation of the system's performance imposed by realistic channels. Hence, we show that our proposed system is robust to temporal variations, thus it is more recommendable the employment of NC-m-MIMO-DPSK in the future communication standards such as 5G. In order to reduce he number of hardware resources required in terms of RF chains, facilitating its implementation in a real system, we propose incorporating differential spatial modulation (DSM). We present and analyze a novel multiuser scheme for NC-m-MIMO combined with DSM with which we can see that the number of antennas is not a affected by the incorporation of DSM, even we have an improvement on the performance with respect to the coherent case. Finally, we study the viability of multiplexing users by constellation schemes against classical multiplexing techniques such as time division multiple access (TDMA). In order to fully characterize the system performance we analyze the block error rate (BLER) and the throughput of a NC-m-MIMO system. The results show a significant advantage regarding the number of antennas for multiplexing in the constellation against TDMA. However, in some cases, the demodulation of multiple users in constellation could require an excessively large number of antennas compared to TDMA. Therefore, it is necessary to properly manage the tradeoff between throughout and the number of antennas, to reach an optimal operational point, as shown in this Thesis.El inmenso uso de contenido multimedia en los dispositivos inalámbricos ha sobrecargado las redes de comunicaciones. Además, la quinta generación (5G) de sistemas de comunicaciones demanda nuevos requisitos para la red de acceso radio, la cual requiere ofrecer capacidades de red mayores y nuevas funcionalidades como comunicaciones ultra fiables y con muy poca letancia (URLLC), comunicaciones vehiculares o aplicaciones como la realidad aumentada. Todo esto ha propiciado una crisis notable en el espectro electromagnético, lo que ha llevado a una necesidad por buscar técnicas alternativas con más eficiencia espectral para acomodar todos los requisitos de las tecnologías de comunicaciones emergentes y futuras. En este contexto, los sistemas multi antena masivos, conocidos como massive MIMO, m-MIMO, han sido propuestos como una solución prometedora que proporciona un incremento substancial de la capacidad de red, convirtiéndose en una de las tecnologías claves para el 5G. Los sistemas m-MIMO elevan enormemente el número de antenas en la estación base, lo que les permite ofrecer alta eficiencia espectral y energética. No obstante, tenemos que tener en cuenta que las actuales tecnologías de comunicaciones emplean técnicas coherentes, las cuales requieren de información del estado del canal y por ello la transmisión de una enorme cantidad de información de señalización. Este inconveniente se ve agravado en el caso del m-MIMO debido al enorme número de antenas. Por ello, la codificación diferencial y la detección no coherente (NC) son una solución alternativa para solventar el problema de m-MIMO en los sistemas coherentes. Esta Tesis se centra en las técnicas de procesado de señal para detección NC junto con m-MIMO, proponiendo nuevos esquemas de constelación y algoritmos de detección NC, donde la información sea transmitida en la diferencia de fase de la señal. Primero, diseñamos nuevas constelaciones para un sistema multi usuario NC en m- MIMO en enlace ascendente (uplink) en canales con desvanecimiento tipo Rayleigh. Estos diseños nos permiten separar las señales de los usuarios en el receptor gracias a la correspondencia unívoca entre la constelación de cada usuario individual y la constelación conjunta recibida en la estación base. Hemos considerado dos enfoques para el diseño en términos de probabilidad de error: cada usuario consigue un rendimiento distinto, mientras que por otro lado, todos los usuarios son capaces de recibir las mismas prestaciones de probabilidad de error. Analizamos el número de antenas necesario para estos diseños y comparamos con el número requerido por otros diseños propuestos en la literatura. Nuestro diseño basado en DPSK requiere un número menor de antenas comparado con los sistemas basados en detección de energía. También comparamos con su homólogo coherente, resultando que NC-m-MIMO basado en DPSK es capaz de superar a los sistemas coherentes con los diseños adecuados. En segundo lugar, para reducir el número de antenas requerido para un rendimiento dado, proponemos incluir un esquema de codificación de canal. Hemos optado por un esquema de modulación codificado por bit entrelazado y decodificación iterativa (BICMID). Hemos empleado la herramienta EXIT chart para el diseño de la codificación de canal, proponiendo un nuevo enfoque para calcular las curvas EXIT de forma NC y basadas en el número de antenas. Los resultados muestran que el número de usuarios servidos por la estación base puede ser incrementado reduciendo un 70% el número de antenas con respecto al caso sin codificación de canal. En particular, para un array de 100 antenas y un diseño que ofrezca iguales prestaciones a todos los usuarios, con un código de tasa 1=2, podemos conseguir la mínima probabilidad de error. En tercer lugar, consideramos escenarios donde el canal tenga una componente predominante de visión directa (LOS) con la estación base modelada mediante un desvanecimiento tipo Rician. Por ejemplo, sistemas inalámbricos de backhaul, entornos rurales o sub urbanos, comunicaciones entre dispositivos (D2D), también cuando nos movemos hacia frecuencias superiores como son en la banda de milimétricas o más recientemente, la banda de terahercios para buscar mayores anchos de banda. Todos estos escenarios están contemplados en el futuro 5G. Los diseños presentados para canales Rayleigh ya no son válidos debido a la componente LOS del canal, por ello presentamos un nuevo diseño de constelación que resuelve el problema de la componente LOS, así como una guía para diseñar nuevas constelaciones. También proponemos un algoritmo asociado al diseñno de la constelación para poder separar a los usuarios en recepción. Además, para contemplar un escenario más realista donde podamos encontrar tanto desvanecimiento Rayleigh como Rice, proponemos agrupar usuarios de ambos grupos, analizando su rendimiento y relación señal a interferencia en la combinación. El adecuado agrupamiento permite unificar el diseño de la constelación para ambos desvanecimientos y por tanto reducir la complejidad en el receptor. También, el número de usuarios multiplicados en la constelación podría ser incrementado, gracias a la mejora en el rendimiento. El cuarto módulo de esta tesis es dedicado a analizar el rendimiento de los diseños propuestos en presencia de canales reales, donde disponemos de variabilidad temporal y en frecuencia. Proponemos usar una métrica que modela las características de la variabilidad temporal y, usando de nuevo la herramienta EXIT, analizamos los sistemas decodificados iterativamente considerando ahora los parámetros prácticos del canal. Este análisis nos permite obtener una estimación de la degradación que sufre el rendimiento del sistema impuesto por canales reales. Los resultados muestran que los sistemas NC-m-MIMO basados en DPSK son muy robustos a la variabilidad temporal por lo que son recomendables para los nuevos escenarios propuestos por el 5G, donde el canal cambia rápidamente. Otra consideración para introducir los sistemas NC con m-MIMO es la problemática de necesitar muchas cadenas de radio frecuencia que llevarían a tamaños de dispositivos enormes. Para reducir este número se propone la modulación espacial. En esta Tesis, estudiamos su uso con los sistemas NC, proponiendo una solución de modulación espacial diferencial para esquemas con múltiples usuarios combinado con NC-m-MIMO. Finalmente, estudiamos la viabilidad de multiplexar usuarios en la constelación frente a usar técnicas clásicas de multiplexación como TDMA. Para caracterizar completamente el rendimiento del sistema, analizamos la tasa de error de bloque (BLER) y el throughput de un sistema NC-m-MIMO. Los resultados muestran una ventaja significativa en cuanto al número de antennas para multiplexar usuarios en la constelación frente al requerido por TDMA. No obstante, en algunos casos, la demodulación de múltiples usuarios en la constelación podría requerir un número de antennas excesivamente grande comparado con la multiplexación en el tiempo. Por ello, es necesario gestionar adecuadamente un balance entre el throughput y el número de antenas para alcanzar un punto operacional óptimo, como se muestra en esta Tesis.Programa Oficial de Doctorado en Multimedia y Comunicaciones por la Universidad Carlos III de Madrid y la Universidad Rey Juan CarlosPresidente: Ana Isabel Pérez Neira.- Secretario: Máximo Morales Céspedes.- Vocal: María del Carmen Aguayo Torre

Quantum information processing approaches in classical systems

The engineering problem of building scalable quantum computers has prompted the development of a rich theory modeling the evolution of quantum systems as well as techniques to preserve quantum information in the presence of noise. Such techniques offer systems-level approaches to the problem of robustly encoding and preserving information and, as a result, see applicability in a wide variety of architectures for computing systems. In this thesis, we visit the mathematical underpinnings of quantum information and apply strategies inspired by quantum information processing to two non-quantum systems to demonstrate advantage. We first describe the construction of a quantum emulation device, an analog electronic system with the same mathematical structure as a gate-based quantum computer, and introduce novel time-domain information encoding methods to increase the computational capacity of the device. We confirm the sustained performance of the improved system by successfully transforming emulated states by randomly selected quantum gates. We then visit similarities between quantum information processing and signal processing in the noncoherent wireless communication setting, the latter being an environment characterized by a lack of instantaneous channel knowledge. We describe the theoretical underpinnings of the noncoherent communication environment from both an information theoretic and signal processing perspective. This leads us to propose a multi-antenna space-time code construction based on a family of quantum error correcting codes known as stabilizer codes. For this code, we derive the optimal decoder in Rayleigh and Ricean fading and benchmark the its performance against coherent and differential coding at comparable rates.Electrical and Computer Engineerin

What is the Value of Joint Processing of Pilots and Data in Block-Fading Channels?

The spectral efficiency achievable with joint processing of pilot and data symbol observations is compared with that achievable through the conventional (separate) approach of first estimating the channel on the basis of the pilot symbols alone, and subsequently detecting the data symbols. Studied on the basis of a mutual information lower bound, joint processing is found to provide a non-negligible advantage relative to separate processing, particularly for fast fading. It is shown that, regardless of the fading rate, only a very small number of pilot symbols (at most one per transmit antenna and per channel coherence interval) should be transmitted if joint processing is allowed.Comment: To appear at IEEE Int. Symposium on Information Theory; 5 page

Subcarrier intensity modulated free-space optical communication systems

This thesis investigates and analyses the performance of terrestrial free-space optical communication (FSO) system based on the phase shift keying pre-modulated subcarrier intensity modulation (SIM). The results are theoretically and experimentally compared with the classical On-Off keying (OOK) modulated FSO system in the presence of atmospheric turbulence. The performance analysis is based on the bit error rate (BER) and outage probability metrics. Optical signal traversing the atmospheric channel suffers attenuation due to scattering and absorption of the signal by aerosols, fog, atmospheric gases and precipitation. In the event of thick fog, the atmospheric attenuation coefficient exceeds 100 dB/km, this potentially limits the achievable FSO link length to less than 1 kilometre. But even in clear atmospheric conditions when signal absorption and scattering are less severe with a combined attenuation coefficient of less than 1 dB/km, the atmospheric turbulence significantly impairs the achievable error rate, the outage probability and the available link margin of a terrestrial FSO communication system. The effect of atmospheric turbulence on the symbol detection of an OOK based terrestrial FSO system is presented analytically and experimentally verified. It was found that atmospheric turbulence induced channel fading will require the OOK threshold detector to have the knowledge of the channel fading strength and noise levels if the detection error is to be reduced to its barest minimum. This poses a serious design difficulty that can be circumvented by employing phase shift keying (PSK) pre-modulated SIM. The results of the analysis and experiments showed that for a binary PSK-SIM based FSO system, the symbol detection threshold level does not require the knowledge of the channel fading strength or noise level. As such, the threshold level is fixed at the zero mark in the presence or absence of atmospheric turbulence. Also for the full and seamless integration of FSO into the access network, a study of SIM-FSO performance becomes compelling because existing networks already contain subcarrier-like signals such as radio over fibre and cable television signals. The use of multiple subcarrier signals as a means of increasing the throughput/capacity is also investigated and the effect of optical source nonlinearity is found to result in intermodulation distortion. The intermodulation distortion can impose a BER floor of up to 10-4 on the system error performance. In addition, spatial diversity and subcarrier delay diversity techniques are studied as means of ameliorating the effect of atmospheric turbulence on the error and outage performance of SIM-FSO systems. The three spatial diversity linear combining techniques analysed are maximum ratio combining, equal gain combining and selection combining. The system performance based on each of these combining techniques is presented and compared under different strengths of atmospheric turbulence. The results predicted that achieving a 4 km SIM-FSO link length with no diversity technique will require about 12 dB of power more than using a 4 × 4 transmitter/receiver array system with the same data rate in a weak turbulent atmospheric channel. On the other hand, retransmitting the delayed copy of the data once on a different subcarrier frequency was found to result in a gain of up to 4.5 dB in weak atmospheric turbulence channel

Near-Instantaneously Adaptive HSDPA-Style OFDM Versus MC-CDMA Transceivers for WIFI, WIMAX, and Next-Generation Cellular Systems

Burts-by-burst (BbB) adaptive high-speed downlink packet access (HSDPA) style multicarrier systems are reviewed, identifying their most critical design aspects. These systems exhibit numerous attractive features, rendering them eminently eligible for employment in next-generation wireless systems. It is argued that BbB-adaptive or symbol-by-symbol adaptive orthogonal frequency division multiplex (OFDM) modems counteract the near instantaneous channel quality variations and hence attain an increased throughput or robustness in comparison to their fixed-mode counterparts. Although they act quite differently, various diversity techniques, such as Rake receivers and space-time block coding (STBC) are also capable of mitigating the channel quality variations in their effort to reduce the bit error ratio (BER), provided that the individual antenna elements experience independent fading. By contrast, in the presence of correlated fading imposed by shadowing or time-variant multiuser interference, the benefits of space-time coding erode and it is unrealistic to expect that a fixed-mode space-time coded system remains capable of maintaining a near-constant BER

Advanced DSP Techniques for High-Capacity and Energy-Efficient Optical Fiber Communications

The rapid proliferation of the Internet has been driving communication networks closer and closer to their limits, while available bandwidth is disappearing due to an ever-increasing network load. Over the past decade, optical fiber communication technology has increased per fiber data rate from 10 Tb/s to exceeding 10 Pb/s. The major explosion came after the maturity of coherent detection and advanced digital signal processing (DSP). DSP has played a critical role in accommodating channel impairments mitigation, enabling advanced modulation formats for spectral efficiency transmission and realizing flexible bandwidth. This book aims to explore novel, advanced DSP techniques to enable multi-Tb/s/channel optical transmission to address pressing bandwidth and power-efficiency demands. It provides state-of-the-art advances and future perspectives of DSP as well

Técnicas de pré-codificação para sistemas multicelulares coordenados

Doutoramento em TelecomunicaçõesCoordenação Multicélula é um tópico de investigação em rápido crescimento e uma solução promissora para controlar a interferência entre células em sistemas celulares, melhorando a equidade do sistema e aumentando a sua capacidade. Esta tecnologia já está em estudo no LTEAdvanced sob o conceito de coordenação multiponto (COMP). Existem várias abordagens sobre coordenação multicélula, dependendo da quantidade e do tipo de informação partilhada pelas estações base, através da rede de suporte (backhaul network), e do local onde essa informação é processada, i.e., numa unidade de processamento central ou de uma forma distribuída em cada estação base. Nesta tese, são propostas técnicas de pré-codificação e alocação de potência considerando várias estratégias: centralizada, todo o processamento é feito na unidade de processamento central; semidistribuída, neste caso apenas parte do processamento é executado na unidade de processamento central, nomeadamente a potência alocada a cada utilizador servido por cada estação base; e distribuída em que o processamento é feito localmente em cada estação base. Os esquemas propostos são projectados em duas fases: primeiro são propostas soluções de pré-codificação para mitigar ou eliminar a interferência entre células, de seguida o sistema é melhorado através do desenvolvimento de vários esquemas de alocação de potência. São propostas três esquemas de alocação de potência centralizada condicionada a cada estação base e com diferentes relações entre desempenho e complexidade. São também derivados esquemas de alocação distribuídos, assumindo que um sistema multicelular pode ser visto como a sobreposição de vários sistemas com uma única célula. Com base neste conceito foi definido uma taxa de erro média virtual para cada um desses sistemas de célula única que compõem o sistema multicelular, permitindo assim projectar esquemas de alocação de potência completamente distribuídos. Todos os esquemas propostos foram avaliados em cenários realistas, bastante próximos dos considerados no LTE. Os resultados mostram que os esquemas propostos são eficientes a remover a interferência entre células e que o desempenho das técnicas de alocação de potência propostas é claramente superior ao caso de não alocação de potência. O desempenho dos sistemas completamente distribuídos é inferior aos baseados num processamento centralizado, mas em contrapartida podem ser usados em sistemas em que a rede de suporte não permita a troca de grandes quantidades de informação.Multicell coordination is a promising solution for cellular wireless systems to mitigate inter-cell interference, improving system fairness and increasing capacity and thus is already under study in LTE-A under the coordinated multipoint (CoMP) concept. There are several coordinated transmission approaches depending on the amount of information shared by the transmitters through the backhaul network and where the processing takes place i.e. in a central processing unit or in a distributed way on each base station. In this thesis, we propose joint precoding and power allocation techniques considering different strategies: Full-centralized, where all the processing takes place at the central unit; Semi-distributed, in this case only some process related with power allocation is done at the central unit; and Fulldistributed, where all the processing is done locally at each base station. The methods are designed in two phases: first the inter-cell interference is removed by applying a set of centralized or distributed precoding vectors; then the system is further optimized by centralized or distributed power allocation schemes. Three centralized power allocation algorithms with per-BS power constraint and different complexity tradeoffs are proposed. Also distributed power allocation schemes are proposed by considering the multicell system as superposition of single cell systems, where we define the average virtual bit error rate (BER) of interference-free single cell system, allowing us to compute the power allocation coefficients in a distributed manner at each BS. All proposed schemes are evaluated in realistic scenarios considering LTE specifications. The numerical evaluations show that the proposed schemes are efficient in removing inter-cell interference and improve system performance comparing to equal power allocation. Furthermore, fulldistributed schemes can be used when the amounts of information to be exchanged over the backhaul is restricted, although system performance is slightly degraded from semi-distributed and full-centralized schemes, but the complexity is considerably lower. Besides that for high degrees of freedom distributed schemes show similar behaviour to centralized ones