Spectra of UWB Signals in a Swiss Army Knife Andrea Ridolfi EPFL, - PowerPoint PPT Presentation

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

O VERVIEW A very general stochastic model for pulse trains – p.2/21

O VERVIEW 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

O VERVIEW 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

T HE M ODEL Shot Noise with Random Excitation ✁ ✄ ✁ ✄ � ✟ ✡ ☞ ✂ ✂ ☎ ✠ ✆ ✆ ☛ ✞ ✝ ✆ – p.3/21

T HE M ODEL Shot Noise with Random Excitation ✁ ✄ ✁ ✄ � ✟ ✡ ☞ ✂ ✂ ☎ ✠ ✆ ✆ ☛ ✞ ✝ ✆ ✟ can be interpreted as a Random Filtering Function – p.3/21

T HE M ODEL Shot Noise with Random Excitation ✄ ✁ ✄ ✁ � ✟ ✡ ☞ ✂ ✂ ☎ ✠ ✆ ✆ ☛ ✞ ✝ ✆ ✟ can be interpreted as a Random Filtering Function ✡ is a sequence of random points ( Point Process ) ✆ – p.3/21

T HE M ODEL 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

T HE M ODEL 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

T HE M ODEL 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

P OWER S PECTRUM General formula for the Power Spectral Density [Ridolfi, 04] ✟ ✄ ✄ ✆ ✞ ✎ ✏ ✝ ✝ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ � ☎ ✟ ☞ � ☛ ☞ ✟ ☞ ✄ ✄ ✡ ✂ ✂ ✂ ✌ ✍ ✂ ☎ ✁ ✠ ☛ ☛ ✄ ✄ – p.4/21

P OWER S PECTRUM General formula for the Power Spectral Density [Ridolfi, 04] ✟ ✄ ✄ ✆ ✞ ✎ ✏ ✝ ✝ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ � ☎ ✟ ☞ � ☛ ☞ ✟ ☞ ✄ ✄ ✡ ✂ ✂ ✂ ✌ ✍ ✂ ☎ ✁ ✠ ☛ ☛ ✄ ✄ where ✝ ✁ ✄ ✁ ✄ ✟ ☞ ✟ ☞ ✂ ✂ is the Fourier transform of w.r.t. , ✂ ☛ ☛ ☞ ☞ and is a random variable distributed as the i.i.d. Marks ✆ – p.4/21

P OWER S PECTRUM General formula for the Power Spectral Density [Ridolfi, 04] ✟ ✄ ✄ ✆ ✞ ✎ ✏ ✝ ✝ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ � ☎ ✟ ☞ � ☛ ☞ ✟ ☞ ✄ ✄ ✡ ✂ ✂ ✂ ✌ ✍ ✂ ☎ ✁ ✠ ☛ ☛ ✄ ✄ where ✝ ✁ ✄ ✁ ✄ ✟ ☞ ✟ ☞ ✂ ✂ is the Fourier transform of w.r.t. , ✂ ☛ ☛ ☞ ☞ and 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

P OWER S PECTRUM General formula for the Power Spectral Density [Ridolfi, 04] ✟ ✄ ✄ ✆ ✞ ✎ ✏ ✝ ✝ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ � ☎ ✟ ☞ � ☛ ☞ ✟ ☞ ✄ ✄ ✡ ✂ ✂ ✂ ✌ ✍ ✂ ☎ ✁ ✠ ☛ ☛ ✄ ✄ where ✝ ✁ ✄ ✁ ✄ ✟ ☞ ✟ ☞ ✂ ✂ is the Fourier transform of w.r.t. , ✂ ☛ ☛ ☞ ☞ and 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

P OWER S PECTRUM 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

P OWER S PECTRUM 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

P OWER S PECTRUM General formula for the Power Spectral Density [Ridolfi, 04] ✟ ✄ ✄ ✆ ✞ ✎ ✏ ✝ ✝ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ � ☎ ✟ ☞ � ☛ ☞ ✟ ☞ ✄ ✄ ✡ ✂ ✂ ✂ ✌ ✍ ✂ ☎ ✁ ✠ ☛ ☛ ✄ ✄ UWB Signals FCC Mask – p.4/21

S IMPLE E XAMPLE : P ULSE M ODULATIONS Pulse Position Modulation ✁ ✄ ✁ ✄ � ✡ ☞ ✂ ✂ � ✁ ☎ ✠ ✠ ✆ ✞ ✝ ✆ where is the pulse shape � ☞ ✡ code the information (positions relative to the regular -grid) ✆ – p.5/21

S IMPLE E XAMPLE : P ULSE M ODULATIONS Pulse Position Modulation ✁ ✄ ✁ ✄ � ✡ ☞ ✂ ✂ � ✁ ☎ ✠ ✠ ✆ ✞ ✝ ✆ It is a Shot Noise with Random Excitation with ✡ ✡ ✁ ☎ ✆ ✁ ✄ ✁ ✄ ✟ ☞ ☞ ✂ ✂ � ☎ ✠ ☛ – p.5/21

S IMPLE E XAMPLE : P ULSE M ODULATIONS Pulse Position Modulation ✁ ✄ ✁ ✄ � ✡ ☞ ✂ ✂ � ✁ ☎ ✠ ✠ ✆ ✞ ✝ ✆ It is a Shot Noise with Random Excitation with ✡ ✡ ✁ ☎ ✆ ✁ ✄ ✁ ✄ ✟ ☞ ☞ ✂ ✂ � ☎ ✠ ☛ Hence ✟ ✁ ✄ ✁ ✁ ✄ ✁ ✁ � ✡ ✂ ✡ ☛ ✡ � � , ✂ ✂ ✁ ☎ ☎ ✠ ✠ ✞ ✝ ✆ – p.5/21

S IMPLE E XAMPLE : P ULSE M ODULATIONS Pulse Position Modulation ✁ ✄ ✁ ✄ � ✡ ☞ ✂ ✂ � ✁ ☎ ✠ ✠ ✆ ✞ ✝ ✆ It is a Shot Noise with Random Excitation with ✡ ✡ ✁ ☎ ✆ ✁ ✄ ✁ ✄ ✟ ☞ ☞ ✂ ✂ � ☎ ✠ ☛ Hence ✟ ✁ ✄ ✁ ✁ ✄ ✁ ✁ � ✡ ✂ ✡ ☛ ✡ � � , ✂ ✂ ✁ ☎ ☎ ✠ ✠ ✞ ✝ ✆ ✝ ✂ ✟ ✆ ✁ ✄ ✁ ✝ ✄ ✟ ☞ ✄ ☎ ✁ ✂ � ✂ � ☎ ☛ – p.5/21

– p.5/21 S IMPLE E XAMPLE : P ULSE M ODULATIONS ✟ ☎ ✄ ✂ ✁ ✝ � ☎ ☎ ✞ ✄ ✟ ☞ ✄ ✄ ✄ ✡ ✄ ✆ ☞ ✁ � ☛ ✂ ✡ ✠ ☎ ✁ ✁ ✝ ✟ ☛ ✄ ✄ ✄ It is a Shot Noise with Random Excitation with ✆ ☎ ✂ ✠ , ✁ ✄ � ✡ ✄ ✁ ✂ ✞ ✁ ✄ ✝ ✂ ✁ ☎ ✆ ✂ ✠ ✁ ✁ ✆ ✠ ✄ ✂ ✂ ✆ ☎ ✁ � ✄ ✞ � ✟ ✝ ✄ ✂ ✄ Pulse Position Modulation ☞ ✂ ✁ ✁ � ✆ ✝ � ✄ ✂ ✟ ✂ ✠ ✡ ☎ ✁ ✁ ✝ � ✁ � � ✞ ✄ ☎ ☎ ☞ ☎ ✡ ✄ ✁ ✄ ☛ ☞ ✂ ✄ ☞ ✂ ✁ ☎ ✁ ☛ ☛ ✝ ✟ ✂ ✂ ✠ ✆ ✁ ✁ ✡ � ☎ ✟ ✝ ✟ ✆ Hence

S IMPLE E XAMPLE : P ULSE M ODULATIONS Pulse Position Modulation ✁ ✄ ✁ ✄ � ✡ ☞ ✂ ✂ � ✁ ☎ ✠ ✠ ✆ ✞ ✝ ✆ It is a Shot Noise with Random Excitation with ✡ ✡ ✁ ☎ ✆ ✁ ✄ ✁ ✄ ✟ ☞ ☞ ✂ ✂ � ☎ ✠ ☛ Hence � ✎ ✏ ✁ ✟ ✟ � ✁ ✄ ☎ ✁ ✝ ✄ ☎ ☎ ✁ ✄ ☎ ✂ � ✁ ✂ ✂ � ✂ ✄ ✂ ✂ ☎ ✁ ✆ ✠ ✟ ✡ ✡ ✞ ✝ ✆ � ✎ ✏ ✟ ✟ ☎ ✁ ✝ ✄ ☎ ☎ ✁ ✄ ☎ � � ✁ ✂ ✡ � ✂ ✄ ✂ ✆ ✠ � ✡ – p.5/21