Matched Filtering for Gravitational Wave Detection#

Matched filtering is the primary technique for detecting gravitational wave signals in noisy detector data. It correlates known template waveforms with the detector data to find signals with high signal-to-noise ratio (SNR).

Overview#

Matched filtering requires:

1. Template waveform (expected signal shape)
2. Conditioned detector data (preprocessed strain)
3. Power spectral density (PSD) of the noise
4. SNR calculation and peak finding

PyCBC supports both time-domain and frequency-domain approaches.

Time-Domain Waveforms#

Generate templates in time domain using get_td_waveform:

python

from pycbc.waveform import get_td_waveform
from pycbc.filter import matched_filter

# Generate time-domain waveform
hp, hc = get_td_waveform(
    approximant='IMRPhenomD',  # or 'SEOBNRv4_opt', 'TaylorT4'
    mass1=25,                  # Primary mass (solar masses)
    mass2=20,                  # Secondary mass (solar masses)
    delta_t=conditioned.delta_t,  # Must match data sampling
    f_lower=20                 # Lower frequency cutoff (Hz)
)

# Resize template to match data length
hp.resize(len(conditioned))

# Align template: cyclic shift so merger is at the start
template = hp.cyclic_time_shift(hp.start_time)

# Perform matched filtering
snr = matched_filter(
    template,
    conditioned,
    psd=psd,
    low_frequency_cutoff=20
)

# Crop edges corrupted by filtering
# Remove 4 seconds for PSD + 4 seconds for template length at start
# Remove 4 seconds at end for PSD
snr = snr.crop(4 + 4, 4)

# Find peak SNR
import numpy as np
peak_idx = np.argmax(abs(snr).numpy())
peak_snr = abs(snr[peak_idx])

Why Cyclic Shift?#

Waveforms from get_td_waveform have the merger at time zero. For matched filtering, we typically want the merger aligned at the start of the template. cyclic_time_shift rotates the waveform appropriately.

Frequency-Domain Waveforms#

Generate templates in frequency domain using get_fd_waveform:

python

from pycbc.waveform import get_fd_waveform
from pycbc.filter import matched_filter

# Calculate frequency resolution
delta_f = 1.0 / conditioned.duration

# Generate frequency-domain waveform
hp, hc = get_fd_waveform(
    approximant='IMRPhenomD',
    mass1=25,
    mass2=20,
    delta_f=delta_f,           # Frequency resolution (must match data)
    f_lower=20                 # Lower frequency cutoff (Hz)
)

# Resize template to match PSD length
hp.resize(len(psd))

# Perform matched filtering
snr = matched_filter(
    hp,
    conditioned,
    psd=psd,
    low_frequency_cutoff=20
)

# Find peak SNR
import numpy as np
peak_idx = np.argmax(abs(snr).numpy())
peak_snr = abs(snr[peak_idx])

Key Differences: Time vs Frequency Domain#

Time Domain ( get_td_waveform)#

• Pros: Works for all approximants, simpler to understand
• Cons: Can be slower for long waveforms
• Use when: Approximant doesn't support frequency domain, or you need time-domain manipulation

Frequency Domain ( get_fd_waveform)#

• Pros: Faster for matched filtering, directly in frequency space
• Cons: Not all approximants support it (e.g., SEOBNRv4_opt may not be available)
• Use when: Approximant supports it and you want computational efficiency

Approximants#

Common waveform approximants:

python

# Phenomenological models (fast, good accuracy)
'IMRPhenomD'      # Good for most binary black hole systems
'IMRPhenomPv2'    # More accurate for precessing systems

# Effective One-Body models (very accurate, slower)
'SEOBNRv4_opt'    # Optimized EOB model (time-domain only typically)

# Post-Newtonian models (approximate, fast)
'TaylorT4'        # Post-Newtonian expansion

Note: Some approximants may not be available in frequency domain. If get_fd_waveform fails, use get_td_waveform instead.

Matched Filter Parameters#

low_frequency_cutoff#

• Should match your high-pass filter cutoff (typically 15-20 Hz)
• Templates are only meaningful above this frequency
• Lower values = more signal, but more noise

Template Resizing#

• Time domain: hp.resize(len(conditioned)) - match data length
• Frequency domain: hp.resize(len(psd)) - match PSD length
• Critical for proper correlation

Crop Amounts#

After matched filtering, crop edges corrupted by:

• PSD filtering: 4 seconds at both ends
• Template length: Additional 4 seconds at start (for time-domain)
• Total: snr.crop(8, 4) for time-domain, snr.crop(4, 4) for frequency-domain

Best Practices#

1. Match sampling/frequency resolution: Template delta_t/ delta_f must match data
2. Resize templates correctly: Time-domain → data length, Frequency-domain → PSD length
3. Crop after filtering: Always crop edges corrupted by filtering
4. Use abs() for SNR: Matched filter returns complex SNR; use magnitude
5. Handle failures gracefully: Some approximants may not work for certain mass combinations

Common Issues#

Problem: "Approximant not available" error

• Solution: Try time-domain instead of frequency-domain, or use different approximant

Problem: Template size mismatch

• Solution: Ensure template is resized to match data length (TD) or PSD length (FD)

Problem: Poor SNR even with correct masses

• Solution: Check that PSD low_frequency_cutoff matches your high-pass filter, verify data conditioning

Problem: Edge artifacts in SNR time series

• Solution: Increase crop amounts or verify filtering pipeline order

Dependencies#

bash

pip install pycbc numpy

References#

• PyCBC Tutorial: Waveform & Matched Filter - Comprehensive tutorial demonstrating waveform generation, matched filtering in both time and frequency domains, and SNR analysis
• PyCBC Waveform Documentation
• PyCBC Filter Documentation