Nonlinear limits to the information capacity of optical fiber communications

The exponential growth in the rate at which information can be communicated through an optical fiber is a key element in the so called information revolution. However, like all exponential growth laws, there are physical limits to be considered. The nonlinear nature of the propagation of light in optical fiber has made these limits difficult to elucidate. Here we obtain basic insights into the limits to the information capacity of an optical fiber arising from these nonlinearities. The key simplification lies in relating the nonlinear channel to a linear channel with multiplicative noise, for which we are able to obtain analytical results. In fundamental distinction to the linear additive noise case, the capacity does not grow indefinitely with increasing signal power, but has a maximal value. The ideas presented here have broader implications for other nonlinear information channels, such as those involved in sensory transduction in neurobiology. These have been often examined using additive noise linear channel models, and as we show here, nonlinearities can change the picture qualitatively.Comment: 1 figure, 7 pages, submitted to Natur

From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals

Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then lowpass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, realtime performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.Comment: 17 pages, 12 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

Restoration of multichannel microwave radiometric images

A constrained iterative image restoration method is applied to multichannel diffraction-limited imagery. This method is based on the Gerchberg-Papoulis algorithm utilizing incomplete information and partial constraints. The procedure is described using the orthogonal projection operators which project onto two prescribed subspaces iteratively. Some of its properties and limitations are also presented. The selection of appropriate constraints was emphasized in a practical application. Multichannel microwave images, each having different spatial resolution, were restored to a common highest resolution to demonstrate the effectiveness of the method. Both noise-free and noisy images were used in this investigation

Perceptual Model-Driven Authoring of Plausible Vibrations from User Expectations for Virtual Environments

One of the central goals of design is the creation of experiences that are rated favorably in the intended application context. User expectations play an integral role in tactile product quality and tactile plausibility judgments alike. In the vibrotactile authoring process for virtual environments, vibra-tion is created to match the user’s expectations of the presented situational context. Currently, inefficient trial and error approaches attempt to match expectations implicitly. A more efficient, model-driven procedure based explicitly on tactile user expectations would thus be beneficial for author-ing vibrations. In everyday life, we are frequently exposed to various whole-body vibrations. Depending on their temporal and spectral proper-ties we intuitively associate specific perceptual properties such as “tin-gling”. This suggests a systematic relationship between physical parame-ters and perceptual properties. To communicate with potential users about such elicited or expected tactile properties, a standardized design language is proposed. It contains a set of sensory tactile perceptual attributes, which are sufficient to characterize the perceptual space of vibration encountered in everyday life. This design language enables the assessment of quantita-tive tactile perceptual specifications by laypersons that are elicited in situational contexts such as auditory-visual-tactile vehicle scenes. Howev-er, such specifications can also be assessed by providing only verbal de-scriptions of the content of these scenes. Quasi identical ratings observed for both presentation modes suggest that tactile user expectations can be quantified even before any vibration is presented. Such expected perceptu-al specifications are the prerequisite for a subsequent translation into phys-ical vibration parameters. Plausibility can be understood as a similarity judgment between elicited features and expected features. Thus, plausible vibration can be synthesized by maximizing the similarity of the elicited perceptual properties to the expected perceptual properties. Based on the observed relationships between vibration parameters and sensory tactile perceptual attributes, a 1-nearest-neighbor model and a regression model were built. The plausibility of the vibrations synthesized by these models in the context of virtual auditory-visual-tactile vehicle scenes was validat-ed in a perceptual study. The results demonstrated that the perceptual spec-ifications obtained with the design language are sufficient to synthesize vibrations, which are perceived as equally plausible as recorded vibrations in a given situational context. Overall, the demonstrated design method can be a new, more efficient tool for designers authoring vibrations for virtual environments or creating tactile feedback. The method enables further automation of the design process and thus potential time and cost reductions.:Preface III Abstract V Zusammenfassung VII List of Abbreviations XV 1 Introduction 1 1.1 General Introduction 1 1.1 Objectives of the Thesis 4 1.2 Structure of the Thesis 4 2. Tactile Perception in Real and Virtual Environments 7 2.1 Tactile Perception as a Multilayered Process 7 2.1.1 Physical Layer 8 2.1.2 Mechanoreceptor Layer 9 2.1.3 Sensory Layer 19 2.1.4 Affective Layer 26 2.2 Perception of Virtual Environments 29 2.2.1 The Place Illusion 29 2.2.2 The Plausibility Illusion 31 2.3 Approaches for the Authoring of Vibrations 38 2.3.1 Approaches on the Physical Layer 38 2.3.2 Approaches on the Mechanoreceptor Layer 40 2.3.3 Approaches on the Sensory Layer 40 2.3.4 Approaches on the Affective Layer 43 2.4 Summary 43 3. Research Concept 47 3.1 Research Questions 47 3.1.1 Foundations of the Research Concept 47 3.1.2 Research Concept 49 3.2 Limitations 50 4. Development of the Experimental Setup 53 4.1 Hardware 53 4.1.1 Optical Reproduction System 53 4.1.2 Acoustical Reproduction System 54 4.1.3 Whole-Body Vibration Reproduction System 56 4.2 Software 64 4.2.1 Combination of Reproduction Systems for Unimodal and Multimodal Presentation 64 4.2.2 Conducting Perceptual Studies 65 5. Assessment of a Sensory Tactile Design Language for Characterizing Vibration 67 5.1.1 Design Language Requirements 67 5.1.2 Method to Assess the Design Language 69 5.1.3 Goals of this Chapter 70 5.2 Tactile Stimuli 72 5.2.1 Generalization into Excitation Patterns 72 5.2.2 Definition of Parameter Values of the Excitation Patterns 75 5.2.3 Generation of the Stimuli 85 5.2.4 Summary 86 5.3 Assessment of the most relevant Sensory Tactile Perceptual Attributes 86 5.3.1 Experimental Design 87 5.3.2 Participants 88 5.3.3 Results 88 5.3.4 Aggregation and Prioritization 89 5.3.5 Summary 91 5.4 Identification of the Attributes forming the Design Language 92 5.4.1 Experimental Design 93 5.4.2 Participants 95 5.4.3 Results 95 5.4.4 Selecting the Elements of the Sensory Tactile Design Language 106 5.4.5 Summary 109 5.5 Summary and Discussion 109 5.5.1 Summary 109 5.5.2 Discussion 111 6. Quantification of Expected Properties with the Sensory Tactile Design Language 115 6.1 Multimodal Stimuli 116 6.1.1 Selection of the Scenes 116 6.1.2 Recording of the Scenes 117 6.1.3 Recorded Stimuli 119 6.2 Qualitative Communication in the Presence of Vibration 123 6.2.1 Experimental Design 123 6.2.2 Participants 124 6.2.3 Results 124 6.2.4 Summary 126 6.3 Quantitative Communication in the Presence of Vibration 126 6.3.1 Experimental Design 127 6.3.2 Participants 127 6.3.3 Results 127 6.3.4 Summary 129 6.4 Quantitative Communication in the Absence of Vibration 129 6.4.1 Experimental Design 130 6.4.2 Participants 132 6.4.3 Results 132 6.4.4 Summary 134 6.5 Summary and Discussion 135 7. Synthesis Models for the Translation of Sensory Tactile Properties into Vibration 137 7.1 Formalization of the Tactile Plausibility Illusion for Models 139 7.1.1 Formalization of Plausibility 139 7.1.2 Model Boundaries 143 7.2 Investigation of the Influence of Vibration Level on Attribute Ratings 144 7.2.1 Stimuli 145 7.2.2 Experimental Design 145 7.2.3 Participants 146 7.2.4 Results 146 7.2.5 Summary 148 7.3 Comparison of Modulated Vibration to Successive Impulse-like Vibration 148 7.3.1 Stimuli 149 7.3.2 Experimental Design 151 7.3.3 Participants 151 7.3.4 Results 151 7.3.5 Summary 153 7.4 Synthesis Based on the Discrete Estimates of a k-Nearest-Neighbor Classifier 153 7.4.1 Definition of the K-Nearest-Neighbor Classifier 154 7.4.2 Analysis Model 155 7.4.3 Synthesis Model 156 7.4.4 Interpolation of acceleration level for the vibration attribute profile pairs 158 7.4.5 Implementation of the Synthesis 159 7.4.6 Advantages and Disadvantages 164 7.5 Synthesis Based on the Quasi-Continuous Estimates of Regression Models 166 7.5.1 Overall Model Structure 168 7.5.2 Classification of the Excitation Pattern with a Support Vector Machine 171 7.5.3 General Approach to the Regression Models of each Excitation Pattern 178 7.5.4 Synthesis for the Impulse-like Excitation Pattern 181 7.5.5 Synthesis for the Bandlimited White Gaussian Noise Excitation Pattern 187 7.5.6 Synthesis for the Amplitude Modulated Sinusoidal Excitation Pattern 193 7.5.7 Synthesis for the Sinusoidal Excitation Pattern 199 7.5.8 Implementation of the Synthesis 205 7.5.9 Advantages and Disadvantages of the Approach 208 7.6 Validation of the Synthesis Models 210 7.6.1 Stimuli 212 7.6.2 Experimental Design 212 7.6.3 Participants 214 7.6.4 Results 214 7.6.5 Summary 219 7.7 Summary and Discussion 219 7.7.1 Summary 219 7.7.2 Discussion 222 8. General Discussion and Outlook 227 Acknowledgment 237 References 237Eines der zentralen Ziele des Designs von Produkten oder virtuellen Um-gebungen ist die Schaffung von Erfahrungen, die im beabsichtigten An-wendungskontext die Erwartungen der Benutzer erfüllen. Gegenwärtig versucht man im vibrotaktilen Authoring-Prozess mit ineffizienten Trial-and-Error-Verfahren, die Erwartungen an den dargestellten, virtuellen Situationskontext implizit zu erfüllen. Ein effizienteres, modellgetriebenes Verfahren, das explizit auf den taktilen Benutzererwartungen basiert, wäre daher von Vorteil. Im Alltag sind wir häufig verschiedenen Ganzkörper-schwingungen ausgesetzt. Abhängig von ihren zeitlichen und spektralen Eigenschaften assoziieren wir intuitiv bestimmte Wahrnehmungsmerkmale wie z.B. “kribbeln”. Dies legt eine systematische Beziehung zwischen physikalischen Parametern und Wahrnehmungsmerkmalen nahe. Um mit potentiellen Nutzern über hervorgerufene oder erwartete taktile Eigen-schaften zu kommunizieren, wird eine standardisierte Designsprache vor-geschlagen. Sie enthält eine Menge von sensorisch-taktilen Wahrneh-mungsmerkmalen, die hinreichend den Wahrnehmungsraum der im Alltag auftretenden Vibrationen charakterisieren. Diese Entwurfssprache ermög-licht die quantitative Beurteilung taktiler Wahrnehmungsmerkmale, die in Situationskontexten wie z.B. auditiv-visuell-taktilen Fahrzeugszenen her-vorgerufen werden. Solche Wahrnehmungsspezifikationen können jedoch auch bewertet werden, indem der Inhalt dieser Szenen verbal beschrieben wird. Quasi identische Bewertungen für beide Präsentationsmodi deuten darauf hin, dass die taktilen Benutzererwartungen quantifiziert werden können, noch bevor eine Vibration präsentiert wird. Die erwarteten Wahr-nehmungsspezifikationen sind die Voraussetzung für eine anschließende Übersetzung in physikalische Schwingungsparameter. Plausible Vibratio-nen können synthetisiert werden, indem die erwarteten Wahrnehmungs-merkmale hervorgerufen werden. Auf der Grundlage der beobachteten Beziehungen zwischen Schwingungs¬parametern und sensorisch-taktilen Wahrnehmungsmerkmalen wurden ein 1-Nearest-Neighbor-Modell und ein Regressionsmodell erstellt. Die Plausibilität der von diesen Modellen synthetisierten Schwingungen im Kontext virtueller, auditorisch-visuell-taktiler Fahrzeugszenen wurde in einer Wahrnehmungsstudie validiert. Die Ergebnisse zeigten, dass die mit der Designsprache gewonnenen Wahr-nehmungsspezifikationen ausreichen, um Schwingungen zu synthetisieren, die in einem gegebenen Situationskontext als ebenso plausibel empfunden werden wie aufgezeichnete Schwingungen. Die demonstrierte Entwurfsme-thode stellt ein neues, effizienteres Werkzeug für Designer dar, die Schwingungen für virtuelle Umgebungen erstellen oder taktiles Feedback für Produkte erzeugen.:Preface III Abstract V Zusammenfassung VII List of Abbreviations XV 1 Introduction 1 1.1 General Introduction 1 1.1 Objectives of the Thesis 4 1.2 Structure of the Thesis 4 2. Tactile Perception in Real and Virtual Environments 7 2.1 Tactile Perception as a Multilayered Process 7 2.1.1 Physical Layer 8 2.1.2 Mechanoreceptor Layer 9 2.1.3 Sensory Layer 19 2.1.4 Affective Layer 26 2.2 Perception of Virtual Environments 29 2.2.1 The Place Illusion 29 2.2.2 The Plausibility Illusion 31 2.3 Approaches for the Authoring of Vibrations 38 2.3.1 Approaches on the Physical Layer 38 2.3.2 Approaches on the Mechanoreceptor Layer 40 2.3.3 Approaches on the Sensory Layer 40 2.3.4 Approaches on the Affective Layer 43 2.4 Summary 43 3. Research Concept 47 3.1 Research Questions 47 3.1.1 Foundations of the Research Concept 47 3.1.2 Research Concept 49 3.2 Limitations 50 4. Development of the Experimental Setup 53 4.1 Hardware 53 4.1.1 Optical Reproduction System 53 4.1.2 Acoustical Reproduction System 54 4.1.3 Whole-Body Vibration Reproduction System 56 4.2 Software 64 4.2.1 Combination of Reproduction Systems for Unimodal and Multimodal Presentation 64 4.2.2 Conducting Perceptual Studies 65 5. Assessment of a Sensory Tactile Design Language for Characterizing Vibration 67 5.1.1 Design Language Requirements 67 5.1.2 Method to Assess the Design Language 69 5.1.3 Goals of this Chapter 70 5.2 Tactile Stimuli 72 5.2.1 Generalization into Excitation Patterns 72 5.2.2 Definition of Parameter Values of the Excitation Patterns 75 5.2.3 Generation of the Stimuli 85 5.2.4 Summary 86 5.3 Assessment of the most relevant Sensory Tactile Perceptual Attributes 86 5.3.1 Experimental Design 87 5.3.2 Participants 88 5.3.3 Results 88 5.3.4 Aggregation and Prioritization 89 5.3.5 Summary 91 5.4 Identification of the Attributes forming the Design Language 92 5.4.1 Experimental Design 93 5.4.2 Participants 95 5.4.3 Results 95 5.4.4 Selecting the Elements of the Sensory Tactile Design Language 106 5.4.5 Summary 109 5.5 Summary and Discussion 109 5.5.1 Summary 109 5.5.2 Discussion 111 6. Quantification of Expected Properties with the Sensory Tactile Design Language 115 6.1 Multimodal Stimuli 116 6.1.1 Selection of the Scenes 116 6.1.2 Recording of the Scenes 117 6.1.3 Recorded Stimuli 119 6.2 Qualitative Communication in the Presence of Vibration 123 6.2.1 Experimental Design 123 6.2.2 Participants 124 6.2.3 Results 124 6.2.4 Summary 126 6.3 Quantitative Communication in the Presence of Vibration 126 6.3.1 Experimental Design 127 6.3.2 Participants 127 6.3.3 Results 127 6.3.4 Summary 129 6.4 Quantitative Communication in the Absence of Vibration 129 6.4.1 Experimental Design 130 6.4.2 Participants 132 6.4.3 Results 132 6.4.4 Summary 134 6.5 Summary and Discussion 135 7. Synthesis Models for the Translation of Sensory Tactile Properties into Vibration 137 7.1 Formalization of the Tactile Plausibility Illusion for Models 139 7.1.1 Formalization of Plausibility 139 7.1.2 Model Boundaries 143 7.2 Investigation of the Influence of Vibration Level on Attribute Ratings 144 7.2.1 Stimuli 145 7.2.2 Experimental Design 145 7.2.3 Participants 146 7.2.4 Results 146 7.2.5 Summary 148 7.3 Comparison of Modulated Vibration to Successive Impulse-like Vibration 148 7.3.1 Stimuli 149 7.3.2 Experimental Design 151 7.3.3 Participants 151 7.3.4 Results 151 7.3.5 Summary 153 7.4 Synthesis Based on the Discrete Estimates of a k-Nearest-Neighbor Classifier 153 7.4.1 Definition of the K-Nearest-Neighbor Classifier 154 7.4.2 Analysis Model 155 7.4.3 Synthesis Model 156 7.4.4 Interpolation of acceleration level for the vibration attribute profile pairs 158 7.4.5 Implementation of the Synthesis 159 7.4.6 Advantages and Disadvantages 164 7.5 Synthesis Based on the Quasi-Continuous Estimates of Regression Models 166 7.5.1 Overall Model Structure 168 7.5.2 Classification of the Excitation Pattern with a Support Vector Machine 171 7.5.3 General Approach to the Regression Models of each Excitation Pattern 178 7.5.4 Synthesis for the Impulse-like Excitation Pattern 181 7.5.5 Synthesis for the Bandlimited White Gaussian Noise Excitation Pattern 187 7.5.6 Synthesis for the Amplitude Modulated Sinusoidal Excitation Pattern 193 7.5.7 Synthesis for the Sinusoidal Excitation Pattern 199 7.5.8 Implementation of the Synthesis 205 7.5.9 Advantages and Disadvantages of the Approach 208 7.6 Validation of the Synthesis Models 210 7.6.1 Stimuli 212 7.6.2 Experimental Design 212 7.6.3 Participants 214 7.6.4 Results 214 7.6.5 Summary 219 7.7 Summary and Discussion 219 7.7.1 Summary 219 7.7.2 Discussion 222 8. General Discussion and Outlook 227 Acknowledgment 237 References 23

Neural networks for optical channel equalization in high speed communication systems

La demande future de bande passante pour les données dépassera les capacités des systèmes de communication optique actuels, qui approchent de leurs limites en raison des limitations de la bande passante électrique des composants de l’émetteur. L’interférence intersymbole (ISI) due à cette limitation de bande est le principal facteur de dégradation pour atteindre des débits de données élevés. Dans ce mémoire, nous étudions plusieurs techniques de réseaux neuronaux (NN) pour combattre les limites physiques des composants de l’émetteur pilotés à des débits de données élevés et exploitant les formats de modulation avancés avec une détection cohérente. Notre objectif principal avec les NN comme égaliseurs de canaux ISI est de surmonter les limites des récepteurs optimaux conventionnels, en fournissant une complexité évolutive moindre et une solution quasi optimale. Nous proposons une nouvelle architecture bidirectionnelle profonde de mémoire à long terme (BiLSTM), qui est efficace pour atténuer les graves problèmes d’ISI causés par les composants à bande limitée. Pour la première fois, nous démontrons par simulation que notre BiLSTM profonde proposée atteint le même taux d’erreur sur les bits(TEB) qu’un estimateur de séquence à maximum de vraisemblance (MLSE) optimal pour la modulation MDPQ. Les NN étant des modèles pilotés par les données, leurs performances dépendent fortement de la qualité des données d’entrée. Nous démontrons comment les performances du BiLSTM profond réalisable se dégradent avec l’augmentation de l’ordre de modulation. Nous examinons également l’impact de la sévérité de l’ISI et de la longueur de la mémoire du canal sur les performances de la BiLSTM profonde. Nous étudions les performances de divers canaux synthétiques à bande limitée ainsi qu’un canal optique mesuré à 100 Gbaud en utilisant un modulateur photonique au silicium (SiP) de 35 GHz. La gravité ISI de ces canaux est quantifiée grâce à une nouvelle vue graphique des performances basée sur les écarts de performance de base entre les solutions optimales linéaires et non linéaires classiques. Aux ordres QAM supérieurs à la QPSK, nous quantifions l’écart de performance BiLSTM profond par rapport à la MLSE optimale à mesure que la sévérité ISI augmente. Alors qu’elle s’approche des performances optimales de la MLSE à 8QAM et 16QAM avec une pénalité, elle est capable de dépasser largement la solution optimale linéaire à 32QAM. Plus important encore, l’avantage de l’utilisation de modèles d’auto-apprentissage comme les NN est leur capacité à apprendre le canal pendant la formation, alors que la MLSE optimale nécessite des informations précises sur l’état du canal.The future demand for the data bandwidth will surpass the capabilities of current optical communication systems, which are approaching their limits due to the electrical bandwidth limitations of the transmitter components. Inter-symbol interference (ISI) due to this band limitation is the major degradation factor to achieve high data rates. In this thesis, we investigate several neural network (NN) techniques to combat the physical limits of the transmitter components driven at high data rates and exploiting the advanced modulation formats with coherent detection. Our main focus with NNs as ISI channel equalizers is to overcome the limitations of conventional optimal receivers, by providing lower scalable complexity and near optimal solution. We propose a novel deep bidirectional long short-term memory (BiLSTM) architecture, that is effective in mitigating severe ISI caused by bandlimited components. For the first time, we demonstrate via simulation that our proposed deep BiLSTM achieves the same bit error rate (BER) performance as an optimal maximum likelihood sequence estimator (MLSE) for QPSK modulation. The NNs being data-driven models, their performance acutely depends on input data quality. We demonstrate how the achievable deep BiLSTM performance degrades with the increase in modulation order. We also examine the impact of ISI severity and channel memory length on deep BiLSTM performance. We investigate the performances of various synthetic band-limited channels along with a measured optical channel at 100 Gbaud using a 35 GHz silicon photonic(SiP) modulator. The ISI severity of these channels is quantified with a new graphical view of performance based on the baseline performance gaps between conventional linear and nonlinear optimal solutions. At QAM orders above QPSK, we quantify deep BiLSTM performance deviation from the optimal MLSE as ISI severity increases. While deep BiLSTM approaches the optimal MLSE performance at 8QAM and 16QAM with a penalty, it is able to greatly surpass the linear optimal solution at 32QAM. More importantly, the advantage of using self learning models like NNs is their ability to learn the channel during the training, while the optimal MLSE requires accurate channel state information

Timing Synchronization and Channel Estimation in Free-Space Optical OOK Communication Systems

Fast and reliable synchronization in free-space optical (FSO) communications is a crucial task that has received little attention so far. Since in these applications the data rate is much higher than in traditional radio-frequency (RF) systems, novel technological constraints may arise in the design of the synchronization algorithms, as for example the need to operate at symbol rate instead with an oversampled data stream. In this work, we consider an FSO link and investigate the problem of channel estimation, symbol timing recovery and frame detection using a known synch pattern. The modulation format is on-off keying (OOK) and the received signal is plagued by a mixture of thermal and shot noise. By applying the least-squares criterion, we derive a novel synchronization scheme that can jointly retrieve all the unknown parameters using symbol-spaced samples. Although designed without taking the noise statistics into account, the estimator performance is assessed in a realistic scenario where shot noise is present. Comparisons are made with the relevant Cramér-Rao bound for the joint estimation of the synchronization parameters and signal-dependent noise variances. Numerical simulations and complexity analysis indicate that the resulting scheme performs satisfactorily with an affordable processing load. Hence, it represents a promising solution for fast synchronization in high-speed FSO communications

Non-Orthogonal Multi-band CAP for Highly Spectrally Efficient VLC Systems

In this work we propose and experimentally demonstrate a novel non-orthogonal multi-band carrier-less amplitude and phase (NM-CAP) scheme for bandlimited visible light communication systems in order to increase the spectral efficiency. We show that a bandwidth saving up to 30% can be achieved thus resulting in 44% improvement in the measured spectral efficiency with no further bit error rate performance degradation compared to the traditional m-CAP scheme. We also show that higher order systems can provide higher bandwidth compression than low order systems. Furthermore, with no additional functional blocks at the transmitter or the receiver the proposed scheme introduces no extra computational complexity.Comment: 6 pages, 5 figure