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CENTRO POLITÉCNICO SUPERIOR UNIVERSIDAD DE ZARAGOZA Chaotic receivers for optical communication systems Ingenier ´ ıa de Telecomunicaci´ on Proyecto de Fin de Carrera Realizado en: Technical University of Denmark Javier Franco Biurrun Julio 2011 Director: Idelfonso Tafur Monroy Dept. Metro-Access & Short Range Systems Technical University of Denmark Ponente: Juan Ignacio Garc´es Gregorio Dpto. de Teor´ıa de la Se˜nal y Comunicaciones C.P.S., Universidad de Zaragoza Centro Polit´ecnico Superior Dpto. de Ingenier´ıa Electr´onica y Comunicaciones C/ Maria de Luna 3, 50018 Zaragoza Edificio Ada Byron http://www.cps.unizar.es Chaotic receivers for optical communication systems Resumen Este proyecto concierne al estudio de la detecci´on de se˜nales enmascaradas en entornos de ruido. Para los escenarios en los que la se˜nal recibida es d´ebil, se presenta una alternativa a los m´etodos de detecci´on determinista m´as com´unmente empleados. Esta alternativa consiste en la implementaci´on de un sistema ca´otico basado en un receptor Duffing de segundo orden. Dicho sistema de recepci´on ca´otico Duffing esta compuesto principalmente por un oscilador no lineal. Su comportamiento variar´a en funci´on de la se˜nal de entrada y transitar´a principalmente entre los estados de orden y caos. Esta caracter´ıstica nos servir´a como punto de partida para la detecci´on de la se˜nal. En esta tesis se ha implementado y evaluado en primer lugar, un modelo de oscilador Duffing para la detecci´on de se˜nales binarias. Adem´as, el autor ha propuesto y testado una modificaci´on del dise˜no convencional del sistema de recepci´on ca´otico Duffing con el prop´osito de mejorar el rendimiento en t´erminos de Bit Error Rate (BER) frente a la Relaci´on Se˜nal a Ruido (SNR). Posteriormente y a trav´es de simulaciones, se han evaluado diversos m´etodos para la detecci´on de s´ımbolos. Con ello se ha pretendido estudiar el rendimiento de cada m´etodo analizando la sensibilidad del BER con respecto al SNR recibido. M´as tarde, utilizando el m´etodo de detecci´on de s´ımbolos ´optimo, se han establecido comparaciones en t´erminos del BER entre los m´etodos de demodulaci´on est´andar y los m´etodos propuestos. Diversos escenarios han sido analizados para diferentes anchos de banda del Filtro Paso Banda (BPF) en recepci´on. Finalmente, se ha llevado a cabo una demostraci´on experimental para escenarios de Radio sobre Fibra (RoF). El formato de modulaci´on considerado ha sido Amplitude-Shift Keying (ASK) para tasas de datos superiores a 1Gbps, con una frecuencia de portadora de 13GHz y todo ello integrado en un montaje de laboratorio sobre fibra ´optica. Se ha podido observar que los resultados experimentales y las simulaciones han resultado en consonancia. ii Contents Tabla de contenidos Chaotic receivers for optical communication systems i 1 Introducci´on 1 1.1 Estado del Arte de los sistemas de recepci´on . . . . . . . . 2 1.2 Objetivo de esta tesis . . . . . . . . . . . . . . . . . . . . 4 1.3 Estructura de la tesis . . . . . . . . . . . . . . . . . . . . . 5 2 Receptor Duffing 7 2.1 Principios fundamentales . . . . . . . . . . . . . . . . . . . 7 2.2 Modelo para la detecci´on de se˜nales . . . . . . . . . . . . 9 3 Implementaci´on del modelo 13 3.1 Modelo mediante Simulink . . . . . . . . . . . . . . . . . . 13 3.2 Modelo mediante Matlab . . . . . . . . . . . . . . . . . . 14 4 Resultados de simulaci´on 17 4.1 Detecci´on de se˜nales 2ASK mediante el oscilador Duffing . 17 4.2 Detecci´on de se˜nales BPSK mediante el oscilador Duffing 19 4.3 Detecci´on de se˜nales BFSK mediante el oscilador Duffing 22 5 Evaluaci´on del rendimiento para la modulaci´on ASK 25 5.1 Descripci´on del sistema completo . . . . . . . . . . . . . . 25 5.2 Resultados de la comparaci´on entre el sistema Duffing y la detecci´on coherente . . . . . . . . . . . . . . . . . . . . 29 5.3 Implementaci´on alternativa del receptor Duffing . . . . . . 31 6 Resultados experimentales 35 6.1 Sistema de recepci´on ca´otico Duffing para una transmisi´on de radio sobre fibra . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Resultado del montaje de laboratorio . . . . . . . . . . . . 38 7 Conclusi´on y trabajo futuro 41 A Duffing receiver 45 A.1 Fundamental principle . . . . . . . . . . . . . . . . . . . . 45 A.2 Bifurcation value calculated via Melnikov method . . . . . 46 iv TABLA DE CONTENIDOS A.3 Signal detection model . . . . . . . . . . . . . . . . . . . . 50 A.4 Detection model for any frequency signal . . . . . . . . . . 53 B Model Implementation 55 B.1 SimulinkModel........................ 55 B.2 MatlabModel......................... 58 C Simulation results 71 C.1 A method for 2ASK signal detection using Duffing Oscillator 71 C.2 A method for BPSK signal detection using Duffing Oscillator 75 C.3 A method for BFSK signal detection using Duffing Oscillator 79 D Performance evaluation for ASK 83 D.1 Transmitter.......................... 83 D.2 Channel............................ 84 D.3 Receiver............................ 85 D.4 Comparative and results: Duffing vs. Coherent detection . 98 D.5 Alternative Duffing receiver implementation . . . . . . . . 103 D.6 Comparative and results: Duffing vs. Envelope detection . 105 E Experimental results 109 E.1 Electrical characterization of the system . . . . . . . . . . 109 E.2 Chaotic Duffing receiving system implementation for Radioover-Fiber transmission . . . . . . . . . . . . . . . . . . . 113 F Integral Implementation 123 F.1 Duffing implemented with Euler Forward method . . . . . 123 F.2 Duffing implemented with Trapezoidal rule method . . . . 123 F.3 Duffing implemented with Fourth-Order Runge-Kutta method124 G Matlab simulations for the three basic modulations 127 G.1 Code for BASK signal detection . . . . . . . . . . . . . . . 127 G.2 Code for BPSK signal detection . . . . . . . . . . . . . . . 128 G.3 Code for BFSK signal detection . . . . . . . . . . . . . . . 129 H Matlab simulations for the performance evaluation for ASK 131 H.1 Code for transmitter and noisy channel . . . . . . . . . . 131 H.2 Code for receiver . . . . . . . . . . . . . . . . . . . . . . . 132 H.3 Code for envelope detection . . . . . . . . . . . . . . . . . 135 TABLA DE CONTENIDOS v H.4 Code for phase variance . . . . . . . . . . . . . . . . . . . 135 H.5 Code for mean method . . . . . . . . . . . . . . . . . . . . 136 H.6 Code for variance method . . . . . . . . . . . . . . . . . . 136 H.7 Code for FFT pattern method . . . . . . . . . . . . . . . 137 H.8 Code for ASK envelope demodulation compared with theoreticalvalues......................... 138 H.9 Code for ASK coherent demodulation compared with theoreticalvalues......................... 141 I Glosario de Acr´onimos 145 vi TABLA DE CONTENIDOS Lista de Figuras 1.1 Visi´on general del escenario h´ıbrido, fibra ´optica - comunicaci´on wireless. El modelo implementa la generaci´on de una se˜nal wireless por m´etodos ´opticos heterodinos, transmisi´on wireless y posteriormente, transporte sobre una fibra ´optica para el procesado en la oficina central. . . . . . 2 1.2 Estructura de la tesis. . . . . . . . . . . . . . . . . . . . . 6 2.1 Diferentes diagramas de fase para el sistema de recepci´on ca´otico Duffing representado sus cuatro estados. El par´ametro δest´a definido a 0.5, el umbral de bifurcaci´on R0(ω) a 0.3765 y Fd= 0.753. ..................... 8 2.2 Diagrama de fase de los distintos estados para situaciones con fuerte ruido o ausencia del mismo. . . . . . . . . . . . 10 2.3 Diagrama de fase en funci´on de las entradas del oscilador Duffing. ............................ 11 2.4 Comportamiento del oscilador Duffing en funci´on del par´ametro ϕ. ............................... 11 3.1 Implementaci´on del oscilador Duffing en Simulink. . . . . 14 4.1 Se˜nal OOK modulada. En color azul aparece la se˜nal transmitida y por tanto, libre de ruido. En color rojo observamos la se˜nal recibida tras el paso por el canal ruidoso. Los par´ametros de la simulaci´on son fc=50.000 π como frecuencia de portadora, amplitud a= 0.2, potencia de ruido 40 y tasa de datos FB= 100Hz. ......... 18 xiv LISTA DE FIGURAS D.19 Comparison between coherence detection method, incoherence detection method and chaotic Duffing receiving system. BPF set at 1GHz, carrier frequency of 13.8GHz, sampling frequency of 80GHz and 1Gbps of bit rate. . . . 101 D.20 Comparison between coherence detection method, incoherence detection method and chaotic Duffing receiving system. BPF set at 2GHz Full Width at Half Maximum (FWHM), carrier frequency of 13.8GHz, sampling frequency of 80GHz and 1Gbps of bit rate. . . . . . . . . 102 D.21 Comparison between coherence detection method, incoherence detection method and chaotic Duffing receiving system. BPF set at 8GHz, carrier frequency of 13.8GHz, sampling frequency of 80GHz and 1Gbps of bit rate. . . . 103 D.22 Classical Duffing oscillator implementation. It is fed by the driver signal and the to-be-detected signal. Distance from chaos state to large periodic state is small. . . . . . . 104 D.23 Modification of the conventional Duffing oscillator implementation. The system is fed by the to-be-detected signal. The driver signal is not needed. Distance from chaos state to large periodic state is larger than in the classical implementation of the Duffing oscillator. . . . . . . . . . . . . . 105 D.24 Comparison between chaotic Duffing receiving system fed with a reference signal and the modification of the conventional Duffing oscillator implementation. BPF set at 1GHz, carrier frequency of 13.8GHz, sampling frequency of 80GHz and 1Gbps of bit rate. . . . . . . . . . . . . . . 106 D.25 Comparison between chaotic Duffing receiving system and envelope detection for different filter widths. Carrier frequency is set at 13.8GHz, sampling frequency at 80GHz and 1Gbps ofbitrate. .................... 108 E.1 Set-up for the electrical characterization of the system. . . 110 E.2 Band-pass filtered signal from the output of Digital Storage Oscilloscope. Bit rate of 400Msps, sampling frequency 20GHz, carrier frequency 600MHz and Vector Signal Analizer output power around −30dBm. ............ 111 E.3 Outputs of the Duffing for 30 dBm of signal power, 40 GHz of sampling frequency, 40 Msps of bit rate and force driver fr= 0.8085. ...................... 111 LISTA DE FIGURAS xv E.4 Outputs of the Duffing for 40 dBm of signal power, 40 GHz of sampling frequency, 40 Msps of bit rate and force driver fr= 0.7816. ...................... 112 E.5 Schematic for a Radio sobre Fibra transmission. Set-up parameters are 1Gbps of bit rate, signal modulated at 13.8GHz for OSNR among 18 and 1dB. Sampling frequency at 80GHz. ...................... 115 E.6 Schematic of the demodulation implemented script. The set-up is composed by three branches. The first simulates a coherent detection receiver. The second branch is the chaotic Duffing receiving system. The third branch in the envelope detection receiver. . . . . . . . . . . . . . . . . . 116 E.7 Comparison between simulated and experimental results for envelope detection receiver and chaotic Duffing receiving system. Communication parameters are 1Gbps of bit rate, signal modulated at 13.8GHz and sampling frequency at 80GHz............................ 119 E.8 Comparison between simulated and experimental results for envelope detection and Duffing methods where the FWHM is five times the signal bandwidth, 5GHz. Communication parameters are 1Gbps of bit rate, signal modulated at 13.8GHz and sampling frequency at 80GHz. . 120 xvi LISTA DE FIGURAS Lista de Tablas 3.1 Comparativa entre el m´etodo de Euler Forward, la regla trapezoidal de integraci´on y la implementaci´on del m´etodo Runge-Kutta para los t´erminos de estabilidad, precisi´on y rendimiento........................... 15 B.1 Numerical results for analyzing the accuracy between Forward Euler integral method, Trapezoidal Rule method and the Runge-Kutta implementation. . . . . . . . . . . . . . 64 B.2 Comparative between Forward Euler integral method, Trapezoidal Rule method and the Runge-Kutta implementation in terms of stability, accuracy and performance. . . . . . . 65 E.1 Equivalence between Energy per Bit to Noise Power Spectral Density Ratio (Eb/No) and OSNR. . . . . . . . . . . 118 xviii LISTA DE TABLAS Cap ´ ıtulo 1 Introducci´on Este proyecto se ha desarrollado en el marco de una Master Thesis de 35 ECTS en el departamento ‘Photonics Engineering Metro-Access and Short Range Systems’ de la Technical University of Denmark (DTU), Lyngby (Copenhagen). Dicho proyecto, previamente a su dep´osito en el Centro Polit´ecnico Superior (C.P.S) de Zaragoza, ha sido entregado, presentado y evaluado en la DTU obteniendo la nota de 12, m´axima calificaci´on en la escala acad´emica danesa y equivalente a un excelente o ‘A’ en el est´andar ECTS. El t´ermino telecomunicaciones hace referencia un amplio grupo de tecnolog´ıas para la transmisi´on de informaci´on sobre cierta distancia. En todas sus formas, el prop´osito fundamental que se persigue es la transmisi´on de la mayor cantidad de datos posible sin ning´un error. En todo sistema existen siempre una serie de limitaciones para alcanzar una comunicaci´on robusta que vienen descritas por el teorema de Shannon-Hartley. En ´el se establece una relaci´on entre las potencias de se˜nal y ruido, la capacidad del canal y el ancho del mismo [2]. Te´oricamente, una cantidad ilimitada de datos podr´ıa ser transmitida libre de errores sobre un ancho de banda infinito y para un canal de comunicaciones libre de ruido. Sin embargo, una comunicaci´on real se ve afectada por estas restricciones de ancho de banda y limitaciones debidas al ruido. En esta tesis considera el caso de detecci´on de una se˜nal de comunicaci´on d´ebil en entornos con una fuerte presencia de ruido. La detecci´on de se˜nales d´ebiles es un concepto ampliamente utilizado en ´ambitos como la navegaci´on por sonido (sonar), las comunicaciones radar, prevenci´on de terremotos, as´ı como el estudio de las radiaciones emitidas y otra serie de aplicaciones de medici´on industrial [3]. Recientemente la detecci´on de se˜nales d´ebiles ha adquirido cierta relevancia en temas de comunicaciones seguras [4]. Este proyecto esta enfocado para el caso de 2 Introducci´on las comunicaciones ´opticas y concretamente para RoF. Se considera un escenario h´ıbrido, en el que la comunicaciones wireless est´a integrada en una red ´optica, como puede verse en la Figura 1.1. La se˜nal wireless es generada por medios ´opticos y transmitida sobre un enlace wireless de Radiofrecuencia (RF). Despu´es de la transmisi´on wireless, la se˜nal recibida junto con el ruido son detectados y transportados sobre una fibra ´optica a la oficina central para su posterior procesado. Se ha considerado la transmisi´on de se˜nales en presencia de ruido, debido al canal de transmisi´on y a la transformaci´on de la onda en una se˜nal digital. Central Office Chaotic Duffing receiving system Symbol detection block Weak signal IM Generation of Wireless signals 1 f r ff  1 Figura 1.1: Visi´on general del escenario h´ıbrido, fibra ´optica - comunicaci´on wireless. El modelo implementa la generaci´on de una se˜nal wireless por m´etodos ´opticos heterodinos, transmisi´on wireless y posteriormente, transporte sobre una fibra ´optica para el procesado en la oficina central. 1.1 Estado del Arte de los sistemas de recepci´on La mayor´ıa de los m´etodos usados para la recuperaci´on de se˜nales en presencia de fuerte ruido alcanzando una correcta demodulaci´on, est´an basado en m´etodos probabil´ısticos. Sin embargo, cuando la se˜nal recibida es d´ebil se requiere el desarrollo de un modelo de distribuci´on probabil´ıstico. Esto conlleva dificultades para aplicaciones pr´acticas como el alto coste computacional [5]. 1.1 Estado del Arte de los sistemas de recepci´on 3 Otro m´etodo empleado en la recuperaci´on de se˜nales basado en la Densidad Espectral de Potencia (PSD) donde se analizan las componentes frecuenciales de la se˜nal. Este m´etodo resulta ´util para identificar las periodicidades de la se˜nal aunque su rendimiento esta limitado cuando la se˜nal se encuentra inmersa en entornos ruidosos [6]. A´un as´ı, puede ser utilizado en combinaci´on con otras t´ecnicas de demodulaci´on para la detecci´on de se˜nales. Profundizaremos en dicho planteamiento en la secci´on 5.1.1. Existen otras alternativas relacionadas con las distribuciones tiempofrecuencia como es el caso de la Transformada R´apida de Fourier (FFT). La limitaci´on de este m´etodo viene impuesta debido a su baja resoluci´on frecuencial, lo que puede conducir a decisiones imprecisas. A lo que hay que a˜nadir que estos planteamientos no resultan los m´as apropiados para el an´alisis de se˜nales no estacionarias ya que no son capaces de revelar la informaci´on inherente almacenada en ellas [7]. Otra alternativa ser´ıa la Transformada de Fourier de Tiempo Reducido (STFT). Como la se˜nal a detectar variar´a a lo largo del tiempo, se utiliza la STFT para determinar la frecuencia sinusoidal y la fase contenida en secciones locales de dicha se˜nal. La STFT junto con la Wavelet Transform (WT) son unas transformadas lineales de representaci´on tiempo-frecuencia. Como transformadas lineales trabajan directamente en el dominio del tiempo y no resultar´an ventajosas para la reducci´on de fuertes ruidos [8]. Las dificultades encontradas para aplicaciones pr´acticas se deben principalmente a las limitaciones de la se˜nal, a la baja resoluci´on frecuencial y a las decisiones imprecisas. Los m´etodos probabil´ısticos, los m´etodos basados en PSD y los m´etodos relacionados con las distribuciones tiempofrecuencia tambi´en fallan en la detecci´on de se˜nales ruidosas, debido a dichas dificultades. De ah´ı la necesidad de implementar un nuevo esquema para la detecci´on de se˜nales basado en los sistemas din´amicos no lineales. La propiedad fundamental de este tipo de sistemas se debe a su elevada Sensibilidad Dependiente de las Condiciones Iniciales (SDIC). De la referencia [9], se muestra como se puede interpretar esta propiedad como una sensibilidad dependiente de los par´ametros y por consiguiente, una peque˜na perturbaci´on de los par´ametros de entrada conllevar´ıa un cambio significativo en el estado de todo el sistema. Este complejo comportamiento de los sistemas deterministas no lineales viene descrito por la teor´ıa del caos [10]. 4 Introducci´on 1.2 Objetivo de esta tesis El prop´osito de esta tesis es analizar el enlace de comunicaciones, centr´andose en la estructura del receptor para se˜nales d´ebiles enmascaradas en un fuerte ruido. En la Figura 1.1 se puede ver el escenario de comunicaciones. Se observa que el receptor esta directamente conectado con la oficina central. En este caso, en el receptor implementa un detector ca´otico a trav´es de la ecuaci´on Duffing. Se ha elegido esta ecuaci´on entre los sistemas cl´asicos no lineales debido a la gran cantidad de documentaci´on previa existente [11]. La naturaleza no lineal de la comunicaci´on objeto de estudio afectar´a a la estructura del receptor, por ello un sistema que resolviese la ecuaci´on Duffing con m´etodos lineales nunca ser´ıa una soluci´on ´optima. Consecuentemente, el modelo de simulaci´on computacional implementado se basa en un an´alisis num´erico. Se conocen t´ecnicas contrastadas de detecci´on que trabajan con niveles negativos de SNR con una elevada precisi´on, [12] y [13]. Este bajo SNR ha sido conseguido como resultado del uso de una elevada frecuencia de muestreo. Este trabajo se centrar´a en la Eb/No, que esta relacionada con la SNR como Eb/No =SNR ·Fs 2Fb, donde Fbes la tasa de bits y Fses la frecuencia de muestreo de la se˜nal. La principal diferencia entre Eb/No y SNR es que Eb/No tiene en cuenta el ancho de banda del ruido (Fs) y el ancho de banda de la se˜nal (Fb). En esta tesis se pretende evaluar te´orica y experimentalmente el rendimiento del receptor Duffing. Adem´as, se requiere analizar la viabilidad del sistema de recepci´on ca´otica Duffing utilizado como alternativa a la detecci´on de envolvente u otros m´etodos de demodulaci´on. Las simulaciones computacionales y los resultados experimentales son presentados para escenarios de RoF. Una vez se haya implementado el sistema de recepci´on ca´otico Duffing, se caracterizar´a dicho sistema para cada una de las t´ecnicas de modulaci´on ASK, Phase-Shift Keying (PSK) y Frequency-Shift Keying (FSK). Posteriormente, se llevar´a a cabo un estudio profundizando en la modulaci´on ASK y asemejando el escenario a una comunicaci´on real. El sistema de 1.3 Estructura de la tesis 5 recepci´on ca´otico Duffing y el m´etodo de detecci´on coherente ser´an analizados y se evaluar´a su rendimiento para diferentes anchos de banda entre 1GHz y 8GHz del filtro de entrada, siendo la tasa de la se˜nal definida a 1GHz para todos los escenarios. El objetivo final es la comparaci´on del rendimiento en t´erminos de BER requerido para diferentes Eb/No entre la detecci´on de envolvente y una modificaci´on del dise˜no convencional del Duffing. La ventaja de este dise˜no es su mayor simplicidad ya que no se necesita ninguna se˜nal de referencia. Las comparaciones son llevadas en las mismas condiciones para ambos sistemas, y para diferentes anchos de banda del BPF. Este BPF ser´a implementado como el primer bloque en el esquema de recepci´on. Finalmente, se pretenden validar los resultados de simulaci´on analizando los correspondientes datos del montaje ´optico de laboratorio. La demostraci´on experimental esta implementada para escenarios donde la velocidad de modulaci´on es de 13GHz de frecuencia de portadora y 1Gpbs de tasa de datos. 1.3 Estructura de la tesis La estructura de este trabajo se ilustra en la Figura 1.2. Esta compuesto por tres bloques principales; antecedentes y planteamiento del problema a resolver, desarrollo de la soluci´on y conclusiones. •Antecedentes y Problema expone una introducci´on b´asica a la detecci´on de se˜nales d´ebiles enmascaradas en entornos ruidos, se presenta el estado del arte y se analiza la aportaci´on de este trabajo en los sistemas de recepci´on. •Implementaci´on y Resultados engloba cuatro cap´ıtulos. En el primero se muestran los pasos a seguir para alcanzar una implementaci´on precisa del modelo de Matlab. En el segundo cap´ıtulo se estudian las tres t´ecnicas de modulaci´on y como se ha de caracterizar al oscilador Duffing para demodular cada una de ellas. En el tercer cap´ıtulo se estudia en profundidad la modulaci´on ASK. En el ´ultimo de estos cap´ıtulos, se realizan diversos montajes de 12 Receptor Duffing El resultado se muestra a continuaci´on: (˙x1=ωx2 ˙x2=ω(x1−x3 1−bx2+ccos ωt)(2.5) Esta nueva ecuaci´on har´a el sistema adecuado para futuras simulaciones en escenarios de RoF. Cap ´ ıtulo 3 Implementaci´on del modelo Una vez se ha explicado te´oricamente el oscilador Duffing en el cap´ıtulo anterior, se pretende construir el modelo de simulaci´on. En una primera aproximaci´on, se implementa dicho modelo a trav´es de Simulink. M´as tarde y a fin de tener un control total del sistema, se define un modelo similar mediante Matlab. Se entrar´a en detalle en el bloque de integraci´on para resolver la ecuaci´on Duffing, as´ı como en las soluciones propuestas. Este cap´ıtulo se explica m´as en profundidad en el Ap´endice B. Se presenta un an´alisis exhaustivo de los m´etodos y procedimientos utilizados, as´ı como una serie de gr´aficas y tablas comparativas justificando todas las decisiones adoptadas. 3.1 Modelo mediante Simulink El uso de Simulink como herramienta de simulaci´on no es algo casual. Es el m´etodo m´as extendido para la implementaci´on del oscilador Duffing como se puede apreciar en la gran mayor´ıa de las publicaciones consultadas, como por ejemplo [14] y [15]. El resultado de dicha implementaci´on puede verse en la figura 3.1. Este sistema trabaja muestra a muestra a partir de la se˜nal de entrada. En el primer bloque (ADD1), la se˜nal de referencia (DRIVER) junto con la se˜nal a detectar (TO-BE-DETECTED SIGNAL) son sumadas a las muestras de salida del ciclo anterior del sistema Duffing. Todo ello es posteriormente amplificado y previa integraci´on, se le resta a la se˜nal de salida de primer orden del sistema. Tras esto, la se˜nal es integrada dos veces (bloques INT) a fin de resolver la ODE de segundo orden de la ecuaci´on. Las salidas OUTPUT X yOUTPUT Y, ser´an utilizadas para 14 Implementaci´on del modelo Figura 3.1: Implementaci´on del oscilador Duffing en Simulink. representar los diagramas de fase. Gracias a estos diagramas es posible conocer el estado en el que se encuentra el sistema en todo momento. Despu´es de una serie de simulaciones y a pesar de que la mayor´ıa de nuestras referencias han obtenido sus resultados a trav´es de la herramienta Simulink, se observa que este m´etodo presenta una serie de desventajas. Al tener sus funciones previamente predefinidas, perdemos gran parte del control sobre la informaci´on con la que se est´a tratando. Adem´as, se necesita una interacci´on continua con Matlab para el procesado de la se˜nal. Es por ello que decidimos construir el modelo haciendo uso ´unicamente de Matlab. 3.2 Modelo mediante Matlab El procedimiento para la implementaci´on del oscilador Duffing en Matlab consiste en descomponer el modelo de Simulink presentado en la figura 3.1, analizando las entradas y salidas de cada bloque. De esta forma es posible calcular los retardos y otras relaciones entre bloques. Una vez se hayan caracterizado todos los bloques, se conectar´an unos con otros. 3.2 Modelo mediante Matlab 15 A lo largo de este proceso aparece un bloque cr´ıtico, el bloque de integraci´on. En ´el se resuelve la ODE de primer orden. El objetivo es definir el algoritmo m´as sencillo posible sin perder precisi´on. Tambi´en se tendr´an en cuenta otros factores como el rendimiento y el coste computacional; ya que para cada iteraci´on la operaci´on se ejecuta varias veces. Para ello han sido testados tres m´etodos diferentes. El primero utiliza el m´etodo integral de Euler. Adem´as se analizan tres posibles variantes de este m´etodo (Ap´endice B.2.1). El segundo m´etodo implementado es la regla trapezoidal (Ap´endice B.2.3). La tercera alternativa es el m´etodo de Runge-Kutta de cuarto orden (Ap´endice B.2.3). Una vez se han implementado los tres m´etodos, definimos unas serie de criterios para elegir el m´etodo m´as adecuado en cada situaci´on. Estos factores son: rendimiento, estabilidad y precisi´on. Los resultados se indican de forma simplificada en la tabla 3.1. Metodo Estabilidad Precisi´on Rendimiento Euler Forward baja baja alto Trapezoidal media media medio Runge-Kutta alta alta bajo Tabla 3.1: Comparativa entre el m´etodo de Euler Forward, la regla trapezoidal de integraci´on y la implementaci´on del m´etodo Runge-Kutta para los t´erminos de estabilidad, precisi´on y rendimiento. Como conclusi´on del an´alisis de estos tres m´etodos podemos afirmar que el m´etodo de Euler (Forward Euler) es el que menor coste computacional requiere y a su vez, es el m´as simple de implementar. Sin embargo, es el m´etodo menos preciso y estable de los tres. La regla trapezoidal es bastante m´as precisa y estable que el m´etodo de Euler. Este m´etodo lo utilizaremos para escenarios donde existe una gran cantidad de datos a procesar y el tiempo de procesado es un factor a tener en cuenta. El m´etodo de Runge-Kutta tiene como desventaja un mayor requerimiento en el tiempo de procesado que otros m´etodos multi-incremento de similares prestaciones. Sin embargo, entre los sistemas analizados en este proyecto, la relativa simplicidad y facilidad de uso compensa su elevado coste computacional. Por lo tanto, este tercer m´etodo ser´a utilizado en 16 Implementaci´on del modelo las principales simulaciones, as´ı como en las caracterizaciones de todo el sistema. Cap ´ ıtulo 4 Resultados de simulaci´on La mayor parte de la documentaci´on sobre osciladores Duffing presenta este sistema como una potente herramienta para la detecci´on de se˜nales d´ebiles. Sin embargo, es muy dif´ıcil encontrar referencias espec´ıficas en la literatura en las que se trabaje con un formato concreto de modulaci´on. Por este motivo se explica en este proyecto como deber´ıa estar definido y caracterizado el oscilador Duffing para poder utilizarlo con los tres formatos b´asicos de modulaci´on. Este cap´ıtulo se encuentra detallado en el Ap´endice C de este proyecto. 4.1 Detecci´on de se˜nales 2ASK mediante el oscilador Duffing Se ha utilizado OOK, la forma m´as simple de la modulaci´on ASK. Tiene la particularidad de que la portadora es nula para la transmisi´on de un cero l´ogico, como se puede observar en la figura 4.1. Como anteriormente se presento en el cap´ıtulo 2, tratamos de asociar el 0 l´ogico con el estado ca´otico del sistema Duffing. Para ello nos aseguramos de que el par´ametro frsea ligeramente inferior al valor del umbral entre ambos estados. Esta explicaci´on est´a ilustrada en la figura 2.3(a). Para el caso de un 1 l´ogico a la entrada del oscilador, la amplitud resultante ser´a la suma de las amplitudes de la se˜nal de referencia con la se˜nal a ser detectada. Esta amplitud total har´a que el oscilador Duffing se desplace desde el estado ca´otico hasta el estado peri´odico, se aprecia en la figura 2.3(b). Esto suceder´a siempre y cuando la fase de la se˜nal este definida fuera del rango indicado en la ecuaci´on (2.4). 18 Resultados de simulaci´on 1 1.5 2 2.5 3 3.5 4 4.5 5 −8 −6 −4 −2 0 2 4 6 8 10 Amplitude [a.u.] Bits transmitted [a.u.] Signal+Noise Signal Figura 4.1: Se˜nal OOK modulada. En color azul aparece la se˜nal transmitida y por tanto, libre de ruido. En color rojo observamos la se˜nal recibida tras el paso por el canal ruidoso. Los par´ametros de la simulaci´on son fc=50.000 πcomo frecuencia de portadora, amplitud a= 0.2, potencia de ruido 40 y tasa de datos FB= 100Hz. La salida del sistema para la detecci´on de la se˜nal OOK se muestra en la figura 4.2. En la primera imagen se presenta el diagrama de fase. Se aprecia como el sistema ha transitado por los estados ca´otico y peri´odico durante toda la transmisi´on. En la segunda imagen, se distingue en color azul la salida de nuestro sistema Duffing. Sobre esta se˜nal, se presentan en color rojo la secuencia de 1’s y 0’s transmitidos. As´ı se puede ver que para la transmisi´on de un 1 l´ogico, el oscilador se encuentra en el estado peri´odico y por lo tanto su salida respecto al tiempo es una se˜nal peri´odica. Para el caso de un 0 transmitido, la se˜nal resultante a la salida del sistema carece de cualquier tipo de periodicidad. Resulta sencillo identificar a simple vista cuando se ha transmitido un 1 o un 0. El c´odigo de Matlab para la detecci´on una se˜nal modulada OOK haciendo uso de un oscilador Duffing se muestra en el Ap´endice G.1. 4.2 Detecci´on de se˜nales BPSK mediante el oscilador Duffing 19 Figura 4.2: Diagrama de fase y salida del Duffing para el caso de detecci´on OOK. En la primera imagen se presenta el diagrama de fase de toda la comunicaci´on. Las ´orbitas exteriores son algo m´as gruesas, ya que han sido recorridas varias veces estando el sistema en el estado peri´odico. Las otras ´orbitas son aleatorias como consecuencia del estado ca´otico. En la segunda imagen, vemos en azul la salida del oscilador Duffing y sobre esa imagen en rojo, una se˜nal que representa la secuencia binaria transmitida. 4.2 Detecci´on de se˜nales BPSK mediante el oscilador Duffing Ahora se va a analizar la modulaci´on PSK. Si se utiliza la configuraci´on anterior para este escenario, el oscilador va a permanecer en el mismo estado durante toda la comunicaci´on. Esto se debe a que la amplitud de la se˜nal de entrada es constante tanto para los 1’s l´ogicos como para los 0’s. Para poder asociar los estados ca´otico y peri´odico con estos 1’s y 0’s l´ogicos, se habr´a de redefinir la fase de nuestra se˜nal. Se fija la amplitud de la se˜nal de entrada por encima del valor umbral Fd, as´ı nos aseguraremos de que el sistema este en el estado peri´odico. Al transmitir un 1 l´ogico, la se˜nal tendr´a la fase ϕ1. Esta fase ha de estar definida en el rango de fases donde se produce transici´on, por lo 20 Resultados de simulaci´on Large scale state Large scale state Chaos state r f a 2 cos1   r f a 2 cos1   0  2  1  2  Figura 4.3: Definici´on de fase para BPSK. Para el escenario en el que los requerimientos de amplitud se cumplen y una se˜nal con fase ϕ1es recibida en el oscilador, el sistema estar´a en el estado peri´odico. Por le contrario, cuando es una se˜nal con fase ϕ2la que se introduce en el oscilador Duffing, el sistema estar´a en el estado ca´otico. Figura 4.4: Se˜nal BPSK modulada. En color azul se representa la se˜nal transmitida con sus transiciones de fase. En color rojo, la se˜nal recibida en el oscilador Duffing. Los par´ametros de la simulaci´on son la frecuencia de portadora fc=50.000 π, amplitud a= 0.2, potencia de ruido 40 y tasa de bit FB= 100Hz. tanto el sistema permanecer´a en el estado peri´odico. Cumpliendo estas 4.2 Detecci´on de se˜nales BPSK mediante el oscilador Duffing 21 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −30 −20 −10 0 10 20 30 Phase diagram x(t) [a.u.] dx(t)/dt [a.u] 0 5 10 15 20 25 30 35 40 45 50 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Output of the Duffing system Samples [a.u.] Amplitude [a.u] Output signal Information transmitted Signal modulated with phase 1 Signal modulated with phase 2 Figura 4.5: Diagrama de fase y salida del oscilador Duffing para la modulaci´on BPSK. En la primera imagen se aprecia el diagrama de fase. En la segunda imagen podemos ver la salida del oscilador Duffing en color azul y sobre esta, una se˜nal que simula la informaci´on transmitida. Debajo de ambas se˜nales, la se˜nal transmitida se presenta en dos colores diferentes. As´ı puede distinguirse entre las dos fases de la se˜nal. dos condiciones se ha asociado el estado peri´odico al 1 l´ogico. La fase ϕ2estar´a definida en el rango en el que no se produce transici´on. De esta forma, las dos condiciones para alcanzar el estado peri´odico no se cumplen y el sistema volver´a al estado ca´otico para el caso del 0 l´ogico. Ambas fases, ϕ1yϕ2, se encuentran definidas en la figura 4.3. En la figura 4.4, se muestra la modulaci´on BPSK antes y despu´es de atravesar el canal ruidoso, colores azul y rojo respectivamente. Los cambios de fase se acent´uan con un circulo blanco a su alrededor. En la figura 4.5, se presenta el diagrama de fase de toda la comunicaci´on. Bajo esto, se ilustra la se˜nal modulada, representada en amarillo y rosa para diferenciar cuando se ha transmitido un 0 o un 1. En color azul esta representada la salida del oscilador Duffing y sobre dicha se˜nal, en color rojo, la secuencia de 1’s y 0’s transmitidos, antes de ser modulada. El c´odigo de Matlab para la detecci´on una se˜nal modulada BPSK usando 28 Evaluaci´on del rendimiento para la modulaci´on ASK threshold 1 threshold 2 one one zero zero Maximum value Sub-maximum Figura 5.4: Umbrales para la decisi´on de s´ımbolo. Un s´ımbolo es interpretado como un 1 l´ogico si su m´aximo se encuentra pr´oximo a threshold 1y su sub-m´aximo pr´oximo a threshold 2. En cualquier otro caso, el s´ımbolo se interpretar´a como un 0. de esos dos valores. Este proceso est´a dividido en cuatro pasos: •Primero, los s´ımbolos cuyo m´aximo excede el primer umbral pero su sub-m´aximo est´a por debajo del segundo umbral se consideran 1’s l´ogicos. Esto se puede interpretar como una se˜nal con una ´unica componente frecuencial. El comportamiento corresponde al espectro de la figura 5.3(a). •Se consideran como 0’s l´ogicos aquellos bits que no cumplen el primer paso. Y adem´as, su m´aximo ha de estar por debajo del primer umbral, as´ı como su sub-m´aximo ha de ser superior al segundo umbral. Se puede ver este caso como una se˜nal que no tiene una componente frecuencial concreta y suficientemente fuerte, pero que a su vez, est´a compuesta por varias frecuencias de no despreciable amplitud. Esto sucede en el espectro de la figura 5.3(b). •En este punto ya se han clasificado los bits m´as evidentes. Aquellos que no son tan claros, porque sus m´aximos y sub-m´aximos no cumplen ambas condiciones, se considerar´an como 1’s l´ogicos si sus valores est´an localizados en un rango en torno al ±11% de cada valor umbral. Principalmente, en este paso se corrigen errores debidos a una decisi´on de umbrales no ´optima. 5.2 Resultados de la comparaci´on entre el sistema Duffing y la detecci´on coherente 29 •Cualquier otro s´ımbolo que no haya sido clasificado se considera como un 0 l´ogico. Despu´es del bloque de decisi´on, la se˜nal se compara con el mensaje original transmitido y se calcula el BER correspondiente. El c´odigo de Matlab desarrollado se puede encontrar adjunto en el Ap´endice H.7. 5.2 Resultados de la comparaci´on entre el sistema Duffing y la detecci´on coherente La comparaci´on entre el sistema de recepci´on ca´otico Duffing y la detecci´on coherente se lleva a cabo utilizando una frecuencia de muestreo de 80GHz, una frecuencia de portadora de 13.8GHz y una tasa de datos de 1Gbps. Se analizan ambos sistemas para el rango de valores de Eb/No que va desde los 0dB hasta los 16dB. La implementaci´on del detector coherente se ilustra mediante bloques en la figura 5.5. From channel Threshold BER <0 > )cos( t  LPF Figura 5.5: Diagrama de bloques de la demodulaci´on coherente. Los resultados m´as significativos se encuentran en la figura 5.2. M´as gr´aficas comparativas aparecen explicadas en el Ap´endice D.4. En la figura 5.6(a) se presenta la comparaci´on directa entre el sistema Duffing, el m´etodo coherente y la detecci´on de envolvente. Se observa como la detecci´on coherente resulta ser el m´etodo m´as efectivo. Por lo tanto nuestro sistema no presenta ninguna mejora para este escenario. Aunque se comporta mejor que el m´etodo de detecci´on de envolvente, no se puede tener en cuenta este resultado ya que no se lleva a cabo en las mismas condiciones. El sistema Duffing, as´ı como el detector coherente, requieren una se˜nal de referencia que el detector de envolvente no utiliza. 30 Evaluaci´on del rendimiento para la modulaci´on ASK En la literatura se ha observado como el oscilador Duffing ha sido utilizado en entornos con tasas elevadas de muestreo y sin ning´un filtrado de banda previo. Eso nos lleva a comparar directamente el sistema de recepci´on ca´otico Duffing con la detecci´on coherente cuando el BPF a la entrada del receptor est´a definido con ocho veces m´as ancho que el ancho de banda de la se˜nal. Los resultados se puede ver en la figura 5.6(b). De ah´ı concluimos que el sistema Duffing tiene un mejor rendimiento para este escenario pero a´un as´ı no es suficiente para alcanzar un BER de 10−3 con 16dB de Eb/No. 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 C o h e r e n c e I n c o h e r e n c e D u f f i n g - l o g ( B E R ) Eb/ N 0 [ d B ] (a) BPF definido a 1GHz 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 - l o g ( B E R ) Eb/ N 0 [ d B ] Coherent C o h e r e n t 8 * F b D u f f i n g D u f f i n g 8 * F b (b) BPF con 8GHz de ancho de banda Figura 5.6: Comparaci´on entre el m´etodo coherente, incoherente y el sistema de recepci´on ca´otico Duffing. Frecuencia de portadora definida a 13.8GHz, frecuencia de muestreo de 80GHz y tasa de datos a 1Gbps. Resumiendo, se observa como el sistema Duffing resulta m´as efectivo en t´erminos de Eb/No que el detector coherente cuando el filtro de entrada del receptor empeora. Se podr´ıa asemejar el comportamiento del oscilador Duffing al de un BPF. Esto puede resultar ventajoso en escenarios donde la frecuencia de portadora no se conoce con precisi´on o en caso de que un Phase-Locked Loop (PLL) no pueda ser implementado. 5.3 Implementaci´on alternativa del receptor Duffing 31 5.3 Implementaci´on alternativa del receptor Duffing En la secci´on anterior se ha visto como el sistema Duffing rend´ıa mejor que el m´etodo de detecci´on de envolvente. Esto es algo l´ogico debido a la mayor complejidad del sistema Duffing. Lo que se pretende ahora es simplificar este sistema para que no requiera ninguna se˜nal de referencia a la entrada del mismo. As´ı el sistema de recepci´on ca´otico Duffing podr´a ser comparado con la detecci´on de envolvente en igualdad de condiciones. Por ello, eliminamos la entrada de la se˜nal de referencia. Entonces, toda la responsabilidad para controlar el sistema se transfiere a la se˜nal a detectar. Por lo tanto, la se˜nal de entrada se encargar´a de hacer transitar al sistema de un estado a otro utilizando ´unicamente su amplitud. a+fr Duffing fr driver To -be -detected signal Chaos state Large-scale periodic state fd fr a Figura 5.7: Implementaci´on cl´asica del oscilador Duffing. Se encuentra alimentado por una se˜nal de referencia y la se˜nal a ser detectada. La distancia entre el estado ca´otico y el estado peri´odico es m´ınima. Observando la figura 5.7, se aprecia la influencia de cada par´ametro en la transici´on. Se ha predefinido el valor del par´ametro frpara que se sit´ue el sistema en el umbral de transici´on entre el caos y la periodicidad. A partir de esto, la amplitud de la se˜nal a ser detectada inducir´a al sistema al estado peri´odico o permanecer´a en el estado ca´otico en funci´on de su valor. 32 Evaluaci´on del rendimiento para la modulaci´on ASK Duffing To -be -detected signal Chaos state Large-scale periodic state fd K*a K Figura 5.8: Modificaci´on propuesta sobre el dise˜no convencional del oscilador Duffing. El sistema esta alimentado ´unicamente por la se˜nal a ser detectada. La distancia entre el estado ca´otico y el estado peri´odico es superior al caso cl´asico. Una vez que hemos eliminado la se˜nal de referencia, insertamos un nuevo bloque de amplificaci´on. Se sit´ua justo antes de alimentar el Duffing (bloque Ken la figura 5.8). A partir de aqu´ı existen dos posibles casos de estudio: •Cuando se recibe un 0, no aparece ninguna se˜nal fuerte a la entrada. Solo habr´a interferencias de s´ımbolos anteriores o ruido del canal. Aunque todo ello sea amplificado no se alcanza un valor considerable para superar el umbral fd. Por lo que el sistema no alcanzar´a el estado peri´odico. •La otra situaci´on se da cuando se recibe un 1 l´ogico. La se˜nal no es muy fuerte por s´ı sola, pero una vez se amplifique, su amplitud superar´a el umbral fdobteniendo una salida peri´odica de nuestro sistema. Los resultados m´as significativos se encuentran en la figura 6.2. Las gr´aficas restantes en este escenario aparecen explicadas en el Ap´endice D.6. Para las simulaciones se utiliza una frecuencia de muestreo de 80GHz, una frecuencia de portadora de 13.8GHz y una tasa de datos de 1Gbps. En la figura 5.9(a) se presenta la comparaci´on directa entre la 5.3 Implementaci´on alternativa del receptor Duffing 33 (a) Comparaci´on para un filtro ideal, BPF definido a 1GHz 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 D u f f i n g i d e a l f i l t e r E n v e l o p e f i l t e r = F s D u f f i n g f i l t e r = F s - l o g ( B E R ) Eb/ N 0 [ d B ] (b) Comparaci´on cuando eliminamos el BPF, ancho de banda equivalente de 80GHz Figura 5.9: Comparaci´on entre el sistema de recepci´on ca´otico Duffing y el m´etodo de detecci´on de envolvente con diferentes anchos de banda del BPF. Frecuencia de portadora definida a 13.8GHz, frecuencia de muestreo de 80GHz y tasa de datos a 1Gbps. modificaci´on del sistema Duffing tradicional y la detecci´on de envolvente. El FWHM del BPF y la se˜nal, tienen el mismo ancho de banda. Se puede ver que aunque el sistema Duffing tenga un mejor comportamiento, las diferencias entre ambos m´etodos son pr´acticamente nulas para un rango de valores de Eb/No, que va entre 0 y 16dB. En la figura 5.9(b), se elimina el filtro BPF y la se˜nal queda limitada por la frecuencia de muestreo. En este caso obtenemos la m´axima ventaja haciendo uso de la modificaci´on del oscilador Duffing. Se requieren 1.5dB menos para alcanzar un BER de 10−3que en el caso de la detecci´on de envolvente. Cabe destacar tambi´en que existe una penalizaci´on de aproximadamente 3dB entre el sistema con un BPF ideal y aquel sin ning´un tipo de BPF. Para concluir, se ha visto c´omo este dise˜no rinde mejor en t´erminos de Eb/No que la detecci´on de envolvente. Cada vez que se ha incrementado el FWHM del BPF, las diferencias en cuanto a rendimiento entre ambos m´etodos han aumentado. 34 Evaluaci´on del rendimiento para la modulaci´on ASK Cap ´ ıtulo 6 Resultados experimentales En este cap´ıtulo se presenta el montaje de laboratorio implementado para verificar si los m´etodos descritos anteriormente pueden ser utilizados para mejorar los resultados ofrecidos por los m´etodos tradicionales Se llevan a cabo dos experimentos. En el primero se pretende caracterizar el´ectricamente el sistema Duffing. Sirve como primera aproximaci´on a un escenario real. Permite anticipar futuros problemas que puedan surgir al trabajar con se˜nales reales. En este caso se conecta el Vector Signal Analizer (VSA) directamente al Digital Storage Oscilloscope (DSO), por lo que apenas existe ruido en la comunicaci´on. Para m´as informaci´on sobre la caracterizaci´on del sistema de recepci´on ca´otico Duffing y los procedimientos seguidos, referirse al Ap´endice E.1. En el segundo experimento se implementa un montaje ´optico y m´as tarde, se procesan los datos. Dicho procesado utiliza la modificaci´on del oscilador Duffing convencional. 6.1 Sistema de recepci´on ca´otico Duffing para una transmisi´on de radio sobre fibra El esquema del montaje de laboratorio detallado se muestra en la figura 6.1. El transmisor es un Teraxion Distributed Feedback Laser (DFB) Narrow Linewidth con un linewidth inferior a 50kHz. La se˜nal PRBS el´ectrica es generada a diferentes tasas con el Pulse Pattern Generator (PPG) Agilent HP 70843A. La se˜nal es amplificada a fin de alimentar el Modulador Mach-Zehnder (MZM). El voltaje de driving del MZM se ajusta para obtener el m´aximo ratio de extensi´on, 17dB. La se˜nal que se obtiene ha sido transmitida a lo largo de la fibra para 36 Resultados experimentales posteriormente a˜nadir ruido. La fuente de ruido blanco es un Erbium- Doped Fiber Amplifier (EDFA). Tras esto, aparece un filtro de 0.8nm de ancho de banda ´optico para limitar ´unicamente el ruido a banda. A continuaci´on, la se˜nal se amplifica de nuevo y se conecta a otro filtro con un ancho de banda ´optico de 0.5nm. La salida del filtro se une directamente al Atenuador ´ Optico Variable (VOA) para mantener el nivel de potencia de la se˜nal constante. El l´aser utilizado como Oscilador Local (LO) es un External Cavity Laser (ECL) con un linewidth inferior a 300kHz. Despu´es se mide la se˜nal con un Power Meter (PM), siendo la potencia del LO −3.2dBm y la potencia de se˜nal −0.9dBm. Finalmente, se conecta la se˜nal a un Fotodiodo (PD). Para este experimento han sido usados dos tipos de PD diferentes. Uno para baja frecuencia con 10GHz de ancho de banda y amplificador interno. El otro PD para alta frecuencia, 40GHz de ancho de banda y amplificaci´on el´ectrica auxiliar. La salida del PD est´a conectada a dos DSO, capaces de capturar 80GSa/s con un ancho de banda de entrada de 30GHz. Las caracter´ısticas de la se˜nal obtenida son las siguiente: 1Gbps de tasa de datos, frecuencia de modulaci´on a 13.8GHz para un OSNR entre 18 y 1dB. La frecuencia de muestreo es 80GHz. Con la se˜nal del laboratorio se alimenta el esquem´atico de la figura 6.2. Este diagrama de bloques viene precedido por un algoritmo de sincronismo de bit (Ap´endice E.2.2). Tras esto, la salida sincronizada se divide en tres ramas: •La primera rama simula un detector coherente con elementos ideales. Su funci´on es la de generar una r´eplica de la se˜nal transmitida. Para ello, se demodulan los primeros 15 bits y se implementa un algoritmo que dar´a una r´eplica de la se˜nal PRBS. •La segunda rama implementa el sistema de recepci´on ca´otico Duffing haciendo uso de la ecuaci´on RK4. Como primer bloque, esta implementado un BPF de ancho de banda variable. •La ´ultima rama es la detecci´on de envolvente. En este caso tambi´en lleva un BPF variable para estudiar como afecta a su rendimiento 6.1 Sistema de recepci´on ca´otico Duffing para una transmisi´on de radio sobre fibra 37 PPG 3 dB 3 dB OSA ASE OSC DSO 5 nm 8 nm 10 dB Power Meter 3 dB Power Meter MZM Power Meter 10 dB Figura 6.1: Esquem´atico para una transmisi´on de RoF. Los par´ametros de la comunicaci´on son 1Gbps de tasa de datos, 13.8GHz de frecuencia de portadora y un OSNR que va de 18 a 1dB. La frecuencia de muestreo est´a definida a 80GHz. 44 Conclusi´on y trabajo futuro •En el campo de las comunicaciones seguras se podr´ıa tener en cuenta el algoritmo ya desarrollado. Su aplicaci´on m´as interesante esta en su uso en el transmisor de un sistema para la codificaci´on de la se˜nal. Una vez esta se˜nal fuese transmitida, se utilizar´ıa el mismo algoritmo en el receptor a fin de decodificar la se˜nal detectada. •Otro punto de estudio interesante es el uso de nuestro sistema en demodulaciones en las que la se˜nal trasmitida haya sido modulada a trav´es de un modulador no lineal. Podr´ıa analizarse si este sistema resulta una alternativa v´alida con un rendimiento mejorado frente a m´etodos existentes. •Implementaci´on en tiempo real para Field-programmable Gate Array (FPGA). Ap´ endice A Duffing receiver The aim of this chapter is to present a mathematical overview about the chaotic Duffing receiving system. We start by defining the basic principles of our second order differential Duffing equation. After this, we describe the procedure to find the bifurcation value which will make our system transit from different states of chaos and stability. Later on in this chapter, we will explain how to characterize this model and which requirements we should fulfil in order to be able to detect weak signals. Finally, we will find the parameter settings required for detecting signals with any frequency. A.1 Fundamental principle Theoretically, chaos can be generated through a first-order differential equation [16]. Nevertheless, the regularity of its output would not result easy to study. This is a risk we cannot afford because the output is a determinant factor to distinguish between noise and signal. For that case, signal and noise would play the same role and would have the same weight in affecting the initial conditions of the chaotic time series [17]. This is the main reason why a second-order ODE is employed to implement the chaotic detector: ¨x+δ˙x+f(x) = fr·s(t) (A.1) where f(x) is a nonlinear restoring force, and fr·s(t) is the driving force. As we mention in the previous chapter; Duffing equation, among all the ODEs, was a good choice for our chaotic detector due to the intensively documentation available. Its associated bifurcation process has already been studied extensively ([18], [19] and [20]). 46 Duffing receiver Thus, we formulate the Duffing equation as follows: ¨x+δ˙x−x+x3=fr·cos(t+θ) (A.2) where the driving force is now defined as a reference signal fr·s(t) = fr·cos(t+θ) with unknown phase θ. So as to simplify future implementations, second order equation above is converted to an equivalent system of first order equations and formula (A.2) can be also written as: (˙x=y ˙y=x−x3+ε(fr·cos(t+θ)−δy)(A.3) where δis the damping ratio, fris the force amplitude and εis a small scaling parameter. A.2 Bifurcation value calculated via Melnikov method The aim of this section is to find numerically the bifurcation threshold. This value causes the transition between states. To achieve that, we start by forcing the scaling parameter to be zero (ε= 0) in equation (A.3)and we obtain an unperturbed system: (˙x=y ˙y=x−x3(A.4) Equation (A.4) yields a Hamilton system [21]. Its Hamiltonian function is defined as: H(x, y) = y2 2−x4 4+x6 6=h(A.5) Let h= 0, the system is composed of two homoclinic 1orbits, Γ0 +,Γ0 − 1In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium. A.2 Bifurcation value calculated via Melnikov method 47 and a point p0= (0,0). Or we can also see it, as one center (0,0) and two hyperbolic saddle points (1,0) and (−1,0). So, the two homoclinic orbits are defined as: q0 +(t)=(√2sech t, −√2sech t·tanh t) q0 −(t) = −q0 +(t)(A.6) From the perspective of q±(0) = (±√2,0) the unperturbative homoclinic orbits can be given as follows: (x0(t) = ±√2sech t y0(t) = ∓√2sech t·tht (A.7) Melnikov function is used to compute the parameter values for which a transverse crossing occurs [22]. This function is related to the separation between the stable and unstable manifolds: M(t0) = +∞ Z −∞ y0(t)[Fcos ω(t+t0)−δy0(t)]dt =−√2F +∞ Z −∞ sechttanh tcos ω(t+t0)dt −2δ +∞ Z −∞ sech2ttanh2tdt (A.8) With reference to the direct integration and residue theory [23], we can get: M(t0;F, δ, ω) = −4δ 3+√2Fsech(πω 2) sin ωt0(A.9) Letting: I1=−4 3, I2=√2πωsech(πω 2). If a chaotic state occurs, M(t0, F, δ, ω) = 0 must have solutions, so we can obtain: I1δ+I2fsin ωt0= 0. Therefore:  I1δ I2f =|sin ωt0| ≤ 1. 48 Duffing receiver And because of: dy(t0) dt0 =√2Fcos(ωt0) cosh(πω/2) (A.10) If dy(t0) dt06= 0,|sin(ωt0)| 6= 1 can be obtained. So we can get: F δ>I1 I2 and we can obtain the following bifurcation threshold for q0 +(t) or q0 −(t): R0(ω) = F δ=4 cosh(πω/2) 3√2πω (A.11) Solving numerically equation A.11, we can know the threshold values for which the system enters a state of chaos, as well into large-scale state when δestablished. For further information in mathematical procedures, refer to [11] and [24]. Above calculations have been done following the these two references. Once the parameter δis fixed, the system will change regularly when varying the parameter F. Using equation (A.11) we can get that the threshold value is 0.7530 , which means that chaos will appear in the system when F > 0.3765 [25]. Depending on the relation between F,δand the threshold value we identify the following states. •When F/δ < R0(ω), in the parameter region stable manifold is disjoint with unstable mainfold in the Poincar´e mapping of the system [25]. Period 1 interior trajectory system in reached, as we can see in the Figure A.1(a). •When F/δ > R0(ω), stable manifold must be joint with unstable manifold and traverse homoclinic point occurs, reciprocal odd-step bifurcation occurs as shown in Figure A.1(b). We refer to this state as bifurcation. •With the increasing of F, system turns into the chaotic motion, as we can see in Figure A.1(c) where the trajectory of the orbits in the phase will tend to fill up a section of the phase space. A.2 Bifurcation value calculated via Melnikov method 49 (a) Homoclinic Orbit, F/δ < R0(ω) (b) Bifurcation, F/δ > R0(ω) (c) Chaos, F/δ > R0(ω) & F < Fd(d) Great Scale Periodic, F > Fd Figura A.1: Different phase plane diagrams of the chaotic Duffing receiving system representing its four different states. The parameter δis fixed at 0.5 and the bifurcation threshold R0(ω) at 0.3765. •When F increases, the oscillator turns from the chaotic motion into the non-equilibrium phase state and making Fincreasing further, F > Fd, the periodic motion takes place, where the phase plane orbit traces out a closed curve, shown in Figure A.1(d). From equation (A.3), analytically predicted threshold values are R0(ω) = 0.41 and Fd= 0.829 through numerically research. Fdwill be a key factor in future implementations. It will define the border between chaos and periodic state and this fact will be our most important tool in order to differentiate between transmitted logical 1’s and logical 0’s. In Table 1, from the literature [3], we can see some experimental results as reference, where it is illustrated the relation between the bifurcation value 50 Duffing receiver Fdand the step size h,γcand hrespectively. The main conclusion obtained is that the smallest the step size, the largest the bifurcation value. This property keeps constant until a minimum step size for which the bifurcation value remain constant and equal to 0.816. A chaotic phase change predicts that weak signal excitation can induce chaotic response. That is why Fshould be set a little smaller than the critical value Fd. This way, the system is put on the verge of changing to the periodic motion but in a critical state. A.3 Signal detection model So far, Duffing has been theoretically characterized. We are going to add the noisy to-be-detected signal s(t) into our system. It represents a small amplitude signal having a little angular frequency difference with respect to the inner driving force. The new Holmes-Duffing equation can be written as: (˙x=y ˙y=x−x3−ky +frcos t+s(t)(A.12) where s(t) = acos((1+∆ω)t+ϕ)+zs.zs represents Gaussian white noise with zero mean and variance z. ∆ωis the angular frequency difference between the inside driving force and the periodic disturbing signal. To configure a Duffing oscillator for weak signal detection, firstly we have to assure the oscillator is on the chaotic motion state. It means to adopt fra bit smaller than the threshold fd, but in the edge transforming to large periodic state. Once the to-be-detected signal is included, its amplitude ‘a’ will contribute to the value of frbecoming fr+aa bit larger than fd. That will change the Duffing oscillator from the previous chaotic state to the stable one. Numerically, what has happened with the total drive force is the following: frcos t+acos((1 + ∆ω)t+ϕ) A.3 Signal detection model 51 re-written as frcos t+acos((1 + ∆ω)t+ϕ) = = (fr+acos(∆ωt +ϕ)) cos t−asin tsin(∆ωt +ϕ) = F(t) cos(t+θ(t)) So we can get:    F(t) = pf2 r+ 2fracos(∆ωt +ϕ) + a2 θ(t) = arctan asin(∆ωt +ϕ) fr+Acos(∆ωt +ϕ)(A.13) Because of af, we can neglect θ(t)≈0, and only F(t) affects the system state. It can also be seen from equation (A.13) that the orbit shift is related to the difference of the phase between the external signal and the reference signal [1]. At this point, we can define two cases to study, depending on the phase difference. First, when ∆ωis null, valid only for simulation experiments and theoretical explanations. Second, and closer to realistic environments, when there is a difference of frequency between reference and to-be-detected signals. Even using a frequency synchronization scheme based on PLL, it would not be possible to avoid it. A.3.1 Phase synchronization There is a second factor to take into account and also a second parameter to calculate in order not to keep the system in the chaotic motion all the time. There is a range of values: π−cos−1a 2fr≤ϕ≤π+ cos−1a 2fr (A.14) where phase variation does not occur, and only if ϕis not in this regime, the phase transition will take place. To have an approximate idea of this range, we simplify equation (A.14) using afand plot a simple graph in Figure A.2. 52 Duffing receiver π π-1 π+1 0 2πφ Phase transition Phase transition No transition Figura A.2: Duffing oscillator behavior depending on ϕ. A.3.2 Small frequency difference Considering equation (A.13) again, if ∆ω6= 0, F(t) will be periodically less than or more than the bifurcation value fd. Figure A.3 represents the vector chart of the total drive force F(t) in both cases. Supposing the inside driving force vector frfixed, the disturbing signal vector will circumrotate very slowly around the former using the angular frequency. •When driver and to-be-detected signal frequencies are close and their directions tend to consistency, the resultant vector becomes the sum of both forces and makes the total driving force amplitude larger than in some time range a+fr> fd. The consequence is that the oscillator transition occurs from chaos state to the large periodic state [26], see graphical description in Figure A.3(a). •The other situation is shown in Figure A.3(b), where force directions tend to deviate from each other and the resultant vector makes the total drive force amplitude smaller than in some time range a+fr< fd. The oscillator will remain in the chaotic motion state. In conclusion, since the frequency of s(t) is the same as that of the referenced signal, the system described by equation (A.12) will cause a phase shift, even if its amplitude is weak. This means that the system will convert the chaotic motion into a large periodic motion. When the frequency of weak signal s(t) is different and far from that of referenced signal, the system will not induce phase shift. Otherwise, phase transition could be done. This property of bifurcation can be used as an indicator for detecting weak signals. When the system is fixed in a critical chaotic state, the signal acos ωt is null and only white noise zs exists, since the A.4 Detection model for any frequency signal 53 (a) F(t) larger than the bifurcation value (b) F(t) smaller than the bifurcation value Figura A.3: The vector relation of the chief instigating force. This figure has been taken from [1] and slightly modified in order to be used for the above explanation. system is immune to a wide range of additive white noise, it remains in the chaotic state. When the signal is detected, even though it is weak, the system will go into periodic state, the phase transition state from chaotic to periodic state is the criterion whether the signal exists. A.4 Detection model for any frequency signal Duffing system in equation (A.3) can only detect low-frequency sinusoidal signals. In order to make the equation (A.3) suitable for detecting highfrequency sine signal and keep the state of chaos [27], it can be modified as follows: ¨x(t) + δ˙x(t)−x(t) + x3(t) = γcos(t+θ) (A.15) Let t=ωξ, then x(t) = x(ωξ) = y(ξ) (A.16) 60 Model Implementation first plot represents the result of the first integral in the system. In blue, it will always be the output of Simulink and over that, it is plotted the output of the integral implemented in Matlab. The second plot shows the second integral consecutive in the system. It is fed by the output of the first integral, so errors could be caused by the previous integral block. Second signals (red or green color) show the evolution of the Euler integrals. It is seen how, in the case of Forward Euler, Matlab output is still converging. By contrast, it does not happen the same for the other two integral approximation methods. They have similar values for the first integral. But in the second plot, there is not any kind of convergence between both signals. This situation is illustrated in the second plot of Figure B.5 for the case of Backward Euler, where signals follow different paths. It is obvious that Euler Forward is the closer method to the Simulink case because it follows the same trajectory. But anyway, for large vector the integral is not accurate enough and in order to prevent future errors from that case when simulations are quite long, we decide to implement a better approximation. Another factors of this method are analyzed in section B.2.4. Matlab code for the implementation can be found in the Appendix F.1. B.2.2 Trapezoidal Rule for Integration Large amount of data will be simulated and errors will appear. In order to solve ODEs more accurately and limit errors as much as possible, we replace Euler method for the Trapezoidal rule. It means a secondorder method instead of a first-order method. From an integral which approximates its value by the area of a rectangle to a method which makes use of the approximation by the area of a trapezoid. Trapezoidal rule has a local error O(h3). The choice of the previous function can be refined to obtain more accuracy with fewer evaluations of the function f(x). We have seen that B.2 Matlab Model 61 using a constant value for the approximation to the function f(x) on each interval leads to a roughly triangular area that either over or under estimates the area beneath f(x). This roughly triangular characteristic suggests the simple refinement of using a straight line to approximate f(x) on each interval [33]. This straight line is the chord joining the points aand bon the curve, as we can see in Figure B.6. The equation for this method can be written as: yn+1 =yn+h∗f(xn)+f(xn+1) 2 Once the method is already implemented and tested, we can see how much precise and stable than Euler Forward is, comparisons are shown in Figure B.8 and table B.1. Trapezoidal Rule is also a more stable method. However, comparing Forward Euler and Trapezoidal Rule, the second one takes much more time to converge [34]. It is shown in Figure B.7 where Duffing oscillator was defined in the large scale periodic. So, the output of the system should be a sine periodic signal form the first sample but it takes around 5000 samples to achieve the convergence where less than 100 were needed in the Forward Euler method implementation. This fact could entail the lost of the first data in the communication using the current method. There is another method that can be implemented in order to improve Trapezoidal Rule method and go a little bit further in the goal of implementing the most accurate system as possible, we can use Runge-Kutta methods. These methods are still one step methods, but they depend on estimations of the solution at different points. Other factors of Trapezoidal Rule method are analyzed and compared in B.2.4. Matlab code for the implementation can be found in F.2. B.2.3 Runge-Kutta Method The main advantages of Runge-Kutta methods are that they are still easy to be implemented, they are very stable, and they are “self-starting” (i.e., unlike multi-step methods, we do not have to treat the first few steps taken by a single-step integration method as special cases). 62 Model Implementation Among all the others, we use the RK4 algorithm to solve the Duffing equation [35]. Four evaluations of f(x) at every timestep will be required. Therefore, the system is a discrete dynamic system by nature. Different step sizes obtain different maps from the same differential equation. The dynamics of all these derived discrete systems are similar, but slightly different from the original continuous system. The common RK4 equation implemented in matlab is the next: y(t0) = y0 k1=f(tn, yn) k2=f(tn+1 2h, yn+1 2hk1) k3=f(tn+1 2h, yn+1 2hk2) k4=f(tn+h, yn+hk3) tn+1 =tn+h yn+1 =yn+1 6h(k1+ 2k2+ 2k3+k4) where y0defines the initial value. Thus, the next value yn+1 in the output will be determined by the present value ynadded to the step size of the interval hand an estimated slope. The slope is a weighted average of slopes: •k1is the slope at the beginning of the interval. •k2is the slope at the midpoint of the interval, using slope k1to determine the value of yat the point tn+1 2husing Euler’s method. •k3is again the slope at the midpoint, but now using the slope k2. •k4is the slope at the end of the interval, using in this case k3. B.2 Matlab Model 63 A weighted average of the slopes is performed, giving more importance to the slopes at the midpoint: slope =1 6(k1+ 2k2+ 2k3+k4) As we know, there is truncation error (also named as discretization error) linked to a Runge-Kutta algorithm. It exists even with infinite precision arithmetic, because it is caused by truncation of the infinite Taylor series to form the algorithm. Discretization error depends on the step size used, and the dependence is especially distinct when the system is strongly nonlinear. As far as our system is concerned, if the step size used is different, the truncation error will bring a distinguishable discrepancy of the critical value fr. Whatever the value of the step size is, the phase transition itself is clear and distinct; what makes the difference is just the value of fr. The truncation error does not mean that the step size is required to be very small to detect chaos onset accurately. The most difficult part of the implementation was the concatenation of the two integrals in our system. Output data from one block were needed as an input for the second one. How it was solved and implemented is shown in Appendix F.3. Other factors of this method are analyzed and compared in section B.2.4. B.2.4 Comparative of integral methods To chose the integral method that will match better in our system, we define as the most important criteria the three following factors: performance, stability, and accuracy. Performance refers to the computational cost of simulations for a given timestep. Stability refers to how well the integrator copes with stiff constraints such as high spring constants before errors become unacceptably large. Accuracy refers to how well the integrator matches the expected result. 64 Model Implementation For stability, systems are tested when the damping ratio δis defined as 0.5. This will be the value used in all the simulations. This choice is supported by the fact that in most of the papers same value is used (e.g., [27] where is named as bor [24], etc). In Figure B.8, it can be observed the behavior of the Forward Euler method, the Trapezoidal Rule method and Runge-Kutta implementation. The one which tends to zero is defined as the most stable. Ordered from least to most stable: Euler, Trapezoidal and RK4. Regarding accuracy, there are also some simulations which show the error with respect to the exact value. Table B.1 illustrates the errors of each case for a simulation. We can state that higher order integrators are most accurate. So, RK4 is the most accurate. Then, it is Trapezoidal Rule method and the least accurate, is the Forward Euler integral method. Time Euler Error Trapezoidal Error RK4 Error 0,4 0,00651 1,79E-05 8,85E-10 0,8 0,03939 1,44E-04 1,98E-09 1,2 0,21938 5,59E-04 2,57E-08 1,6 0,902 4,85E-04 1,12E-07 2 2,8816 3,36E-03 3,57E-07 Tabla B.1: Numerical results for analyzing the accuracy between Forward Euler integral method, Trapezoidal Rule method and the Runge-Kutta implementation. Performance is the last criterion to evaluate. We run the same simulation for the three methods and therefore, we can classify them from slow to faster: RK4, Trapezoidal and Euler. RK4 method resulted four times slower than Euler method. To summarize, Table B.2 indicates the ranking of each integrator in terms of stability, accuracy and performance. As conclusion, Forward Euler method requires the lowest computational power and it is the easiest to implement. However, it is the least accurate and stable. Trapezoidal method is much more accurate than Euler method and also more stable. This system will be considered for scenarios B.2 Matlab Model 65 Method Stability Accuracy Performance Euler Forward low low high Trapezoidal medium medium medium Runge-Kutta high high low Tabla B.2: Comparative between Forward Euler integral method, Trapezoidal Rule method and the Runge-Kutta implementation in terms of stability, accuracy and performance. where a high amount of data is processed and the computational time becomes important. In the last case we analyze the Runge-Kutta method, its primary disadvantage is that requires significantly more computation time than multi-step methods of comparable accuracy and mainly compared to the developed methods above, and it does not easily yield good global estimations of the truncation error. However, for the systems under investigation in this thesis, the advantage of the relative simplicity and ease of use of RK4 method far outweighs the disadvantage of their relatively high computational cost. It will be used in most of the simulations as well as in the main Duffing characterization. 66 Model Implementation (a) Phase diagram in periodic state, no input noise (b) Phase diagram in periodic state, noisy input (c) System output in the periodic state free of noise (d) System output in the periodic state with noisy input Figura B.2: Large scale periodic state - Matlab Simulink results. B.2 Matlab Model 67 (a) Phase diagram in chaos, no input noise (b) Phase diagram in chaos, noisy input (c) System output in chaos free of noise (d) System output in chaos with noisy input Figura B.3: Chaotic state - Matlab Simulink results. 68 Model Implementation Figura B.4: Forward Euler Integration method. The first plot represents the result of the first integral in the system using Forward Euler. The second plot is the output of a second Forward Euler integral, fed by the output of the first Forward Euler integral block. Simulink outputs are plotted in color blue, they represent the most accurate solution. Red color signal represents the output of the first integral and green color signal shows the second integral, both obtained through Matlab simulation. B.2 Matlab Model 69 Figura B.5: Backward Euler Integration. The first plot represents the result of the first integral in the system using Backward Euler. The second plot is the output of a second Backward Euler integral, fed by the output of the first Backward Euler integral block. Simulink outputs are plotted in color blue, they represent the most accurate solution. Red color signal represents the output of the first integral and green color signal shows the second integral, both obtained through Matlab simulation. Figura B.6: Trapezoidal Rule, f(x) (blue) is approximated by a linear function (red). Image obtained from http://wikipedia.org 76 Simulation results and large scale periodic would not be possible. Due to this fact, we have to redefine the Duffing in order to reach both states making the system able to differentiate between logical ‘1’ and logical ‘0’. Large scale state Large scale state Chaos state r f a 2 cos1   r f a 2 cos1   0  2  1  2  Figura C.5: Phase definition for BPSK. For the scenario in which amplitude requirements were fulfilled, if a signal with phase ϕ1feeds the Duffing oscillator, the system will keep in the large scale periodic state. By contrast, when the signal with phase ϕ2goes into the Duffing oscillator, the system will remain in the chaotic state. The new idea lies in the smart definition of two different phases. One of these should be in the regimen that fulfil Duffing oscillator requirements (ϕ1) and the other should be out (ϕ2). When the signal with phase ϕ1 feeds the Duffing oscillator, the system will reply as usual. It will keep in the large scale periodic state because phase and amplitude requirements are fulfilled. By contrast, when the signal with phase ϕ2goes into the Duffing oscillator, only the amplitude requirement will be satisfied and consequently, system will remain in the chaotic state. This concept is explained graphically in Figure C.5. The general form for BPSK follows the equation: sb(t) = q2Eb Tbcos(2πfct+ π(1 −n)), n = 0,1.This yields two phases and for our case, binary data propagates with the following signals: φ1(t) = r2 Ts cos(2πfct+ϕ1) φ2(t) = r2 Ts cos(2πfct+ϕ2) C.2 A method for BPSK signal detection using Duffing Oscillator 77 Figura C.6: BPSK modulated signal. In blue color is the transmitted signal where it can be observed the phase transitions. In red color, the received signal after the channel. Parameters of the carried simulation are the carrier frequency at fc=50.000 π, amplitude a= 0.2, noise power of 40 and bit rate FB= 100Hz. For the simulation we have implemented a basic set-up similar to the one in Figure C.2, but in this case signal it is modulated with the equations above. Then, strong noise involves the transmitted signal. Transmission parameters are also the same that in previous section: fc= 50.000 π, amplitude a= 0.2, noise power of 40 and bit rate FB= 100Hz. In Figure C.6, it is depicted the signal before going though the channel (in blue) and the same signal after the channel (in red). Phase changes in the signal are highlighted with a white circle around them. In Figure C.7, it is plotted the phase diagram in the first image. In the second, it is shown the output of the system. In blue the output of the Duffing, in red transmitted logical 1’s and logical 0’s. Just below that, it is the transmitted signal in two different colors. Each one represents a different phase in the signal. 78 Simulation results −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −30 −20 −10 0 10 20 30 Phase diagram x(t) [a.u.] dx(t)/dt [a.u] 0 5 10 15 20 25 30 35 40 45 50 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Output of the Duffing system Samples [a.u.] Amplitude [a.u] Output signal Information transmitted Signal modulated with phase 1 Signal modulated with phase 2 Figura C.7: Phase diagram and output of the Duffing for BPSK detection. In the first one, the phase diagram is shown. The second plot shows in blue color the output of the Duffing oscillator and over that, in red color, it is a line which simulates the transmitted binary information. Below these signals, it is the transmitted signal in two different colors. Each color represents a different phase in the signal. By the Figure C.7, we can intuitionally see that the envelope waves of phase shift are smooth, hence we can say the system is in the periodic state; and if the envelope waves of phase shift have a great fluctuation, we can say that the system is in the chaotic state. Matlab code for the BPSK signal detection using Duffing oscillator can be found in Appendix G.2. C.3 A method for BFSK signal detection using Duffing Oscillator 79 C.3 A method for BFSK signal detection using Duffing Oscillator Last but not least, the modulation to analyze is FSK. This technology has many merits such high power efficiency, convenient realization, reliable control and low cost. That is why is widely applied in digital communication fields [36]. In our experiment we will use a pair of discrete frequencies to transmit binary information (f1and f2). We are going to fulfil all the specifications for f1, so driver frequency will be really close and phase will be also fixed in the right regime (see Figure A.2). All this will make Duffing oscillator goes into the periodic state. The other frequency f2will not match with the reference signal, so periodic state could not be reached, keeping the Duffing oscillator in the chaotic state. 2 3 4 5 6 7 8 9 10 −4 −3 −2 −1 0 1 2 3 4 Amplitude [a.u.] Bits transmitted [a.u.] Signal+Noise Signal Figura C.8: BFSK modulated signal. In blue color is the transmitted signal where it can be observed the two different frequencies. In red color, the received signal after the channel. The two carrier frequencies are f1= 59.000Hz and f2= 60.000Hz. Sampling frequency is fs= 24.000.000Hz and code rate 600 bits. Also for this scenario, the values frand fdare really determinant. As in the of OOK modulation, they have to be close to each other (see 80 Simulation results Figure C.1). For the simulation we have slightly followed [13] and we defined the same parameters. The two carrier frequencies are f1= 59.000Hz (corresponding to 1) and f2= 60.000Hz (corresponding to 0). Sampling frequency is fs= 24.000.000Hz and code rate 600 bits, having the next relation between frequencies fs/fb= 40.000. Figure C.8 follows the structure than the previous plots when noisy signal was shown. We can see how the two different frequencies compose the transmitted signal. In the plot below (Figure C.9) is illustrated how Duffing behaves with this kind of modulation. −3 −2 −1 0 1 2 3 −40 −30 −20 −10 0 10 20 30 40 Phase diagram x(t) [a.u.] dx(t)/dt [a.u] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 105 −3 −2 −1 0 1 2 3Output of the Duffing system Samples [a.u.] Amplitude [a.u] Output signal Information transmitted Signal at frequency f1 Signal at frequency f2 Figura C.9: Phase diagram and output of the Duffing for BFSK detection. In the first one, the phase diagram is shown. The second plot shows in blue color the output of the Duffing oscillator and over that, in red color, it is a line which simulates the transmitted binary information. Below these signals, it is the transmitted signal in two different colors. Each color represents a different frequency in the signal. The method here explained is able to distinguish between two frequencies. However, there are some proposals to implement a system which is able to distinguish an unknown frequency signal, [25] and [29]. The idea is to design a Duffing oscillator array to scan the incoming signal. Whenever the Duffing oscillator reaches the intermittent chaos, the frequency of the C.3 A method for BFSK signal detection using Duffing Oscillator 81 detected signal is close to the frequency of the system. This should be the basic theory to detect weak signals using Duffing oscillator arrays. Matlab code for BFSK signal detection using Duffing oscillator can be found in Appendix G.3. 82 Simulation results Ap´ endice D Performance evaluation for ASK The main goal in this chapter is to establish in which cases a chaotic Duffing receiving system can be applied, in which cases it is worse or better than the conventional methods. As a Duffing oscillator is one part of the demodulation set-up, we also test which option for subsequent symbol decision match better just after the Duffing oscillator in order to get the lowest BER possible. To accomplish this assessment, five different methods for symbol decision are implemented and tested in the following sections. We developed two scripts. The first one is in charge of the transmitter implementation and noise loading, the other is in charge of the reception and subsequent demodulation. The two scripts are linked by a file. This file is created by the transmitter and it is similar to those obtained from a DSO. It contains its sampled output bits, which are fed as input for the receiver model. This configuration will become useful for future experimental applications where we will have only the file obtained from the DSO and our receiver. D.1 Transmitter The first block in the transmitter is the PRBS signal generator. It is a random sequence, that is, the value of the first elements is independent of the values of any of the other elements, similar to real random sequences. But it is ‘pseudo’ because it is deterministic and after Nelements it starts to repeat itself. Nin our simulations will be 215 −1 or 27−1. PRBS signal generator block feeds the ASK modulator block. There, 84 Performance evaluation for ASK PRBS signal is upsampled and filtered. Upsample is carried in order to increase the N´umero de muestras por s´ımbolo (Nss) in the demodulation. Filter can be a raised-cosine, which is a filter frequently used for pulseshaping 1in digital modulation due to its ability to minimize Interferencia Intersimb´olica (ISI). Before transmission, baseband signal is modulated in to a carrier frequency; as shown in Figure D.1. Carrier frequencies used in our study are around 15 and 25GHz. The scheme for all this explained above can be seen in Figure D.1. PRBS Raised Cosine ASK generation To channel AWG noise Quantization noise To receiver Transmitter Channel Figura D.1: Blocks diagram of transmitter and channel model. D.2 Channel The noisy channel is implemented in the same script as the transmitter but it takes up a different part in the communication. They can be found in Appendix H.1. Two noises affects the channel: 1Pulse shaping is the process of changing the waveform of transmitted pulses. Its purpose is to make the transmitted signal better suited to the communication channel by limiting the effective bandwidth of the transmission. By filtering the transmitted pulses this way, the intersymbol interference caused by the channel can be kept in control. In RF communication, pulse shaping is essential for making the signal fit in its frequency band. D.3 Receiver 85 •AWG noise is the first of them. It is a random signal with a flat power spectral density and Gaussian amplitude distribution that will be added to the modulated signal [37]. We are going to study all values within a range from 15 to −5dB of Eb/No. •Quantization Noise is a type of distortion that occurs when an analog waveform is encoded into a digital signal [38]. It will appear in the laboratory when the signal was acquired from the channel by the DSO. It has been included in the simulation in order to be the most accurate possible. D.3 Receiver The first stage in the receiver is a BPF. Its main function is to remove the out-of-band noise. It is a fifth order butterworth filter. This option was chosen among all other filters because it has the flattest passband meaning which is very good at simulating the passband of an ideal filter. Duffing RK4 Envelope method From channel Phase variance method Mean method Variance method FFT pattern method BPF Figura D.2: Blocks diagram of receiver. After this, all the chaotic Duffing receiving set-up is implemented. First, we use an interpolation block [39] made up by an upsample plus a BPF. The purpose of that is to reduce the stepsize, getting more samples per symbol. It will make Duffing oscillator output more accurate. This block could be dispensable. In a simulation environment, increasing the sampling frequency and fixing the bit rate should be enough; that would increase the number of samples per symbol. In a real environment it is not an easy task due to the limitation of the devices, the most we will 92 Performance evaluation for ASK we will have approximately null mean for the case of transmitted logical 1’s. And for logical 0’s, the arithmetic mean will be a large value. After that, threshold decision is carried through the histogram and finally BER is calculated. All the set-up is shown in Figure D.9. The mean method is composed of a block where the mean of each bit is calculated, threshold block, symbol decision and BER calculation. From Duffing Nss*(num_symbols- 1)+1:Nss*num_symbols 1:Nss Nss+1:2*Nss …. Mean Mean Mean Threshold BER <0 > Figura D.9: Blocks diagram mean method. The main advantage of this method is its simplicity. Duffing oscillator response has an acceptable accuracy compared with the current methods. The worst point is the difficulty to fulfil all the requirements above in order to have similar outputs with Figure D.7. Matlab code can be found in Appendix H.5. D.3.4 Variance method We would like to extent the previous method of symbol decision for all kind of Duffing oscillator outputs. Signals with small amplitude in the chaotic state, as the one shown in Figure C.4, are more common in detection than the conventional Duffing oscillator output, e.g. Figure D.8. In this new proposal, we will continue splitting the Duffing oscillator output by symbols. It will be eliminated about 7% of the first and last samples of every symbol to assure stability. However, in this case we will apply the variance to every symbol instead of the mean. Variance will be used as a measurement to quantify how far a set of samples are spread out from each other. D.3 Receiver 93 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Number of samples Amplitude [a.u.] Output signal Information transmitted Output Variance method Figura D.10: Demodulation using variance method. The signal in blue represents the output of our Duffing oscillator. The signal in green is the transmitted PRBS signal and the signal in red color shows the variance value for every symbol. Figure D.10 shows in blue color the output of the Duffing oscillator and how it is the variance for each case. When the Duffing oscillator is in the large periodic state, variance value is higher than it is in the chaotic state. Signal in blue is, as usual, the output of our Duffing system. Green signal is the transmitted PRBS signal. It represents transmitted logical 1’s or 0’s. The signal we will use for symbol decision is the one in red color, it shows the variance value for every symbol. As we can see in the Figure D.10, results for ones and zeros do not have a complicated fixed value decision process. It is worthy to say that in this case we do not feed the threshold decision block with those values and we do not compare them with the threshold. The most interesting issue is to study the relation of one value with the previous one. In such case, we can really appreciate differences among the two kinds of transmitted symbols. Thus, we create a new vector which stores the difference between values, related to that it will be easier to establish a threshold value. This method is the most accurate until now, it takes long time to process 94 Performance evaluation for ASK a signal in comparison with those methods before but it is an accurate method to work with the output of the Duffing oscillator. From Duffing Nss*(num_symbols-1)+1:Nss*num_symbols 1:Nss Nss+1:2*Nss …. Variance Variance Variance Threshold BER <0 > 7% 7% 7% 7% Figura D.11: Blocks diagram variance method. The set-up for this method is depicted in Figure D.11. The variance method is composed of a block where the variance of each bit is calculated, threshold block, symbol decision and BER calculation. The main difference between the plot in Figure D.9 and the current set-up, is the replacement of the mean block for a different one where the variance is calculated. By contrast, Matlab code differs substantively and it is attached at the Appendix H.6. D.3.5 FFT pattern method By now, the four developed and tested methods explained above, have worked with the properties of the Duffing oscillator output in the time domain. Now, we are going to characterize the signal in the frequency domain. As we know, signal in the large periodic state has some kind of periodicity and this fact should be reflected on its FFT. Chaos state has to be characterized at some point, perhaps signal is composed by the mixture of different frequencies or it has a constant spectrum without any frequency standing out. To clear up these question, we split the Duffing output signal in symbol and we apply FFT to every part. FFT of transmitted logical 1’s and 0’s are shown in Figure D.12 and Figure D.13. From there, we are going to extract some conclusions. D.3 Receiver 95 Evaluating Figure D.12, it can be seen how a maximum peak appears. From that frequency to baseband there is no any other peak in the spectrum. In the other direction, the only significative frequencies are the integer multiples of the signal, called harmonics. −4 −3 −2 −1 0 1 2 3 4 x 1010 −40 −30 −20 −10 0 10 20 30 Frequency [Hz.] |X(f)| [dB] 1 maximum 27.0565 dB Figura D.12: Spectrum of transmitted logical 1. In the case of Figure D.13, signal has more frequencies than Figure D.12. We can find a maximum in baseband. The second maximum is located at the same frequency at which the transmitted logical ‘1’ had its peak. So, this frequency is also strong in the chaos signal but lower than for the first case. There are also a lot of frequencies, between the two maximums, present in the spectrum with a significative amplitude. Figura D.13: Spectrum of transmitted logical 0. What we are going to implement is an algorithm which studies the frequency where logical ‘1’s have a maximum (reference peak) and stores 96 Performance evaluation for ASK that value in a vector. From that frequency, we will scan all the frequencies down to baseband. The goal is to find a second frequency peak or pattern which proves the existence of more strong frequencies in that range (we will name the new peak as sub-maximum). Amplitude value of these sub-maximums will be also stored in another vector. Theoretically, it should be enough defining only one maximum, one threshold and studying only one point per symbol to decide between ‘1’ or ‘0’. But experimentally, we have noticed a high improvement in the detection accuracy when two different reference points per symbol are taken. Therefore two conditions, two thresholds, have to be analyzed instead of only one. So far, two histograms should be created from the two vectors. One with the stored maximum and the other with the stored sub-maximum peaks. Subsequently threshold values for each case can be calculated. Once we have the two thresholds, one for maximums and other for sub-maximums, we implement a decision block. threshold 1 threshold 2 one one zero zero Maximum value Sub-maximum Figura D.14: Threshold system for symbol decision. Symbol is interpreted as logical ‘1’ if its maximum is around the threshold 1 and its sub-maximum around the threshold 2. Symbol is interpreted as logical ‘0’ in any other case. How decision is taken is illustrated in Figure D.14. This block consists in four steps: 1. In the first one, symbols which its maximum exceed the first thresh- D.3 Receiver 97 old and at the same time, its sub-maximum is below the second threshold are considered as logical ‘1’. This could be interpret as a signal with only one strong frequency component and nothing else around. Figure D.12 has a similar behavior. 2. In the case that first condition is not fulfilled. They will be consider as logical ‘0’, symbols with maximum smaller than the first threshold and sub-maximum larger than the second threshold. That is interpreted as symbols which does not have an specified strong periodicity signal but still have important signals in other frequencies. As happens in Figure D.13. 3. We have already classified most evident symbols. For the third case we analyze those in doubt. Their values do not fulfil any of the conditions above but are close to do it. Those symbols which are located in a range around the ±11% of each threshold value, will be considered as logical ‘1’. It includes those symbols which only fulfilled one of the thresholds and were close to the other. This case corrects errors produced due to a non-optimal definition of threshold values. 4. In the last case, we take as logical ‘0’ any other value that were not included in the previous conditions: when sub-maximum is extremely high being the maximum larger than its value, when maximum is not large enough but sub-maximum is correct and so on. After the decision block, signal is compared with transmitted message and BER is calculated. All the set-up can be seen in Figure D.15. The FFT pattern method is composed of a block which calculates the FFT, two different threshold blocks, symbol decision for each threshold and BER calculation. After numerous simulations, this FFT pattern method is the one selected among methods (coherent and incoherent detection) due to its accurately. Variance method and FFT pattern method have a similar performance until 4dB. From this value to 15dB, FFT pattern has achieved in a better performance of around 0.7dB. Matlab code of the current method can be found in Appendix H.7. 98 Performance evaluation for ASK From Duffing Nss*(num_symbols- 1)+1:Nss*num_symbols 1:Nss Nss+1:2*Nss …. Threshold 1 BER <0 > FFT FFT FFT Maximum Submaximum Maximum Submaximum Maximum Submaximum Threshold 2 Figura D.15: Blocks diagram FFT pattern method. D.4 Comparative and results: Duffing vs. Coherent detection In order to be able to compare our demodulator with the coherent and envelope detection methods, we are going to implement the coherent and the incoherent demodulators. They have to match with the transmitter code (Appendix H.1), so there will be two extra blocks in our code. −5 0 5 10 15 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/N0 (dB) BER Theoretical values Simulated values Figura D.16: Envelope detection. Comparative between theoretical results and our simulation results. Envelope detector, as incoherent method, will follow a similar scheme such at that shown in Figure D.4. The only different is from where the signal comes, this time it will come directly from the channel, instead of D.4 Comparative and results: Duffing vs. Coherent detection 99 coming from the Duffing output. To guarantee a properly implementated method, we compare simulated BER in our system with theoretical curves. Comparison is plotted in Figure D.16 and it demonstrates how similar results are obtained. It is also important to realize than simulation only goes until 12dB of Eb/No, after that value there are no errors. It is due to the high number of simulated bits needed to find an error in the receiver. For example, in case of 15dB of Eb/No, it would be needed to process 10 million of bits and computer resources are limited for such a case. Matlab code for the whole communication can be found in Appendix H.8. From channel Threshold BER <0 > )cos( t  LPF Figura D.17: Set-up of coherent demodulation. It is also implemented a coherent demodulator [41]. It is composed of mixer, a LPF, a downsampling block, a threshold block and BER calculation. The set-up is shown in Figure D.17. In this case we multiply signal from the channel by a sine signal at the same frequency. Phases of both signals have to be matched as well. Output is low pass filtered. We have also implemented a script to compare simulated and theoretical results. It happens the same as in the envelope detection where simulation is limited by the number of simulated symbols, as we can see in Figure D.18. Code implemented for all this above can be located in Appendix H.9. After coherent and incoherent models have been implemented, we are going to compare their performance with our chaotic Duffing receiving system. Communication will be configured in the next way: •Frequency sampling: 80GHz. 100 Performance evaluation for ASK −5 0 5 10 15 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/N0 (dB) BER Simulated values Theoretical values Figura D.18: Coherence detection. Comparative between theoretical results and our simulation results. •Carrier frequency: 13.8GHz. •Bit rate: 1Gbps. As we said in section D.3, we try to get a high number of samples per symbol. That will be advantageous for the Duffing oscillator in the way the more samples, the more immunity against noise will be acquired. It is not the only reason, also reference papers define this relation between carrier and sampling frequency as large as we have done, [12] and [15]. In Figure D.19, it is depicted the result of the first comparison. Filters in the reception are implemented as ideal filters, their bandwidth is set as the signal width. Coherent detection results in the most effective method. Duffing oscillator response is just located in the middle of both classical methods. For this first test, we do not observe any advantage in our system. Chaotic Duffing receiving system has a better performance in terms of Eb/No than envelope detection. However, this comparison is not carried at the same conditions. The Duffing oscillator and the coherent method are using a reference signal in the same frequency range as the transmitted. For D.4 Comparative and results: Duffing vs. Coherent detection 101 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 C o h e r e n c e I n c o h e r e n c e D u f f i n g - l o g ( B E R ) Eb/ N 0 [ d B ] Figura D.19: Comparison between coherence detection method, incoherence detection method and chaotic Duffing receiving system. BPF set at 1GHz, carrier frequency of 13.8GHz, sampling frequency of 80GHz and 1Gbps of bit rate. this reason, both are more complicated to implement than the incoherent method and thus, they should achieve better results. In the next simulation, we increase the bandwidth of the BPF in the reception until to 2GHz FWHM. This wide of the filter could be closer to a real environment where filters are wider than theoretically needed. The idea to proceed this way came from the fact that Duffing oscillator has been used with high oversampling rates and no filter in reception while detecting weak signals [15], as we mentioned in the introduction of this Thesis. For that reason, we are going to study along some simulations its behavior for different bandwidths of the receiver filter. New bandwidth will be two times the signal bandwidth, 2GHz FWHM. Results are depicted in Figure D.20. For this situation we skip envelope 108 Performance evaluation for ASK (a) Comparison for an ideal filter. BPF set at 1GHz. 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 D u f f i n g i d e a l f i l t e r E n v e l o p e f i l t e r = 2 * F b D u f f i n g f i l t e r = 2 * F b - l o g ( B E R ) Eb/ N 0 [ d B ] (b) Comparison with filter = 2*Fb. BPF set at 2GHz. 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 D u f f i n g i d e a l f i l t e r E n v e l o p e f i l t e r = 3 * F b D u f f i n g f i l t e r = 3 * F b - l o g ( B E R ) Eb/ N 0 [ d B ] (c) Comparison with filter = 3*Fb. BPF set at 3GHz. 0 2 4 6 8 1 0 1 2 1 4 1 6 76 5 4 3 2 1 D u f f i n g i d e a l f i l t e r E n v e l o p e f i l t e r = F s D u f f i n g f i l t e r = F s - l o g ( B E R ) Eb/ N 0 [ d B ] (d) Comparison when there is no reception system. Equivalent BPF 80GHz. Figura D.25: Comparison between chaotic Duffing receiving system and envelope detection for different filter widths. Carrier frequency is set at 13.8GHz, sampling frequency at 80GHz and 1Gbps of bit rate. Ap´ endice E Experimental results Experimental set-up and results will be presented in this chapter to verify whether methods described previously can be used to obtain improved results. In the first section, there is a first experiment with the aim to characterize electrically our Duffing system for future experiments. In the second section we analyze experimentally our simulation results. For that, we implement a set-up in the laboratory and we process offline the captured data. Demodulation will be carried with the new design of Duffing system. E.1 Electrical characterization of the system This simulation has the purpose to get a first approximation about chaotic Duffing receiving system performance with real data, test and decide which PRBS signal would be more convenient or which output power will influence the system positively. That is why the implemented set-up is really simple, as we can see in Figure E.1. It is composed by a VSA directly connected to a DSO. There is not any external noise source added to the system, the very little amount of noise comes from the output of the VSA. As we said, there is not any interest in a correct demodulation at this point, we will focus on the phase diagram of our system and how difficult distinguishing between logical 0’s and logical 1’s from the output signal is. Configuration parameters in the set-up are defined as follows: 110 Experimental results Digital Storage Oscilloscope Vector Signal Analizer Figura E.1: Set-up for the electrical characterization of the system. •Bit rate of the signal will be fixed as 400Msps for the entire experiment. •Sampling frequency will be defined either 20GHz or 40GHz. It will lead in 500 or 1000 samples per symbol, respectively. The idea continue being to define a system as simple as possible and with this sampling rate, we would not need any upsampling block before Duffing. •Carrier frequency is set at 600MHz. •Different PRBS signals will be tested. PRBS pattern: 2n−1 (n: 9,11 and 15). •Different VSA output power will be also tried (0 dBm, −20 dBm, −30 dBm and −40 dBm). After that, there are also some variables in the script that will remain fixed for the entire experiment. To assure phase transition, the phase of the driver will be defined as ϕ= 0. As in previous scripts, damping ratio will be set at 0.5 and the limit value between chaos and periodicity will be fd= 0.829, so depending on the input power we will define the force amplitude frin the range (0.7852,0.8137). In Figure E.2 there are plotted 20 symbols of the signal acquired by the DSO after being filtered by a fourth order butterworth BPF. It becomes apparent the clarity of the signal. Signal from Figure E.2 is then introduced into the Duffing oscillator. Output can be seen at Figure E.3(b) and phase diagram at Figure E.3(a). Signals with different powers have been tested and no problems turned up. E.1 Electrical characterization of the system 111 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Samples Amplitude [V.] Figura E.2: Band-pass filtered signal from the output of Digital Storage Oscilloscope. Bit rate of 400Msps, sampling frequency 20GHz, carrier frequency 600MHz and Vector Signal Analizer output power around −30dBm. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 dx(t)/dt [a.u.] x(t) [a.u.] (a) Phase diagram 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Samples Amplitude [a.u.] (b) Duffing output Figura E.3: Outputs of the Duffing for 30 dBm of signal power, 40 GHz of sampling frequency, 40 Msps of bit rate and force driver fr= 0.8085. For cases where frequency sampling was 20GHz and 40GHz, it can be easily noticed when a binary zero or a binary one was transmitted in both cases. From previous simulations we discovered a better behavior in Duffing response when the number of samples per symbol was above 400. But testing two different values that would not present any problem (500 and 1000 samples per symbol), we wanted to quantize if there was any significative difference between those two different values or on the 112 Experimental results contrary, accuracy was guaranteed above 400 samples in any case. And we conclude that the second affirmation was fulfilled and above a certain number of samples, there is not any significative improvement. Another conclusion for this first experiment is that Duffing setting is required for any variation of power in the incoming signal. The value frcan be calculated automatically in the code and it will be valid, but it is always advisable to test nearby values in order to specify the best one. It also means for future experiment that different SNR would need a different configuration of the parameter fr. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Samples Amplitude [a.u.] (a) ϕ= 0.007 ∗π 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Samples Amplitude [a.u.] (b) ϕ= 0.012 ∗π Figura E.4: Outputs of the Duffing for 40 dBm of signal power, 40 GHz of sampling frequency, 40 Msps of bit rate and force driver fr= 0.7816. Same problem as above happens to drive phase ϕ. It is also connected to fr, as we can see in A.3.1. If fris modified, ϕshould be also adjusted. In the case of not doing it we could incurrent in a loss of power for transmitted ones and they could be interpreted as zeros. This lost of power (lost of periodicity) is shown in Figure E.4(b) comparing with Figure E.4(a). In the previous chapter it was mentioned a problem in the detection of signal when it transits quickly and continuously from binary one to binary zero or viceversa. We have used different PRBS signals in order to check this fact. When n= 15 is supposed than there will be more consecutive ones or zeros than for the case of n= 7, obtaining less transitions. For all those cases, Duffing reacts in the same way and there is not any PRBS signal which produces a better performance in our system. E.2 Chaotic Duffing receiving system implementation for Radio-over-Fiber transmission 113 E.2 Chaotic Duffing receiving system implementation for Radio-over-Fiber transmission This second experiment is much more complex than our first attempt. It has been built in order to simulate an entire optical communication where its output will be demodulated and offline processed. E.2.1 Set-up implementation The detailed schematic showing the experimental setup used is presented in Figure E.5. The transmitter is a Teraxion DFB Narrow Linewidth Laser with a linewidth of less than 50kHz. Electrical PRBS signal at different rates are generated with Agilent HP 70843A PPG. Signals are subsequently amplified to feed the MZM. The driving voltage of the MZM has been adjusted to obtain the highest extension ratio, 17dB. The output signal is transmitted through the fiber. Noise is added to the received signal. EDFA as a white noise source and filtered with 0.8nm bandwidth optical filter with which we will keep only in-band noise, whereas out-of-band noise will be removed. Signal is amplified again and connected to a new filter having 0.5nm bandwidth. The output of the filter is attached to a VOA in order to keep the power of the optical signal in a constant level. LO laser is ECL with a linewidth less than 300kHz. Signal is measured with a PM at the output, being LO power −3.2dBm and signal power −0.9dBm. Finally, signal is fed to a PD. Two different PDs were used in the experiment. One for low frequency with a 10GHz bandwidth and internal amplifier and a second one for high frequency, 40GHz and external electrical amplifier. The output of the PD are attached to two LeCroy WaveMaster 830Zi-A DSO capable of capturing 80GSa/s with a 30GHz bandwidth input. The measurements gathered with the previous set-up are: •1Gbps of bit rate modulated at 13.8GHz for OSNR among 18 and 114 Experimental results 1dB in the cases of having or not a high linewidth. Sampling frequency at 80GHz. •1Gbps of bit rate modulated at 26.4GHz for OSNR among 18 and 1dB. Sampling frequency at 80GHz. •2Gbps of bit rate modulated at 26.4GHz for OSNR among 18 and 1dB. Sampling frequency at 80GHz. •5Gbps of bit rate modulated at 26.4GHz for OSNR among 18 and 1dB. Sampling frequency at 80GHz. •10Gbps of bit rate modulated at 24.8GHz for OSNR among 18 and 1dB. Sampling frequency at 80GHz. Captured data are later on transferred to a computer and offline processed on Matlab. The results obtained are outlined in the following sections. E.2 Chaotic Duffing receiving system implementation for Radio-over-Fiber transmission 115 PPG 3 dB 3 dB OSA ASE OSC DSO 5 nm 8 nm 10 dB Power Meter 3 dB Power Meter MZM Power Meter 10 dB Figura E.5: Schematic for a Radio sobre Fibra transmission. Set-up parameters are 1Gbps of bit rate, signal modulated at 13.8GHz for OSNR among 18 and 1dB. Sampling frequency at 80GHz. 116 Experimental results E.2.2 Offline demodulation Unlike simulations in the previous chapter, we have to put an extra effort to determine which sample is the first for every symbol. We did not have this problem before in the symbol synchronization because transmission was also under our control. Until now, we have just taken a part of the generated PRBS signal, no matter if it was the beginning of the communication or not. For every captured signal, their symbols have a fixed Nss. Our goal is to find the location of that first sample in one of the first symbols. From that, we can demodulate the signal as we were doing previously. To achieve this synchronization we implement one block previous to the BPF. It squares the input signal and from its output we search for a maximum in the first transmitted symbols. From all the samples which compose a symbol, that maximum will be in Nss 2, so that position means that we are located in the middle of a transmitted one. We reject the next Nss 2samples and since then, synchronization for next symbols is assured. BER From channel BPF Duffing RK4 Threshold )cos( t  LPF FFT pattern method <0 > PRBS generation <0 > Thr. 1 Thr. 2   2  BPF Threshold Recovery signal <0 > BPF variable Recovery signal BER Figura E.6: Schematic of the demodulation implemented script. The set-up is composed by three branches. The first simulates a coherent detection receiver. The second branch is the chaotic Duffing receiving system. The third branch in the envelope detection receiver. Later on, signal goes through three different paths and from them we obtain the curves Eb/No vs. BER. Set-up can be graphically seen in Figure E.6. E.2 Chaotic Duffing receiving system implementation for Radio-over-Fiber transmission 117 1. Ideal coherent detection branch is composed by a perfect matched BPF and connected to a block which demodulates signal. After all the process, recovered signal will be the most accurate output possible to obtain. Thus, we will take its first 15 bits and thanks to that, we will be able to regenerate the transmitted PRBS signal. That will be used further to compare PRBS signal with recovery signals from the two methods of demodulation described just below. 2. Duffing branch is the Duffing set-up used previously in numerous simulations in sections D.5 and D.6. In the entrance of its branch is defined a tuneable BPF, it will be used to compare its response for different width values. Several test will be carried out with and without the reference signal. 3. Envelope detection branch is similar to the set-up implemented in section D.4. It will be the easiest method to compute. Its output will be also studied for scenarios where BPF is not always perfectly matched. E.2.3 Experimental results of the set-up The most interesting data is the one where bit rate is 1Gbps and carrier frequency about 13.8GHz. It is because the Nss for this case is 80. In the last versions of our code this value could be taken as acceptable with any upsampling block. But even if we increase the Nss, less extra samples were needed. That is why it will be the most convenient measurement from the laboratory. Another point of interest resides in the comparison between the OSNR measured from the laboratory and the Eb/No used in the previous chapter for simulations. The first alludes to the optical domain and is the ratio between the signal power and the noise power in a given bandwidth. For our case a reference bandwidth of 0.1 nm is used. The second stage is to measure the signal to noise ratio for a digital communication system. It is measured at the input of the receiver and it is used as the basic reference of how strong the signal is. In order to merge the two different measures in a same plot we have to 124 Integral Implementation 7driver = cos (2* pi *F_RF *(1- diff_freq )*t_up + phase_duff ); 8 9% var2 10 omega =2* pi *F_RF ; 11 12 k =0.3; % default value k = 0.3; 13 r =0.3; % default value k = 0.6; 14 15 number_points = length ( yfilt_upsample ); 16 17 w2b =0; w2c =0; 18 xd = -1;x = -1; 19 h =1/ Fs_up ; 20 salida = zeros (1 , number_points ); 21 salida_2 = zeros (1, number_points ); 22 tv = zeros (1 , number_points ); 23 24 for i =1: length ( signal ) -1 25 26 xdd =-k*w*xd +w ^2*(x-x ^3+ r* driver (i) +20* signal (i)); % out sin integrar 27 28 w1b=xdd+ w2b ; 29 w2b=xdd+ w1b ; 30 31 xd= w1b /(2* Fs_up ); % out tras una integral 32 33 w1c=xd+ w2c ; 34 w2c=xd+ w1c ; 35 36 x=w1c /(2* Fs_up); % out tras las 2 integrales 37 38 salida (1 ,i)=xd ; 39 salida_2 (1 ,i)=x; 40 tv (1 ,i)= t_up (i); 41 42 end F.3 Duffing implemented with Fourth-Order Runge- Kutta method 1phase_duff = 0* pi ; 2diff_freq = 1; 3ymod_duff = cos (2* pi *F_RF* diff_freq * t_up + phase_duff ); 4 5driver = ymod_duff; signal_in = yfilt_upsample ; 6max_signal = max ( signal_in ); 7signal = signal_in ./ max_signal ’; 8 9% var2 F.3 Duffing implemented with Fourth-Order Runge-Kutta method125 10 omega =2* pi *F_RF ; 11 K1 = omega ^2; 12 K2 =1* omega ; 13 K3 =(1/ omega ); 14 15 %var ADD 16 out_add = zeros (1 , length ( t_up )); 17 18 %var INTEG 19 salida = zeros (1 , length ( t_up )); 20 salida_2 = zeros (1 , length ( t_up )); 21 aux4 = zeros (1 , length ( t_up )); 22 h =1/ Fs_up ; 23 24 % var3 25 gamma = 0.5; % DAMPING COEFFICIENT -> changeless 26 fr = 0;% -1/ max ( ynoisy ); % fd = 0.8274; 27 A = 4; 28 29 for i =1: length ( signal ) -1 30 31 x1 = salida_2 (i); 32 y1 = salida (i); 33 f1 = - gamma * K2 * y1 +K1*(x1 - x1 ^3+ fr * driver (i)+A * signal (i)); 34 35 x2 = salida_2 (i )+y1 *( h /2); 36 y2 = salida (i) +f1 *( h /2); 37 f2 = - gamma * K2 * y2 +K1*(x2 - x2 ^3+ fr * driver (i)+A * signal (i)); 38 39 x3 = salida_2 (i )+y2 *( h /2); 40 y3 = salida (i) +f2 *( h /2); 41 f3 = - gamma * K2 * y3 +K1*(x3 - x3 ^3+ fr * driver (i +1) +A* signal (i +1)); 42 43 x4 = salida_2 (i)+ y3*h; 44 y4 = salida (i)+ f3 *h; 45 f4 = - gamma * K2 * y4 +K1*(x4 - x4 ^3+ fr * driver (i +1) +A* signal (i +1)); 46 47 salida_2 (i +1) = salida_2 (i) +(( h /6) *(y1 +2* y2 +2* y3 +y4 )); 48 salida (i +1) = salida (i) +((h /6) *( f1 +2* f2 +2* f3+ f4)); 49 50 end 126 Integral Implementation Ap´ endice G Matlab simulations for the three basic modulations G.1 Code for BASK signal detection 1clear all 2close all 3clc 4 5k =0.3; % default value k = 0.3; 6r =0.6; % default value k = 0.6; 7 8D =0.1; 9w=10; 10 w2b =0; w2c =0; 11 xd = -1;x = -1; 12 tfinal =50; 13 fs=1000; 14 h =1/ fs; 15 number_points = fs * tfinal ; 16 salida = zeros (1 , number_points ); 17 salida_2 = zeros (1, number_points ); 18 tv = zeros (1 , number_points ); 19 bits = round ( rand (1 , tfinal /10) ); 20 mod = kron (bits , ones (1, fs *10)); 21 noise_total = 0; 22 23 t = linspace (0 , tfinal -h, number_points ); 24 phase = 1* pi; 25 signal = mod .* cos (w*t+ phase ); 26 driver = cos(w*t); 27 noise = 2* randn (1 , number_points ); 28 29 for i=1: number_points 30 t=(i -1)*h; 31 signal = mod(i)* cos (w*t(i)); 32 driver = cos(w*t(i)); 33 noise (i) = A; 34 noise_total = noise_total + abs( noise(i)) .^2; 35 xdd =-k*w*xd +w ^2*(x-x ^3+ r* driver (i)+ noise (i)+ signal (i) ); % out sin integrar 36 w1b=xdd+ w2b ; 37 w2b=xdd+ w1b ; 38 xd= w1b /(2* fs); % out tras una integral 128 Matlab simulations for the three basic modulations 39 w1c=xd+ w2c ; 40 w2c=xd+ w1c ; 41 x=w1c /(2* fs); % out tras las 2 integrales 42 salida (1 ,i)=xd ; 43 salida_2 (1 ,i)=x; 44 tv (1 ,i)=t(i); 45 end 46 47 pot_mod = sum (abs ( mod) .^2) / length ( mod); 48 pot_noise = noise_total / number_points ; 49 SNR = 10* log10 ( pot_mod / pot_noise ) G.2 Code for BPSK signal detection 1clear all 2close all 3clc 4 5k =0.3; % default value k = 0.3; 6r =0.6; % default value k = 0.6; 7 8 9w=10; 10 w_neg = w; 11 w2b =0; w2c =0; 12 xd = -1;x = -1; 13 tfinal =500; 14 fs=1000; 15 h =1/ fs; 16 number_points = fs * tfinal ; 17 salida = zeros (1 , number_points ); 18 salida_2 = zeros (1, number_points ); 19 tv = zeros (1 , number_points ); 20 21 bits = round ( rand (1 , tfinal /10) ); 22 bits_neg = not( bits); 23 mod = kron (bits , ones (1, fs *10)); 24 mod_neg = kron (bits_neg , ones (1 ,fs *10) ); 25 noise_total = 0; 26 27 t = linspace (0 , tfinal -h, number_points ); 28 phase = 1* pi; 29 signal_f1 = mod .* cos (w*t+ phase ); 30 signal_f2 = mod_neg .* cos ( w_neg * t+ phase + pi ); 31 signal = signal_f1 + signal_f2 ; 32 driver = cos(w*t); 33 noise = 2* randn (1 , number_points ); 34 35 for i=1: number_points 36 G.3 Code for BFSK signal detection 129 37 noise_total = noise_total + abs( noise(i)) .^2; 38 xdd =-k*w*xd +w ^2*(x-x ^3+ r* driver (i)+ noise (i)+ signal (i) ); % out sin integrar 39 w1b=xdd+ w2b ; 40 w2b=xdd+ w1b ; 41 xd= w1b /(2* fs); % out tras una integral 42 w1c=xd+ w2c ; 43 w2c=xd+ w1c ; 44 x=w1c /(2* fs); % out tras las 2 integrales 45 salida (1 ,i)=xd ; 46 salida_2 (1 ,i)=x; 47 tv (1 ,i)=t(i); 48 49 end; G.3 Code for BFSK signal detection 1clear all 2close all 3clc 4 5k =0.3; % default value k = 0.3; 6r =0.6; % default value k = 0.6; 7 8w=10; 9w_neg = 14; 10 w2b =0; w2c =0; 11 xd = -1;x = -1; 12 tfinal =500; 13 fs=1000; 14 h =1/ fs; 15 number_points = fs * tfinal ; 16 salida = zeros (1 , number_points ); 17 salida_2 = zeros (1, number_points ); 18 tv = zeros (1 , number_points ); 19 20 bits = round ( rand (1 , tfinal /10) ); 21 bits_neg = not( bits); 22 mod = kron (bits , ones (1, fs *10)); 23 mod_neg = kron (bits_neg , ones (1 ,fs *10) ); 24 noise_total = 0; 25 26 t = linspace (0 , tfinal -h, number_points ); 27 phase = 1* pi; 28 signal_f1 = mod .* cos (w*t+ phase ); 29 signal_f2 = mod_neg .* cos ( w_neg * t+ phase ); 30 signal = signal_f1 + signal_f2 ; 31 driver = cos(w*t); 32 noise = randn (1, number_points); 33 130 Matlab simulations for the three basic modulations 34 for i=1: number_points 35 t=(i -1)*h; 36 signal = mod(i)* cos (w*t(i)); 37 driver = cos(w*t(i)); 38 noise (i) = A; 39 noise_total = noise_total + abs( noise(i)) .^2; 40 xdd =-k*w*xd +w ^2*(x-x ^3+ r* driver (i)+ noise (i)+ signal (i) ); % out sin integrar 41 w1b=xdd+ w2b ; 42 w2b=xdd+ w1b ; 43 xd= w1b /(2* fs); % out tras una integral 44 w1c=xd+ w2c ; 45 w2c=xd+ w1c ; 46 x=w1c /(2* fs); % out tras las 2 integrales 47 salida (1 ,i)=xd ; 48 salida_2 (1 ,i)=x; 49 tv (1 ,i)=t(i); 50 end Ap´ endice H Matlab simulations for the performance evaluation for ASK H.1 Code for transmitter and noisy channel 1clear all ; 2close all ; 3clc; 4 5%Bit and smapling Rates 6Fs =80 e9; % Sampling frequency 7Ts =1/ Fs ; 8 9Fb = 80 e6; 10 Tb =1/ Fb ; 11 12 F_RF= 600 e6 ; % Modulation frequency 13 14 Nss=Fs/Fb; 15 num_symb = 1 e3 ; % Number symbols 16 ts = num_symb / Fb; % Length of simulation 17 18 %Time and Frequency axis 19 t= linspace (0 ,( ts -1/ Fs ), num_symb * Nss) ’; % temporal axe 20 fplot = linspace (-Fs /2 ,Fs/2 , Nss* num_symb )’; % for plotting ( fftshift signal ) 21 22 % Signal generation 23 PRBS = load ( ’ autentico_PRBS1 5_2 ^15. txt ’); 24 PRBS=PRBS’; 25 rep_PRBS = ceil ( num_symb /( length (PRBS ))); 26 bits1 = repmat (PRBS ,[ rep_PRBS 1]); 27 bits1 = bits1 (1: num_symb ); 28 clear PRBS ; 29 30 % Upsample 31 y_up = upsample ( bits1 , Nss); 32 y_up = y_up (1: length (t)); 33 34 % Raised cosine 35 delay = 3; 132 Matlab simulations for the performance evaluation for ASK 36 [num , den ] = rcosine (1, Nss , ’fir / sqrt ’ ,1 , delay ); 37 yinf2 = filter (num ,den , y_up ); 38 yinf2 = yinf2 ./ max( max ( yinf2)); 39 yinf2 = yinf2 (3* Nss: end ); 40 41 42 % Modulation 43 y_cos = cos (2* pi *F_RF .*t (1: end -3* Nss +1) ); 44 ymod = yinf2 ’.* y_cos ; 45 46 % Noise awg 47 SNR = 0; 48 pot_ymod = sum (abs ( ymod ) .^2) / length ( ymod); 49 pot_ymod_db = 10* log10 ( pot_ymod ); 50 noise_power_db = pot_ymod_db - SNR ; 51 noise_power = 10^( noise_power_db *0.1) ; 52 noise = randn ( length ( ymod ) ,1) * sqrt ( noise_power ); 53 ynoisy = ymod + noise ; 54 Eb_No = 10* log10(Fs /(2* Fb))+ SNR 55 Eb_No_pp1 = 10* log10 (1/(2* noise_power )) 56 % ynoisy = awgn (ymod ,SNR ,’ measured ’); 57 58 % Quantization Noise 59 Ei = ynoisy ; 60 norm =max ([ Ei ]); 61 Ei2 = ( Ei ./ norm ); 62 pn = {’mode ’, ’ roundmode ’, ’overflowmode ’, ’format ’}; 63 pv = {’fixed ’, ’ ceil ’, ’saturate ’, [6 5]}; % Quantification 8 bits A/D 64 q = quantizer (pn , pv ); 65 EiQ = num2int (q, Ei2) ./(2^7) ; 66 ynoisy_quant = EiQ; 67 68 69 file = ’ channel . mat ’; 70 ymod_tx = ymod; 71 % save(file , ’ynoisy ’, ’SNR ’, ’Nss_tx ’, ’bits1 ’, ’ num_symb ’, ’ ymod_tx ’, ’yinf2 ’, ’y_cos ’); 72 save( file , ’Fs ’, ’Fb ’, ’F_RF ’, ’SNR ’, ’num_symb ’, ’bits1 ’, ’ yinf2 ’, ’y_cos ’, ’ ymod_tx ’, ’ynoisy ’); H.2 Code for receiver 1clear all ; 2close all ; 3clc; 4 5file1 = ’ channel . mat ’; 6load( file1 , ’Fs ’, ’Fb ’, ’F_RF ’, ’SNR ’, ’num_symb ’, ’bits1 ’, ’ yinf2 ’, ’y_cos ’, ’ ymod_tx ’, ’ynoisy ’); H.2 Code for receiver 133 7 8Ts = 1/ Fs; 9Tb = 1/ Fb; 10 ts = num_symb / Fb ; 11 Nss = Fs/Fb ; 12 13 %Time and Frequency axis 14 t = linspace (0 ,( ts -1/ Fs),num_symb* Nss ) ’; % temporal axe 15 fplot = linspace (-Fs/2 , Fs /2 , Nss * num_symb ) ’; % for plotting ( fftshift signal ) 16 17 % Band pass filter reception 18 Wo_rx = 0.92; 19 [ B_rx , A_rx ] = besself (1 , Wo_rx ); 20 yfilter_rx = filter ( B_rx , A_rx , ynoisy ); 21 ynorm_rx = yfilter_rx ./ max( max( yfilter_rx )); 22 yout_rx = ynorm_rx - mean ( ynorm_rx ); 23 24 % % butterworth filter 25 Wn = [F_RF -Fb F_RF+Fb ]*(2/ Fs); 26 [ b_duff , a_duff ] = butter (4 , Wn ); 27 ynoisy2 = filter ( b_duff , a_duff , ynoisy ); 28 ynoisy2 = ynoisy2 ./ max ( max( ynoisy2 )); 29 30 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 31 % 1 -> duffing 32 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 33 Nss_tx = Nss; 34 35 % Upsample ( interpolation ) 36 Nss_duff = Fs/Fb; 37 Fs_up = Nss_duff * Fs /( Nss_tx ); 38 ynoisy_up = upsample ( ynoisy , Nss_duff / Nss_tx ); 39 ts_up = length ( ynoisy_up ) /( Nss_duff * Fb ); 40 t_up = linspace (0 ,( ts_up -1/ Fs_up ), length ( ynoisy_up ))’; 41 42 43 % Band pass filter ( interpolation ) 44 Wn_upsample = [ F_RF -( Fb *1) F_RF +( Fb *1) ]*(2/ Fs_up ); 45 [ b_upsample , a_upsample ] = butter (5 , Wn_upsample ); 46 yfilt_upsample = filter ( b_upsample , a_upsample , ynoisy_up ); 47 yfilt_upsample = yfilt_upsample./max(yfilt_upsample); 48 49 phase_duff = 0.5* pi; 50 diff_freq = 1; 51 ymod_duff = cos (2* pi * F_RF * diff_freq * t_up + phase_duff ); 52 53 % var1 54 % c = 0.81; % c =0.77; %0.82673 55 % A = 1; % A =0.2; %0.2 56 57 driver = ymod_duff; signal_in = ynoisy ; 58 % driver_in = ymod_duff; signal_in = yfilt_upsample ;