Detection and recovery of weak signals - Lim, May Tan (2000)
An efficient noise dithering procedure is demonstrated for measuring weak doublet spectra with a Fourier transform interferometer where the weak interferograms are sampled by a 1-bit analog-to-digital converter. In the absence of noise, no information is obtained regarding the doublet spectrum because the modulation (AC) term s(x) of the interferogram is detectable which happens when ls(x) is less than the instrumental detection limit Β for all path difference x values. Extensive numerical experiments are carried out to test the recovery of the following interferogram which represents the doublet spectrum: s(x) = (s₀ / 2) exp (-π²x² /β) [cos (2π f₁x) + cos (2π f₂x)], for different s₀ linewidth factor β and <f> = (f₁ + f₂) /2. Even for short observation time (sampling periods) of s(x), the resonant frequencies can be located at high accuracy over a wide range of <f> and β values. Signal-to-noise ratios greater than 50 are also obtained for the power spectra. Rapid and accurate recovery of the lost high-frequency components in the undersampled representation of a bandlimited signal s(x) is demonstrated using the simplex projection method (SPM). the spectral extrapolation technique is effective if: 1) Fourier spectrum S(f) of s(x) contains features that are exhibited regularly within the signal bandwidth, and 2) Fourier transform {S(m)} of the undersampled representation of s(x) contains sufficient information about the said regularities. The SPM is utilized to determine the various features contained in {S(m)} and to establish their possible pattern of appearance. The performance of the recovery procedure is tested as a function of the sampling rate. Two test signals with distinctly different Fourier spectrum profiles are considered: 1) Interferogram of a spectral doublet, and 2) Four-point object. In both cases, the bandwidth of s(x) is known a priori and used to determine the number of unknown to be solved. For an undersampled interferogram that contains only 54% of the energy of the spectral doublet, 42 unknown components have been calculated to decrease the normalized mean-square error of the interpolated signal by 75% relative to the undersampled data. The extrapolation technique is shown to be robust to the presence of additive noise in S(f). The SPM is also demonstrated on the Raman spectrum of CCl₄