First of all, it is important to discuss how to evaluate the TF representation for undersampled signals. For considered instant *n*, we are looking for available samples from set **Q**={*n*_{i},*i*=1,…, *K*} that are within the window:

$$ {n}_i\mid {n}_i\in \left[n-h/2,n+h/2\right) $$

Then, the STFT output can be calculated as:

$$ {\mathrm{STFT}}_h\left(n,\omega \right)=\mathrm{T}\left\{x\left({n}_i\right)\exp \left(- j\omega \left({n}_i-n\right)\right)\mid {n}_i\in \right[n-h/2,n+h/2\left)\right\}\kern0.84em h\in \mathbf{H}. $$

Operator T{} is selected here to average available modulated samples within the considered window. It can be treated as the L-filter STFT version. The STFTs are evaluated for various window lengths due to the different behavior of the corresponding IF estimators related to bias and variance for the applied window width. We intend to have a set of STFTs with diverse elements (wide windows - robust to the noise influence but can experience emphatic bias, narrow windows - small bias, and emphatic noise influence and all cases between) to be able to achieve reachable performance under various circumstances.

The IF estimation is performed by maximizing STFT (3). For a small ratio available to samples required by Nyquist criterion (*K*/*N*) and narrow windows, there can be no available samples in the neighborhood *n*_{i} ∈ [*n* − *h*/2, *n* + *h*/2). It is adopted that the IF estimate is zero in this region but in practice, it could be applied an interpolation using available IF estimates to further improve results. Several IF samples are selected in each trial of the RANSAC. The IF (or signal) is reconstructed using these samples. However, due to several reasons (missing samples, outliers in the IF estimate, etc.), the reconstructed IF (or signal) can be inaccurate. Therefore, the random samples selection should be repeated multiple times, and the most accurate among reconstructed signals/IF estimates is selected based on an appropriate criterion. A window width from set **H** and IF estimate samples \( {\hat{\omega}}_h(n) \) are randomly selected in each trial of the proposed RANSAC procedure. Polynomial parameters of the signal phase are obtained using polynomial regression/interpolation. The STFT-based IF estimation of higher-order PPSs exhibits bias even for signals sampled according to the Nyquist criterion [26]. Then, the O’Shea refinement is applied to improve the accuracy of the PPS coefficients estimation toward the Cramer-Rao lower bound. In the case of undersampled signals, refinement is even more important since it can improve relatively inaccurate estimates without outliers what cannot be done for IF estimate samples with outliers.

Now, we can summarize the proposed RANSAC algorithm in the subsequent subsections. Firstly, a pseudo-code of the proposed technique is given followed by a detailed explanation of the crucial algorithm steps.

### 3.1 Pseudocode

#### 3.1.1 RANSAC algorithm with the O’Shea refinement for parameter estimation of undersampled signals

**Input:** PPS signal *x*(*n*_{i}) *i*=1,…, *K* of order *M* where samples are given in increasing order *n*_{i}≤*n*_{i+1}.

*Step I Evaluation of the STFTs with various window widths and obtaining corresponding IF estimates.*

__For__ *h*∈**H**

$$ {\mathrm{STFT}}_h\left(n,\omega \right)=\mathrm{T}\left\{x\left({n}_i\right)\exp \left(- j\omega \left({n}_i-n\right)\right)\mid {n}_i\in \right[n-h/2,n+h/2\left)\right\}\kern0.84em h\in \mathbf{H} $$

$$ {\hat{\omega}}_h(n)=\arg \underset{\omega }{\max}\mid {\mathrm{STFT}}_h\left(n,\omega \right)\mid . $$

__EndFor__

*Step II Initialization of criterion functions:*

$$ {J}_{\mathrm{max}}=0\kern1em {J}_{\mathrm{max}}^{\prime }=0. $$

*Step III Trials with random window length and random IF samples* (Λ trials in total).

__For__ λ=1:Λ

*Step III.A Random selection*

Randomly select *h*_{η}∈**H** and IF samples \( {\hat{\omega}}_{\eta}\left({n}^j\right)\kern0.84em \eta \in \mathbf{H}\kern0.6em {n}^j\in \mathbf{Q}\kern0.36em ,\kern0.36em {n}^j<{n}^{j+1},j\in \left[1,M\right] \).

*Step III.B Rough parameter estimation*

Perform polynomial regression/interpolation of the IF estimation samples:

$$ \left\{{\hat{a}}_1^{\lambda },\dots, {\hat{a}}_M^{\lambda}\right\}={\left({\boldsymbol{\Gamma}}^T\boldsymbol{\Gamma} \right)}^{-1}{\boldsymbol{\Gamma}}^T{\left[{\hat{\omega}}_{\eta}\left({n}^1\right)\kern0.36em {\hat{\omega}}_{\eta}\left({n}^2\right)\kern0.36em \cdots {\hat{\omega}}_{\eta}\left({n}^M\right)\right]}^T $$

(4)

where **Γ** is *M*×*M* matrix with elements γ_{i,j}=*j*(*n*^{i})^{j-1} and *i*,*j*∈[1,*M*]. This is a rough estimate in the QML-algorithm terminology.

*Step III.C O'Shea refinement*

Dechirping with parameters available from the rough stage:

$$ {x}^{\prime}\left({n}_i\right)=x\left({n}_i\right)\exp \left(-j\sum \limits_{m=1}^M\frac{{\hat{a}}_m^{\lambda }{\left({n}_i\right)}^m}{m}\right). $$

Phase unwrapping:

$$ \hat{\phi}\left({n}_i\right)=\mathrm{unwrap}\left\{\mathrm{phase}\left\{{x}^{\prime}\left({n}_i\right)\right\}\right\}. $$

(5)

Polynomial regression of the signal phase:

$$ \left\{\delta {\hat{a}}_0^{\lambda },\delta {\hat{a}}_1^{\lambda },\dots, \delta {\hat{a}}_M^{\lambda}\right\}={\left({\boldsymbol{\Xi}}^T\boldsymbol{\Xi} \right)}^{-1}\boldsymbol{\Xi} {\left[\hat{\phi}\left({n}_1\right)\kern0.24em \hat{\phi}\left({n}_2\right)\kern0.36em \cdots \hat{\phi}\left({n}_K\right)\right]}^T, $$

(6)

where matrix **Ξ** has (*M*+1)×*K* elements given as \( {\xi}_{i,j}={n}_i^j,\kern0.36em i\in \left[1,K\right],\kern0.36em j\in \left[0,M\right]. \)

*Step III.D Fine estimate is obtained as*:

$$ {{\hat{a}}}_m^{{\prime}\lambda }={\hat{a}}_m^{\lambda }+\delta {\hat{a}}_m^{\lambda}\kern0.6em m=1,2,\dots, M $$

$$ {{\hat{a}}}_0^{{\prime}\lambda }=\delta {\hat{a}}_0^{\lambda }. $$

*Step III.E Evaluation of criterion function* (for estimates with and without refinement) [31, 32]:

$$ J\left(\lambda \right)=\left|\sum \limits_{i=1}^{N_c}x\left({n}_i\right)\exp \left(-j\sum \limits_{m=1}^M\frac{{\hat{a}}_m^{\lambda }{\left({n}_i\right)}^m}{m}\right)\right| $$

$$ {J}^{\prime}\left(\lambda \right)=\left|\sum \limits_{i=1}^{N_c}x\left({n}_i\right)\exp \left(-j\sum \limits_{m=1}^M\frac{{{\hat{a}}}_m^{{\prime}\lambda }{\left({n}_i\right)}^m}{m}\right)\right|. $$

__If__ *J*(λ)>*J*, set *J*=*J*(λ), and update estimates as \( \left\{{\hat{a}}_1,\dots, {\hat{a}}_M\right\}=\left\{{\hat{a}}_1^{\lambda },\dots, {\hat{a}}_M^{\lambda}\right\} \). __EndIf__

__If__ *J*'(λ)>*J*'set *J*'=*J*'(λ) and update \( \left\{{\hat{a}}_0^{\prime },{\hat{a}}_1^{\prime },\dots, {\hat{a}}_M^{\prime}\right\}=\left\{{{\hat{a}}}_0^{{\prime}\lambda },{{\hat{a}}}_1^{{\prime}\lambda },\dots, {{\hat{a}}}_M^{{\prime}\lambda}\right\} \). __EndIf__*.*

__EndFor__

**Output:** The estimated set of phase coefficients \( \left\{{\hat{a}}_0^{\prime },{\hat{a}}_1^{\prime },\dots, {\hat{a}}_M^{\prime}\right\} \) or \( \left\{{\hat{a}}_1,\dots, {\hat{a}}_M\right\} \) can be used to reconstruct signal \( x(n)=\hat{A}\exp \left(-j\sum \limits_{m=1}^M\frac{{\hat{a}}_m{n}^m}{m}\right) \) where the amplitude is estimated as

$$ \hat{A}=\mathrm{mean}\left\{\left|x\left({n}_i\right)\exp \left(-j\sum \limits_{m=1}^M\frac{{\hat{a}}_m{\left({n}_i\right)}^m}{m}\right)\right|\right\}. $$

### 3.2 Comments on the algorithm

Due to the large percentage of missing samples in recordings, it can be expected a significant percentage of outliers in the IF estimate. Therefore, the RANSAC procedure is employed to select, among numerous wrong estimates, accurate IF estimate samples. Parameters of the signal phase are estimated allowing accurate signal reconstruction [26, 27]. The STFT is used in the QML-RANSAC algorithm for the IF estimation in the rough algorithm stage. Here, due to missing samples evaluation is performed with the modified (robust) STFT form in *Step I*.

Two criterion functions are initialized in *Step II*. The first is evaluated without the O’Shea refinement and used only for reference to demonstrate the refinement stage impact on the algorithm accuracy. Both criterion functions evaluated in *Step III.E* correspond to the ML optimization. Polynomial regression is applied in rough (4) and fine (6) estimation stages. However, it should be noted that in the rough stage it is performed only on *M* randomly selected samples of the IF estimate, while in the fine stage it is performed on all available *K* signal phase samples. Weight coefficients in matrices **Γ** and **Ξ** are different since evaluations in (4) and (6) are performed for different functions (IF and phase).

There are also some issues in the O’Shea refinement algorithm worth to be discussed. They are related to two important steps in the refinement procedure: filtering and phase unwrapping. The signal filtering is commonly performed in the refinement stage of the algorithm in both the QML and QML-RANSAC to reduce noise-like effects from the estimate and from this point of view it could be useful:

$$ {\hat{x}}^{\prime}\left({n}_i\right)=\frac{1}{2l+1}\sum \limits_{k=-l}^l{x}^{\prime}\left({n}_{i+k}\right). $$

(7)

However, it can introduce additional errors distorting dechirped signal *x*^{′}(*n*_{i}) used in the further procedure. These errors can be emphatic when neighbor signals are far from each other. In our experiments, filtering is not applied (set *l*=0 in (7)) but it remains for further research on how it can be utilized especially for signals corrupted by a significant amount of additive noise. A similar problem holds with the phase unwrapping since it can be inaccurate for a low number of available samples (low *K*/*N* ratio). However, the phase unwrapping is applied in our experiments as in [26, 28] with good results but some enhancement of this procedure is probably possible remaining for further research.