Spectra of UWB Signals in a Swiss Army Knife

Spectra of UWB Signals in a Swiss Army Knife Andrea Ridolfi EPFL, Switzerland joint work with Pierre Brémaud, EPFL (Switzerland) and ENS Paris (France) Laurent Massoulié, Microsoft Cambridge (UK) Martin Vetterli, EPFL (Switzerland) Moe Z. Win, MIT (USA) p.1/21

OVERVIEW A very general stochastic model for pulse trains p.2/21

OVERVIEW A very general stochastic model for pulse trains Large family of Pulse Modulated Signals (UWB communications) Multipath Faded Pulse Sequences Biological Signals Network Traffic p.2/21

OVERVIEW A very general stochastic model for pulse trains Large family of Pulse Modulated Signals (UWB communications) Multipath Faded Pulse Sequences Biological Signals Network Traffic The model is Modular (features can be subsequently added) It holds in the Spatial Case (e.g. locations of mobile units) It easily allows to compute the Power Spectrum Spectral contribution of different features appear clearly and separately. p.2/21

THE MODEL Shot Noise with Random Excitation p.3/21

THE MODEL Shot Noise with Random Excitation can be interpreted as a Random Filtering Function p.3/21

THE MODEL Shot Noise with Random Excitation can be interpreted as a Random Filtering Function is a sequence of random points (Point Process) p.3/21

THE MODEL Shot Noise with Random Excitation can be interpreted as a Random Filtering Function is a sequence of random points (Point Process) is a sequence of i.i.d. random parameters (Marks) p.3/21

THE MODEL Shot Noise with Random Excitation can be interpreted as a Random Filtering Function is a sequence of random points (Point Process) is a sequence of i.i.d. random parameters (Marks) is called Marked Point Process p.3/21

THE MODEL Shot Noise with Random Excitation Randomly filtered Point Process not limited to the Poisson case First order characteristics (for free!): Campbell s theorem Second order characteristics: Covariance Measure (see [Daley & Vere-Jones, 1988 & 2002]). p.3/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] p.4/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] where and is the Fourier transform of w.r.t., is a random variable distributed as the i.i.d. Marks p.4/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] where and is the Fourier transform of w.r.t., is a random variable distributed as the i.i.d. Marks is the power spectral (pseudo) density of the point process (Bartlett spectrum) p.4/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] where and is the Fourier transform of w.r.t., is a random variable distributed as the i.i.d. Marks is the power spectral (pseudo) density of the point process (Bartlett spectrum) is the average number of point per unit of time (average intensity of the point process) p.4/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] Swiss Army Knife structure: by appropriately choosing and we can obtain several exact Spectral Expressions: p.4/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] Swiss Army Knife structure: by appropriately choosing and we can obtain several exact Spectral Expressions: UWB Transmissions Multipath Faded Pulse Trains Neuronal Sequences Network Traffic p.4/21

POWER SPECTRUM General formula for the Power Spectral Density [Ridolfi, 04] UWB Signals FCC Mask p.4/21

SIMPLE EXAMPLE: PULSE MODULATIONS Pulse Position Modulation where is the pulse shape code the information (positions relative to the regular -grid) p.5/21

SIMPLE EXAMPLE: PULSE MODULATIONS Pulse Position Modulation It is a Shot Noise with Random Excitation with p.5/21

SIMPLE EXAMPLE: PULSE MODULATIONS Pulse Position Modulation It is a Shot Noise with Random Excitation with Hence, p.5/21

SIMPLE EXAMPLE: PULSE MODULATIONS Pulse Position Modulation It is a Shot Noise with Random Excitation with Hence, p.5/21

SIMPLE EXAMPLE: PULSE MODULATIONS Pulse Position Modulation It is a Shot Noise with Random Excitation with Hence, p.5/21

SIMPLE EXAMPLE: PULSE MODULATIONS Pulse Position Modulation It is a Shot Noise with Random Excitation with Hence p.5/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter where is the pulse shape and code the information represent the clock jitter p.6/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter It is a Shot Noise with Random Excitation with, i.e. p.6/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter It is a Shot Noise with Random Excitation with, i.e. Hence, p.6/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter It is a Shot Noise with Random Excitation with, i.e. Hence, p.6/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter It is a Shot Noise with Random Excitation with, i.e. Hence, p.6/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter It is a Shot Noise with Random Excitation with, i.e. Hence p.6/21

SIMPLE EXAMPLE: PULSE MODULATIONS Let s add some features: Pulse Position and Amplitude Modulation with jitter It is a Shot Noise with Random Excitation with, i.e. Hence Modularly added features appear explicitly and separately. p.6/21

MORE PULSE MODULATIONS.. more modelling and more (free) exact spectral densities Pulse interval modulations Pulse modulations with Time-Hopping Pulse modulations with Direct-Sequences Pulse modulations with random losses Pulse modulations with random distortions.. any combination of the above p.7/21

MORE PULSE MODULATIONS.. more modelling and more (free) exact spectral densities Pulse interval modulations Pulse modulations with Time-Hopping Pulse modulations with Direct-Sequences Pulse modulations with random losses Pulse modulations with random distortions.. any combination of the above a matter of choosing the appropriate and which can be modularly complexified ad libitum., p.7/21

MULTIPATH FADED PULSE TRAINS Pulse Train over a Multipath Fading Channel multipath fading channel p.8/21

MULTIPATH FADED PULSE TRAINS Pulse Train over a Multipath Fading Channel multipath fading channel Double Cluster (indoor) model of [Saleh & Valenzuela, 1987] pulse primary replicas (principal reflections) secondary replicas (secondary scattering) time p.8/21

MULTIPATH FADED PULSE TRAINS p.9/21 Pulse Train with Principal Replicas where is the pulse shape is a Marked Point Process

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal Replicas where is the pulse shape is a Marked Point Process are the Propagation Delays, assumed to be i.i.d. w.r.t. is a sequence of i.i.d. Point Processes p.9/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal Replicas where is the pulse shape is a Marked Point Process are the Propagation Delays, assumed to be i.i.d. w.r.t. is a sequence of i.i.d. Point Processes are the Gains. p.9/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal Replicas and Random Gains p.10/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal Replicas and Random Gains where are random parameters, i.i.d. w.r.t. to both indexes (Marks) is a sequence of i.i.d. Marked Point Processes are the random Gains p.10/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal Replicas and Random Gains Let p.10/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal Replicas and Random Gains Let Then.. a Shot Noise with Random Excitation. p.10/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal and Secondary Replicas and Random Gains p.11/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal and Secondary Replicas and Random Gains Pulse Train p.11/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal and Secondary Replicas and Random Gains Secondary Replicas (Secondary Scattering) of the Pulse Train i.i.d. Marked Point Processes p.11/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal and Secondary Replicas and Random Gains Primary Replicas (Principal Reflections) of the Pulse Train p.11/21

MULTIPATH FADED PULSE TRAINS Pulse Train with Principal and Secondary Replicas and Random Gains Secondary Replicas (Secondary Scattering) of Principal Reflections,, i.i.d. Marked Point Processes. p.11/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation p.12/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation where p.12/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation and where p.12/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation Remarks: The Double Cluster nature of the point is captured by the Marks (Cluster Point Processes are indeed Shot Noises [Bremaud et al, 2005]) p.13/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation Remarks: The Double Cluster nature of the point is captured by the Marks (Cluster Point Processes are indeed Shot Noises [Bremaud et al, 2005]) No Poisson assumption on the Pulses and the Replicas (Poisson assumption on the pulses is unrealistic) p.13/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation Remarks: The Double Cluster nature of the point is captured by the Marks (Cluster Point Processes are indeed Shot Noises [Bremaud et al, 2005]) No Poisson assumption on the Pulses and the Replicas (Poisson assumption on the pulses is unrealistic) Fading can be taken into account using The Marks (Pulse Distortion) p.13/21

MULTIPATH FADED PULSE TRAINS.. a Shot Noise with Random Excitation Remarks: The Double Cluster nature of the point is captured by the Marks (Cluster Point Processes are indeed Shot Noises [Bremaud et al, 2005]) No Poisson assumption on the Pulses and the Replicas (Poisson assumption on the pulses is unrealistic) Fading can be taken into account using The Marks (Pulse Distortion) Modulating Process (Fast Fading) (see the proceedings for more details). p.13/21

POWER SPECTRUM p.14/21 Recall

POWER SPECTRUM p.14/21 Recall where now with

POWER SPECTRUM p.15/21 Therefore

POWER SPECTRUM Therefore Characteristics of the Pulse Train, e.g. of the Pulse Modulation p.15/21

POWER SPECTRUM Therefore Characteristics of the Multipath Fading Channel. p.15/21

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL Double Poisson Cluster model of [Saleh & Valenzuela, 1987] p.16/21

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL Double Poisson Cluster model of [Saleh & Valenzuela, 1987] The primary and secondary replicas are Poisson Point Processes for every, form a Poisson Point Process for every and, form a Poisson Point Process p.16/21

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL Double Poisson Cluster model of [Saleh & Valenzuela, 1987] The primary and secondary replicas are Poisson Point Processes for every, form a Poisson Point Process for every and, form a Poisson Point Process The gains are deterministic p.16/21

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL Double Poisson Cluster model of [Saleh & Valenzuela, 1987] The primary and secondary replicas are Poisson Point Processes for every, form a Poisson Point Process for every and, form a Poisson Point Process The gains are deterministic We do not assume the Pulse Train to be Poisson p.16/21

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL Double Poisson Cluster model of [Saleh & Valenzuela, 1987] Then and p.16/21

SOFTWARE TOOLBOX The Swiss Army structure of the spectral formula (modularity) allowed to easily develop a Matlab Software Toolbox (GNU GPL) lcavwww.epfl.ch/software The software is by default UWB oriented (UWB Pulse Modulated Signals). Acknowledgments: We are grateful to Justin Salez for coding in Matlab our theoretical results; G.M. Maggio for the precious advice in defining default parameters of UWB transmissions. p.17/21

NUMERICAL RESULTS We consider a PPM transmitted signal with the following characteristics Symbol period of ns Binary symbols coded into ns and ns Gaussian derivative pulse shape, V, width ns over a Double Cluster Poisson multipath fading channel characterized by ns ( ) with ns ( ) with. p.18/21

NUMERICAL RESULTS 120 dbm/hz 140 160 180 200 0 1 2 3 4 5 6 Hz x 10 9 PPM p.19/21

NUMERICAL RESULTS 120 dbm/hz 140 160 180 200 0 1 2 3 4 5 6 Hz x 10 9 PPM with primary replicas p.19/21

NUMERICAL RESULTS 120 dbm/hz 140 160 180 200 0 1 2 3 4 5 6 Hz x 10 9 PPM with primary and secondary replicas p.19/21

REMARKS Approach based on a Shot Noise with Random Excitation Provides a very general yet tractable model Provides exact general expressions of the Power Spectra It unifies, simplifies and extends previous results and provides new ones It allows to tackle very complicated pulse signal scenarios It easily allows a software implementation. p.20/21

MORE DETAILS A. Ridolfi, Power Spectra of Random Spikes and Related Complex Signals. With Application to Communications. Ph.D. thesis, EPFL, Lausanne, Switzerland, 2004. P. Brémaud, L. Massoulié and A. Ridolfi. Power Spectra of Random Spike Fields & Related Processes. To appear in Journal of Applied Probability, December 2005. A. Ridolfi and M. Z. Win. Ultrawide Bandwidth Signals as Shot-Noise: a Unifying Approach. To appear in IEEE Journal on Selected Areas in Communications, special issue on UWB wireless communications - theory and applications, 2005. A. Ridolfi and M. Z. Win. Power Spectra of Multipath Faded Pulse Trains. IEEE International Symposium on Information Theory - ISIT 2005. A. Ridolfi and M. Z. Win. Spectrum of Random Pulse Trains Received via Multipath Channels. Technical report, to be submitted, 2005. lcavwww.epfl.ch/ ridolfi andrea.ridolfi@epfl.ch p.21/21