Ajout de correction.

Arthur HUGEAT
1 parent ec91065abc
Showing 2 changed files with 104 additions and 24 deletions Side-by-side Diff
ifcs2018_journal.tex
ifcs2018_journal_reponse.tex
@@ -405,16 +405,18 @@
 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
  
-This model is non-linear and even non-quadratic, as $F$ does not have a known
-linear or quadratic expression. We introduce $p$ FIR configurations
-$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
-% r2.12
+{\color{red}
+This model is non-linear since we multiply some variable with another variable
+and it is even non-quadratic, as $F$ does not have a known
+linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
 This variable must be defined by the user, it represent the number of different
 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
-functions from GNU Octave).
-We define binary
+functions from GNU Octave). So $C_{ij}$ and $\pi_{ij}^C$ become constant and
+we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations
+thanks our rejection criterion. We also defined binary
 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
 and 0 otherwise. The new equations are as follows:
+}
  
 \begin{align}
 a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
  
@@ -427,13 +429,46 @@
 respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
  
-% r2.13
-This modified model is quadratic since we multiply two variables in the
-equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
-The Gurobi
+{\color{red}
+However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
+$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
+linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size
+we define $0 < \pi_i^- \leq 128$ which is the maximal data size that we can process.
+}
+Moreover the Gurobi
 (\url{www.gurobi.com}) optimization software is used to solve this quadratic
 model, and since Gurobi is able to linearize, the model is left as is. This model
 has $O(np)$ variables and $O(n)$ constraints.
+
+% This model is non-linear and even non-quadratic, as $F$ does not have a known
+% linear or quadratic expression. We introduce $p$ FIR configurations
+% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
+% % r2.12
+% This variable must be defined by the user, it represent the number of different
+% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
+% functions from GNU Octave).
+% We define binary
+% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
+% and 0 otherwise. The new equations are as follows:
+%
+% \begin{align}
+% a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
+% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
+% \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
+% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
+% \end{align}
+%
+% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
+% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
+% Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
+%
+% % r2.13
+% This modified model is quadratic since we multiply two variables in the
+% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
+% The Gurobi
+% (\url{www.gurobi.com}) optimization software is used to solve this quadratic
+% model, and since Gurobi is able to linearize, the model is left as is. This model
+% has $O(np)$ variables and $O(n)$ constraints.
  
 Two problems will be addressed using the workflow described in the next section: on the one
 hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
@@ -74,7 +74,7 @@
 %REVIEWERS' COMMENTS:
  
 \documentclass[a4paper]{article}
-\usepackage{fullpage,graphicx}
+\usepackage{fullpage,graphicx,amsmath}
 \begin{document}
 {\bf Reviewer: 1}
  
@@ -85,7 +85,7 @@
 On page 2,  "...allowing to save processing resource..." could be improved.       % r1.1
 }
  
-The sentence was split and now reads ``number of coefficients irrelevant: processing 
+The sentence was split and now reads ``number of coefficients irrelevant: processing
 resources are hence saved by shrinking the filter length.''
  
 {\bf
@@ -95,7 +95,7 @@
 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
  
-{\bf 
+{\bf
 On page 2, the whole paragraph "The first step of our approach is to model..."   % r1.3
 could be improved.
 }
@@ -103,7 +103,7 @@
 Indeed this paragraph has be written again and now reads as\\
 ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
 the filtering part of the signal processing chain, we have not included the PRN generator or the
-ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 
+ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
 The filtering can be done in two ways, either by considering a single monolithic FIR filter
 requiring many coefficients to reach the targeted noise rejection ratio, or by
 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
@@ -197,8 +197,8 @@
 bandwidth.
 }
  
-See above: the absolute value within the passband will reject filters with 
-excessive ripples, including excessive attenuation, within the passband. 
+See above: the absolute value within the passband will reject filters with
+excessive ripples, including excessive attenuation, within the passband.
  
 {\bf
 In addition, I suggest to address the following points:                           % r2.4
  
@@ -211,12 +211,11 @@
 have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
 the first output, an IIR will start providing outputs only 5 time steps after the initial
 input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
-response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue 
+response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
 in a openloop phase noise characterization instrument, the large
 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
 is not considered as an issue as would be in a closed loop system in which lag aims at being
-minimized to avoid oscillation conditions.
-''
+minimized to avoid oscillation conditions.''
  
 {\bf
 - Fig. 4: the Author should motivate in the text why it has been chosen           % r2.5
@@ -243,8 +242,8 @@
 coefficient resolution?
 }
  
-We have now stated in the beginning of the document that ``we have not included the PRN generator 
-or the ADC in the model: the input data size and rate are considered fixed and defined by the 
+We have now stated in the beginning of the document that ``we have not included the PRN generator
+or the ADC in the model: the input data size and rate are considered fixed and defined by the
 hardware.'' so indeed the input datastream resolution is considered as a given.
  
 {\bf
  
  
  
  
  
@@ -268,21 +267,67 @@
 - Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose       % r2.12 - fait
 these particular configurations? Are they chosen automatically?
 }
+See below: we have added a better description of $p$ during the transformation explanation.
+``we introduce $p$ FIR configurations.
+This variable must be defined by the user, it represent the number of different
+set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
+functions from GNU Octave)''
  
-JMF : repondre
-
 {\bf
 - Page 4, line 31: how does the delta function transform model from non-linear    % r2.13 - fait
 and non-quadratic to a quadratic?}
  
+The first model is non-quadratic but when we introduce the $p$ configurations,
+we can estimate the function $F$ by computing
+the rejection for each configuration, so the model become quadratic because we have
+some multiplication between variables. With the definition of $\delta_{ij}$ we can
+replace the multiplication between variables by multiplication with binary variable and
+this one can be linearise as follow:\\
+$y$ is a binary variable \\
+$x$ is a real variable bounded by $X^{max}$ \\
+\begin{equation*}
+  m = x \times y \implies
+  \left \{
+  \begin{split}
+    m & \geq 0 \\
+    m & \leq y \times X^{max} \\
+    m & \leq x \\
+    m & \geq x - (1 - y) \times X^{max} \\
+  \end{split}
+  \right .
+\end{equation*}
+Gurobi does the linearization so we don't explain this step to keep the model more
+simple. However, to improve the transformation explanation we have rewrote the
+paragraph ``This model is non-linear and even non-quadratic...''.
+
 JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
 je ne comprends pas comment ca repond a la question
  
+AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
+
+Le problème n'est pas linéaire car nous multiplions des variables
+entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
+des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
+quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
+quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
+$\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
+de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
+cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
+nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
+qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
+linéarisation pour nous.
+
+
 {\bf
 - Captions of figure and tables are too minimal.                                  % r2.14
+}
+We have change the captions of fig 10-16.
+
+{\bf
 - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c)   % r2.15 - fait
 of a single figure. Same for fig. 13-16.
 }
+We add two sub figure to group the fig.10-12 and fig. 13-16
  
 {\bf
 - Please increase the number of averages for the spectrum. Currently the noise    % r2.16 - fait
@@ -290,7 +335,7 @@
 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
 }
  
-Indeed averaging had been omitted during post-processing and figure generation: we 
+Indeed averaging had been omitted during post-processing and figure generation: we
 are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra
 now exhibit sub-dBpk-pl line thickness.
...	...	@@ -405,16 +405,18 @@
405	405	$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
406	406	Finally, equation~\ref{eq:init} gives the number of bits of the global input.
407	407
408		-This model is non-linear and even non-quadratic, as $F$ does not have a known
409		-linear or quadratic expression. We introduce $p$ FIR configurations
410		-$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
411		-% r2.12
	408	+{\color{red}
	409	+This model is non-linear since we multiply some variable with another variable
	410	+and it is even non-quadratic, as $F$ does not have a known
	411	+linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
412	412	This variable must be defined by the user, it represent the number of different
413	413	set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
414		-functions from GNU Octave).
415		-We define binary
	414	+functions from GNU Octave). So $C_{ij}$ and $\pi_{ij}^C$ become constant and
	415	+we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations
	416	+thanks our rejection criterion. We also defined binary
416	417	variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
417	418	and 0 otherwise. The new equations are as follows:
	419	+}
418	420
419	421	\begin{align}
420	422	a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
421	423
...	...	@@ -427,13 +429,46 @@
427	429	respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
428	430	Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
429	431
430		-% r2.13
431		-This modified model is quadratic since we multiply two variables in the
432		-equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
433		-The Gurobi
	432	+{\color{red}
	433	+However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
	434	+$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
	435	+linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size
	436	+we define $0 < \pi_i^- \leq 128$ which is the maximal data size that we can process.
	437	+}
	438	+Moreover the Gurobi
434	439	(\url{www.gurobi.com}) optimization software is used to solve this quadratic
435	440	model, and since Gurobi is able to linearize, the model is left as is. This model
436	441	has $O(np)$ variables and $O(n)$ constraints.
	442	+
	443	+% This model is non-linear and even non-quadratic, as $F$ does not have a known
	444	+% linear or quadratic expression. We introduce $p$ FIR configurations
	445	+% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
	446	+% % r2.12
	447	+% This variable must be defined by the user, it represent the number of different
	448	+% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
	449	+% functions from GNU Octave).
	450	+% We define binary
	451	+% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
	452	+% and 0 otherwise. The new equations are as follows:
	453	+%
	454	+% \begin{align}
	455	+% a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
	456	+% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
	457	+% \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
	458	+% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
	459	+% \end{align}
	460	+%
	461	+% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
	462	+% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
	463	+% Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
	464	+%
	465	+% % r2.13
	466	+% This modified model is quadratic since we multiply two variables in the
	467	+% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
	468	+% The Gurobi
	469	+% (\url{www.gurobi.com}) optimization software is used to solve this quadratic
	470	+% model, and since Gurobi is able to linearize, the model is left as is. This model
	471	+% has $O(np)$ variables and $O(n)$ constraints.
437	472
438	473	Two problems will be addressed using the workflow described in the next section: on the one
439	474	hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
...	...	@@ -74,7 +74,7 @@
74	74	%REVIEWERS' COMMENTS:
75	75
76	76	\documentclass[a4paper]{article}
77		-\usepackage{fullpage,graphicx}
	77	+\usepackage{fullpage,graphicx,amsmath}
78	78	\begin{document}
79	79	{\bf Reviewer: 1}
80	80
...	...	@@ -85,7 +85,7 @@
85	85	On page 2, "...allowing to save processing resource..." could be improved. % r1.1
86	86	}
87	87
88		-The sentence was split and now reads ``number of coefficients irrelevant: processing
	88	+The sentence was split and now reads ``number of coefficients irrelevant: processing
89	89	resources are hence saved by shrinking the filter length.''
90	90
91	91	{\bf
...	...	@@ -95,7 +95,7 @@
95	95	Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
96	96	Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97	97
98		-{\bf
	98	+{\bf
99	99	On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3
100	100	could be improved.
101	101	}
...	...	@@ -103,7 +103,7 @@
103	103	Indeed this paragraph has be written again and now reads as\\
104	104	``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
105	105	the filtering part of the signal processing chain, we have not included the PRN generator or the
106		-ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
	106	+ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
107	107	The filtering can be done in two ways, either by considering a single monolithic FIR filter
108	108	requiring many coefficients to reach the targeted noise rejection ratio, or by
109	109	cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
...	...	@@ -197,8 +197,8 @@
197	197	bandwidth.
198	198	}
199	199
200		-See above: the absolute value within the passband will reject filters with
201		-excessive ripples, including excessive attenuation, within the passband.
	200	+See above: the absolute value within the passband will reject filters with
	201	+excessive ripples, including excessive attenuation, within the passband.
202	202
203	203	{\bf
204	204	In addition, I suggest to address the following points: % r2.4
205	205
...	...	@@ -211,12 +211,11 @@
211	211	have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
212	212	the first output, an IIR will start providing outputs only 5 time steps after the initial
213	213	input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
214		-response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
	214	+response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
215	215	in a openloop phase noise characterization instrument, the large
216	216	numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
217	217	is not considered as an issue as would be in a closed loop system in which lag aims at being
218		-minimized to avoid oscillation conditions.
219		-''
	218	+minimized to avoid oscillation conditions.''
220	219
221	220	{\bf
222	221	- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5
...	...	@@ -243,8 +242,8 @@
243	242	coefficient resolution?
244	243	}
245	244
246		-We have now stated in the beginning of the document that ``we have not included the PRN generator
247		-or the ADC in the model: the input data size and rate are considered fixed and defined by the
	245	+We have now stated in the beginning of the document that ``we have not included the PRN generator
	246	+or the ADC in the model: the input data size and rate are considered fixed and defined by the
248	247	hardware.'' so indeed the input datastream resolution is considered as a given.
249	248
250	249	{\bf
251	250
252	251
253	252
254	253
255	254
...	...	@@ -268,21 +267,67 @@
268	267	- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait
269	268	these particular configurations? Are they chosen automatically?
270	269	}
	270	+See below: we have added a better description of $p$ during the transformation explanation.
	271	+``we introduce $p$ FIR configurations.
	272	+This variable must be defined by the user, it represent the number of different
	273	+set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
	274	+functions from GNU Octave)''
271	275
272		-JMF : repondre
273		-
274	276	{\bf
275	277	- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
276	278	and non-quadratic to a quadratic?}
277	279
	280	+The first model is non-quadratic but when we introduce the $p$ configurations,
	281	+we can estimate the function $F$ by computing
	282	+the rejection for each configuration, so the model become quadratic because we have
	283	+some multiplication between variables. With the definition of $\delta_{ij}$ we can
	284	+replace the multiplication between variables by multiplication with binary variable and
	285	+this one can be linearise as follow:\\
	286	+$y$ is a binary variable \\
	287	+$x$ is a real variable bounded by $X^{max}$ \\
	288	+\begin{equation*}
	289	+ m = x \times y \implies
	290	+ \left \{
	291	+ \begin{split}
	292	+ m & \geq 0 \\
	293	+ m & \leq y \times X^{max} \\
	294	+ m & \leq x \\
	295	+ m & \geq x - (1 - y) \times X^{max} \\
	296	+ \end{split}
	297	+ \right .
	298	+\end{equation*}
	299	+Gurobi does the linearization so we don't explain this step to keep the model more
	300	+simple. However, to improve the transformation explanation we have rewrote the
	301	+paragraph ``This model is non-linear and even non-quadratic...''.
	302	+
278	303	JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
279	304	je ne comprends pas comment ca repond a la question
280	305
	306	+AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
	307	+
	308	+Le problème n'est pas linéaire car nous multiplions des variables
	309	+entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
	310	+des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
	311	+quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
	312	+quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
	313	+$\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
	314	+de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
	315	+cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
	316	+nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
	317	+qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
	318	+linéarisation pour nous.
	319	+
	320	+
281	321	{\bf
282	322	- Captions of figure and tables are too minimal. % r2.14
	323	+}
	324	+We have change the captions of fig 10-16.
	325	+
	326	+{\bf
283	327	- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait
284	328	of a single figure. Same for fig. 13-16.
285	329	}
	330	+We add two sub figure to group the fig.10-12 and fig. 13-16
286	331
287	332	{\bf
288	333	- Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait
...	...	@@ -290,7 +335,7 @@
290	335	differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
291	336	}
292	337
293		-Indeed averaging had been omitted during post-processing and figure generation: we
	338	+Indeed averaging had been omitted during post-processing and figure generation: we
294	339	are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra
295	340	now exhibit sub-dBpk-pl line thickness.
296	341