1. Matched Filtering Junwei Cao (曹军威) and Junwei Li (李俊伟)
Tsinghua University
Gravitational Wave Summer School Kunming, China, July 2009 1

2. Weak Signals, Loud NoiseIn
Signals are weak
How to dig out GW signals
If we know, at least to some level of accuracy, the form of h(t)

3. A Sample Filtering
Multiply s(t) by h(t)

4. A Sample Filtering (contd.)
T can’t be infinite
Not necessary to have h0>n0
For example,
τ0=1 ms, T=1 year

5. Optimal Filtering To obtain the highest possible SNR Define
SNR definition (in amplitude)
: Filter function

6. S & N

7. S/N
Using eq. 1.7
Therefore

8. What is the Filter K(t)
Define a scalar product
Then

9. The Solution Search for the “vector” unit norm
So, the matched filter is

10. The Optimal Value of SNR
Insert eq into 2.11
Insert ea into 2.10

11. Bursts
What are bursts?
From astrophysical phenomena, e.g. supernova explosions, final merging of a neutron star-neutron star binary system
Large energy, short time, typically less than a second
Hard to know the form of
Use order-of-magnitude estimate 11

12. Narrow-band Detectors
Only sensitive to frequencies in a bandwidth , centered around
cannot change much
Approximate with
Eq becomes 12

13. Broad-band Detectors Bandwidth Order-of-magnitude estimate
: characteristic value of over
: characteristic value of over
A detailed form of bursts is important 13

14. A Model for Bursts A sine-Gaussian : the amplitude : the duration
Fourier transform
If , near , basically Gaussian
Maximum of the Gaussian: 14

15. Estimate Minimum h0 When ,  relatively flat
When ,  sharply peaked around
For narrow-band detectors, plug eq into eq. 2.16
For broad-band detectors, plug eq into eq. 2.17 15

16. Root-sum-square Amplitude
defined by
For sine-Gaussian 16

17. An Exercise
The total energy radiated by the source in GWs, assuming a sine-Gaussian waveform
A sine-Gaussian burst (1ms duration, 1kHz ) which release 1/100 solar masses in the Virgo galaxy cluster, which is at r~14Mpc
Requirement
Approximate and 17

18. Answer

19. Thank You !
Junwei Cao 19