João Guerreiro

Rui Dinis

Paulo Montezuma

On Quasi-Optimum Detection of Nonlinearly Distorted OFDM Signals

Keywords: OFDM signals, nonlinear distortion effects, optimum receiver, Euclidean distance

In this paper we considered OFDM schemes that somehow have a nonlinear operation on their transmission chain. Contrary to the conventional OFDM implementations, we consider the distortion term that come from that operation as something useful that has information on the transmitted signals. To efficiently harvest this information, we develop an optimum-based receiver that presents good performance improvements without introduce very high complexities. The performance of this sub-optimal receiver is investigated under different types of channel and considering receive diversity.

Introduction

OFDM (Orthogonal Frequency Division Multiplexing) [START_REF] Cimini | Analysis and Simulation of a Digital Mobile Channel Using Orthogonal Frequency Division Multiplexing[END_REF] schemes support the physical layer of several wireless communications standards such as DVB [START_REF]Framing structure, channel coding and modulation for digital terrestrial television[END_REF], LTE [START_REF]3rd Generation Partnership Project: Technical Specification Group Radio Access Network; Physical Layers Aspects for Evolved UTRA[END_REF] and WiMAX [START_REF]IEEE Standard for Local and Metropolitan Area Networks -Part 16: Air Interface for Fixed Broadband Wireless Access Systems[END_REF]. The reasons behind their popularity reside mainly on their facility to cope with frequency-selective channels without the need for complex equalization processes and their simplicity of implementation. However, OFDM signals have a critical issue: they present a very high Peak-to-Average Power Ratio (PAPR). If nothing is done, this will cause severe amplification difficulties and strong nonlinear distortion effects at the transmitter output. As the use of an high Input Back-Off (IBO) is not a good solution due to the inherent energy inefficiency, several techniques that aim to reduce the envelope fluctuations of OFDM signals have been proposed in the literature [START_REF] Rahmatallah | Peak-To-Average Power Ratio Reduction in OFDM Systems: A Survey And Taxonomy[END_REF]. Between them, the simplest one involves the use of an envelope clipper followed by a frequency-domain filter (FDF), which allows an efficient PAPR reduction without introduce intolerable complexities and compromise the spectral efficiency [START_REF] Dinis | A Class of Nonlinear Signal Processing Schemes for Bandwidth-Efficient OFDM Transmission with Low Envelope Fluctuation[END_REF]. However, although the clipped signals are linearly amplified with ease, they will also present nonlinear distortion effects since they are generated through a nonlinear operation.

It is well known that the samples of an OFDM signal are well modeled by a Gaussian distribution, specially when a high number of subcarriers is considered. This approximation allows the use of the Bussgang theorem [START_REF] Rowe | Memoryless Nonlinearities with Gaussian Input: Elementary Results[END_REF] that states that a nonlinearly distorted OFDM signal can be divided into two uncorrelated components: an useful term that is proportional to the transmitted signal and a second one that represents the nonlinear distortion. Considered as noise in the typical OFDM implementations, the distortion term is seen as something that degrades the performance. To avoid this, there are receivers that try to estimate and cancel this distortion [START_REF] Gusmão | Iterative Receiver Techniques for Cancellation of Deliberate Nonlinear Distortion in OFDM-Type Transmission[END_REF], but their usefulness are very limited specially at high Signal-to-Noise Ratios (SNRs). In this work we considered not only to accept the nonlinear distortion but also to use it as something that can effectively improve the performance of the nonlinear OFDM schemes. In fact, the distortion term has useful information on the transmitted signals that can be used to improve the performance [START_REF] Montezuma | Should we Avoid Nonlinear Effects in a Digital Transmission System?[END_REF]. The best way to use this information is to consider an optimum receiver that makes an estimate of the transmitted signal based on the Euclidean Distance, which is evaluated using the two terms of the Bussgang Theorem. The research question can be stated as: can the nonlinear OFDM schemes perform better than the linear ones with the use of optimum receivers? To answer this question, we adopted a research approach based on the study of the relation between the Euclidean distance and the average bit energy with and without nonlinear distortion effects, which allows to conclude about the expected Bit Error Rate (BER). Surprisingly, it was verified that the nonlinear OFDM schemes can have better performances than the linear ones. As the complexity of the optimum receiver is very high even considering a small number of subcarriers, we propose a reduced-complexity receiver that in spite of not performing a full optimum detection can also achieve excellent performances.

The paper is divided in the following sections: In Sec.2 we present the relationship between this work and the collective awareness systems (CAS). Sec. 3 concerns about the communication scenario that is used as the basis of this work as well as the principal characteristics of nonlinearly distorted OFDM signals modeled with a Gaussian approximation. In Sec. 4 the optimum detection is introduced and the potential performance of the optimum receiver is presented. Sec. 5 presents a sub-optimal received and shows its performance results under different scenarios. Finally, Sec. 6 concludes this paper and presents some directions of future work.

Relationship to Collective Awareness Systems

CAS are today one of the biggest technological challenges in the digital world. The idea of everyone to communicate and use collective knowledge to decide in real-time presupposes solid communication mechanisms that can provide a good Quality of Service (QoS) without high energy consumption. Behind the real-time decisions, it is easily to see that wireless communications have a very important role. This work aims to present improvements in the quality of wireless communications that use OFDM modulations, which is the case of the most part of the standards, by reducing the energy consumption that is need to achieve a specific error rate and hence improving the energy efficiency of the equipments that compose the collective awareness systems. In this section we characterize the considered nonlinear OFDM scenario by describing the signals along the transmission chain depicted in Fig. 2. We represent the baseband data signal with the vector ܁ = [ܵ ܵ ଵ . .. ܵ ேெିଵ ] ் ∈ ℂ ேெ , where ܰ is the number of subcarriers considered in each OFDM block and ܯ is the oversampling factor. Each complex data symbol ܵ is selected from an ℳ-order QAM constellation (e.g. (

Nonlinear OFDM Schemes

) 1
If high values of ܰ are considered (let's say ܰ ≥ 32), the imaginary and the real parts of the time-domain samples represented by ܛ can be modeled by a Gaussian distribution. Representing ̅ݏ ூ and ̅ݏ ொ as the considered Gaussian random variables, we can write

 ௦̅ ݏ( ூ ) = ଵ ఙ√ଶగ exp ቀ- (௦ ିఓ) మ ଶఙ మ ቁ, (2)
and ݏ( ூ ) = ݏ( ொ ) . The absolute value of ܛ is represented by

܀ = |ܛ| = [ܴ ܴ ଵ . .. ܴ ேெିଵ ] ் ∈ ℝ ேெ and modeled by ܴ ത , which is a
Rayleigh-distributed random variable with its PDF given by

 ோ ത (ܴ) = ோ ఙ మ exp ቀ- ோ మ ଶఙ మ ቁ ,)ܴ(ݑ (3)
with ܴ(ݑ ത ) denoting the unitary step function and ߪ ଶ = var(̅ݏ ூ ) = var(̅ݏ ொ ), denoting the variance of the real and imaginary parts that is assumed to be equal. Under the Gaussian approximation, the Bussgang theorem [START_REF] Rowe | Memoryless Nonlinearities with Gaussian Input: Elementary Results[END_REF] states that the nonlinearly distorted signal represented by

ܡ = )|ܛ|(݂ = )܀(݂ = ݕ[ ݕ ଵ . .. ݕ ேெିଵ ] ் ∈ ℂ ேெ can be decomposed in two terms, i.e., ܡ = ܛߙ + ,܌ (4)
where ܛ is the time-domain version of the modulated signal, ܌ = [݀ ݀ ଵ . .. ݀ ேெିଵ ] ் ∈ ℂ ேெ is the distortion term and ߙ is a scaling factor given by

ߙ = ॱ[ܴ ݊ (ܴ ݊ )] ॱ[ܴ ݊ మ ] . (5)
The nonlinearity considered in our transmission chain is assumed to be memoryless. These nonlinearities can be modeled by the Saleh model [START_REF] Saleh | Frequency-Independent and Frequency-Dependent Nonlinear Models of TWT Amplifiers[END_REF] that characterizes the output of a nonlinear device using its AM-AM and AM-PM conversion functions, both receiving the absolute value of the signal as input (and represented by )⋅(ܣ and Θ(⋅) , respectively). Using this model, we can also write the ݊-th output sample as

݂(ܴ ) = ܴ(ܣ )exp(݆(arg(ݏ ) + Θ(ܴ )), (6)
with arg(ݏ ) being the original phase of the ݊-th time-domain sample. The considered memoryless nonlinearity is an ideal envelope clipper with normalized clipping level

ݏ ெ /ߪ, ݂(ܴ ) = ൜ ܴ , ܴ ≤ ݏ ெ /ߪ ݏ ெ exp(݆arg(ݏ )), ܴ > ݏ ெ /ߪ. (7)
Note that in this case ݂(ܴ ) = ܴ(ܣ )exp(݆arg(ݏ )), since we don't have phase distortion. The use of a normalized clipping level is related to the random nature of our signals. A low ݏ ெ /ߪ means that the random samples of the signal will enter in the nonlinear region very often. Making use of the DFT definition of (1), we can express the frequency version of (4) by

܇ = ܡ۴ = ܁ߙ + ۲, (8)
where ۲ = ܦ[ ܦ ଵ . .. ܦ ேெିଵ ] ் ∈ ℂ ேெ is the frequency-domain version of the distortion term that can be modeled by ܦ ഥ , which is a Gaussian random variable with zero mean as shown in [START_REF] Dinis | A Class of Nonlinear Signal Processing Schemes for Bandwidth-Efficient OFDM Transmission with Low Envelope Fluctuation[END_REF]. In order to maintain a good spectral efficiency and the bandwidth of the original transmitted signal, a frequency-domain filter (FDF) must be used to eliminate the out-of-band radiation inherent to the nonlinear operation. In our signal processing scheme, the filtering operation is represented as a multiplication by the diagonal matrix ۵, that is defined as

۵ = diag ൭ 0. . . . .0 ᇣᇤᇥ (ெିଵ)ே/ଶ 1. . . . .1 ᇣᇤᇥ ே 0. . . . .0 ᇣᇤᇥ (ெିଵ)ே/ଶ ൩൱. (9)
The filtered signal is given by

܇ = .܇۵ (10)
At the reception side (after the channel effect), we have the time-domain block

ܢ = ݖ[ ݖ ଵ . .. ݖ ேெିଵ ] ் ∈ ℂ ேெ with the corresponding frequency-domain block ܈ = [ܼ ܼ ଵ . .. ܼ ேெିଵ ] ் ∈ ℂ ேெ defined by ܈ = ܇۶ + ۼ = ۶۵۴(۴ ିଵ )܁ + ۼ = ܁۵۶ߙ + ۶۵۲ + ,ۼ (11)
where ۼ = [ܰ ܰ ଵ . .. ܰ ேெିଵ ] ் ∈ ℂ ேெ represents the noise components and

۶ = diag([ܪ ܪ ଵ . .. ܪ ேெିଵ ] ் ), (12)
represents the channel frequency responses.

Optimum Detection and its Potential Performance

The presence of nonlinear distortion effects on the transmitted signals is typically seen as something undesirable. However, the optimum detection can take advantage of the distortion introduced by the nonlinearity for performance improvements. To understand this, let us first recall that the optimum receiver estimate is obtained by minimizing the least square error (LSE) between the received signal ܈ and a possible transmitted signal, ܇۶ = ۶۴݂(۴ ିଵ .)܁ To evaluate the LSE between two vectors we can compute the squared Euclidean norm between them, which makes possible to write that the optimum estimate is given by

܁ = min ܁ ห|܈ -۶۴݂(۴ ିଵ |)܁ห ଶ , (13)
which, in fact, is the minimum Euclidean distance between ܇۶ = ۶۴݂(۴ ିଵ )܁ and .܈ Note that to choose the best sequence, all the possible transmitted signals must be evaluated. Here the term "evaluated" means that the possible transmitted sequence must be submitted to the same signal processing chain of the received signal. Let us also recall that what strongly conditions the BER of a communication system is the ratio between the average bit energy, ܧ , and the minimum squared Euclidean Distance between the transmitted signals, ࣞ ଶ . In the presence of nonlinear distortion effects, this ratio is higher and, consequently, we can have potential performance improvements. To verify this, let us consider two nonlinearly distorted OFDM signals that differ in one symbol, ܇ (ଶ) = ۴݂(۴ ିଵ ܁ (ଶ) ) and ܇ (ଵ) = ۴݂(۴ ିଵ ܁ (ଵ) ) associated to the modulated data signals ܁ (ଶ) and ܁ (ଵ) , respectively. The Euclidean distance between ܁ (ଶ) and ܁ (ଵ) is

ࣞ ଶ = ቚห܁ (ଶ) -܁ (ଵ) หቚ ଶ = ∑ ே ୀଵ หܵ (ଶ) -ܵ (ଵ) ห ଶ . (14)
With normalized QPSK constellations (i.e. ܵ = ±1 ± ݆), we have ࣞ ଶ = ܧ4 , with

ܧ = ||܇|| మ ୪୭ మ (ℳ)ே ℳୀସ = 1. (15)
Considering that the signals are nonlinearly distorted and using (8), the Euclidean Distance is given by

ࣞ ே ଶ = ቚห܇ (ଶ) -܇ (ଵ) หቚ ଶ = ∑ ெே ୀଵ หߙ൫ܵ (ଶ) -ܵ (ଵ) ൯ + ܦ (ଶ) -ܦ (ଵ) ห ଶ , (16)
and typically we have ࣞ ே ଶ > ܧ4 . In these conditions we can define an asymptotic gain given by

࣡ = 10log ଵ ቀ ࣞ ಿಽ మ ࣞ మ ቁ. (17)
In From Fig. 2, it is clear that gain is almost higher than one. Moreover, the stronger the nonlinear distortion effects the higher is the magnitude of the gain. For instance, with ߤ = 1 and ݏ ெ /ߪ = 1.0, we have an asymptotic gain around 1.4 dB, but the gain can even reach 2.7 dB for ݏ ெ /ߪ = 0.5 and the same value of ߤ. From the figure it is also clear another important aspect: the average value of the gain is proportional to ߤ, i.e., ॱ[࣡] ఓୀ = ܽॱ[࣡] ఓୀଵ , which unveils that even in a coded scenario where ߤ depends on the free distance of the code and is typically higher than one, we will also have potential performance improvements. To verify the asymptotic gains associated to the optimum detection in nonlinear OFDM schemes, we obtain the BER using the following approximation (that is valid considering that the minimum Euclidean distance is dominated by sequences that differ in only ߤ = 1 symbols)

ܲ = ∑ ܲ(࣡ = ࣡ )ܳ ൬ට࣡ ଶா ್ ே బ ൰, ( 18
)
with ࣡ being the value that the gain can takes (measured in linear units) and ܲ(࣡ = ࣡ ) is the probability of the gain ࣡ be equal to ࣡ . In Fig. 2 ൰. As is expected from the gains distribution, when the nonlinear distortion effects are stronger the potential improvements in the BER are higher too. At ܲ = 10 ିଷ and ݏ ெ /ߪ = 0.5 the asymptotic gain relative to the linear case is around 2.6 dB. When we consider a FDF to remove the out-of-band energy inherent to the nonlinear operation, the gains are lower, which is an expected result since less subcarriers are considered in the computation of the Euclidean distance, i.e., ࣞ ே ଶ is computed only on the ܰ in-band subcarriers. However, even considering filtered sequences, the asymptotic gain can reach a value around 1.3 dB.

Sub-optimal Receiver and its Performance Results

The potential performance improvements unveiled in the previous section are inherent to the optimum detection, where the optimum receiver selects its estimate by comparing the received signal with all the possible transmitted signals. The computational load of this method is clearly too high, even considering a moderate number of subcarriers and small constellations. In this section we propose a sub-optimal receiver whose the main idea is to compare the received signal with only a set of all the possible transmitted signals. The main purpose of this receiver is achieve performance improvements closest the ones obtained by the optimum detection but, at the same time, reduce drastically the complexity associated with this type of detection. It makes uses of the fact that typically the transmitted signal differs in a few bits from the received signal and, thus, it is likely that we have the optimal sequence within the set of the tested ones without make all the possible comparisons. Our sub-optimal receiver starts its decision process by taking the hard decisions associated to the received signal. After that, it changes the first bit, modulates the resulting sequence and submits it to the same blocks whereby the received signal was submitted. Then, the Euclidean distance between the resultant signal and the received signal is evaluated. If this distance reduces, the bit modification is maintained, if not, the bit returns to its original value. This process is repeated for all the log ଶ ܰ)ܯ( bits that compose the data sequence. Moreover, it and can restart ܭ times, since the sub-optimum estimate is modified during the process. Clearly, the complexity of this algorithm is very reduced when compared to the full optimum receiver, since it will test only log ଶ (ℳ)ܰܭ sequences. An important metric to evaluate the performance of our sub-optimal receiver is the difference between its performance and the performance of the optimum receiver. However, the very high complexity of this receiver (it must analyze 2 ୪୭ మ (ℳ)ே sequences before taking a decision) makes its simulation inviable. To obtain an approximate optimum performance, we considered to apply the behavior of the sub-optimal receiver but starting with the transmitted sequence instead of the hard-decisions sequence. Then, the selected sequence is the sequence between: the hard-decisions sequence, the transmitted sequence, or one of their modifications that has the lower Euclidean distance relatively to the received signal. Fig. 3 presents the BER performance of the sub-optimal receiver described above. The considered OFDM signal has ܰ = 64 useful subcarriers with QPSK constelladions and an oversampling factor ܯ = 4. The complex data symbols were mapped under a Gray mapping rule. The nonlinear device corresponds to an ideal envelope clipping with a normalized clipping level ݏ ெ /ߪ = 1.0. We considered an ideal AWGN channel and ܰ ோ௫ -order diversity at reception. From Fig. 4 it is clear that the sub-optimal receiver improves the performance of conventional nonlinear OFDM schemes. Without diversity and at ܲ = 10 ିଷ , the sub-optimal receiver presents a gain of approximately 1.3 dB relatively to the linear OFDM. When ܰ ோ௫ = 2 the gain is around 1.1 dB. It is also important to note the low difference between the sub-optimal receiver performance and the approximate optimum performance. This means that even testing only a small part of all the possible transmitted signals, the performance is still close to the one potentially obtained by the optimum receiver. In Fig. 4 it is shown the BER of the sub-optimal receiver considering a XTAP channel with ܰ ோ௬ = 32 rays of uncorrelated Rayleigh fading and ܰ ோ௫ -order diversity at reception. From Fig. 5, it is clear that independently of the diversity order, in a frequency-selective channel the gains are higher than in a ideal AWGN channel. For instance, at ܲ = 10 ିଶ , the gain the sub-optimal receiver is around 5.4 dB relatively to a linear OFDM transmission and 10.3 dB relatively to the conventional nonlinear OFDM receiver. These higher gains can be explained due to the diversity associated to the selectivity of the channel that can be enhanced by the diversity inherent to the nonlinear distortion effects. It is also important to remark that the approximate performance of the optimum receiver is much better than the performance of our sub-optimal receiver which unveils that in a frequency-selective channel an increased complexity means better performances, however, even with a reduced complexity receiver, we have remarkable improvements relatively to the conventional nonlinear OFDM schemes.

Conclusions and Further Work

In this paper we considered the nonlinear distortion effects as useful information that can improve the performance of OFDM schemes. The adequate receiver to explore this information is the optimum receiver. However, its complexity is too high for practical applications. To overcome this problem, we present a sub-optimal receiver whose the performance is very close to the one that can be obtained by the optimum receiver but has much lower complexity. We include results with diversity and conclude that the sub-optimal detection also allows a performance improvement in these type of scenarios.

The guidelines for future work can pass to theoretically derive the potential gains of the optimum (and sub-optimum) receivers under AWGN and frequency-selective channels.

Another important open question is to theoretically quantify what is the impact of the filtering operation in the potential gains of the optimum-based detection.

Fig. 1 .

 1 Fig. 1. Basic nonlinear OFDM scheme.

  4-QAM). The time-domain version ܁ is obtained through an Inverse Discrete Fourier Transform (IDFT), i.e., ܛ = ۴ ିଵ ܁ = ݏ[ ݏ ଵ . .. ݏ ேெିଵ ] ் ∈ ℂ ேெ , where ۴ represents the Discrete Fourier Transform (DFT) matrix with the (݊,݊′) element defined by ܨ

Fig. 1

 1 is shown the probability density function (PDF) of (17) without considering an FDF (i.e., the diagonal of ۵ is zeros) and considering OFDM signals with ܰ = 64 useful subcarriers distorted by an ideal envelope clipper with normalized clipping level ݏ ெ /ߪ. It is assumed an ideal AWGN channel (i.e., ܪ| | = 1 ∀ ݇ ) and ߤ different symbols between ܁ (ଶ) and ܁ (ଵ) .

Fig. 2 .

 2 Fig. 2. Distribution of the asymptotic gain for different values of ݏ ெ /ߪ and ߤ.

  it is shown the BER obtained using (18) and considering OFDM signals with ܰ = 64 subcarriers and different normalized clipping levels ݏ ெ /ߪ.

Fig. 3 .

 3 Fig. 3. Approximate BER for different values of ݏ ெ /ߪ with and without FDF. From the Fig. 3 it is clear that in the case of linear OFDM, ܲ(࣡ = 1) = 1 which means that ࣞ ே ଶ = ࣞ ଶ = ܧ4 and the approximate BER of (18) is equal to the theoretical linear OFDM BER given by ܲ = ܳ ൬ට ଶா ್ ே బ

Fig. 4 .

 4 Fig. 4. Sub-Optimal receiver and approximate pptimum receiver BER in an ideal AWGN channel considering ܰ ோ௫ -order diversity.

Fig. 5 .

 5 Fig. 5. Sub-Optimal receiver and Approximate Optimum receiver BER in a frequency-selective channel considering ܰ ோ௫ -order diversity.

1 This work was supported by the FCT/MEC projects CTS PEst-OE/EEI/UI0066/ 2011, IT PEst-OE/EEI/LA0008/ 2013, OPPORTUNISTIC-CR PTDC/EEA-TEL/115981/ 2009, ADCOD PTDC/EEA-TEL/099973/2008, ADIN PTDC/EEI-TEL/2990/2012 and Femtocells PTDC/EEA-TEL/120666/2010 as well as by grant SFRH/BD/90997/2012.


Slavisa Tomic
email: s.tomic@campus.fct.unl.pt

Marko Beko
email: mbeko@uninova.pt

Rui Dinis
email: rdinis@fct.unl.pt

Miroslava Raspopovic
email: miroslava.raspopovic@metropolitan.ac.rs

Distributed RSS-based Localization in Wireless Sensor Networks with Asynchronous Node Communication

Keywords: Wireless localization, wireless sensor network (WSN), received signal strength (RSS), second-order cone programming problem (SOCP), cooperative localization, distributed localization

In this paper we address the node localization problem in large-scale wireless sensor networks (WSNs) by using the received signal strength (RSS) measurements. According to the conventional path loss model, we first pose the maximum likelihood (ML) problem. The ML-based solutions are of particular importance due to their asymptotically optimal performance (for large enough data records). However, the ML problem is highly non-linear and non-convex, which makes the search for the globally optimal solution difficult. To overcome the non-linearity and the non-convexity of the objective function, we propose an efficient second-order cone programming (SOCP) relaxation, which solves the node localization problem in a completely distributed manner. We investigate both synchronous and asynchronous node communication cases. Computer simulations show that the proposed approach works well in various scenarios, and efficiently solves the localization problem. Moreover, simulation results show that the performance of the proposed approach does not deteriorate when synchronous node communication is not feasible.

Introduction

Wireless sensor network (WSN) consists of a large number of low-power sensor nodes that have some sensing, processing and communication capabilities. WSNs find application in various areas, and the capability to accurately locate all sensor nodes in the network is essential for many of them (e.g. monitoring, military operations, rescue missions, etc.). In general, sensor nodes can be classified as anchor and target (source) nodes [START_REF] Destino | Positioning in Wireless Networks: Noncooperative and Cooperative Algorithms[END_REF]. The positions of anchor nodes are known a priori (usually measured by global positioning system (GPS) or manually), while the positions of target nodes are yet to be determined. For economic or other practical reasons, only a small fraction of nodes are set to be anchor nodes, hence, an efficient algorithm for node localization is necessary for WSNs.

Instead of the use of GPS system, which is very expensive and limited to outdoor environments, algorithms that rely on distance measurements between neighboring nodes are emerging recently [START_REF] Ouyang | Received Signal Strength-based Wireless Localization via Semidefinite Programming: Noncooperative and Cooperative schemes[END_REF], [START_REF] Wang | A New Approach to Sensor Node Localization Using RSS Measurements in Wireless Sensor Networks[END_REF], [START_REF] Wang | On Received-Signal-Strength Based Localization with Unknown Transmit Power and Path Loss Exponent[END_REF], [START_REF] Tomic | RSS-based Localization in Wireless Sensor Networks using SOCP Relaxation[END_REF], [START_REF] Vaghefi | Cooperative Received Signal Strength-Based Sensor Localization With Unknown Transmit Powers[END_REF]. Depending on the available hardware, current distance-based algorithms extract the distance information from time-of-arrival (TOA), time-difference-of-arrival (TDOA), angle-of-arrival (AOA) or received signal strength (RSS) measurements [START_REF] Biswas | Semidefinite Programming for Ad Hoc Wireless Sensor Network Localization[END_REF]. Localization based on RSS measurements requires the least processing and communication (the least energy), and no specialized hardware [START_REF] Patwari | Location Estimation in Sensor Networks[END_REF], which makes it an attractive low-cost solution for the localization problem.

Localization algorithms can be executed in a centralized or a distributed fashion. The former approach assumes the existence of a fusion center which coordinates the network and performs all computational processing. This approach leads to large energy and bandwidth consumption, with a bottleneck around the fusion center and the computational complexity of such an approach grows with the increase of a number of nodes in the network. In many practical scenarios, it is not efficient or not possible for the nodes to share their private objective functions with a central processor or with each other [START_REF] Sundhar Ram | Distributed Subgradient Projection Algorithm for Convex Optimization[END_REF], which makes the distributed localization a more preferable solution. Even though it is sensitive to error propagation, distributed concept is energy-efficient, has low-computational complexity and high-scalability [START_REF] Destino | Positioning in Wireless Networks: Noncooperative and Cooperative Algorithms[END_REF]. In such an approach, the communication is possible only between the neighboring nodes, and the data associated with each node is always processed locally.

Recent RSS-based localization algorithms use a centralized concept to solve the sensor nodes localization problem [START_REF] Ouyang | Received Signal Strength-based Wireless Localization via Semidefinite Programming: Noncooperative and Cooperative schemes[END_REF], [START_REF] Wang | A New Approach to Sensor Node Localization Using RSS Measurements in Wireless Sensor Networks[END_REF], [START_REF] Wang | On Received-Signal-Strength Based Localization with Unknown Transmit Power and Path Loss Exponent[END_REF], [START_REF] Tomic | RSS-based Localization in Wireless Sensor Networks using SOCP Relaxation[END_REF], [START_REF] Vaghefi | Cooperative Received Signal Strength-Based Sensor Localization With Unknown Transmit Powers[END_REF]. A distributed approach for solving the RSS localization problem with synchronized node communication was introduced in [START_REF] Tomic | Distributed RSS-based Localization in Wireless Sensor Networks Using Convex Relaxation[END_REF], [START_REF] Cota-Ruiz | A Distributed Localization Algorithm for Wireless Sensor Networks Based on the Solution of Spatially-Constrained Local Problems[END_REF]. However, synchronized node communication requires the use of more sophisticated hardware in the network, which raises the cost of the network implementation.

In this work, we consider a large-scale RSS-based target localization problem, and we provide a solution that is completely based on a distributed approach. We investigate both the synchronous and asynchronous node communication scenarios, and propose a novel algorithm based on second-order cone programming (SOCP) relaxation. We first formulate the maximum likelihood (ML) optimization problem, which is highly non-linear and non-convex. To overcome these difficulties, we propose a SOCP relaxation approach to transform the original ML problem into a convex one, which can then be efficiently solved by interior-point algorithms [START_REF] Boyd | Convex Optimization[END_REF].

Relationship to Collective Awareness Systems

Sensor nodes in WSNs are deployed over a monitored area in order to acquire the desired information (such as temperature, wind speed, pressure, etc.). Nowadays, the basic concept of WSNs is used for the Internet-of-Things (IoT) in which all devices, objects and environments are connected through Internet to form the so-called smart environments [START_REF] Jiang | A Distributed RSS-Based Localization Using a Dynamic Circle Expanding Mechanism[END_REF]. Such systems are capable of harnessing collective intelligence for promoting innovation and taking better-informed and sustainability-aware decisions.

Being able to accurately estimate object's position is a key factor in a number of sensor network applications (such as energy-efficient routing, target tracking and detection). Combining the location information with other information collected inside the network enables us to link objects, people and knowledge and develop intelligent systems (collective awareness systems). Such systems may improve safety and efficiency in everyday life, since each individual device can make betterinformed and substantially-aware decisions to respond faster and better to the changes in dynamical environments (search and rescue missions, logistics in warehouses, etc.). Furthermore, such systems enhance sustainable growth, since they support new forms of social and business innovation.

Problem Formulation

We consider a p-dimensional (p≥ 2) WSN with M + N nodes, where M and N are the number of target and anchor nodes, respectively. The locations of the nodes are denoted as , … , , ା , … , ା . The considered WSN can also be seen as a connected graph, ܩ = (ܸ, ,)ܧ where V and E represent the set of nodes (vertices) and the set of node connections (edges) in the graph, respectively. Due to lifetime of the network or other physical limitations, each node has a limited communication range, ܴ. An edge exists between two nodes, i and j, if and only if they are within the communication range of each other, i.e. ܧ = ൛(݅, ݆) ∶ ฮ -ฮ ≤ ܴൟ. The set of neighbors of a target node i is defined as ߗ = ሼ݆ ∶ (݅, ݆) ∈ ܧሽ, and each neighboring node j is seen as an anchor node in the localization process by the i-th target node.

We assume that each target node i is given an initial estimation of its position, () .

For ease of expression, we define = ሾ ଵ , ଶ , … , ሿ as the 2 × M matrix of all target positions in the WSN; hence, () contains all initial estimations of the target positions.

Node i measures the received power from the signal transmitted by its neighboring node j, ܲ , which, under the log-normal shadowing, can be modeled as (in dBm) [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF], [START_REF] Sichitiu | Localization of wireless sensor networks with a mobile beacon[END_REF]

ܲ = ܲ -10ߛ log ଵ ฮ -ฮ ݀ + ݒ , ∀(݅, ݆) ∈ ,ܧ (1)
where ܲ is the power measured at a short reference distance ݀ ൫ฮ -ฮ ≥ ݀ ൯, γ is the path loss exponent, and ݒ is the log-normal shadowing term between the i-th target node and its neighbor node j, modeled as a zero-mean Gaussian random variable with variance ߪ ଶ , i.e. ݒ ~ ࣨ൫0, ߪ ଶ ൯.

Based on the measurements from (1), we derive the maximum likelihood (ML) estimator as

= argmin 1 ߪ ଶ (,)∈ா ቈܲ -ܲ + 10ߛ log ଵ ฮ -ฮ ݀ ଶ .
(

) 2
The least squares (LS) problem defined in ( 2) is non-linear and non-convex, hence, finding the globally optimal solution is difficult, since there may exist multiple local optima. Following [START_REF] Tomic | RSS-based Localization in Wireless Sensor Networks using SOCP Relaxation[END_REF], we show that the RSS measurement model in (1) can be approximated into a convex optimization problem, which can be solved by interior point algorithm [START_REF] Boyd | Convex Optimization[END_REF], and obtain the global solution.

For the sake of simplicity, in the further text we assume that ߪ ଶ = ߪ ଶ , ∀(݅, ݆) ∈ .ܧ Furthermore, we assume that the transmit power, ܲ ் , of each node in the network is equal, i.e. all nodes have identical ܲ and ܴ.

Distributed Localization Using SOCP Relaxation

Note that the objective function in (2) depends only on the positions and pairwise measurements between the neighboring nodes. This means that we can portion the objective function in (2) and perform the minimization independently by each target node, using only local information gathered from its neighbors. Instead of having a sink, which collects and processes the information from all nodes in WSN, we can divide the optimization problem into smaller sub-problems which can be carried out locally by each target node. This kind of problem execution is particularly suitable for large scale networks, since the number of nodes inside the network has no major impact on the neighborhood fragments, and hence, the computational complexity remains the same (no significant changes) as more nodes are added in the network [START_REF] Destino | Positioning in Wireless Networks: Noncooperative and Cooperative Algorithms[END_REF].

The price to pay for using this kind of approach is the increased energy consumption due to higher node communication, since distributive localization algorithms require repetition of the following phases:

• Communication phase: nodes in the network transmit their estimated position to their neighboring nodes. • Computation phase: each target node computes its position estimation based on the information gathered in the communication phase. If the information exchange between nodes is always performed at the beginning of each iteration, we say that the node communication is synchronous [START_REF] Srirangarajan | Distributed Sensor Network Localization Using SOCP Relaxation[END_REF]. However, due to some physical imperfections (e.g. internal clock of the nodes), synchronous node communication may not be feasible in practice. This is the reason why we also investigate asynchronous node communication. In such communication, a randomly chosen target node performs an update of its position estimation based on the available information from its neighbors at that particular moment (computation phase) and transmits the updated information to its neighbors thereupon (communication phase). This process is then repeated until each node reaches the maximum number of iterations, ܭ ௫ .

In the k-th iteration of the computation phase of our algorithm, each target node i solves a SOCP relaxation of the following problem:

ෝ () = argmin 1 ߪ ଶ ܲ -ܲ + 10ߛ log ଵ ฮ -ෝ (ିଵ) ฮ ݀ ൩ (,)∈ா ଶ , (3)
where ෝ (ିଵ) denotes the estimated position of the neighboring node j in the (k-1)the iteration. In the following text we will describe a SOCP relaxation method which approximates the problem in (3) into a convex one.

SOCP Relaxation

Approximating (1) as

ܲ ≈ ܲ -10ߛ log ଵ ቛ ି ೕ (ೖషభ) ቛ ௗ బ , ∀(݅, ݆) ∈ ,ܧ we get ߙ ฮ - (ିଵ) ฮ ଶ ≈ ݀ ଶ , (4)
where ߙ = 10 ು ೕ -ು బ ఱം . According to (4), the following LS estimation problem can be formulated 1 :

() = argmin ቀߙ ฮ - (ିଵ) ฮ ଶ -݀ ଶ ቁ ଶ . (,)∈ா (5
Problem in ( 6) is a SOCP problem, which can efficiently be solved by the CVX package [START_REF] Srirangarajan | Distributed Sensor Network Localization Using SOCP Relaxation[END_REF] for specifying and solving convex programs. We will refer to [START_REF] Vaghefi | Cooperative Received Signal Strength-Based Sensor Localization With Unknown Transmit Powers[END_REF] as "SOCP" in the further text. 1 We can rewrite (1) as

ೕ ି బ ହఊ + log ଵ ቛ ି ೕ (ೖషభ) ቛ మ ௗ బ మ = ௩ ೕ ହఊ , which corresponds to ߙ ቛ - (ିଵ) ቛ ଶ = ݀ ଶ 10 ೡ ೕ
ఱം . For sufficiently small noise, we can apply the first-order Taylor series expansion to the right-hand side of the previous expression to obtain ߙ ቛ -

(ିଵ) ቛ ଶ = ݀ ଶ ቀ1 + ୪୬ ଵ ହఊ ݒ ቁ, i.e. ߙ ቛ - (ିଵ) ቛ ଶ = ݀ ଶ + ߝ , where ߝ = ݀ ଶ ୪୬ ଵ ହఊ
ݒ is a zero-mean Gaussian random variable with variance

(୪୬ ଵ) మ ௗ బ ర ఙ మ ଶହఊ మ , i.e., ߝ ~ ࣨ ቀ0, (୪୬ ଵ) మ ௗ బ ర ఙ మ ଶହఊ మ ቁ.
Clearly, the corresponding LS estimator is given by (5).

Complexity Analysis

The trade-off between the computational complexity and the estimation accuracy is the most important feature of any localization algorithm, since it determines its applicability potential. Here, we consider only the worst case asymptotic complexity of an algorithm.

According to [START_REF] Pólik | Interior Point Methods for Nonlinear Optimization, Nonlinear Optimization[END_REF], the worst case complexity of the proposed "SOCP" approach is ܯ • ܭ ௫ • ࣩ((maxሼ|ߗ |ሽ) ଷ ), where |ߗ | represents the cardinality of the set Ω , for ݅ = 1, … , .ܯ As we can see from the above result, the worst case complexity of a distributive localization algorithm mainly depends on the neighborhood fragments (the biggest one). If the number of nodes in WSNs is increased, it will not significantly affect the size of the neighborhood fragments, which makes the distributed algorithms a desirable solution in highly-dense or large-scale networks.

Simulation Results

In this section, we present the computer simulation results in order to evaluate the performance of the proposed approach. The considered algorithms were solved by using the MATLAB package CVX [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF], where the solver is SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF].

Nowadays, flexibility and adaptability of a network are very important features in practical applications; hence we consider a random deployment of the nodes. All nodes were randomly deployed inside a square region of length ܤ = 30 m in each Monte Carlo ܯ( ) run. In order to make the comparison fair, we first obtained ܯ = 500 nodes positions, and we applied the proposed approaches for those scenarios. In each ܯ run, we made sure that the network graph is connected. The path loss exponent is ߛ = 3, the reference distance ݀ = 1 m, the reference power ܲ = -10 dBm, and the communication range of a node ܴ = ܤ 5 ⁄ m. We assumed that the initial estimation of the target positions, () , is in the intersection of the diagonals of the square area, and that ܭ ௫ = 200. We assume that one iteration step is completed after ܯ nodes compute and transmit their position estimations, for both synchronous and asynchronous node communication. As the performance metric we used the normalized root mean square error (NRMSE), defined as

NRMSE =ට ଵ ெ ெ ∑ ∑ ฮ -ෝ ฮ ଶ ெ ୀଵ ெ ୀଵ , (6)
where ෝ denotes the estimate of the true location of the j-th target in the i-th Monte Carlo run, .

The NRMSE performance of the proposed method versus number of iterations for synchronous and asynchronous node communication, and variable ܰ is depicted in Fig. 1. From Fig. 1 we can see improvement of NRMSE performance as the number of iterations is increased, as expected. Further, we can see that NRMSE performance improves as ܰ is increased, as expected. Lower estimation accuracy is achieved with asynchronous than with synchronous node communication in the early phase of the algorithm. However, we can see that the asymptotical performance of the proposed approach is the same for both asynchronous and synchronous node communication. Furthermore, Fig. 1 exhibits that all major improvements in the performance occur until approximately 80 iterations, and we can conclude that our algorithm converges after this number of iterations, for the considered scenarios. The NRMSE performance of the proposed method versus number of iterations for synchronous and asynchronous node communication, and variable ܯ is depicted in Fig. 2. We can see from Fig. 2 that NRMSE performance of the proposed approach does not degrade as more target nodes are added in the network. As it was anticipated, estimation accuracy is weaker in the early phase of the algorithm for higher ,ܯ and the proposed algorithm converges slower for this setting. Fig. 2 exhibits better NRMSE performance of the proposed approach with synchronized than with asynchronous node communication after the first few iterations. However, asymptotical performance is the same for both types of node communication.

Conclusions

In this we investigated the RSS-based sensor localization problem in largescale WSNs, which we solved in a completely distributed fashion. We considered both synchronous and asynchronous node communication scenarios, and we proposed a distributed algorithm that is based on SOCP relaxation technique. Due to practical demands, such as flexibility and adaptability of the network, randomly generated WSN were taken into consideration. Simulation results show that the proposed approach efficiently solves the sensor localization problem for different settings. Moreover, simulation results show that, even though synchronous node communication is preferred over asynchronous when only few iteration steps are allowed, the asymptotical performance of the proposed approach does not suggest any preference between the mentioned types of node communication.

  = ݔ‖ ‖ ଶ , and ࢠ = ൣݖ ൧, where ݖ = ߣ ฮ -(݅, ݆) ∈ .ܧ Introduce an epigraph variable ,ݐ and apply second-order cone constraint (SOCC) to obtain a convex optimization problem: , ݆) ∈ ,ܧ ‖2ࢠ; ݐ -1‖ ≤ ݐ + 1, ‖2 ; ݕ -1‖ ≤ ݕ + 1.

Fig. 1 .

 1 Fig. 1. Comparison of the performance of the proposed method for synchronous and asynchronous node communication: NRMSE versus number of iterations for variable ܰ.

Fig. 2 .

 2 Fig. 2. Comparison of the performance of the proposed method for synchronous and asynchronous node communication: NRMSE versus number of iterations: (a) ܯ = 40, (b) ܯ = 60.

Acknowledgments. This work was partially supported by Fundação para a Ciência e Tecnologia under Projects PEst-OE/EEI/UI0066/2011, PTDC/EEATEL/099973/ 2008-ADCOD, PTDC/EEA-TEL/115981/2009-OPPORTUNISTIC-CR, PTDC/EEI-TEL/2990/2012-ADIN and PTDC/EEA-TEL/120666/2010-Femtocells, and Ciência 2008 Post-Doctoral Research grant. M. Beko is also a collaborative member of Instituto de Sistemas e Robótica -Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, and SITILabs/ILIND/COFAC: R&D Unit of Informatics Systems and Technologies, Instituto Lusófono de Investigação e Desenvolvimento (ILIND), COFAC, Lisboa, Portugal.


Miguel Duarte

Antonio Furtado

M Luis

Luis Bernardo

Rui Dinis

Rodolfo Oliveira

Practical Assessment of Energy-Based Sensing through Software Defined Radio Devices

Keywords: Software Defined Radio, Energy Sensing, System Performance

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Cognitive Radio (CR) has been proposed as an effective answer to alleviate the increasing demand for radio spectrum [START_REF] Yucek | A survey of spectrum sensing algorithms for cognitive radio applications[END_REF]. CR nodes, usually denominated Secondary Users (SUs) due to its non-licensed operation, must be aware of the activity of the licensed users, denominated Primary Users (PUs), in order to dynamically access the spectrum without causing them harmful interference.

Spectrum Sensing (SS) aims at detecting the availability of vacant portions (holes) of spectrum and has been a topic of considerable research over the last years [START_REF] Yucek | A survey of spectrum sensing algorithms for cognitive radio applications[END_REF]. It plays a central role in CR systems. The traditional SS techniques include Waveformbased sensing (WBS) [START_REF] Zahedi-Ghasabeh | Spectrum Sensing of OFDM Waveforms Using Embedded Pilots in the Presence of Impairments[END_REF], a coherent technique that consists on correlating the received signal with a priori known set of different waveform patterns; Matched Filter-based sensing (MFBS) [START_REF] Bouzegzi | Matched Filter Based Algorithm for Blind Recognition of OFDM Systems[END_REF], an optimal sensing scheme where the received signal is also correlated with a copy of the transmitted one; and Cyclostationaritybased sensing (CBS) [START_REF] Al-Habashna | Cyclostationarity-Based Detection of LTE OFDM Signals for Cognitive Radio Systems[END_REF], a technique that exploits the periodic characteristics of the received signals, i.e., carrier tones, pilot sequences, etc. MFBS assumes prior knowledge of the primary's signal, while WBS assumes that the received signal matches with one of the patterns previously known. This means that these sensing techniques are not feasible in some bands, where several communication technologies may operate without a priori knowledge. On the other hand, CBS is impracticable for signals that do not exhibit cyclostationarity properties.

Energy-based sensing (EBS) [START_REF] Urkowitz | Energy Detection of Unknown Deterministic Signals[END_REF], [START_REF] Ghasemi | Optimization of Spectrum Sensing for Opportunistic Spectrum Access in Cognitive Radio Networks[END_REF] is the simplest spectrum sensing technique and its main advantage is related with the fact that it does not need any a priori knowledge of PU's signal. At the same time, it is well known that EBS can exhibit low performance in specific comparative scenarios [START_REF] Bhargavi | Performance comparison of energy, matched-filter and cyclostationarity-based spectrum sensing[END_REF], or when noise's variance is unknown or very large. EBS has been studied in several CR scenarios, namely on local and cooperative sensing schemes [START_REF] Yucek | A survey of spectrum sensing algorithms for cognitive radio applications[END_REF]. More recently, several EBS schemes adopting sub-Nyquist sampling have been proposed, which are advantageous in terms of the sensing duration [START_REF] Tian | Compressed Sensing for Wideband Cognitive Radios[END_REF].

This work evaluates the performance of the EBS technique in a real Software Defined Radio (SDR) platform. EBS' theoretical probabilities of detection and false alarm are compared with practical ones obtained with the SDR system. The comparison indicates that the theoretical probabilities are successfully validated.

Relationship to Collective Awareness Systems

As it is well known, collective systems massively rely on communication that is most of the times supported by wireless links. Wireless communication technologies allow high level of device's mobility, and are indicated for scenarios where mobility is required. Simultaneously, wireless technologies avoid the use of physical (wired) connections, being an effective solution for collective awareness systems relying on mobile players. As mentioned in the introduction, Cognitive Radio was proposed as a solution to alleviate the increasing demand for radio spectrum and, consequently, as more radio spectrum becomes available more wireless devices can be used. Consequently, CR will help the development of novel collective awareness systems by increasing the number of wireless devices that may access the network.

Energy-based Sensing

This work considers a cognitive radio network with a pair of PUs accessing the channel and a pair of SUs that access the channel in an opportunistic way. The considered scheme is similar with the one presented in [START_REF] Luis | Towards a Realistic Primary Users' Behavior in Single Transceiver Cognitive Networks[END_REF]. SUs are equipped with a single radio transceiver. However, because SUs are unable to distinguish SUs and PUs' transmissions, SU's operation cycle includes the sensing and transmission periods, which facilitates the synchronization of the sensing task. Sensing and transmission period durations are represented by ܶ ௌ ௌ and ܶ ௌ respectively, as illustrated in Fig. 1. given by the channel sampling period adopted in the spectrum sensing task. The first ܰ ௌ slots define the sensing period duration, and the remaining ones ሺܰ ௌ + 1 toܰ ் ሻ represent the transmission period duration. To distinguish between occupied and vacant spectrum bands, SUs sample the channel during the sensing period ܶ ௌ ௌ , and for each sample k two hypotheses can be distinguished

ℋ : ݔሺ݇ሻ = ݓሺ݇ሻ ℋ ଵଵ : ݔሺ݇ሻ = ݓሺ݇ሻ + ݏሺ݇ሻ ݇ = 1,2, … , ܰ ௌ ݇ = 1,2, … , ܰ ௌ , (1)
where ݏሺ݇ሻ denotes the signal transmitted by the PUs, with distribution ࣨሺߤ ௦ , ߪ ௦ ଶ ሻ. ݓሺ݇ሻis assumed to be a zero-mean variable with variance ߪ ଶ = 1, representing additive Gaussian white noise. The received signal is given by

ܻ~൜ ࣨሺߤ , σ ଶ ሻ, ℋ ࣨሺߤ + ߤ ௦ , σ ଶ + σ ଶ ௦ ሻ, ℋ ଵଵ . (2)
Therefore, for a single SU, the probability of detection ܲ ℋ భభ and the probability of false alarm ܲ ி ℋ బబ are represented by

ܲ ℋ భభ = ܲݎሺܻ > ߛ|ℋ ଵଵ ሻ = ࣫ ቆ ߛ -ሺߤ + ߤ ௦ ሻ σ ଶ + σ ଶ ௦ ቇ (3)
ܲ ி ℋ బబ = ܲݎሺܻ > ߛ|ℋ ሻ = ࣫ ൬ ߛ -ߤ σ ଶ ൰, (4)
where ࣫ሺ. ሻ is the tail probability of the standard normal distribution.

In the testbed L time samples are used to determine the fast Fourier Transform (FFT).

The FFT is represented in L frequency bins and ܮ/ݏܰ FFTs are computed during the sensing period (ܶ ௌ ௌ ). During this period, the average energy received per sample, ܻ , is defined as

ܻ = ଵ ே௦ ∑ ܺ ே௦/ ୀଵ , ( 5
)
where ܺ is the sum of the power of each individual FFT bin, given by

ܺ = ∑ ቚݔ ݁ ିଶగ ಽ ቚ ଶ ୀଵ , ݇ = 1, . . . .ܮ (6)
The decisions performed in the testbed are according to the following conditions,

ܥ = 0, ℋ ଵଵ |ܻ < ߛ ܥ ଵ = 1, ℋ ଵଵ |ܻ > ߛ (7)
ܤ = 0, ℋ |ܻ < ߛ ܤ ଵ = 1, ℋ |ܻ > ߛ,
where ܥ ଵ contributes for the measured probability of detection, while ܤ ଵ represents the case of false alarm.

System Implementation

Testbed

The EBS was implemented using the GR (GNURadio) software architecture [START_REF]GNURadio -Free Software Toolkit for[END_REF] and the USRP (Universal Software Radio Peripheral) hardware platform [10]. These tools are compliant with the notion of SDR (Software Defined Radio), mostly due to their flexibility when it comes to DSP (Digital Signal Processing). This flexibility opens up options when dealing with Digital Signal Processing. In this case, a range of FFT sizes can be chosen, instead of a fixed one like in most DSP devices. This allows for a variation in the ܰ ௌ ܰ ் ⁄ ratio.

The setup consisted of two USRP B100 devices connected to one computer. The USRPs are connected by a coaxial cable and two 30dB attenuators to achieve the desired signal-to-noise (SNR) values. One USRP implements a Primary User while the other behaves as the Secondary user, sensing for the presence of the first one. The basic blocks of the GR system are written in C++. These blocks link directly with the USRP and its FPGA. Python is used as a "glue" to connect these blocks and as a controller to the system flow (issuing start and stop commands). The script used to build the EBS was entirely written in python, also making use of python math library tools.

EBS Structure

A conventional EBS consists of a low pass filter, an A/D converter, a square-law device and an integrator (see Fig. 1). The detector model used in this work consists of an A/D converter followed by an L-Point FFT [START_REF] Ghasemi | Optimization of Spectrum Sensing for Opportunistic Spectrum Access in Cognitive Radio Networks[END_REF].The FFT provides the windowing needed to filter out unnecessary frequency bands. This data is then averaged (see Fig. 3), and the sensing output is determined according to ܥ ܥ/ ଵ or ܤ ܤ/ ଵ conditions (eq. ( 7) ).

Primary User Signal

The Primary signal is assumed to obey a Normal Distribution with higher amplitude than the noise distribution for sake of simplicity. The signal generation is achieved using the blocks in Fig. 5.

Comparison Results

Calibration and Parameter Calculation

This subsection validates the noise ݓሺ݇ሻ and PU's signal ݏሺ݇ሻ . The USRP's parameters used in the validation are shown in Table 1.

Practical Probabilities

Taking the x-axis values of Fig. 6 as a hypothetical range for the decision threshold, the probabilities ܲ and ܲ ி were computed from the conditions expressed in equation [START_REF] Bhargavi | Performance comparison of energy, matched-filter and cyclostationarity-based spectrum sensing[END_REF]. Basically, ܲ was computed as the ratio of the number of times the condition ܥ ଵ was observed over the total number of decisions. ܲ ி was similarly computed as the ratio of the number of times the condition ܤ ଵ was observed over the total number of decisions. These probabilities are shown in Fig. 7, where the achieved results are compared with the theoretical ones. As can be seen, the registered practical values approach the theoretical ones. This is, however, for a very long sensing period (512 ms). Fig. 8 shows the results achieved for a shorter sensing period ( ܰ ௌ =512000, which corresponds to ܶ ௌ ௌ = .)ݏ݉2.15 In this case the practical results underperform the case of longer sensing period, but the practical results roughly follow the theoretical trend.

Performance Evaluation

With the knowledge of the signal and noise statistics, a threshold can be defined so that the desired conditions (a certain pair of ܲ and ܲ ி ) can be achieved. By inspecting the theoretical curves for the probabilities, a threshold was defined where the ܲ ி was set to 5%. Since ܲ ி only depends on noise power, and not on noise plus signal, its value should remain constant for any SNR value. ܲ does not follow the same rationale, since it depends on SNR (eq. ( 3)). The following test evaluates the detection performance for a fixed decision threshold (3.7E-07) and varying the SNR from -10 to 2 dB. Different sensing periods were also tested, from 0.0125s to 0.512s. The obtained results are shown in Fig 9. As can be seen, the statistic test for sensing period durations longer than 0.05s start to be redundant, since they present almost the same performance. At 0dB we observe that the detector performance is high, and for ܶ ௌ ௌ > ݏ50.0 the miss-detection probability is lower than 10%. Due to lack of space we do not present false alarm probabilities for the same scenario, but for sensing period durations longer than 0.05s we have observed that the false alarm probability is always lower than 5%, which confirms the expected performance for the adopted decision threshold.

6

Conclusions

In this work we have addressed the assessment of energy detection for cognitive radio systems. We started to characterize the performance of an energy detector through practical USRP devices. The theoretical performance was successfully validated through practical results. For higher SNR values, the detection probability is high even for short sensing periods, achieving detection probabilities that approach 1. For lower SNR values a higher sensing time is required but even in these cases substantial performance can be achieved.

Fig. 1 .

 1 Fig. 1. SU's frame structure representing SU's operation cycle. The SU's frame, ܶ ி ௌ = ܶ ௌ ௌ + ܶ ௌ , contains ܰ ் slots, where each slot duration is

Fig. 2 .

 2 Fig. 2.Typical Energy-based Detector.

Fig. 3 .

 3 Fig. 3. FFT Energy-Based Detector.

Fig. 4 .

 4 Fig. 4. GNURadio Blocks.

Fig. 5 .

 5 Fig. 5. PU signal generator.

  ) and Signal plus Noise (ݓሺ݇ሻ + ݏሺ݇ሻ) signals were sampled during a 10 second interval. The histogram of the received energy is plotted in Fig.6. In this figure the green curve approximates the noise distribution, while the red one approximates the noise plus signal distribution. Since the overlapping area of the approximate distributions is negligible, a decision threshold can be defined to identify the two conditions with low error.

Fig. 6 .

 6 Fig. 6. Histogram of Signal and Signal plus Noise data and respective normal approximations. P is the number of occurrences.

Fig. 7 .

 7 Fig. 7. Theoretical and practical probabilities ܲ and ܲ ி (ܶ ௌ ௌ = .)ݏ215.0

Fig. 8 .

 8 Fig. 8. Theoretical and Real Probabilities (ܶ ௌ ௌ = 0.0512ݏሻ.

Fig. 9 .

 9 Fig. 9. ܲ curve for SNR [-10,2] dB.

Table 1 .

 1 USRP Configuration

Acknowledgments. This work was partially supported by the Portuguese Science and Technology Foundation under the projects PTDC/EEA-TEL/115981/2009, PTDC/EEA-TEL/120666/2010, PTDC/EEI-TEL/2990/2012, PEst-OE/EEI/UI0066/2011, PEst-OE/EEI/LA0008/2011, SFRH/BD/68367/2010 and SFRH/BD/68367/2010.


Hugo Serra

Rui Santos-Tavares

Nuno Paulino
email: nunop@uninova.pt

A Top-Down Optimization Methodology for SC Filter Circuit Design Using Varying Goal Specifications

Keywords: Computer-aided design, genetic algorithm, filter design, switchedcapacitor

The design of Switched-Capacitor (SC) filters can be an arduous process, which becomes even more complex when the high gain amplifier is replaced by a low gain amplifier or a voltage follower. This eliminates the virtual ground node, requiring the compensation of the parasitic capacitances during the design phase. This paper proposes an automatic procedure for the design of SC filters using low gain amplifiers, based on a Genetic Algorithm (GA) using hybrid cost functions with varying goal specifications. The cost function first uses equations to estimate the filter transfer function, the gain and settling-time of the amplifier and the RC time constants of the switches. This reduces the computation time, thus allowing the use of large populations to cover the entire design space. Once all specifications are met, the GA uses transient electrical simulations of the circuit in the cost functions, resulting in the accurate determination of the filter's transfer function and allowing the accurate compensation of the parasitic capacitances, obtaining the final design solution within a reasonable computation time.

Introduction

Analog filters are very important blocks in several electronic systems such as RF transceivers or sigma delta modulators. They allow selecting signals with different frequencies and eliminating unwanted signals.

The scaling-down of transistors in advanced deep-submicron CMOS technologies results in the reduction of the intrinsic gain (g m /g ds ) [START_REF] Perez | Performance enhanced op-amp for 65nm CMOS technologies and below[END_REF] and increases the variability, making the design of high gain high bandwidth opamps increasingly difficult. This limitation has large impact in the performance of filter circuits.

To avoid the difficulty of designing high gain amplifiers, SC filters can be implemented using low gain amplifiers [START_REF] Serra | A switched-capacitor biquad using a simple quasi-unity gain amplifier[END_REF]. However, this approach removes the circuit's virtual ground node. Without this node, parasitic insensitive SC branches cannot be used and the filter's transfer function becomes sensitive to the effects of parasitic s, which have to be compensated during the design phase of the filter.

This paper presents an automatic procedure for the compensation of the parasitic capacitances introduced by parasitic sensitive branches and a general design methodology for the design of SC filters, using low gain amplifiers, based on a GA using hybrid cost functions with varying goal specifications. The exact transfer function of a SC filter can be obtained by simulating the circuit's impulse response. This, however, results in a large computation time that limits the maximum size of the population of the GA. An alternative is to estimate the filter's transfer function using equations, resulting in a low computation effort and allowing the evaluation of a larger population in a reasonable time. This approach allows the design space of the filter to be completely explored. Once a solution is found, it is then possible to reduce the population size in the GA, obtaining the final design solution using the more computation intensive and more accurate simulation based cost functions. During this step, the optimization will also calculate the final values of the capacitances taking into consideration the parasitic capacitance values. This hybrid cost function (equation based/simulation based) allows a larger design space to be explored and obtaining a design solution with a good accuracy, while still having a low computation time.

Relationship to Collective Awareness Systems

Collective awareness systems are information and communication technology systems that allow the interaction of people, knowledge, and objects through a network. The development of these systems, which require faster, smaller, and less power consuming circuits, while performing more complex operations, is only possible using advanced nanometer CMOS technologies. Filter circuits are essential blocks in front-end transceivers that allow the data collection and also support the communication between the agents involved in a network. This paper describes a topdown optimization methodology for the design of SC filter's using low gain amplifiers. Using this optimization methodology, it is possible to quickly study new SC filters in order to find new topologies capable of addressing the more demanding specifications required for collective awareness systems.

Proposed Design Methodology

A software add-on was developed to implement the proposed methodology. This addon was integrated into an existing optimization software platform [START_REF] Santos-Tavares | Time-Domain Optimization of CMOS Amplifiers -Based on Distributed Genetic Algorithms[END_REF], [START_REF] Figueiredo | A two-stage fully differential inverter-based self-biased CMOS amplifier with high efficiency[END_REF], [START_REF] Santos-Tavares | Optimum sizing and compensation of two-stage CMOS amplifiers based on a time-domain approach[END_REF], based on the open-source circuit simulator, Ngspice [START_REF] Nenzi | Ngspice users manual version 25[END_REF]. Initially, a population consisting of N randomly created chromosomes is generated based on the maximum and minimum allowed values of each component, which defines the design space. The fitness of each chromosome is then calculated, based on the desired specifications, and sorted. A new population is created based on the genetic material of the best chromosomes and complemented with selection, cross-over and mutation operations among the remaining chromosomes. This process is repeated during a number of generations or until the desired specifications are met. The fitness is calculated using the exponential-based equations presented in [START_REF] De Melo | Design methodology for sigma-delta modulators based on a genetic algorithm using hybrid cost functions[END_REF].

The methodology uses a multi-step approach to obtain the optimized filter design. In the first step, the SC filter is optimized from an ideal standpoint using the equation that describes its transfer function. Once the desired specifications for the filter are met, the second step starts, where the tool focuses on optimizing the amplifier circuit at transistor level, in order to have enough gain and closed-loop bandwidth to replace the ideal gain obtained from the top-level optimization. After this step, it is necessary to compensate the effects of the amplifier's input and input to output parasitic capacitances in the SC filter. In the third step, the switches are optimized in order to have a RC time constant that is small enough to charge the capacitors in half the clock period. When these specifications are met the parasitic capacitances of the switches are compensated through a transient simulation, in Ngspice, to determine the fitness of each SC circuit at transistor level. During this step, the design space (the component's values allowed variations) is restricted to be around the best values obtained in the previous steps. After this final optimization step, the resulting component's values take into consideration all the parasitic effects in the circuit. In order to facilitate the convergence of the GA during the first step, the cost function uses variable goal specifications. By starting the optimization with an "easier" goal and then converging the goal to the desired specifications, it is possible speed up the convergence process and obtaining a solution in fewer generations.

A 2nd Order Low-pass SC Filter

A 2 nd order low-pass biquad filter (Fig. 1) is presented as a proof-of-concept for the proposed design method. This architecture is based on the continuous low-pass Sallen-Key topology replacing the resistors with parallel SC branches. Capacitors C 2 , C B and C B2 were added to the architecture to facilitate the compensation of the parasitic capacitances in each node.

First Step -Top-level Optimization

In the top level optimization, all the filter components are considered ideal (capacitors, switches and amplifier). The chromosome used in this step, which is shown in Table 1, contains the values of the capacitors and the desired gain for the amplifier. Once the values of the capacitors are generated, they are scaled down based on the smallest capacitor and the desired minimum value. The optimization will then try to find a solution that best fits the design specifications, while minimizing the gain value, because this will make the optimization of the amplifier circuit, in the second step, easier, as this value will be used as a design specification. The optimization will also try to minimize the total value of the capacitors.

Top level C R1 C R2 C 1 C 2 C B C B2 G Amplifier W 1 L 1 W 2 L 2 W 3 L 3 W 4 L 4 W 5 L 5 I D V cm Switches W s1 W s2 W s3 W s4 L s
The design specifications in this step are given in the frequency domain (Fig. 2), and are evaluated based on the ideal transfer function of the filter, which was obtained from a charge conservation perspective, considering that the filter's output is sampled at the end of phase Φ 1 . The passband specification is controlled by the indicators F passlow and F cutoff , in which the signals attenuation must be delimited between A max1 and A max2 , i.e., these four indicators are used to obtain the desired low frequency gain of the filter and move the poles to frequencies around or above F cutoff . To place the poles as close as possible to the cutoff frequency, two additional indicators are used (F specific and A min1 ). By indicating the desired attenuation at a specific frequency we are forcing the poles to move to the cutoff frequency. The indicators F s /2 and A min2 are used to reinforce the pole placement at the cutoff frequency. In this optimization step, the evaluation of the chromosomes is based on a variable goal equation (1), i.e., the specifications (goals) of the indicators (A min1 and A min2 ) will start with an initial value (Ind I ) and converge as the generations progress, following an exponential evolution, to the final value (Ind F ). The speed at which the indicator converges to the final value is controlled by the variable w. The variables G C and G S represent the current generation and the generation at which the indicator reaches its final value at the latest, respectively. Fig. 2 shows how the indicator A min1 behaves when using the variable goal equation. While converging from Ind I to Ind F , the optimizer will find the chromosome that satisfies the passband specification and that has an attenuation at F specific closest to the variable goal equation, at the current generation. Once the variable goal equation reaches the final value (G C = G S ), the optimizer will find the chromosome that satisfies the passband specification and that has the smallest A min1 at F specific . By temporarily softening the specifications we are increasing the probability of converging to the best solution.

Variable Goal Specification = Ind + ሺInd ୍ -Ind ሻe ିହ ୵ሺୋ ౙ /ୋ ౩ ሻ ሺ1 1 1 1ሻ

Second Step -Amplifier Optimization

In this step, the amplifier circuit is designed to have the gain value obtained in the previous step and optimized following the time-domain methodology that was presented in [START_REF] Santos-Tavares | Optimum sizing and compensation of two-stage CMOS amplifiers based on a time-domain approach[END_REF], which ensures that if a given settling error is reached within the desired settling time, then the amplifier has enough open-loop gain, closed-loop bandwidth, and slew-rate. This methodology simplifies the circuit's evaluation and helps the optimization process to converge faster. To calculate the settling time it is necessary to obtain the closed-loop circuit transfer functions during phase Φ 2 and Φ 1 (Fig. 3a and Fig. 3b), and apply the inverse Laplace transform multiplied by a unity input step (1/s). The closed-loop transfer function can be obtained by replacing the amplifier block G(s) with its equivalent medium frequency small signal model. Both transfer functions need to be calculated because charge flows to the filter's output in both clock phases. V CR2 Φ 1 and V C2 Φ 2 represent the charge stored in capacitor C R2 during phase Φ 1 and in capacitor C 2 during phase Φ 2 , respectively, and can be calculated from [START_REF] Serra | A switched-capacitor biquad using a simple quasi-unity gain amplifier[END_REF], where V amp represents the amplitude of the input signal.

V େଶ మ = C ୖଵ V ୟ୫୮ /ሺC ଵ + C ଶ + C ୖଵ ሻ ሺ2 2 2 2ሻ V ୈଶ భ = ሺC ଵ + C ଶ ሻV େଶ మ /ሺC ଵ + C ଶ + C ୖଶ ሻ
The amplifier circuit used is shown in Fig. 4. Due to its differential configuration, this amplifier is capable of achieving gains larger than one. The chromosome for this step (Table 1) contains the width and length of the amplifier's transistors, the biasing currents and the input common-mode voltage. The design specifications which are evaluated during this step are the average between the amplifier's input and output common mode voltages, which is evaluated based on DC Operating Point (OP) simulation; the closed-loop gain, which is calculated from the step response with s = 0; the settling time during phase Φ 2 and Φ 1 , also calculated from the step response; and the amplifier's current and area.

During this optimization step, the amplifier's input and the input to output parasitic capacitances are calculated based on a DC OP simulation. Once the specifications are met the corresponding parasitic capacitances are stored, to be later compensated into capacitor C B and C B2 . Considering the amplifier circuit from Fig. 4, the compensated capacitor values can be obtained from [START_REF] Santos-Tavares | Time-Domain Optimization of CMOS Amplifiers -Based on Distributed Genetic Algorithms[END_REF], where G 1 and G 2 represent the voltage gain between the gate and drain and the voltage gain between the gate and the source of transistor M 1 , respectively.

C ଶ ᇱ = C ଶ -C ୠଶ -C ୱଶ ሺ3 3 3 3ሻ C ᇱ = C -C ୠଵ -ሺ1 + G ଵ ሻC ୢଵ -ሺ1 + G ଶ ሻC ୱଵ -C ୢଶ -C ହ

Third Step -Switch Sizing Optimization

In the third step, the switches dimensions are optimized. The chromosome for this step (Table 1) contains the widths and lengths of the transistors operating as switches.

Even though it is possible to use the same methodology as in the previous step and obtain the exact expressions for the time constants, it is computationally heavy and should only be used when first order RC approximations do not give sufficiently accurate results. In this filter the only switch that requires the computationally heavy method is switch S 2 , because it is the switch most susceptible to the charge effect from the amplifier at transistor level. The settling time of this switch can be obtained from the closed-loop circuit shown in Fig. 3a considering node V c2 Φ2 as the output instead of V out Φ2 . The remaining RC time constants are calculated using first order approximations. The fitness equation of this step is based on the RC time constants/settling time and the overall switch area.

Clock-boosted phases were used to reduce the switches conductance non-linearity and overall value. During this step the value of the switches parasitic capacitances are calculated based on a DC OP simulation and stored, so that they can be used in the next step to compensate their effect on the filter.

Fourth Step -Top-Level Adjustment

The last optimization step is used to validate the equation-based design through electrical simulation and to fine-tune the capacitor values. The filter transfer function is now evaluated by running an electrical transient time simulation of the impulse response of the complete filter circuit. Since this procedure is very computational intensive and takes some time, it is difficult to use large populations or a large number of generations in this step. Due to this limitation, the design space is restricted to be around the values of the best chromosome obtained previously. Moreover, the capacitor values are adjusted to take into consideration the parasitic capacitances previously calculated. Unlike the calculation of the amplifier's parasitic capacitances which is relatively straightforward, due to their value being constant for a fixed voltage, the value of the switch's parasitic capacitances can have a variation of over 5 times depending if the switch is ON or OFF, making the compensation process more complex. Equation (4) was implemented to overcome this problem. The first generation chromosomes are all identical apart from a random compensation factor (CF) which varies between 0 and 5. From this initial population, it is possible to converge to an optimized solution very rapidly, i.e., within a few generations.

C ୡ୭୫୮ = C ୬୭୫ -CF C ୮ୟ୰ , 0 < ܨܥ < 5 ሺ4ሻ

Simulation Results

The biquadratic low-pass filter shown in Section 4 was designed in a standard 1.2 V 130 nm CMOS technology. The filter specifications used in this section are shown in Table 3. All simulations obtained from the optimizer were validated in Spectre using BSIM3V3.2.4 models of the transistors and MIMCAPS capacitor models (including parasitics).

In the first step, several optimizations were performed with and without the use of the varying goal specifications, on average, without using the varying goal specifications; it took the optimizer 96 generations to find a chromosome with fitness above 90%, and only 58 generations when using the varying goal specifications during the first 30 generations. In both cases, each generation was composed of 2500 chromosomes. The values obtained in this step were C R1 = 54.44 fF, C R2 = 50.00 fF, C 1 = 4530.80 fF, C 2 = 99.24 fF, C B = 473.95 fF, C B2 = 131.64 fF, and G = 1.

Table 2 shows the amplifier specifications, the simulation results and the best chromosome optimized in the second step, obtained after only 10 generations of 2500 chromosomes each. The results show that the continuous time equation-based methodology gives similar results to the ones obtained through Cadence. Based on the DC OP simulation of the amplifier using the best chromosome, the values of the parasitic capacitances were calculated and compensated into capacitors C B and C B2 . The filter's switches were also optimized in order to have time constants small enough to charge the capacitors in less than half a clock period. The length chosen for the switches was fixed at L = 120 nm and the optimized widths obtained were W1 = W3 = W4 = 240 nm and W2 = 300 nm. The optimizer took 15 generations of 2500 chromosomes to find the smallest widths for the switches while maintaining times constants small enough to charge the capacitors in less than half a clock period.

Table 3 shows the results obtained during each step of the optimization process. From rows (B) and (C), it can be concluded that the equation method for the automatic compensation of the amplifier's parasitic capacitances is very accurate. From (E) and (F), it can be seen that the open source simulator Ngspice gives very similar results to Cadence. In the last step, after 12 generations, the capacitor values obtained were C R1 = 53.14 fF, C R2 = 47.78 fF, 1 = 4754.51 fF, C 2 = 97.29 fF, C B = 459.30 fF, C B2 = 27.71 fF. Although a solution that satisfies all the specifications can be found after a single generation in the top-level adjustment, using equation ( 4), the optimization process continued for another 11 generations in order to try to find an even better solution than the one obtained after the first generation.

Conclusion

A design methodology for SC filters based on a GA using hybrid cost functions with varying goal specifications was proposed. The cost functions first use equations to obtain approximate solutions, and once all specifications are met, the GA uses the results of transient electrical simulations of the circuit in the cost functions, resulting in accurate transfer functions and allowing the compensation of parasitic capacitances within a reasonable amount of generations. Furthermore, it was concluded that using variable goal specifications reduces even further the amount of generations needed to obtain a solution within the design specifications.

Fig. 1 .

 1 Fig. 1. Low-pass SC biquad filter in single-ended configuration.

Fig. 2 .

 2 Fig. 2. Filter's variable goal specifications.

Fig. 3 .

 3 Fig. 3. Equivalent closed-loop circuit during clock phase (a) Φ 2 and (b) Φ 1 .

Fig. 4 .

 4 Fig. 4. Voltage-combiner amplifier with source degeneration.

Table 1 .

 1 Chromosomes used in each optimization step by the GA.

Table 2 .

 2 Amplifier specifications, simulation results, and best chromosome.

		Norm. closed-loop gain	Settling-Time Φ2 (%)	Settling-Time Φ1 (%)
	Specifications	3.8808		< 75.0			< 75.0
	Optimizer		3.8808		65.13			59.92
	Cadence		3.8808		65.20			55.20
	W1 (µm)	W2 (µm)	W3 (µm)	W4 (µm)	W5 (µm)	Power (µW)	CBcomp (fF)
	0.37	62.29	39.19	90.5	0.38	261.38	468.67
	L1 (nm)	L2 (nm)	L3 (nm)	L4 (nm)	L5 (nm)	Vcmi (mV)	CB2comp (fF)
	520	240	350	450	220	450		26.99

Table 3 .

 3 Specifications and simulation results of the low-pass SC filter.

	Fpasslow	Fcutoff	Fspecific	Fs / 2
	10 kHz 1 MHz 6 MHz 50 MHz


Fábio Passos
email: f.passos@campus.fct.unl.pt

Helena Mª

Fino
email: hfino@fct.unl.pt

Elisenda Roca

Single-Objective Optimization Methodology for the Design of RF Integrated Inductors

Keywords: Integrated Inductors, RF, Design, Optimization etc

Designing integrated inductors for RF applications is quite a challenging task due to the necessity of minimizing the parasitic effects arising from using today's technologies. The multiplicity of non-ideal effects to be minimized makes imperious the use of optimization-based design methodologies. In this paper a model-based optimization methodology is considered as a way of offering the designer the possibility to obtain inductors with maximum quality factor. The inductor model accounts for square, hexagonal or octagonal topologies. Furthermore tapered inductors are also accounted for in the proposed model. The use of the inductor model reduces the optimization time significantly. The validity of the results obtained is checked against electromagnetic (EM) simulations. As an application example, the particular case for the design of inductors for 2.4 GHz is illustrated.

Introduction

Integrated Inductors are a very important element on modern radio frequency (RF) design [START_REF] Ali | Electromagnetics for High-Speed Analog and Digital Communication Circuits[END_REF]. During the past few years, the design of integrated inductors has attracted much interest in both IC design and electronic design automation communities. In order to design an integrated inductor three main parameters must be considered by the designer [START_REF] Zhan; Sapatnekar | Optimization of integrated spiral inductors using sequential quadratic programming[END_REF]. One of the most critical parameter, if not the most critical one, is inductance value. Afterwards, quality factor (Q) and self-resonance frequency (SRF) [START_REF] Mohan | The Design, Modeling and Optimization of On-chip Inductor and Transformer Circuits[END_REF] are to be taken into consideration.

Given an inductor design, different approaches may be followed in order to evaluate these parameters. The most common and widely used are: electromagnetic (EM) simulations, surrogate models and lumped element models.

EM simulations are time consuming, i.e. a simulation of a single inductor in a visual environment with ADS Momentum [START_REF]Agilent technologies[END_REF] can last up to eight hours. However this is the most accurate simulation possible when designing inductors. Surrogate models i.e. [START_REF] Nieuwoudt | Variability-Aware Multilevel Integrated Spiral Inductor Synthesis[END_REF] are mathematically extensive, computing exhausting and they are not physically understandable, and they introduce errors due to the mathematical expressions. On the other hand, lumped element models are based on physical expressions, which allow a much easier interpretation of the models [START_REF] Passos | A Wideband Lumped-Element Model for Arbitrarily Shaped Integrated Inductors[END_REF]. Furthermore, as they rely on geometrical and technological parameters, design migration to new technologies is a straightforward process. In the end, these lumped element models are the option that allows the best trade-off between accuracy and simulation complexity. In order to obtain an inductor with desired impedance or the maximum quality factor possible, optimization based simulations must be used. It is possible to understand that if EM simulations and surrogate models are time prohibitive for only one inductor, to integrate these evaluation tools into optimization loops, is not a practicable solution. These leave designers with only one solution that is the usage of lumped element models. In this work the common π-model is used to model integrated inductors. After presenting the main contribution of the work to the implementation of collective Awareness systems, the inductor model is briefly described. Then, the integration of the inductor model into an optimization tools is presented. For the optimization a selection based differential evolution algorithm (SDBE) is used. Validity of the proposed methodology is performed by comparing the results of the optimization process for the design of an integrated inductor for 2.4 GHz against EM simulations. Finally conclusions are driven.

Relationship to Collective Awareness Systems

Collective Awareness systems link objects, people and knowledge in order to foster new forms of social and business innovation. The implementation of such systems relies on networks supporting the interconnections of a large number of heterogeneous cooperating devices. The development of systems has been made possible due to the rapid evolution of electronic technologies, which enable the implementation of ever more complex functions, in smaller and more rapid and less consuming circuits. To cope with the necessity of minimizing the power consumption of such systems, new design methodologies must be adopted so that the ever more stringent specifications may be attained. In the particular case of communications services, e.g. wireless communications devices, integrated inductors are becoming widely used in either Voltage Controlled Oscillators, LNAs and other RF blocks. The mains contribution of this paper is the definition of a single-objective modelbased optimization methodology for the design of integrated inductors. The main added value relies on the possibility for designing square, hexagonal or octagonal, tapered/no tapered inductors.

Integrated Inductor Model

The simple p-model is the most widely used for characterizing integrated inductor. Its applicability, however is limited to frequencies in the order of 1GHz. F The model used in this work is based on the segment model approach [START_REF] Koutsoyannopoulos | A generic CAD model for arbitrarily shaped and multi-layer integrated inductors on silicon substrates[END_REF]. The inductor is divided into segments and then each segment is characterized with the π lumped-element circuit shown in Fig. 1. This figure illustrates a typical square integrated inductor, where n is the number of turns, w, the width of the metal turn, s, the spacing between metal turns, D out , the outer diameter and, finally, D in , the inner diameter. The series branch of this model consists of L s , R s and C s . The series resistance, R s arises from metal resistivity of the inductor and is closely related to the quality factor, being a key issue for inductor modeling. The series feed-forward capacitance, C p , is usually considered as the overlap capacitance between the spirals and the underpass metal lines, also called C o [START_REF] Yue | Physical modeling of spiral inductors on silicon[END_REF]. However, as the minimum feature size of CMOS process continues to shrink, the spacing between metal spirals, s, can be reduced to a value similar to the distance to the underpass. Therefore the coupling capacitance between metal lines, C s , can increase its significance in the total capacitance of the device, and therefore in the self-resonance frequency (SRF). C p is then given by the sum of C s and C o . The capacitance C ox represents the oxide capacitance between the spiral and the substrate. The silicon substrate is modeled with C sub and R sub [START_REF] Lee | The Design of CMOS Radio-Frequency Integrated Circuits[END_REF]. A detailed explanation on how to evaluate L s may is presented in [START_REF] Passos | Analythical Characterization of Variable Width Integrated Spiral Inductors[END_REF]. For the remaining elements analytical expressions may be obtained in [START_REF] Passos | Lumped Element Model for Arbitrarily Shaped Integrated Inductors -A Statistical Analysis[END_REF].

For the sake of simplicity some physical aspects were not taken into account in the analytical characterization of the model elements. For example, the model does not take into account Eddy currents [START_REF] Yue | Physical modeling of spiral inductors on silicon[END_REF], thus for small inner diameters, the model is less accurate. Also, with a wider metal, the parasitic capacitances and fringing capacitances tend to increase and again, induce higher errors. Since this model is to be integrated into an optimization based design tool, the validity of the model along the overall design space was performed through a statistical analysis [START_REF] Passos | Lumped Element Model for Arbitrarily Shaped Integrated Inductors -A Statistical Analysis[END_REF].

Optimization of Integrated Inductors

This section discusses the optimization of integrated inductors in a 0.35 µm CMOS technology for 2.4 GHz. In this work an octagonal layout is simulated because it uses less metal to draw turns than square or hexagonal layouts, and can therefore achieve higher quality factors. This section presents a single-objective optimization using the selection based differential evolution (SBDE) algorithm is presented. The results of the optimization processes are validated against EM simulations. For inductor optimization, common design variables are the geometric parameters (i.e. number of turns, the inner diameter, the turn width and the spacing between turns) and the objectives are the inductor performances at a certain frequency (i.e. equivalent inductance, quality factor, self-resonance frequency and area). For the specific case of inductor optimization the SRF may not be explicitly defined as an objective. The guaranty that the SRF is well above the operating frequency may be obtained through the definition of some specific constraints such as the imposition that the inductor has to be in the plan bandwidth zone, i.e., the frequency band where the inductance value remains within 10% from the DC value [START_REF] Sieiro | Synthesis of compact planar inductors in LTCC technology[END_REF].

In the case of the single objective, if several objectives envisaged into one optimization process, it is possible to build an objective function reflecting the weighted sum of the several objective functions. The general constraints imposed in the optimization processes are:

• The inductance value is limited to only 5% error from the inductance desired at 2.4 GHz, • The inductance curve has to be in the plain bandwidth zone, • The quality factor has to be before the peak value, which means that the quality factor has to be in the positive slope region of the quality factor.

The single-objective optimization was made with the objective of finding an inductor with a given inductance value, while maximizing the quality factor and minimizing the area. This was possible through the weighted objective function shown in ( 1)

݁ݒ݅ݐ݆ܾܱܿ݁ = 7.0(-(݊݅ܯ • ܳ) + 0.3 • )ܽ݁ݎܣ (1)
Several simulations were made in order to obtain inductors with a desired inductance for 1nH to 5nH. All optimizations were done with 300 individuals and 300 iterations and the simulation time was about 300 seconds per simulation. The inductors obtained were then simulated with ADS Momentum, in order to check the validity of the results. The comparison values and curves are shown for each of the inductors obtained. The physical parameters of the inductors are given in Table 2, as well as the values for inductance and quality factor at 2.4 GHz. All the physical parameters are given in µm, and the inductance in nH. The relative error (ε) is given in %. The physical parameters also had to be limited so that the model is accurate over the design space considered. So, the geometric parameters were limited according to the values shown in Table 1. Layout and fabrication processes impose limits on the precision of the physical dimensions, and so not all values in a range can be used, only some discrete values. These values are defined by the grid of the optimization process and are related to each technology. It is possible to conclude that the model can be used in RF frequency range in the design space where the optimization process was developed. The errors observed in the optimization process are usually quite acceptable. It is also possible to observe that the inductor that has the highest error has the Din and the w close to its boundaries, which increases the error values.

Conclusions and Future Work

In this work an efficient lumped-element model to characterize integrated inductors is presented. The model uses analytical expressions to evaluate the lumped elements values. The model is able to predict the inductance and the quality factor values with relative accuracy. The model was also integrated into an optimization process and the results validated against EM simulation. It has been proved that the model is suitable for RF circuit design and its best feature is the ability to save time in the design process. As future work, the model can be fitted to smaller frequency ranges in order to obtain higher accuracy. The complexity of the model can also be increased, adding elements to the equivalent circuit.

Fig. 1 .

 1 Fig.1. Square inductor, its physical parameters and the π model.

Table 1 .

 1 Restrictions on the inductor physical parameters.

	Parameter	Minimum Maximum
	n	2	7
	ܖܑ (µm)	100	250
	w (µm)	5	10

Table 2 .

 2 Comparison between inductance and quality factor for optimized inductors.

	n w		s			ε			ε
	1 nH 2 10	123	2.5	1.04	1.01	2.97	8.80	9.07	2.98
	2 nH 3 10	106	2.5	2.10	1.87	12.3	11.17 9.85	13.4
	3 nH 3 10	144	2.5	2.80	2.66	5.26	11.32 10.35 9.37
	4 nH 3 8.03 185	2.5	3.77	3.71	1.62	10.24 1.10	1.39
	5 nH 4 5.63 143	2.5	5.03	4.90	2.65	9.21	9.55	3.56