% MANUSCRIPT NO. TUFFC-09469-2019.R1
% MANUSCRIPT TYPE: Papers
% TITLE: Filter optimization for real time digital processing of radiofrequency signals: application to oscillator metrology
% AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, Pierre-Yves; Friedt, Jean-Michel

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{\bf\Large
Rebuttal letter to the review \#2 of the manuscript entitled\\

``Filter optimization for real time digital processing of radiofrequency
signals: application to oscillator metrology''
}

by A. Hugeat \& al.
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% REVIEWERS' COMMENTS:
% Reviewer: 1
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{\bf
\noindent
Comments to the Author

 The Authors have implemented all Reviewers’ remarks except the one related to the criterion that, in my opinion, is the most important one. By considering ``the minimal rejection within the stopband, to which the sum of the absolute values within the passband is subtracted to avoid filters with excessive ripples, normalized to the bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)'' (please, find a way to state criterions more clearly), the Authors get filters with very different behaviors in pass band and, consequently, their comparison loses its meaning.

 In practice, the Authors use a good method based on a bad criterion, and this point weakens a lot the results they present.

 In phase noise metrology, the target is an uncertainty of 1 dB, even less. In this regard, I would personally use a maximum ripple in pass band of 1 dB (or less), while, in some cases, the filters presented in the Manuscript exceed 10 dB of ripple, which is definitely too much.

 The Authors seem to be reactive in redoing the measures and it does not seem a big problem for them to re-run the analysis with a better criterion. The article would gain a lot, because, in addition to the methodology, the reader could understand if it is actually better to put a cascade of small filters rather than a single large filter that is an interesting point.

 To help the Authors in finding a better criterion (``...finding a better criterion to avoid the ripples in the passband is challenging...''), in addition to the minimum rejection in stop band, I suggest to specify also the maximum ripple in pass band as it is done, for example, in fig. 4.10, pg. 146 of Crochierie R. E. and Rabiner L. R. (1983) ``Multirate Digital Signal Processing'', Prentice-Hall (see attach). This suggestion, in practice, specify the maximum allowed deviation from the transfer function modulus of an ideal filter: 1 in pass band and 0 in stop band. As a result, it should solve one of the Authors’ concerns: ``Selecting a strong constraint such as the sum of absolute values in the passband is too selective because it considers all frequency bins in the passband while the stopband criterion is limited to a single bin at which rejection is poorest…'' since both pass and stop bands are considered in the same way.

 I understand that the Manuscript is devoted to present a methodology (``In this article we focus on the methodology, so even if our criterion could be improved, our methodology still remains and works independently of rejection criterion.'').  Please, remember that a methodology is a solution to a class of problems and the example chosen to present the methodology plays a key role in showing to the reader if the method is valid or not. Here the example problem is represented by the synthesis of a decimation filter to be used in phase noise metrology. Many of the filters presented by the Authors in figures 9 and 10 as the output of this methodology are not suitable to be used in this context, since, for example, some of them have an attenuation as high as 50 dB in DC (!) that poses severe problems in interpreting the phase noise power spectral densities. What is the cause of this fail? The methodology or the criterion?

In my opinion, it is mandatory to correct the criterion and to re-run the analysis for checking if the methodology works properly or not.
In the end, I suggest to publish the Manuscript After Minor Revisions.
}

\noindent
Our answer: 

We are grateful for the opportunity provided by the reviewer to implement the constructive
comment provided in the review. Indeed we have thoroughly reviewed our investigation by implementing
a threshold criterion on the ripple level in the passband of the filters considered in the analysis.
By selecting a 1~dB maximum ripple level in the passband, the transfer functions indeed
closely match the targeted shape, as expected from an optimization analysis. The conclusion
of the initial paper are not changed but the numerical values of all tables as well as all figures
have been updated accordingly, as highlighted in red in the submitted manuscript. All
datasets have been re-computed with the updated criterion.

While the methodology provides the same results as proposed in the initial manuscript,
a significant fraction of the initial filter set is discarded by the hard threshold rejection
criterion, thus reducing the search space and hence reducing the search time. The conclusion
about the computation duration has been updated accordingly since on the one hand all possible
filter combinations could now be analyzed, and a maximum number of four cascaded filters meets
the optimum solution with no additional improvement when adding a fifth stage. Hence, the search
is now exhaustive for solving the considered problem. Such a statement has been added in the
conclusion

The DC component cancellation was an erroneous analysis of the filter transfer function when normalizing
the output power to the input power: removing the DC component erroneously led to this unexpected
drop of the filter transfer function close to 0~Hz. All charts have been updated accordingly
after correcting for this mistake.
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\hfill{\mbox{Yours sincerely, A. Hugeat}
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