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

SLIDE 1

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

SLIDE 2

OVERVIEW

A very general stochastic model for pulse trains

– p.2/21

SLIDE 3

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

SLIDE 4

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

SLIDE 5

THE MODEL

Shot Noise with Random Excitation

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄

– p.3/21

SLIDE 6

THE MODEL

Shot Noise with Random Excitation

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✟

can be interpreted as a Random Filtering Function

– p.3/21

SLIDE 7

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

SLIDE 8

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

SLIDE 9

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

SLIDE 10

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

SLIDE 11

POWER SPECTRUM

General formula for the Power Spectral Density [Ridolfi, 04]

✁

✁ ✂ ✄ ☎ ✄ ✄ ✄ ☎ ✆ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☞ ✌ ✍ ✎ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ✏

– p.4/21

SLIDE 12

POWER SPECTRUM

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

SLIDE 13

POWER SPECTRUM

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

SLIDE 14

POWER SPECTRUM

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

SLIDE 15

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

SLIDE 16

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

SLIDE 17

POWER SPECTRUM

General formula for the Power Spectral Density [Ridolfi, 04]

✁

✁ ✂ ✄ ☎ ✄ ✄ ✄ ☎ ✆ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☞ ✌ ✍ ✎ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ✏

UWB Signals FCC Mask

– p.4/21

SLIDE 18

SIMPLE EXAMPLE: PULSE MODULATIONS

Pulse Position Modulation

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✁ ✡ ✠ ☞ ✆ ✄

where

is the pulse shape

☞ ✆

code the information (positions relative to the regular

✡

grid)

– p.5/21

SLIDE 19

SIMPLE EXAMPLE: PULSE MODULATIONS

Pulse Position Modulation

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✁ ✡ ✠ ☞ ✆ ✄

It is a Shot Noise with Random Excitation with

✡ ✆ ☎ ✁ ✡ ✟ ✁ ✂ ☛ ☞ ✄ ☎

✁

✂ ✠ ☞ ✄

– p.5/21

SLIDE 20

SIMPLE EXAMPLE: PULSE MODULATIONS

Pulse Position Modulation

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✁ ✡ ✠ ☞ ✆ ✄

It is a Shot Noise with Random Excitation with

✡ ✆ ☎ ✁ ✡ ✟ ✁ ✂ ☛ ☞ ✄ ☎

✁

✂ ✠ ☞ ✄

Hence

✠

✁ ✂ ✄ ☎

✁

✡ ✟ ✆ ✝ ✞ ✂ ✁ ✂ ✠ ✁ ✁ ✡ ✄

,

☛ ☎

✁

✡

– p.5/21

SLIDE 21

SIMPLE EXAMPLE: PULSE MODULATIONS

Pulse Position Modulation

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✁ ✡ ✠ ☞ ✆ ✄

It is a Shot Noise with Random Excitation with

✡ ✆ ☎ ✁ ✡ ✟ ✁ ✂ ☛ ☞ ✄ ☎

✁

✂ ✠ ☞ ✄

Hence

✠

✁ ✂ ✄ ☎

✁

✡ ✟ ✆ ✝ ✞ ✂ ✁ ✂ ✠ ✁ ✁ ✡ ✄

,

☛ ☎

✁

✡ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ☎ ✝

✁

✂ ✄

✁

✂ ✟ ✄ ☎ ✆

– p.5/21

SLIDE 22

SIMPLE EXAMPLE: PULSE MODULATIONS

Pulse Position Modulation

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✁ ✡ ✠ ☞ ✆ ✄

It is a Shot Noise with Random Excitation with

✡ ✆ ☎ ✁ ✡ ✟ ✁ ✂ ☛ ☞ ✄ ☎

✁

✂ ✠ ☞ ✄

Hence

✠

✁ ✂ ✄ ☎

✁

✡ ✟ ✆ ✝ ✞ ✂ ✁ ✂ ✠ ✁ ✁ ✡ ✄

,

☛ ☎

✁

✡ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ☎ ✝

✁

✂ ✄

✁

✂ ✟ ✄ ☎ ✆ ☎ ✆ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ✞ ☎ ✝

✁

✂ ✄

✆

✁ ✠ ✁ ✂ ✄ ✂ ✄ ☎ ✆ ✄ ✄ ✄ ✝ ✟ ✁ ✂ ☛ ☞ ✄ ✄ ✄ ✄ ✟ ✞ ☎ ☎ ✝

✁

✂ ✄ ☎ ✟

– p.5/21

SLIDE 23

SIMPLE EXAMPLE: PULSE MODULATIONS

Pulse Position Modulation

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✁ ✡ ✠ ☞ ✆ ✄

It is a Shot Noise with Random Excitation with

✡ ✆ ☎ ✁ ✡ ✟ ✁ ✂ ☛ ☞ ✄ ☎

✁

✂ ✠ ☞ ✄

Hence

✁

✁ ✂ ✄ ☎ ☎ ✝

✁

✂ ✄ ☎ ✟ ☎

✆

✁ ✁ ✂ ✄ ✂ ✄ ☎ ✟

✡

✟ ✆ ✝ ✞ ✂ ✎ ✂ ✠ ✁ ✡ ✏ ✡

✡

☎ ✝

✁

✂ ✄ ☎ ✟ ✎

✠

☎

✆

✁ ✁ ✂ ✄ ✂ ✄ ☎ ✟ ✏

– p.5/21

SLIDE 24

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

SLIDE 25

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

SLIDE 26

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

SLIDE 27

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

SLIDE 28

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

SLIDE 29

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

SLIDE 30

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

SLIDE 31

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

SLIDE 32

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

SLIDE 33

MULTIPATH FADED PULSE TRAINS

Pulse Train over a Multipath Fading Channel

✁

✂ ✄

multipath fading channel

✁

✂ ✄

– p.8/21

SLIDE 34

MULTIPATH FADED PULSE TRAINS

Pulse Train over a Multipath Fading Channel

✁

✂ ✄

multipath fading channel

✁

✂ ✄

Double Cluster (indoor) model of [Saleh & Valenzuela, 1987]

time pulse primary replicas (principal reflections) secondary replicas (secondary scattering)

– p.8/21

SLIDE 35

MULTIPATH FADED PULSE TRAINS

Pulse Train with Principal Replicas

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡ ✆ ✝ ✞

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

✏

where

is the pulse shape

✁ ✡ ✆ ☛ ☞ ✆ ✄

is a Marked Point Process

– p.9/21

SLIDE 36

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

SLIDE 37

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

SLIDE 38

MULTIPATH FADED PULSE TRAINS

Pulse Train with Principal Replicas and Random Gains

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡ ✆ ✝ ✞

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

– p.10/21

SLIDE 39

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

SLIDE 40

MULTIPATH FADED PULSE TRAINS

Pulse Train with Principal Replicas and Random Gains

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡ ✆ ✝ ✞

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

Let

☞

✆

☎ ✄ ☞ ✆ ☛

✆

✆ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄ ☎

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

– p.10/21

SLIDE 41

MULTIPATH FADED PULSE TRAINS

Pulse Train with Principal Replicas and Random Gains

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡ ✆ ✝ ✞

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

Let

☞

✆

☎ ✄ ☞ ✆ ☛

✆

✆ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄ ☎

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

Then

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄

.. a Shot Noise with Random Excitation .

– p.10/21

SLIDE 42

MULTIPATH FADED PULSE TRAINS

Pulse Train with Principal and Secondary Replicas and Random Gains

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡ ✆ ✝ ✞

✁

✂

✄

✂ ✠ ✡ ✆ ✠ ✡ ✁ ✆ ✂ ✄ ✄

☛

☞ ✆ ✆ ☎ ✁ ✄ ✡ ✁ ✆ ✂ ✄ ✄

☛

☞ ✁ ✆ ✂ ✄ ✄

✆

✡ ✆ ✝ ✞

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

✡ ✆ ✝ ✞

✁

✂

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

✠

✡ ✁ ✆ ✂

✄
☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

☎ ✁ ✄ ✡ ✁ ✆ ✂

✄
☛

☞ ✁ ✆ ✂

✄
✆

– p.11/21

SLIDE 43

MULTIPATH FADED PULSE TRAINS

Pulse Train with Principal and Secondary Replicas and Random Gains

✁

✂ ✄ ☎ ✆ ✝ ✞

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡ ✆ ✝ ✞

✁

✂

✄

✂ ✠ ✡ ✆ ✠ ✡ ✁ ✆ ✂ ✄ ✄

☛

☞ ✆ ✆ ☎ ✁ ✄ ✡ ✁ ✆ ✂ ✄ ✄

☛

☞ ✁ ✆ ✂ ✄ ✄

✆

✡ ✆ ✝ ✞

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

✡ ✆ ✝ ✞

✁

✂

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

✠

✡ ✁ ✆ ✂

✄
☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

☎ ✁ ✄ ✡ ✁ ✆ ✂

✄
☛

☞ ✁ ✆ ✂

✄
✆

Pulse Train

– p.11/21

SLIDE 44

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

SLIDE 45

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

SLIDE 46

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

SLIDE 47

MULTIPATH FADED PULSE TRAINS

.. a Shot Noise with Random Excitation

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄

– p.12/21

SLIDE 48

MULTIPATH FADED PULSE TRAINS

.. a Shot Noise with Random Excitation

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄

where

☞

✆

☎ ✎ ☞ ✆ ☛

✆

☛

✁

✆ ✂

☛

✁ ✁ ✂ ✄ ✏

– p.12/21

SLIDE 49

MULTIPATH FADED PULSE TRAINS

.. a Shot Noise with Random Excitation

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄

where

☞

✆

☎ ✎ ☞ ✆ ☛

✆

☛

✁

✆ ✂

☛

✁ ✁ ✂ ✄ ✏

and

✟ ✁ ✂ ✠ ✡ ✆ ☛ ☞

✆

✄ ☎

✁

✂ ✠ ✡ ✆ ☛ ☞ ✆ ✄ ✡

✁

✂

✄

✂ ✠ ✡ ✆ ✠ ✡ ✁ ✆ ✂ ✄ ✄

☛

☞ ✆ ✆ ☎ ✁ ✄ ✡ ✁ ✆ ✂ ✄ ✄

☛

☞ ✁ ✆ ✂ ✄ ✄

✆

✡

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

✡

✁

✂

✁

✂

✎

✂ ✠ ✡ ✆ ✠ ✡

✆

✄

✠

✡ ✁ ✆ ✂

✄
☛

☞ ✆ ✏ ☎

✎

✡

✆

✄

☛

☞

✆

✄

✏

☎ ✁ ✄ ✡ ✁ ✆ ✂

✄
☛

☞ ✁ ✆ ✂

✄
✆

– p.12/21

SLIDE 50

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

SLIDE 51

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

SLIDE 52

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

SLIDE 53

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

SLIDE 54

POWER SPECTRUM

Recall

✁

✁ ✂ ✄ ☎ ✄ ✄ ✄ ☎ ✆ ✝ ✟ ✁ ✂ ☛ ☞

✄

✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☞ ✌ ✍ ✎ ✝ ✟ ✁ ✂ ☛ ☞

✄

✏

– p.14/21

SLIDE 55

POWER SPECTRUM

Recall

✁

✁ ✂ ✄ ☎ ✄ ✄ ✄ ☎ ✆ ✝ ✟ ✁ ✂ ☛ ☞

✄

✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☞ ✌ ✍ ✎ ✝ ✟ ✁ ✂ ☛ ☞

✄

✏

where now

✝ ✟ ✁ ✂ ☛ ☞

✄

☎ ✝

✁

✂ ☛ ☞ ✄

✄

✂ ☛ ✄

☛
✁

✆ ✆

with

✄

✂ ☛ ✄

☛
✁

✆ ✆ ☎

✡
✁

✂

✁

✂ ✟ ✄ ☎ ✁

✂

☎

✁

✡

☛

☞

✄

✡

✁

✂

✁

✂ ✟ ✄ ☎ ✁ ✄ ☎ ✆ ✝ ☎ ✁ ✄ ✡ ✁ ✄ ✄

☛

☞ ✁ ✄ ✄

✆

✡

✁

✂

✁

✂ ✟ ✄ ☎ ✁

✂

☎

✁

✡

☛

☞

✄
✁

✂

✁

✂ ✟ ✄ ☎ ✁ ✄ ✂ ✆ ✝ ☎ ✁ ✄ ✡ ✁

✄
☛

☞ ✁

✄
✆
– p.14/21

SLIDE 56

POWER SPECTRUM

Therefore

✁

✁ ✂ ✄ ☎ ☎ ☎

✝
✁

✂ ☛ ☞ ✄ ✁ ☎ ✟ ✄ ✄ ✄ ☎ ✆ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☎ ✆ ☎ ✝

✁

✂ ☛ ☞ ✄ ☎ ✟ ✞ ☎ ✆ ✄ ✄ ✄ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✄ ✄ ✄ ✟ ✞ ✠ ☛ ☎ ☎

✝
✁

✂ ☛ ☞ ✄ ✁ ☎ ✟ ✄ ✄ ✄ ☎ ✆ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✞ ✄ ✄ ✄ ✟

– p.15/21

SLIDE 57

POWER SPECTRUM

Therefore

✁

✁ ✂ ✄ ☎ ☎ ☎

✝
✁

✂ ☛ ☞ ✄ ✁ ☎ ✟ ✄ ✄ ✄ ☎ ✆ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☎ ✆ ☎ ✝

✁

✂ ☛ ☞ ✄ ☎ ✟ ✞ ☎ ✆ ✄ ✄ ✄ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✄ ✄ ✄ ✟ ✞ ✠ ☛ ☎ ☎

✝
✁

✂ ☛ ☞ ✄ ✁ ☎ ✟ ✄ ✄ ✄ ☎ ✆ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✞ ✄ ✄ ✄ ✟

Characteristics of the Pulse Train, e.g. of the Pulse Modulation

– p.15/21

SLIDE 58

POWER SPECTRUM

Therefore

✁

✁ ✂ ✄ ☎ ☎ ☎

✝
✁

✂ ☛ ☞ ✄ ✁ ☎ ✟ ✄ ✄ ✄ ☎ ✆ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✞ ✄ ✄ ✄ ✟

✠

✁ ✂ ✄ ✡ ☛ ☎ ✆ ☎ ✝

✁

✂ ☛ ☞ ✄ ☎ ✟ ✞ ☎ ✆ ✄ ✄ ✄ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✄ ✄ ✄ ✟ ✞ ✠ ☛ ☎ ☎

✝
✁

✂ ☛ ☞ ✄ ✁ ☎ ✟ ✄ ✄ ✄ ☎ ✆ ✝

✄

✂ ☛ ✄

☛
✁

✆ ✆ ✞ ✄ ✄ ✄ ✟

Characteristics of the Multipath Fading Channel.

– p.15/21

SLIDE 59

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL

Double Poisson Cluster model of [Saleh & Valenzuela, 1987]

– p.16/21

SLIDE 60

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

SLIDE 61

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

SLIDE 62

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

SLIDE 63

EXAMPLE: CLASSICAL MULTIPATH FADING CHANNEL

Double Poisson Cluster model of [Saleh & Valenzuela, 1987] Then

☎

✁

✂ ☛ ✁

☛
✁

✄ ✄ ✁ ☎

✡

☛

☎
✁

✂ ✄ ✡ ☛ ✁ ✝ ☎ ✁ ✁ ✂ ✄ ✡ ☛

☛

✁

☎
✁

✂ ✄ ✝ ☎ ✁ ✁ ✂ ✄

and

☎ ✆ ☎

✁

✂ ☛ ✁

☛
✁

✄ ✄ ☎ ✟ ✞ ☎

✡

☛

✁

✄ ✄

☎
✁

✂ ✄ ✄ ✄ ✟ ✄ ✂ ☛ ✁ ✁ ✄ ✄ ✝ ☎ ✁ ✁ ✂ ✄ ✄ ✄ ✟ ✄ ✂ ✡ ✄ ✄

✡

☛ ✁ ✝ ☎ ✁ ✁ ✂ ✄ ✄ ✄ ✟ ☛

✁

✄ ✄

☎
✁

✂ ✄ ✄ ✄ ✟ ✡ ☎ ☎

✁

✂ ☛ ✁

☛
✁

✄ ✄ ✁ ☎ ✟

– p.16/21

SLIDE 64

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

SLIDE 65

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
ver a Double Cluster Poisson multipath fading channel characterized by

☛

☎
✁

✁

✂

✂ ✆

✂

✁ ✝ ✄

(

✁
✂

✂

ns

✁ ✂

)

☎

✁

✂ ✄ ☎

✁

✞

✟

with

✁

✠

☎

✁ ✂ ✆

✂

✁ ✝ ☛ ✁ ☎

✁

✁ ✡ ✆

✂

✁ ✝ ✄

(

✁

✡

ns

✁

)

☎ ✁ ✁ ✂ ✄ ☎

✁

✞ ✄ ✟

with

✁

✠ ✁ ☎ ✂ ✂ ✆

✂

✁ ✝

.

– p.18/21

SLIDE 66

NUMERICAL RESULTS

1 2 3 4 5 6 x 10

9

−200 −180 −160 −140 −120 Hz dBm/Hz

PPM

– p.19/21

SLIDE 67

NUMERICAL RESULTS

1 2 3 4 5 6 x 10

9

−200 −180 −160 −140 −120 Hz dBm/Hz

PPM with primary replicas

– p.19/21

SLIDE 68

NUMERICAL RESULTS

1 2 3 4 5 6 x 10

9

−200 −180 −160 −140 −120 Hz dBm/Hz

PPM with primary and secondary replicas

– p.19/21

SLIDE 69

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

SLIDE 70

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