ℓ_p-STFT: A Robust Parameter Estimator of a Frequency Hopping Signal for Impulsive Noise

Round 1

Reviewer 1 Report

The lp-STFT algorithm is proposed in this paper for robust and accurate parameter estimation of frequency hoping signals. Parameters introduced in equation 1 need more explanation. Diagrams in figure 1 need measurement units. Diagrams 4 and 5 prove the robustness of the lp-STFT to the accuracy
of parameter estimation in strong impulsive noise. In my point of view the manuscript is suitable for publication.

Author Response

We would like to express our gratitude to the associate editor for handling our paper and the anonymous reviewers for their useful suggestions and criticism. The comments are well taken and the manuscript has been revised accordingly. Our responses to the comments are given as follows.

 

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Sparse representation for frequency estimation in non-stationnary structure is not new, it has been widely studied. However, the article provides a suitable framework for the application of IRLS, and allows the estimation of a single frequency polluted by complex distribution of impulsive noises.

I suggest the author to provide additional references using sparse-time frequency representation that may be suitable for this case; for instance :

Kowalski, M. (2009). Sparse regression using mixed norms. Applied and Computational Harmonic Analysis, 27(3), 303-324

Gardner, T. J., & Magnasco, M. O. (2006). Sparse time-frequency representations. Proceedings of the National Academy of Sciences, 103(16), 6094-6099.

Usually, IRLS converges quickly (sometimes after 2 iteration). How many iterations are required for this problem? Can the authors provide convergence curve as an example ?

Author Response

We would like to express our gratitude to the associate editor for handling our paper and the anonymous reviewers for their useful suggestions and criticism. The comments are well taken and the manuscript has been revised accordingly. Our responses to the comments are given as follows.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors have introduced an innovative time frequency estimator which is robust to the impulsive noise. According to my understanding, the major contributions are as follows: 

converting the conventional 1D STFT estimation processing into a 2D matrix form Based on the above mentioned 2D form, a LS variant is applied to solve the unknown signal, namely the time frequency spectrum. In the meantime, the authors choose the lp norm instead of the L2 norm in LS objective function. The motivation is to suppress the influence of outliers, i.e. impulsive noise. 

However, this manuscript has not convinced me in its current form due to the following reason:

there are many errors in the notations and equations, e.g.: Line 87, I cannot see that the phi in Equation (1) is a Gaussian distribution if alpha = 2.  Line 107 and 108, the matrix Y is defined as set of column vectors y_1, y_2, ..., y_N. However, incorporating the definition of y_N  the definition of matrix X, I cannot yield out the results in Equation (5). Line 111, in the exponent of elements in vector a_K(w_k) it should be "w_k", not the "w". Line 138, it is an estimator of x_n. Thus, the left side of the equation misses an subscription "n". Moreover, on the right side, a(w_k) is a K*1 vector, and x_n is K*1 vector. They cannot be directly multiplied without transposing. Line 142, on the right side of Equation (11), a(w_k)*x_k is a vector, and y_k is a scalar. A subtraction cannot be carried out between a scalar and vector. Line 147, the matrix W_k has identical entries on its diagonal line. Is it seriously wanted ? Line 169, the definition of DWSD is not clear. What is the meaning of "SP"? How are the time variables t_1 and t_2 correlated to the time variable t ? What does the "w" denote ?  I guess the w in equation 18 and 19 means window function. Then, how can I incorporate the window function into the algorithm in Table 1? In Line 177m the notation in equation (21) is not allowed in a serious scientific work. The authors have to very clearly define how the 1st order difference is carried out. The current expression is similar to a numerical 2nd derivation.... In Line 177, without an illustration with plot, it is hard to comprehend what the authors' idea here is.  Line 197, which variable in the previous section does the window length 251 and 51 correspond to?  The authors alleged that the lp norm is superior to l2 norm in dealing with outliers. I would like to see the comparison in the revision. The authors should verify them in their test framework. 

Author Response

We would like to express our gratitude to the associate editor for handling our paper and the anonymous reviewers for their useful suggestions and criticism. The comments are well taken and the manuscript has been revised accordingly. Our responses to the comments are given as follows.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

I appreciate the significant improvement made by the authors in this revision. However, I still have some points regarding to the section 3.1. 

 

In equation 5, I assume that the matrix A is some kind of inverse discrete Fourier transform matrix. But its dimension is M*K instead of K*K in usual case. It means in each column of matrix X, an M-Point IDFT is fulfilled with an input of K-point spectrum. If M does not equals to K, it is not a valid IDFT transform! Generally, if we want generate an M-Point IDFT result out of a K-point spectrum, zero-padding is required.  The transition from equation 11 to equation 12 is not plausible to me. The m-th element y_m in vector y_M equals to the product of row vector A_m (the m-th row of matrix A) multiplying with column vector x_K. In equation 12, a summation operator over k is missing before "a_M(w_k)x_k" in my opinion. 

Author Response

We would like to express our gratitude to the associate editor for handling our paper and the anonymous reviewers for their useful suggestions and criticism. The comments are well taken and the manuscript has been revised accordingly. Our responses to the comments are given as follows.

Author Response File: Author Response.pdf