# Statistical Properties of Energy Detection for Spectrum Sensing by Using Estimated Noise Variance

*Reviewer 1:*Anonymous

*Reviewer 2:*Anonymous

*Reviewer 3:*Anonymous

**Round 1**

*Reviewer 1 Report*

The paper presents thorough analytical results about the energy detection. The results are convincing. Maybe some practical use cases could make the paper more interesting.

Unfortunately, there are several such typos and mistakes in the manuscript which I can not interpret; especially I have problem with the references.

There are some references which contain ? mark, see e.g. line 20, 35, 72, 73, 212.

Again I don't know why is Theorems 4 and 5 mentioned twice in lines 77,79, and also in Conclusions.

There are such references which I can not interpret, for example references like "gamma+distribution", "lambda-fa-tilde", "lambda-d-tilde" and I could list them over several lines. Are they notation for variables or references to equations? OK, I suppose what they are referenced to, but not obvious and not a usual form.

The structure of the statement of theorems is not clear. For example in page 7 there is "Proof of Theorem 4" But what is the statement? For me, a theorem has two parts: the statement (some short sentences with equations) and the proof. Here the statement is missing. Maybe there are some statements scattered in the above text, but the theorem should be a clear and short statement. This is problem not only in the case of Theorem 4 but in the case of other Theorems and Lemmas.

By the way: why are there two proofs for theorem 4, 5 and Lemma 5. What are Theorems 1-2-3? If you have theorem 4 and 5, you should have theorem 1-2-3. If there are, please note them. If there is an Example 3, I think there should be Example 1-2, but I can not find such notations. What is remark 4?

The above remarks make me feel like I would read a part of a thesis and I missed the other parts of it. Did any problems happen after the paper was uploaded? The new version of the paper must be revised very carefully.

The figures don't have captions, and are very small. The graphs also should have clear notations of axes.

There are some sentences which are grammatically not correct:

"Thus, a question arises here:how far it is between Pf a and EˆP f a."

"By stirling" --> "By Stirling"

I thing the word "discovery" should be replaced by other word.

"Again, from another angle"

"In view of the facts"

"from average point of view"

Summing up, I thing there are several typos and editing problems. These errors must be corrected.

*Author Response*

The paper presents thorough analytical results about the energy detection. The results are convincing. Maybe some practical use cases could make the paper more interesting.

Unfortunately, there are several such typos and mistakes in the manuscript

which I can not interpret; especially I have problem with the references. There

are some references which contain ? mark, see e.g. line 20, 35, 72, 73, 212.

R: The references have been now correctly cited. Some typos and mistakes,

caused by some document format (tex source le format), have also been cor-

rected.

Again I don't know why is Theorems 4 and 5 mentioned twice in lines 77,79,

and also in Conclusions.**Response**: *The theorem number mentioned in line 77 should be 1 and 3, and in line 79be 5 and 6. These typos are caused by tex le format error, which have nowbeen corrected.*

There are such references which I can not interpret, for example references

like "gamma+distribution", "lambda-fa-tilde", "lambda-d-tilde" and I could

list them over several lines. Are they notation for variables or references to

equations? OK, I suppose what they are referenced to, but not obvious and not

a usual form.**Response**: *These are labels of equation number, which are caused by missing denitionin the beginning of the tex le. These equation numbers have now been correctlycited in this revised version.*

The structure of the statement of theorems is not clear. For example in page 7

there is "Proof of Theorem 4" But what is the statement? For me, a theorem has

two parts: the statement (some short sentences with equations) and the proof.

Here the statement is missing. Maybe there are some statements scattered in

the above text, but the theorem should be a clear and short statement. This is

problem not only in the case of Theorem 4 but in the case of other Theorems

and Lemmas.

**Response**:

*The format of theorem statement has now been corrected. The theorem block*

is missing in the former version due to missing denition in the beginning of

the tex le.

is missing in the former version due to missing denition in the beginning of

the tex le.

By the way: why are there two proofs for theorem 4, 5 and Lemma 5. What are

Theorems 1-2-3? If you have theorem 4 and 5, you should have theorem 1-2-3.

If there are, please note them. If there is an Example 3, I think there should be

Example 1-2, but I can not nd such notations. What is remark 4?

**Response**:

*The number of theorems and examples has now been corrected.*

The above remarks make me feel like I would read a part of a thesis and I missed

the other parts of it. Did any problems happen after the paper was uploaded?

The new version of the paper must be revised very carefully.

**Response**:

*The problems mentioned above are all caused by the uncorrected setting of*

blocks of theorem, example, lemma, etc, in the beginning of the tex le.

blocks of theorem, example, lemma, etc, in the beginning of the tex le.

The gures don't have captions, and are very small. The graphs also should

have clear notations of axes.

**Response**:

*The gures are now bigger. The axes have now been mentioned clearly in the*

caption descriptions.

caption descriptions.

There are some sentences which are grammatically not correct:

"Thus, a question arises here: how far it is between Pf a and E?P f a."

**Response**:

*The sentence has been changed to "Thus, a question arises: what is the*

dierence between Pfa and E ^ Pfa."

dierence between Pfa and E ^ Pfa."

"By stirling" --> "By Stirling"

**Response**: *Corrected*.

I thing the word "discovery" should be replaced by other word.

"Again, from another angle"

"In view of the facts"

"from average point of view"**Response**: *The word "discovery" is replaced by "observation". And the following 3 shortphrases are respectively replaced by "Again", "By the facts", "In an averagesense".*

Summing up, I thing there are several typos and editing problems. These errors

must be corrected.

**Response**:

*In this revised version, some typos and mistakes (e.g., theorems and lemmas*

number and etc), caused by some document format (tex source le format), have

been corrected.

number and etc), caused by some document format (tex source le format), have

been corrected.

*Reviewer 2 Report*

It is not so clear what is the source of x(t) in Figure 1, and what is the knowledge implied for the rest of the model.

There are numerous typos regarding the use of the template. For example, many citations are missing (a question mark is instead in their position). The same stands for references to theorems on page 3. In many lines, there are references to values and formulas that do not follow the template and are not understood. Please follow the template for the figures and the captions of them. Also, finish appropriately any proof in the manuscript; check guidelines in the journal template. Provide more discussion regarding the figures.

It is not understood why you provide so many proves for the same Theorem, at least as denoted in the manuscript. Most of the paper is a collection of proves.

What is the actual impact for the readers of the journal? How can this be exploited actually in the industry? What are the requirements for its efficiency? These are questions that need to be answered for the readers.

*Author Response*

It is not so clear what is the source of x(t) in Figure 1, and what is the knowledge

implied for the rest of the model.**Response**: *The signal x(t) is the signal to be detected. If the noise variance and theSNR(signal-noise-ratio) are all known, it is well-solved detection issue. In thispaper, we consider the case that the noise variance is estimated by N noiseobservation and the SNR is provided.*

There are numerous typos regarding the use of the template. For example,

many citations are missing (a question mark is instead in their position). The

same stands for references to theorems on page 3. In many lines, there are

references to values and formulas that do not follow the template and are not

understood. Please follow the template for the gures and the captions of them.

Also, nish appropriately any proof in the manuscript; check guidelines in the

journal template. Provide more discussion regarding the gures.

**Response**:

*Some typos and mistakes (e.g., references, equation numbers, theorems, g-*

ures and lemmas number and etc), caused by some document format (tex source

le format) have been corrected in this revised version.

ures and lemmas number and etc), caused by some document format (tex source

le format) have been corrected in this revised version.

It is not understood why you provide so many proves for the same Theorem, at

least as denoted in the manuscript. Most of the paper is a collection of proves.

**Response**:

*The uncorrected denition for the block of theorem has now been corrected.*

The number of theorem has now been corrected.

The number of theorem has now been corrected.

What is the actual impact for the readers of the journal? How can this be

exploited actually in the industry? What are the requirements for its eciency?

These are questions that need to be answered for the readers.**Response**: *Let us show a simple example to make sense of the theoretical results. LetM = 60, N = 30, if we want to set a threshold by the standard energy detectionmethod (replacing the noise variance by its estimation) to guarantee E ^ Pfa =0:05, the preassigned false alarm rate to set the threshold should be cautious andit is found by Theorem 1 that it should be approximately P0fa = 0:00033955,which is much smaller than the desired rate 0:05. This is partially caused bythe estimated noise variance ^2 = 1NPNk=1 v2k. However, if the sample numberN increases to 100, the preassigned false alarm rate could be P0fa = 0:0129. Asthe sample number N increases, the preassigned rate approaches to 0:05 itself.We thus believe that the theoretical results may have real application in energydetection when using estimated noise variance in some circumstances.*

*Reviewer 3 Report*

Energy detection is a popular technique that has been considered by a lot of researchers. The idea of the paper is not that interesting as even if we select the threshold accurately the energy detection performance is low compared with cyclostationary features detection and matched filter. The authors have to write a very solid argument to convince the reviewers of their idea. In addition, how you paper is different from [1], in which the authors measured the level of noise and accordingly set the threshold. They showed that the performance of energy detection can be improved slightly.

[1] Arjoune et al. Spectrum sensing: enhanced energy detection technique based on noise measurement. CCWC2018. DOI: 10.1109/CCWC.2018.8301619

The list of references is short and it does not reflect that the authors did a broad review of litterature. Please consider reviewing recent references and select high quality references to build upon your research.

The authors should write a very strong introduction in order to improve their paper.

*Author Response*

Energy detection is a popular technique that has been considered by a lot of

researchers. The idea of the paper is not that interesting as even if we select

the threshold accurately the energy detection performance is low compared with

cyclostationary features detection and matched lter. The authors have to write

a very solid argument to convince the reviewers of their idea. In addition, how

you paper is dierent from [1], in which the authors measured the level of noise

and accordingly set the threshold. They showed that the performance of energy

detection can be improved slightly.

[1] Arjoune et al. Spectrum sensing: enhanced energy detection technique based

on noise measurement. CCWC2018. DOI: 10.1109/CCWC.2018.8301619**Response**: *Though the mentioned paper consider the using the estimated noise varianceto construct the threshold for detection, the construction is directly by replacingthe exact noise variance by estimated noise variance in the standard thresholdformula. The eect caused by this replacement has not been analyzed. In thispaper we analyzed the eect caused by this replacement (see Theorem 1 and 3 forthe expectation of E[ ^ Pfa] and E[^~ Pfa]) and based on these two theorems, much accurate thresholds are derived for detection.*

The list of references is short and it does not re

ect that the authors did a broad

review of litterature. Please consider reviewing recent references and select high

quality references to build upon your research.

**Response**:

*Two more papers related to the topic have now been cited.*

The authors should write a very strong introduction in order to improve their paper.

**Response**:

*The paragraph 2 (model description) and 5 (a simple demonstration example*

of new threshold is presented) of the Introduction section have been rewritten

to be more clear and to stress the key contribution of the established theoretical

results. Fig. 1 has also been re-generated to show the whole framework of the

detection process. It seems to us the motivation of this paper has been well-

presented.

of new threshold is presented) of the Introduction section have been rewritten

to be more clear and to stress the key contribution of the established theoretical

results. Fig. 1 has also been re-generated to show the whole framework of the

detection process. It seems to us the motivation of this paper has been well-

presented.

**Round 2**

*Reviewer 2 Report*

The authors have managed to address the review comments. They should consider however connect this work with an actual application, in terms of the problem, topic, and the required means to offer a competitive solution.

*Author Response*

Response: This paper is a theoretical analysis rather than a real technical application.

The paragraph 3 of page 2 has been re-organized to stress the encountered problem

for detection using estimated noise. A simple example is also presented to

show the potential application in real spectrum sensing. Let M = 60, N = 30,

if we want to set a threshold by the standard energy detection method (replacing

the noise variance by its estimation) to guarantee E ˆ Pfa = 0:05, the preassigned

false alarm rate to set the threshold should be cautious and it is found by Theorem

1 that it should be approximately P0

fa = 0:00033955, which is much smaller

than the desired rate 0:05. However, if the sample number N increases to 100,

the preassigned false alarm rate could be P0

fa = 0:0129. As the sample number

N increases, the preassigned rate approaches to 0:05 itself.

*Reviewer 3 Report*

The paper addresses a timely subject which is cognitive radio. Specifically, spectrum sensing issues related to energy detector whose performance is dependent on the threshold selection. The authors propose a method to obtain more suitable CFAR thresholds for energy detection. There are still some issues that the authors have to address before this paper can be accepted.

First, the introduction is not well written. The authors reviewed only 15 research paper of which only a few are solid work. The authors needs to improve the literature review and the selection of their citation. Only three paper are from 2018 and 2019, the rest is 10 years old. There are some important recent work that the authors need to consider. Here are some suggestions for the authors.

Arjoune, Y.; Kaabouch, N. A Comprehensive Survey on Spectrum Sensing in Cognitive Radio Networks: Recent Advances, New Challenges, and Future Research Directions. Sensors 2019, 19, 126.

Ali, W. Hamouda, "Advances on Spectrum Sensing for Cognitive Radio Networks: Theory and Applications", IEEE Commun. Surv. Tutorials, pp. 1-1, 2016.

D. M. M. Plata, A. G. A. Reátiga, "Evaluation of energy detection for spectrum sensing based on the dynamic selection of detection-threshold", Procedia Eng., vol. 35, pp. 135-143, 2012.

In addition to the citation problem of the paper, the way the introduction is written is another issue that weaken the paper. Please avoid using equations in the introduction, your mathematical model should be described in the methodology section.

How the work presented in this paper related to the following papers:

Arjoune, et al. Spectrum sensing: Enhanced energy detection technique based on noise measurement, 2018 IEEE 8th Annual Computing and Communication Workshop and Conference (CCWC), DOI: 10.1109/CCWC.2018.8301619

D. R. Joshi, D. C. Popescu, O. A. Dobre, "Adaptive spectrum sensing with noise variance estimation for dynamic cognitive radio systems", Conference on Information Sciences and Systems (CISS), pp. 1-5, 2010.

J. Wu, T. Luo, G. Yue, "An Energy Detection Algorithm Based on Double-Threshold in Cognitive Radio Systems", International Conference on Information Science and Engineering, pp. 493-496, 2009.

In Figure 5, why the blue line does not have a legend?

References should be formatted according to the MDPI authors’ guideline.

Figure 1 should be placed in the methodology section

The authors can introduce a new section and call it notation in which place the

Why example 1 and Example 2 are written in italic?

Line 19, why the format of the citation is different from the rest?

*Author Response*

...Here are some suggestions for the authors.

Arjoune, Y.; Kaabouch, N. A Comprehensive Survey on Spectrum Sensing in

Cognitive Radio Networks: Recent Advances, New Challenges, and Future Research

Directions. Sensors 2019, 19, 126.

Ali, W. Hamouda, ”Advances on Spectrum Sensing for Cognitive Radio Networks:

Theory and Applications”, IEEE Commun. Surv. Tutorials, pp. 1-1,

2016.

D. M. M. Plata, A. G. A. Retiga, ”Evaluation of energy detection for spectrum

sensing based on the dynamic selection of detection-threshold”, Procedia Eng.,

vol. 35, pp. 135-143, 2012.* Response*: The three papers have now been cited in first paragraph of the introduction

section.

In addition to the citation problem of the paper, the way the introduction is

written is another issue that weaken the paper. Please avoid using equations in

the introduction, your mathematical model should be described in the methodology

section.

**: The equations are now moved to section 2.**

*Response*How the work presented in this paper related to the following papers:

Arjoune, et al. Spectrum sensing: Enhanced energy detection technique based

on noise measurement, 2018 IEEE 8th Annual Computing and Communication

Workshop and Conference (CCWC), DOI: 10.1109/CCWC.2018.8301619

D. R. Joshi, D. C. Popescu, O. A. Dobre, ”Adaptive spectrum sensing with

noise variance estimation for dynamic cognitive radio systems”, Conference on

Information Sciences and Systems (CISS), pp. 1-5, 2010.

J. Wu, T. Luo, G. Yue, ”An Energy Detection Algorithm Based on Double-

Threshold in Cognitive Radio Systems”, International Conference on Information

Science and Engineering, pp. 493-496, 2009.

**: The three papers have now been cited in the first paragraph of page 2. The first**

*Response*two paper use the estimated noise variance for detection, they did not analyze

the effect caused by using the estimation variance. The third paper proposed a

double threshold detection method using exact noise variance to separate three

cases of detection: spectrum free, spectrum occupied and not certain.

In Figure 5, why the blue line does not have a legend?** Response**: The blue line is the identical line, i.e., the E ˆ Pfa is equivalent to the value of

Pfa.

References should be formatted according to the MDPI authors guideline.** Response**: The references have now been presented in the MDPI’s format.

Figure 1 should be placed in the methodology section

**: Figure 1 has now been placed in the methodology section.**

*Response*The authors can introduce a new section and call it notation in which place the

**: The notations have now been presented in the end of section 2.**

*Response*Why example 1 and Example 2 are written in italic?

**: This format is caused by the the default format of the MDPI document under**

*Response*the setting the block beginfExampleg and endfExampleg, just like the theorem block.

Line 19, why the format of the citation is different from the rest?

**: Now changed to the same format.**

*Response*