Signal-Alignment

Algorithms to align 1D signals by deriving the offset using different cross-correlation methods.

Alignment via chi-squared minimization

The first alignment technique is labeled "align_spectra". This function shifts a signal by resampling it via linear interpolation, the edges of the signal are ignored because the algorithm focuses on aligning two signals within a region of interest. The shifted signal is then compared to a "template" signal where a chi-squared minimization occurs to derive the optimal offset between the two. Each signal (template and shifted) is normalized (mean=1) prior to shifting so that the algorithm focuses mainly on comparing the shape of the data. One caveat about this techqnique, it is prone to finding local minimum unless priors are heavily constrained since it uses a gradient-based optimization method.

Alignment via Fourier space

The second alignment routine is called "phase_spectra" and uses a phase correlation in fourier space (https://en.wikipedia.org/wiki/Phase_correlation). It applies the same technique as that link except in 1D instead of 2D. After independent testing this algorithm behaves in a similar manner as the chi-squared minimization except at a lower precision. The only caveat about the phase correlation is if you cross-correlate your data at the native resolution you can only achieve integer precision. To achieve sub-pixel/ sub-integer precision, I've added an option to upscale the data prior to the cross-correlation. By upscaling the data N times you can achieve a precision of 1/N as far as the shift value goes. That means if N=100, then the smallest shift value you can get is 0.01.

Here is an image of the alignment algorithm in use for real data. I was interested in comparing the shift between two signals that happen to be astrophysical spectra at two different wavelength regions.

Example

import numpy as np
from scipy import signal
from scipy.ndimage import shift
import matplotlib.pyplot as plt

from signal_alignment import phase_align, chisqr_align

if __name__ == "__main__":

    NPTS = 100
    SHIFTVAL = 4
    NOISE = 1e-2 # can perturb offset retrieval from true
    print('true signal offset:',SHIFTVAL)

    # generate some noisy data and simulate a shift
    og = signal.gaussian(NPTS, std=4) + np.random.normal(1,NOISE,NPTS)
    shifted = shift( signal.gaussian(NPTS, std=4) ,SHIFTVAL) + np.random.normal(1,NOISE,NPTS)

    # align the shifted spectrum back to the real
    s = phase_align(og, shifted, [10,90])
    print('phase shift value to align is',s)

    # chi squared alignment at native resolution
    s = chisqr_align(og, shifted, [10,90], init=-3.5,bound=2)
    print('chi square alignment',s)

    # make some diagnostic plots
    plt.plot(og,label='original data')
    plt.plot(shifted,label='shifted data')
    plt.plot(shift(shifted,s,mode='nearest'),ls='--',label='aligned data') 
    plt.legend(loc='best')
    plt.show()

Use cases

These algorithms were created for the scientific manuscript: Ground-Based Spectroscopy of XO-2b using a Systematic Wavelength Calibration to correct for time-dependent wavelength shifts as a result of atmospheric differential refraction.
Additionally, these techniques are used as part of the on-chip guiding system for the NESSI instrument at Palomar Observatory (See SPIE proceeding: The New NESSI – Refurbishment of an NIR MOS for characterizing exoplanets using the Hale Telescope)

Citation

If you use these algorithms please cite the article below

Pearson K. A., et al., 2019, AJ, 157, 21

Here is an example bibtex

@article{Pearson2019,
  author={Kyle A. Pearson and Caitlin A. Griffith and Robert T. Zellem and Tommi T. Koskinen and Gael M. Roudier},
  title={Ground-based Spectroscopy of the Exoplanet XO-2b Using a Systematic Wavelength Calibration},
  journal={The Astronomical Journal},
  volume={157},
  number={1},
  pages={21},
  url={http://stacks.iop.org/1538-3881/157/i=1/a=21},
  year={2019}
}