A New Wavelet Denoising Method for Experimental Time-Domain Signals: Pulsed Dipolar Electron Spin Resonance

Abstract

We adapt a new wavelet-transform-based method of denoising experimental signals to pulse-dipolar electron-spin resonance spectroscopy (PDS). We show that signal averaging times of the time-domain signals can be reduced by as much as 2 orders of magnitude, while retaining the fidelity of the underlying signals, in comparison with noiseless reference signals. We have achieved excellent signal recovery when the initial noisy signal has an SNR ≳ 3. This approach is robust and is expected to be applicable to other time-domain spectroscopies. In PDS, these time-domain signals representing the dipolar interaction between two electron spin labels are converted into their distance distribution functions P(r), usually by regularization methods such as Tikhonov regularization. The significant improvements achieved by using denoised signals for this regularization are described. We show that they yield P(r)'s with more accurate detail and yield clearer separations of respective distances, which is especially important when the P(r)'s are complex. Also, longer distance P(r)'s, requiring longer dipolar evolution times, become accessible after denoising. In comparison to standard wavelet denoising approaches, it is clearly shown that the new method (WavPDS) is superior.

Figure 1

Block diagrams for determining distance distribution P(r) from the dipolar signal by Tikhonov regularization using L-curve criterion: (A) standard approach; (B) new approach after WavPDS denoising.

Figure 2

(A) Block diagram of a standard wavelet denoising method. (B) Block diagram of the new wavelet denoising method. (C) Block diagram of WavPDS. S(t) and S′(t) are the noisy and denoised signal, respectively; D_j and D′_j are the noisy and denoised Detail components at the jth Decomposition level, respectively; A_j and A′_j are the noisy and denoised Approximation components at the jth Decomposition level, respectively; DWT and IDWT represents discrete wavelet transform and inverse wavelet transform, respectively. (Parts A and B of Figure 2 are reprinted with permission from ref . © 2016 IEEE.)

Figure 3

Comparison of distance distributions with different λ values for Tikhonov regularization of the model signal. The model signal was generated from a Gaussian distribution centered at 5 nm with a standard deviation of 0.3 nm. (Note: MEM was used to constrain P(r) ≥ 0.)

Figure 4

Model data, unimodal distribution: blue, model signal (reference); red, noisy signal; black, WavPDS denoised signal. The model signal was generated from a Gaussian distribution centered at 5 nm with a standard deviation of 0.3 nm. White Gaussian noise was added to generate the noisy signals at SNRs 30, 10, and 3. (A) Comparison of noisy and denoised signals. (B) Comparison of the reference signal with the denoised signal obtained by WavPDS. (C) Distance distributions from noisy, denoised, and reference signals.

Figure 5

Model data, bimodal distribution: blue, model signal (reference); red, noisy signal; black, WavPDS denoised signal. The model signal was generated from two Gaussian distributions centered at 4 and 5 nm and each with a standard deviation of 0.3 nm. White Gaussian noise was added to generate the noisy signals at SNRs 30, 10, and 3. (A) Comparison of noisy and denoised signals. (B) Comparison of the reference signal with the denoised signal obtained by WavPDS. (C) Distance distributions from noisy, denoised, and reference signals.

Figure 6

Model data, results of longer evolution time at higher noise (low SNR) compared to shorter evolution time at lower noise (high SNR): blue, model signal (reference); red, noisy signal; black, WavPDS denoised signal. (A) Model signal at evolution time 8 µs generated from bimodal distance distribution. (B) Bimodal distance distribution generated from two Gaussian distributions centered at 5.3 and 6 nm, each with standard deviation 0.3 nm, and peak heights of 0.8 and 1, respectively. (C) Comparison of noisy, denoised, and Model signals at 3 µs evolution time for the bimodal distance distribution. (D) Distance distributions from noisy, denoised, and model signals at 3 µs evolution time. (E) Comparison of noisy, denoised, and model signals at 8 µs evolution time for the bimodal distance distribution. (F) Distance distributions from noisy, denoised, and model signals at 8 µs evolution time.

Figure 7

Experimental data, unimodal distribution: blue, reference signal; red, noisy signal; black, WavPDS denoised signal. The experimental signal was generated from T4 Lysozyme spin-labeled at mutant 44C/135C with 63 µM concentration at acquisition times 952, 112, and 14 min (section 2.E). (A) Comparison of noisy and denoised signals. (B) Comparison of reference signal with denoised signals after applying WavPDS. (C) Distance distributions from noisy, denoised, and reference signals. The denoised signal at 952 min was used as the reference.

Figure 8

Experimental data, bimodal distribution: blue, reference signal; red, noisy signal; black, WavPDS denoised signal. The experimental signal was generated from the T4 lysozyme spin-labeled admixture of mutants 8C/44C and 44C/135C at concentrations of 44 and 47 µM, respectively, at acquisition times 360, 48, and 8 min. (A) Comparison of noisy and denoised signals. (B) Comparison of denoised signals after WavPDS with reference. (C) Distance distributions from noisy, denoised, and reference signals. The denoised signal at 360 min was used as the Reference.