# Maximum a posteriori (MAP)-based tag estimation method for dynamic framed-slotted ALOHA (DFSA) in RFID systems

- Jinchul Choi
^{1}and - Chaewoo Lee
^{1}Email author

**2012**:268

https://doi.org/10.1186/1687-1499-2012-268

© Choi and Lee; licensee Springer. 2012

**Received: **5 March 2012

**Accepted: **17 July 2012

**Published: **23 August 2012

## Abstract

Radio frequency identification (RFID) is a non-contact technology that uses radio frequency electromagnetic fields to transfer data from a tag attached to an object, for the purposes of automatic identification and tracking. One of the common problems that arise in any RFID deployment is the collision between tags which reduces the efficiency of the RFID system. Dynamic framed-slotted ALOHA (DFSA) is one of the most popular approaches to resolve the tag collision problem. In DFSA, each tag randomly selects one of the time slots of a frame and transmits its data at the slot. Unless the tag successfully transmits its data to a reader, it will try again in the next frame. It is widely known that the optimal performance of framed-slotted ALOHA is achieved when the frame size (i.e., number of time slots) is equal to the number of tags to be identified. So, a reader dynamically adjusts the next frame size according to the number of tags. Thus, it is important to accurately estimate the number of tags. In this article, we propose an accurate maximum a posteriori (MAP)-based tag estimation method with low computational complexity. The idea behind our method is to more accurately determine the most potential number of tags which draws the observed results on the basis of both a posteriori probability and *a priori* probability. Simulation results show that our method improves the accuracy of tag estimation and the speed of tag identification.

## Keywords

## Introduction

Radio frequency identification (RFID) systems that identify tagged objects via near/far-field wireless communications to realize ubiquitous computing are drawing much attention. The operation of RFID systems often involves a situation in which numerous tags are simultaneously placed in the interrogation zone of a single reader. The tags may collide with each other, leading to retransmission of tag data that brings about a waste of bandwidth and an increase in the total delay. To resolve the tag collision problem, a number of tag anti-collision algorithms have been proposed[1]. The primary concern in the algorithms is how to read multiple tags as fast and as reliably as possible.

Tag anti-collision algorithms are mainly grouped into tree-based[2] and ALOHA-based[3] algorithms. Tree-based algorithms work by repeatedly splitting the group of colliding tags into two disjoint subsets. The subsets become smaller and smaller until the number of tags within a subset reduces to one, in which case the tag would be uniquely identified. However, as the number of tags increases, the performance of tree-based algorithms decreases. This is because the colliding tags are successively grouped into two subsets, and each subset may still contain many tags resulting in collisions. The tree-based algorithms have been studied extensively in the literature[4–6].

*U*of FSA with

*N*tags and

*L*slots in a frame is given by

*L*, given the number of tags

*N*in the interrogation range. Figure1 shows the well-known upper bound of the throughput of

*e*

^{−1}that is characteristic for slotted ALOHA and also applies to FSA. The maximum throughput occurs when the frame size equals to the number of tags, i.e.,

*L*=

*N*. Thus, for high performance, it is desirable that the frame size is dynamically adjusted according to the number of unread tags; the mechanism is referred to as Dynamic Framed-Slotted ALOHA (DFSA)[3]. However, DFSA necessitates the reader accurately estimating the number of unread tags to decide an appropriate frame size. To deal with this, various methods to estimate the number of unread tags have been studied in the literature[9–20].

In this article, we propose an accurate and simple maximum a posteriori (MAP)-based tag estimation method for DFSA in RFID systems. In the proposed scheme, we derive a probability mass function (PMF) that describes the relative probability of detection results occurring at a given number of tags and then, based on the derived PMF and the prior tag distribution (if it is postulated), determine the most potential number of tags which draws the detection results observed in a read cycle as the optimal estimate. However, this method may result in a heavy computational load due to the wide search range of tag quantity. To deal with this problem, we propose a simple iterative algorithm based on Newton’s method. In our simulations, comparison with several conventional tag estimates shows that the proposed iterative algorithm has lower computational complexity and less error.

The rest of the article is organized as follows. We analyze several important tag estimation methods in the following section. In Section “The proposed tag estimation mechanism”, we propose an MAP-based tag estimation with low computational complexity. Simulation results are shown in Section “Numerical results”. Finally, we give our concluding remarks in Section “Conclusion”.

## Related study

The lower bound method[10] is obtained through the observation that a collision involves at least two different tags. Suppose that after carrying out an FSA in which the frame size is set to *F*, the reader can observe *s*_{
i
} idle slots, *s*_{
s
} success (or singly occupied) slots, and *s*_{
c
}collision slots, where *s*_{
i
} + *s*_{
s
} + *s*_{
c
}= *F*. Then, the lower bound method simply estimates the number of tags as 2*s*_{
c
}. On the other hand, the Schoute method evaluates the expected number of tags per collision slot as 2.39 on the basis of the results in[9]. The authors of[11, 12] proposed tag estimation methods based on the Schoute method for RFID anti-collision using FSA. According to these studies, if the frame size is assumed to be the number of tags, the expected value of collision tags is 2.39*s*_{
c
}. Although the lower bound and the Schoute methods are simple and easy to implement, their assumptions are difficult to confirm in practice. Thus, using the expected value to estimate tag quantity may cause significant error and increase read delay[13].

*n*tags are to be read and a frame of

*F*time slots. For a given time slot, the number of tags allocated to the slot is binomially distributed with

*n*Bernoulli trials and 1/

*F*occupied probability. Thus, the probability of finding

*r*tags in the slot is given by

*F*is sufficiently large, (2) can be approximated by a Poisson distribution with mean

*n*/

*F*. Accordingly, the probabilities of idle, successful transmission, and collision for the slot is given by

*s*

_{ i },

*s*

_{ s }, and

*s*

_{ c }are given by

In[15, 16], the authors proposed the Lottery Frame (LoF) scheme to reduce the multiple-reading by multiple readers with overlapping interrogation regions. LOF adopts the ZE of[14] and dynamically adjusts the system load factor (the ratio of the tag quantity to the frame size) to eliminate the multiple-reading.

*n*, then the Chen method’s estimate denoted by${\hat{n}}_{\text{Chen}}$ is given by

Then, assuming that the numbers of idle and success slots observed in a read cycle are identical tot heir expectations, the Chen method estimates the number of tags by substituting the expectations with the observations from an FSA. Though the Chen method and the three binomial estimators are simple, some cases (i.e., *s*_{
i
}= 0, *s*_{
s
}= 0, *s*_{
c
}= 0) need special treatment and their accuracies are not satisfactory since the outcomes observed in a read cycle seldom correspond with their expectations[18].

*s*

_{ i }

*s*

_{ s }

*s*

_{ c }) and the expected value vector to estimate the number of tags

*n*for which the distance becomes minimal using the exhaustive search. The Vogt method’s estimate denoted by${\hat{n}}_{V\mathit{\text{ogt}}}$ is given by

The Vogt method improves the accuracy of tag estimation compared with the lower bound and the Chen methods. However, the Vogt method requires rather large computational load because it works with no information about the search space.

*β*

_{ k }and

*γ*

_{ k }denote the ratio of the frame size to the tag quantity and the number of tags per collision slot after the

*k*th iteration, respectively. According to[18], the sequence in 8 tends to converge for a reasonable threshold. Then, the Eom method’s estimate denoted by${\hat{n}}_{\text{Eom}}$ is given by

where${\gamma}_{k}^{\ast}$ denote the limit value at the *k*^{∗}th iteration (starting from *β*_{1} = *∞*, *γ*_{1} = 2). The Eom method gives a rather accurate estimate when the frame size is large. However, as the frame size decreases, its estimate error increases.

In general, a reader in an RFID system usually reads tags only once in a frame. In this context, Bayesian tag estimate which is available with few observation samples can be an appropriate approach. In[21], Rivest proposes a Bayesian transmission strategy for an FSA broadcast system. Based on Rivest’s work, Floerkemeier also applies the Bayesian transmission strategy to a framed aloha RFID system in[22, 23]. These works mainly concern about how to minimize the posterior expected value of a risk function (i.e., the posterior expected risk), such as using Bayesian rule to update the probability distribution of tag quantity and adjusting the frame size. However, the tag estimation method has not been discussed in these references.

On the other hand, in[24], using different risk (or loss) functions, the authors propose three Bayesian tag estimation methods: Bayesian mean-square, Bayesian absolute-error, and Bayesian posterior-probability estimation method. According to their simulations, the accuracy of Bayesian posterior-probability estimate (BPE) is the best among their methods. BPE is the same form as the maximum likelihood estimation (MLE) in[13], thus their accuracies are identical. Although BPE and MLE have better estimate performance than other existing methods, there is substantial room for improvement in estimation accuracy because they are only grounded on a posterior probability.

In this article, we propose an accurate MAP-based tag estimation method with low computation complexity. The idea behind our method is to more accurately determine the most potential number of tags which draws the observed results on the basis of both a posteriori probability and *a priori* probability. Furthermore, we propose a simple iterative algorithm based on Newton’s method and reduce computational complexity.

## The proposed tag estimation mechanism

### The MAP-based tag estimate

Consider an RFID system in which the tag anti-collision problem is avoided by the DFSA algorithm. Suppose there is a detection result *s* = {*s*_{
i
},*s*_{
s
},*s*_{
c
}} observed in a read cycle, coming from a distribution with an unknown PMF *f*_{0}depending on the number of tags. It is however surmised that function *f*_{0} belongs to a certain family of distributions {*f*(*s*|*θ*),*θ*∈Θ}, where Θ denotes the domain of the distributable number of tags, so that *f*_{0} = *f*(*s*|*θ*_{0}). The value *θ*_{0} is unknown and is referred to as the *true value* of parameter *θ*. We want to find an estimate$\widehat{\theta}$ which would be as close to the true value *θ*_{0}as possible.

*g*over

*θ*exists. This allows us to treat

*θ*as a random variable as in Bayesian statistics. Then, the posterior distribution of

*θ*is given by

where *g* is PMF of *θ*. This is a straightforward application of Bayes’ theorem.

*θ*

_{0}by finding a value of

*θ*that maximizes

*f*(

*θ*|

*s*). Thus, the optimal estimate of

*θ*can be derived as the mode of the posterior distribution of this random variable:

*θ*and therefore plays no role in the optimization. Consequently, (11) can be simplified as

*s*occurring when the number of tags is given by

*θ*, i.e.,

*f*(

*s*|

*θ*). All the time slots in a read cycle only belong to one of three possible outcomes such as idle, success, or collision slot. Thus we can conclude that the three outcomes are mutually exclusive and collectively exhaustive (MECE). Then,

*f*(

*s*|

*θ*) is simplified by Bayes’ theorem and the property of MECE:

Equation (13) holds because Pr(*s*_{
c
}|*s*_{
i
},*s*_{
s
},*θ*) = 1.

*s*

_{ i }|

*θ*) and Pr(

*s*

_{ s }|

*s*

_{ i },

*θ*). Pr(

*s*

_{ i }|

*θ*) means the probability that

*θ*tags select their respective slots among

*F*slots leaving exactly

*s*

_{ i }idle slots. Assuming that

*θ*=

*n*, using (2) and (3) we have

*s*

_{ s }|

*s*

_{ i },

*θ*) means the probability that

*θ*tags select their respective slots among (

*F*−

*s*

_{ i })slots with exactly

*s*

_{ s }slots being singly occupied. Since the (

*F*−

*s*

_{ i }) slots belong to either success or collision slots, the probability that one of the slots is singly occupied is$\frac{{p}_{s}}{{p}_{s}+{p}_{c}}$. Thus we have

holds because *s* _{
i
} + *s* _{
s
} + *s* _{
c
}= *F* and *p* _{
i
} + *p* _{
s
} + *p* _{
c
}= 1.

*f*(

*s*|

*θ*) as follows.

According to (12), using both a prior distribution *g*(*θ*) and a posteriori distribution *f*(*s*|*θ*), we can determine the best estimate$\widehat{\theta}$ which is the most potential number of tags. From a statistical point of view, the method of MAP estimation is considered to be more robust (with some exceptions) and yields estimates with good statistical properties[25]. Furthermore, the proposed MAP-based tag estimate is very suitable for estimation under few observation samples[26].

The MAP-based tag estimate requires empirical data about tag detection and a prior distribution over tag quantity *θ* to obtain an optimal estimate. A prior distribution is often the purely subjective assessment of an experienced expert. Nevertheless, there are several ways to collect the prior. One reasonable approach is to make the prior a normal distribution with expected value equal to the present tag quantity, with variance equal to the variance of the past several tag quantities. This approach has a property in common with many priors, namely, that the posterior from one problem (present tag quantity) becomes the prior for another problem (next tag quantity)[27]. Other approach is to choose a conjugate prior when they can, to make calculation of the posterior distribution easier. The details of how to collect a prior distribution are shown in[28].

If an informative prior (which expresses specific, definite information about tag quantity) is not postulated or cannot be collected, we use a uniform prior distribution (that is, a constant function). This is based on the principle of indifference, which assigns equal probabilities to all possibilities[29]. Then, the MAP estimate of *θ* coincides with MLE in[13] and BPE[24], and thus the accuracy of our estimation method is equal to those of MLE and BPE.

Except for the accuracy of estimation, one obvious concern of both estimates is a tag quantity range over which the maximum probability needs to be searched. If the range is wide, the estimates’ computational complexity will be high. To find extremum, our estimate and MLE basically needs to search *N* − *s*_{
s
} + 2*s*_{
c
} times in the search range of tag quantity (i.e., from *θ* = *s*_{
s
} + 2*s*_{
c
}to *θ* = *N*). In this article, we are based on the results of[24], we reduce the computational complexity by narrowing the search range. Consequently, our method needs to search$|{\hat{\theta}}_{M\phantom{\rule{0.3em}{0ex}}A\phantom{\rule{0.3em}{0ex}}P}-\overline{\theta}|+2$ times where$\overline{\theta}=\text{ln}\left({s}_{i}/F\right)/\text{ln}\left(1-1/F\right)$.

On the other hand, if we need to use a uniform prior distribution due to non-informative prior, we can reduce the complexity more. In next section, we propose an iterative algorithm to determine the estimate in less computational complexity when informative prior is not given or a uniform prior distribution is used.

### Iterative algorithm to reduce computational complexity

*S*be a vector of three random variables with a probability distribution with the function

*f*depending on a parameter

*x*. Assuming that the prior distribution of

*x*is uniform, the function

*x*, is called the likelihood function (of

*x*, given the outcome

*s*of

*S*)[25]. Let

*l*(

*x*) be a probability distribution with the likelihood function$\mathcal{L}$ depending on a parameter

*x*. From (16) and (17), we have

where *s*_{
c
} ≥ 1 and *s*_{
i
} + *s*_{
s
} ≥ 1. If *s*_{
c
}is zero, we can easily conclude that all the tags were read. On the other hand, if both *s*_{
i
} and *s*_{
s
}are zero, we can conclude that all the tags were collided with each other. As the number of tags increases, the chances of this happening grow higher. The estimate from our method must be the maximum number of tags that the RFID system can read. Thus we only consider a read cycle where *s*_{
c
} ≥ 1 and *s*_{
i
} + *s*_{
s
} ≥ 1, hence *x* ≥ 2 and *F* ≥2.

Suppose$\{x\in \mathbb{R}|2\le x\le N\}$ where *N* is the maximum number of tags that the RFID system can tolerate. It is obvious that *l* is continuous on the interval because *F* ≠ 0. Thus the maximum value is undoubtedly one among *l*(2),*l*(*N*) or the extreme values of *l*. Since the values of *l*(2) and *l*(*N*) can be simply obtained from (18), we focus on the extreme values.

*l*

^{ ′ }(

*x*), i.e.,

*∂*

*∂x*

*l*(

*x*), exists,

*l*has a local maximum or minimum at some number

*c*in [2,

*N*when

*l*

^{ ′ }(

*c*) = 0[30]. To find the first-order derivative of

*l*it is easier to differentiate the logarithm of the function rather than the function itself, hence we first take the natural logarithm on both sides. Then, after implicit differentiation we have

*l*

^{ ′ }(

*x*) = 0,$\frac{\partial}{\partial x}\text{ln}l\left(x\right)$ should be zero because

*l*(

*x*) > 0 for all

*x*. Thus we will focus on the roots of

*∂*

*∂x*ln

*l*(

*x*). It is not easy to obtain the roots using elementary algebra, so we apply a numerical approach in finding the roots of general polynomial equation. Let

*h*(

*x*) is continuous on the interval [2,

*N*] because${t}^{x}+\frac{x}{F}{t}^{x-1}\ne 1$. Thus we have

*h*

^{ ′ }(

*x*) is continuous on the interval. Thus, we can use Newton’s method which gives better approximations to the roots of a real-valued function[31]. As proved in the Appendix,

*h*(

*x*) has only a single root. Assume that there exists a root

*q*∈ [2,

*N*] where

*h*(

*q*) = 0. According to Newton’s method, if

*h*

^{ ′ }(

*q*) ≠ 0, the sequence${\left\{{q}_{k}\right\}}_{k=0}^{\infty}$ defined by the iteration

*q*with any initial estimate

*q*

_{0}∈[2,

*N*. Nevertheless, it is desirable to choose one as close as possible to the root to reduce the iteration times. To give a better initial estimate, we use the Chen method[17] mentioned in Section “Related study”. In addition, the real number of tags is at least

*s*

_{ s }+ 2

*s*

_{ c }. Thus, we set the initial estimate as

*x*

_{0}using (23), we start the iteration based on

*x*

_{ k }until the difference between

*x*

_{ k }and

*x*

_{k−1}is less than a threshold

*ε*

_{ TH }, i.e.,

Let *x*^{∗} be the converged value of *x*_{
k
}. According to the proof in the Appendix, *h*(*x*) is strictly decreasing and has only a single root, i.e., *x*^{∗}. Note that *l*^{
′
}(*x*) = *l*(*x*)*h*(*x*) and *l*(*x*) > 0 on the interval [2,*N*]. Since *h*(*x*) ≥ 0 where *x* ≤ *x*^{∗}, we can conclude that function *l* monotonically increases on the interval [2,*x*^{∗}]. On the contrary, *l* monotonically decreases on the interval [*x*^{∗},*N*] because *h*(*x*) ≤ 0 where *x* ≥ *x*^{∗}. Consequently, *l* attains the maximum value at *x*^{∗}. We determine, therefore, the estimate$\widehat{\theta}=\left[{x}^{\ast}\right]$ that maximizes the likelihood of detection results observed in a read cycle.

### Computational complexity analysis

Among conventional tag estimates, the simpler ones are the lower bound method, the Schoute method, the three binomial mean estimators (i.e., ZE, SE, CE) and the Chen method. Their estimate can be simply obtained. On the other hand, to obtain the value of conditional probability Pr(*θ*|*s*) from *θ* = *s*_{
s
} + 2*s*_{
c
} to *θ* = *N*, MLE needs to enumerate the conditional probability *N* − *s*_{
s
}− 2*s*_{
c
} + 1 times. Similarly, the Vogt method needs to search *N*−*s*_{
s
}− 2*s*_{
c
} + 1 times over the search tag range to find extremum. The Eom method uses an iterative algorithm and has less complexity than the Vogt method or MLE. However, a more detailed complexity analysis is likely intractable[18]. Unlike the methods above, by narrowing the search range of tag quantity, BPE reduces its computational complexity. According to[24], BPE needs to search$|{\hat{n}}_{\text{BPE}}-\overline{n}|+2$ where${\hat{n}}_{\text{BPE}}$ denote the Bayesian tag estimate and$\overline{n}=\text{ln}\left({s}_{i}/F\right)/\text{ln}\left(1-1/F\right)$.

*a*(

*x*) with

*c*-digit precision, provided that a good initial approximation is known, is O((log

*c*)

*F*(

*c*) where

*F*(

*c*) is the cost of calculating

*a*(

*x*)/

*a*

^{ ′ }(

*x*) with

*c*-digit precision. On the other hand, when an informative prior distribution is postulated, the complexity of our method is similar to that of BPE. The computational complexity of the primary estimates is listed in Table1.

**Computational complexity**

Estimation method | Computational complexity |
---|---|

The Vogt method | O( |

MLE | O( |

BPE | $\text{O}\left(|{\widehat{n}}_{\text{BPE}}-\overline{n}|+2\right)$ |

MAP (with a uniform prior) | O((log |

MAP (with a non-uniform prior) | $\text{O}\left(|{\widehat{n}}_{\text{MAP}}-\overline{n}|+2\right)$ |

## Numerical results

Computer simulations are performed to check and extend the analytical results of the previous section. We consider a single reader and a number of tags under error-free channel environment. The prior distribution for tag quantity is assumed a normal distribution with mean *μ* and variance *σ*^{2}. We set the frame size and the tag quantity of each read cycle to generate from the prior normal distribution and obtain detection result *S* from the read cycle. Then, we perform the proposed tag estimation method. Finally, we obtain the average estimate after completing 1,000 cycles. In our method, *ε*_{TH} is set to 0.001.

*μ*from 8 to 256 or from 2 to 64, respectively. The variance

*σ*

^{2}is set to 0.05

*μ*or 0.1

*μ*, respectively. Note that when a uniform prior distribution is considered, our method is actually MLE[13] and has the same results as MLE and BPE. Thus, we do not give the estimation results of our method which incorporates a uniform prior distribution in our simulations. In addition, the performance of SE[14] is insufficient and we do not give its results.

When the number of tags is considerably less than the frame size, Lower bound method is the most efficient among the methods. However, as the number of tags increases, its estimation errors increase steeply. Among the existing methods, BPE and MLE which have the similar curves to the Eom method show reasonable performance. On the other hand, the accuracies of other estimates such as ZE, CE, Chen, Schoute and Vogt methods are not satisfactory. Figures3 and4 present that the proposed method provides more precise estimation than the other methods except the region in which the number of tags is comparatively less than the frame size.

*N*and are quite high (i.e., more than

*N*/2). In addition, the accuracies of simple estimates such as the lower bound method, the Schoute method, the three binomial mean estimators and the Chen method are insufficient. Thus, we do not give the results of them in Figure5. Figure5 presents that the complexity of our iterative algorithm is less than that of BPE. When a non-uniform prior distribution is postulated, the complexity of our method is a little bit high than that of BPE. Nevertheless, our method gives more accurate estimate than BPE and MLE as shown in Figures3 and4.

*μ*varies from 100 to 1000. In Figure6, we compare the RFID system efficiency, which is defined as the ratio of the number of success slots to the number of slots used to detect all the tags. Figure6 shows that our tag estimation method gives the highest system efficiency compared to the others. It also demonstrates that the performance of DFSA can be nearly improved to the maximum efficiency of Slotted ALOHA (about 36.8%) using our tag estimation method.

## Conclusion

In this article, we have proposed an accurate MAP-based tag estimation method for DFSA in RFID systems and improved the proposed method to reduce computational complexity. The proposed method can be used to obtain a point estimate of an unobserved tag quantity on the basis of empirical detection data and an informative prior. From our theoretical derivation and simulation results, we draw the conclusions as follows.

The proposed MAP-based tag estimation improves the accuracy of tag estimation and requires less time slots to read out all tags than the other existing methods. When an informative prior is not postulated or a uniform prior is used, our method can reduce computational complexity using the proposed iterative algorithm. Nevertheless, our method produces the same estimates as MLE and BPE. When the precise number of tags is not given, which is usual in practice, our method can achieve accurate estimation performance. Therefore, our method can become a suitable candidate for such a mechanism and can be useful for engineers to implement the DFSA algorithm in a real RFID system.

## Appendix

### Proof that *h(x)* has only a single root

The function *h* is continuous on the closed interval [2,*N*. According to the intermediate value theorem[30], if *h*(2)*h*(*N*) < 0 and *h* is a strictly monotonic function, *h* has only a single root in the interval. To prove that *h* has only a single root, we show that *h*(2) > 0, *h*(*N*) < 0, and *h*^{
′
}(*x*) < 0. Note that we only consider a read cycle where *s*_{
c
}≥ 1, *s*_{
i
} + *s*_{
s
} ≥ 1, hence *F* ≥ 2.

*h*(2) > 0. Let

*h*(2) ≤ 0. Then, we have

*s*

_{ c }(

*F*−1). Then, we have

*F*, the left-hand side of (28) is minimized when

*s*

_{ c }= 1 because ln

*t*< 0. Substituting

*s*

_{ c }with one, the left-hand side of (28) is given by −

*F*ln

*t*. Let

*m*(

*F*) has following properties.

- (i)
*m*(2) > 1. - (ii)
$\underset{F\to \infty}{\text{lim}}m\left(F\right)=1$ because $\underset{F\to \infty}{\text{lim}}{\left(1+\frac{1}{F-1}\right)}^{F}=e$.

- (iii)
Assuming

*m*is a continuous function, we have ${m}^{\prime}\left(F\right)=-\left(\text{ln}\left(1-\frac{1}{F}\right)+\frac{1}{F-1}\right)$. Since ${m}^{\prime}\left(F\right)<0$ where*F*≥ 2,*m*(*F*) is strictly decreasing. Since frame size*F*is finite in practice, taken the three properties together, we have*m*(*F*) > 1. This contradicts the proposition that*h*(2) ≤ 0. Therefore, it can be concluded that*h*(2) > 0 where*s*_{ c }≥ 1.

*h*(

*N*) < 0.

*h*(

*N*) is given by

*N*is sufficiently large, both

*t*

^{ N }and$\frac{N}{F}{t}^{N-1}$ can be approximated to zero. Thus, we have

Since ln*t* < 0 and *s*_{
i
} + *s*_{
s
}≥ 1, it can be concluded that *h*(*N*) < 0 where *N* is sufficiently large.

*h*

^{ ′ }(

*x*) < 0 on the interval [2,

*N*]. From (21) we have

*n*(

*x*) is larger than or equal to zero, then

*h*

^{ ′ }(

*x*) is negative at all

*x*. Now we prove that

*n*(

*x*) ≥ 0 by showing that

*n*(

*x*) is strictly increasing and

*n*(2) ≥ 0. It is obvious that

*n*(

*x*) is continuous on the interval. Thus we have

*F*ln

*t*> 1, thus we have

*F*ln

*t*< −1. Since

*t*

^{x−1}< 1,

*F*ln

*t*+

*t*

^{x − 1}is less than zero. It can, therefore, be concluded that

*n*

^{ ′ }(

*x*) > 0, i.e.,

*n*(

*x*) is strictly increasing. Next we show that

*n*(2) is positive at all

*F*. Let

*x*be two and

*F*be the function

*n*’s variable and allowed to vary freely. Let

*r*(

*F*) be a function that describes the value of

*n*(2) at varying

*F*. Then, we have

*r*(

*F*) has following properties.

- (i)
*r*(2) > 0. - (ii)
$\underset{F\to \infty}{\text{lim}}r\left(F\right)=0$.

- (iii)Assuming
*r*is a continuous function, we have ${r}^{\prime}\left(F\right)=\frac{2}{F(F-1)}\left\{F\text{ln}t+1+\text{ln}t+\frac{1}{2F}t\right\}$. The derivation of*r*^{ ′ }(*F*) is as follows. Since $t=1-\frac{1}{F},\frac{\partial t}{\partial F}=\frac{1}{{F}^{2}}$. Thus, we have $\frac{\partial r\left(F\right)}{\partial F}=\frac{1}{{F}^{2}}\frac{\partial r\left(F\right)}{\partial t}$. Then,*r*^{ ′ }(*F*) is given by$\begin{array}{c}{r}^{\prime}\left(F\right)=\frac{1}{{F}^{2}}\frac{\partial r\left(F\right)}{\partial t}\\ \phantom{\rule{9em}{0ex}}=\frac{1}{{F}^{2}}\left\{\frac{F+1}{t}\text{ln}t+\frac{\left(F+1\right)\text{ln}t+2}{t}+\frac{1}{F}\right\}\hfill \\ \phantom{\rule{9em}{0ex}}=\frac{1}{{F}^{2}}\frac{1}{t}\left\{\left(F+1\right)\text{ln}t+\left(F+1\right)\text{ln}t+2+\frac{1}{F}t\right\}\hfill \\ \phantom{\rule{9em}{0ex}}=\frac{2}{F(F-1)}\left\{\left(F+1\right)\text{ln}t+1+\frac{1}{2F}t\right\}\hfill \\ \phantom{\rule{9em}{0ex}}=\frac{2}{F(F-1)}\left\{F\text{ln}t+1+\text{ln}t+\frac{1}{2F}t\right\}.\hfill \end{array}$(34)

From *F* ln*t* < −1, we have *F* ln*t* + 1 < 0. This inequality can be transformed into$\text{ln}t<-\frac{1}{F}$. Since$\frac{t}{2}<1$, we have$\text{ln}t<-\frac{1}{F}<-\frac{1}{2F}t$. Thus *r*^{
′
}(*F*) is negative for all *F*. Taken three properties together, it can be concluded that *r*(*F*) > 0, i.e., *n*(2) > 0. Consequently, we have that *n*(*x*) > 0. Therefore, we can conclude that *h*^{
′
}(*x*) < 0 on the interval [2,*N*].

From the preceding three steps, it can be concluded that *h*(*x*) has only a single root in the interval [2,*N*] where *s*_{
c
} ≥ 1 and *s*_{
i
} + *s*_{
s
} ≥ 1.

## Declarations

### Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2007605).

## Authors’ Affiliations

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