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Heterodyne detection with LISA. for gravitational waves parameters estimation. Nicolas Douillet. Outline. (1) : LISA (Laser Interferometer Space Antenna (2) : Model for a monochromatic wave (3) : Heterodyne detection principle (4) : Some results on simulated data analysis

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heterodyne detection with lisa

Heterodyne detection with LISA

for gravitational waves parameters estimation

Nicolas Douillet

  • (1): LISA (Laser Interferometer Space


  • (2): Model for a monochromatic wave
  • (3): Heterodyne detection principle
  • (4): Some results on simulated data analysis
  • (5): Conclusion & future work
lisa configuration
LISA configuration

- Heliocentric orbits,

free falling spacecraft.

  • LISA center of mass
  • Follows Earth, delayed
  • from a 20° angle.
  • 60° angle between LISA
  • plan and the ecliptic plan.

- LISA arm’s length: 5. 109 m to detect gravitational waves with frequency in: 10-4   10-1 Hz

- LISA periodic motion -> information on the direction of the wave.

motivations for lisa
Motivations for LISA

Existing ground based detectors such as VIRGO and LIGO are « deaf » in low

frequencies (  < 10 Hz).

Limited sensitivity due to « seismic wall » (LF vibrations transmitted by the

Newtonian fields gradient)

  • A space based detector

allows to get rid of this


  • Possibility to detect

very low frequency

gravitional waves.

monochromatic waves
Monochromatic waves

Sources: signals coming from coalescing binaries

long before inspiral step. Frequency  considered

as a constant.

h+ / h: amplitude following + / x polarization

+ / : directional functions

Gravitational wave causes

perturbations in the metric


Effect (amplified) of a

Gravitational wave on a ring

of particles:

+ polarization

x polarization

model for a monochromatic wave 1
Model for a monochromatic wave(1)

Unknown parameters:

 (Hz): source frequency

 (rad): ecliptic latitude

 (rad): ecliptic longitude

 (rad): polarization angle

 (rad): orbital inclination angle

h (-): wave amplitude

 (rad): initial source phase

LISA response to the incoming GW:

T : LISA period (1 year)

model for a monochromatic wave 2
Model for a monochromatic wave (2)


Amplitude modulation (envelope)

Shape depends on source location: (, )


pattern beam functions 1
Pattern beam functions (1)

Change ofreference frame for

and pattern beam functions.

: polarization angle

Spacecraft n° in LISA triangle.

envelope heterodyne detection 1
Envelope heterodyne detection (1)


(1): Fundamental frequency (0) search

Detect the maximum in the spectrum of the product between source signal (s) and a template signal (m) which frequency lays in the range:

Frequency precision is reached with a nested search.


envelope heterodyne detection 2
Envelope heterodyne detection (2)

(3): Shift spectrum (offset zero-frequency) by heterodyning at , then low-pass filtering

(Filter above )

8 lateral bands: [0; 7] (empirical) -> compromise between accepted noise

level and maximum frequency needed to rebuild the envelope ( = 1/ T)

(2): Envelope reconstruction

Fourier sum

correlation optimization 1
Correlation optimization (1)

Correlation surface between template and experimental envelope

correlation optimization 2
Correlation optimization (2)
  • Principle: correlation maximization between signal envelope end envelope
  • template (or mean squares minimization).
  • (2) Method: gradient convergence and quasi-Newton optimization methods.
  • (3) Conditions: already lay on the convex area which contains the maximum.
sources mix


Sources mix

Possible to distinguish between n

sources since their fundamental

frequencies are spaced enough

(sidebands don’t cover each other):



Envelope detection (1)




Envelope detection (2)

symmetries ambiguities

LISA main symmetry

E(-, + ) = E(, )

Correlation symmetry

Corr(, ) = Corr(-,+ )

Symmetries & ambiguities
symmetries 1
Symmetries (1)

Some parameters remains difficult to estimate due to the high number of the

envelope symmetries on the parametersand.


symmetries 2
Symmetries (2)

Ie -> risks of being stuck on correlation secondary maxima in N dimensions

space (varied topologies resolution problem).

how to remove sky location uncertainty 1
How to remove sky location uncertainty (1)

Choice between (,) and ( -, +) depends on the sign of the product

If  is the colatitude (ie   [0;  ]), and when t=0

From the source signal, we compute the quantity

hence the sign of  and 


Source localization

Simulated data from LISA data analysis community

statistics on sky location angles

Max error: polar source ( = /2)

Max sensitivity: source direction  to LISA plan

( ~ /6)

Statistics on sky location angles (,)

 = f()

 = f()

noise robustness tests static source
Noise robustness tests (static source)

True value

Estimations (180 runs on the noise)

typical errors on estimated parameters



Typical errors on estimated parameters

Average relative errors for   /3

compare two parameters estimation techniques template bank vs mcmc
Compare two parameters estimation techniques: template bank vs MCMC

(1): Matching templates (template bank and scan parameters space till reaching correlation maximum -> systematic method)

- Advantages: ● easy/friendly programmable

● quite good robustness

- Limitations: ● N dimensions parameters space. (memory space and computation time expensive)

● difficulties to adapt and apply this method for more complex waveforms

(2): MCMC methods, max likelihood ratio: motivations

(statistics & probability based methods)

- Advantages: ● No exhaustive scan of the parameters space (dim N).

● muchlower computing cost and smaller memory space

- Limitations: ● Careful handling: high number parameters to tune in the algorithm (choice of probability density functions of the parameters)

conclusion and future work
Conclusion and future work

- Encouraging results of this method (heterodyne detection) on monochromatic waves. Could still to be improved however.

- Continue to develop image processing techniques for trajectories segmentation (chirp & EMRI) in time-frequency plan. (level sets, ‘active contours’ methods import from medical imaging and shape optimization)

  • Combining this methods (graphic first estimation of parameters) with Monte-Carlo Markov Chains algorithms (numeric finest estimation) allows in a way to ‘‘ log-divide’’ the dimensions of the parameters space (N5 + N2 instead of N7 for example).