Claude Berrou

Sandrine Vaton

Michel J Ez

Catherine Douillard

Computing the Minimum Distance of Linear Codes by the Error Impulse Method

Keywords: minimum distance, error oor, linear code, convolutional, concatenated, turbo code

A new method for computing the minimum distances of linear error-correcting codes is proposed and justi ed. Unlike classical techniques that rely on exhaustive o r partial enumeration of codewords, this new method is based on the ability of the Soft-In decoder to overcome Error Impulse input patterns. It is shown that the maximum magnitude of the Error Impulse that can be corrected by the decoder is directly related to the minimum distance. This leads to a very fast algorithm to obtain minimum distances of any linear code whatever the block size and the code rate considered. In particular, the method can be advantageously worked out for turbo-like concatenated codes.

Introduction

The asymptotic performance of a linear block error correcting code at very low error rates is completely determined by its minimum distance d min and by the multiplicity n(d min ) if one considers the Frame Error Ratio (FER).

Consider a C(n k) linear block code (k is the number of information bits, n is the size of the codewords). Denote by d min its minimum distance for any distance d d min , denote by n(d) its multiplicity ( n umber of codewords with weight d). On the Gaussian channel with maximum-likelihood (ML) decoding the Frame Error Ratio (FER) can be upperbounded by the union bound: (1)

where erfc(x) is the complementary error function: erfc(x) = 2 = p R 1 x exp(;t 2 ) dt.

At l o w error rates, this upper bound is very tight moreover, as the coe cients erfc(x 1=2 ) decrease exponentially This work will be presented at ISIT'02, Lausanne, Switzerland, July 2002. with x, the FER can be approximated, at low error rates, by the rst term of the union bound: FER ' 1 2 n(d min )erfc((d min R Eb N0 ) 1=2 )

(2)

When designing a code that operates at very low error rates, it is very di cult to determine its performance by Monte Carlo simulations. It is highly desirable to propose some techniques to determine the minimum distance d min and its multiplicity n(d min ).

For algebraic codes such as BCH codes or Reed Solomon codes, d min is a parameter that is speci ed prior to the design of the code. For elementary convolutional codes, the Viterbi algorithm 4] can be practically used to estimate the minimum distance, but this algorithm cannot be used to estimate the minimum distance of concatenated codes and in particular of Turbo Codes (TCs).

It is then important to develop some fast algorithms to compute the minimum distance d min of concatenated codes and, in particular, of TCs. This will make it possible to design e cient TCs without intensive computations. This will for example make it possible to design good component codes and good interleavers in record time. Some authors have suggested a simpli ed approach t o determine the minimum distance d min of TCs: looking for the minimum weight d (2) min of the codewords generated by weight 2 information words 8]. With statistical interleaving, when the interleaver size tends to in nity, t h e codewords of weight d min are generated by information words of weight 2 with a high probability. But for non statistical interleavers, this method only provides an upper bound on the minimum distance: d min d (2) min , and in general this bound is not tight.

Another method has been proposed recently to compute the minimum distance d min , its information bit multiplicity w(d min ) and its multiplicity n(d min ) for parallel and serially concatenated convolutional codes 7]. This method does not make any assumption about the weight of the input sequence. It is based on the notion of constrained subcode i.e. a subset of a code de ned via constraints on the edges of its trellis. It consists of an iterative construction of the information words that generate codewords of minimum weight. This method keeps track of the di erent \candidate" information words that are under construction in a stack. For large interleavers, the memory requirements of the method can be very large.

The method that we propose in this paper is based on the notion of the Error Impulse Response of the decoder. The Error Impulse Response of the iterative Soft-In/ Soft-Out (SISO) decoding algorithm was introduced in 2]. It has also been used in 5]. The proof of our method is based on a reasoning in the Euclidean space R n . For the proof, we suppose that the decoder is a maximum likelihood (ML) decoder on the Gaussian channel and that the modulation is BPSK or QPSK. In practice, the turbo decoder on the Gaussian channel is used.

Principle of the method

We use the following notations:

x = ( ;1 ;1 ;1) is the word associated with the \all zero" codeword by the modulation. y = ( ;1 ;1 ;1 ;1 + A i ;1 ;1) is the input to the decoder. A i is a positive real number called the error impulse. i is the position of error. y is decoded according to the ML criterion on the Gaussian channel. The decoded codeword x is such that:

< x y > < z y > 8z 2 C (3)
where < > is the scalar product and C is the code. Theorem 1 If there exists a position of error i and an error impulse A i such that the decoded codeword is not the \all zero" codeword, then

A i min z2C=zi=+1 w H (z) (4)
where the minimum is taken over all codewords z such that z i = + 1 and where w H (z) is the Hamming weight of z.

Proof:

Assume that x 6 = x.

Theorem 3 for any error position i, there exists a positive error impulse A i such that A i = m i n fA i = x 6 = xg = m a x fA i = x = xg = min z2C zi=+1 w H (z) [START_REF]3rd Generation Partnership Project. Multiplexing and Channel Coding (FDD)[END_REF] The minimum distance of the code is :

d min = min i A i (13)

Proof

It results from Th. (1) and (2) that

maxfA i = x = xg min z2C zi=+1 w H (z) minfA i = x 6 = xg (14)
Suppose that maxfA i = x = xg 6 = m i n fA i = x 6 = xg. There exists A i such t h a t m a x fA i = x = xg < A i < minfA i = x 6 = xg. For this A i , either x 6 = x which is contradictory to A i < minfA i = x 6 = xg, or x = x which is contradictory to A i > maxfA i = x = xg. Therefore, the inequalities in -Eq.-( 14) are equalities and there exists A i such that: A i = m i n fA i = x 6 = xg = m a x fA i = x = xg = min z2C zi=+1 w H (z)

Practical issues

In most cases, the code is cyclic, which means that if a circular shift is applied to a codewo r d i t i s s t i l l a c o d e w ord.

In this case, it is su cient to test one position of error i to obtain the minimum distance. This is, for example, the case for convolutional codes if one neglects possible side e ects due to termination.

For concatenated codes, when iterative SISO decoding is not available, the method cannot be applied because one-step decoding is largely suboptimal.

For turbo like (serially or parallel) concatenated codes the method can be applied with an iterative SISO decoder, although it has not been proved that turbo decoding is ML. This pattern of noise (no noise at all except in position i where A i is large) is very improbable on the Gaussian channel. It is not sure that the turbo decoder maximizes < y z > since the quasi optimality of turbo decoding has been proven only by simulation with realistic patterns of noise.

Nevertheless the experiment s h o ws good agreement b etween the minimum distances of TCs obtained with this algorithm and the asymptote obtained by s i m ulation. The appropriate SISO decoder for each component c o d e i s t h e max-log-MAP algorithm 6]. This algorithm does not require the knowledge of the channel parameters. The extrinsic information should be passed from one component decoder to the other without any alteration (no attenuation, no saturation). The number of iterations needed for convergence depends on the expected value of d min .

It may v ary from a few tens to several hundreds for very large minimum distances. For example, in some cases, when d min is around 20, the estimated d min with 256 iterations is one point up on the value obtained with 128 iterations.

For the parallel concatenation of recursive systematic codes with interleaving, some restrictions should be made to the algorithm given in Section (3.1). In this case the error impulse A i should be applied only on the systematic bits, but not on the parity bits.

When examining a turbo like concatenated code, the number ofsymbols to be tested is related to the periodic properties of the code. In the case of a classical turbo code, this number, denoted T, is the lowest common multiple of the period of the interleaver and of the period of the puncturing pattern, if any. The former depends on the permutation model that was adopted in the design of the interleaver. When no periodicity is observable (for instance when the permutation function was obtained by some pragmatic computer-based research), T must be maximum (i.e. T = k).

Multiplicity o f d min

Because the method does not explicitly provide the multiplicity of codewords of weight d min and higher, we m ust introduce some further hypotheses to use the asymptote on the FER given in -Eq.-(2). These assumptions seem to be in good agreement w i t h the performance obtained by s i m ulation, although they are irrelevant for some other codes, such as product codes.

Hyp. 1: there is only one codeword z with weight A i and such t h a t z i = +1.

Hyp. 2: all the distances A i obtained for the whole set of positions i (1 i k) concern distinct codewords (there is no overlapping).

The former hypothesis is optimistic, while the latter is pessimistic. Both together, as con rmed in the examples given in the next Section, provide a good estimate of the FER, that can thus be calculated as:

FER ' 1 2 X i=1 k erfc( r RA i Eb N0 ) (17)
Note that if the spectral thinning is pronounced the error oor will be completely determined by the minimum distance d min = m i n i A i and its multiplicity n(d min ) w h i c h

UMTS/3GPP standard

A TC has been speci ed in the UMTS/3GPP 1] standard for personal communications. This standard uses binary 8-state RSC codes as component codes. The error impulse method has been tested on this turbo code for a relatively small interleaver: k = 640 bits. In this case the spectral thinning is not extremely pronounced because the interleaver size is moderate. Therefore, we have t a k en into account not only the codewords of weight d min but also codewords with weights d d min . There is good agreement b e t ween the performance simulated, and the performance estimated by the Error Impulse method. A loss of 0.2{0.3 dB can be observed this loss has been observed by many contributors 2] 9] and may be due to the suboptimality of the decoder.

Conclusion

A p o werful tool for the design of linear codes, and in particular of turbo codes, has been introduced and justi ed. Whatever block sizes and coding rates, it gives the possibility to forecast performance at very low error rates in seconds or minutes. It may be used for instance as a fundamental algorithm in the search for good permutations in the construction of turbo codes. Nevertheless, further investigation has to be conducted to understand better the behavior of turbo decoders facing Error Impulse sequences.

  We i d e n tify any c o d e w ord with a sequence of +1s and ;1s. The Hamming weight o f a c o d e w ord is the number of +1's in this codeword.

  codeword z 6 = x has at least one position i such t h a t z i = +1. Therefore d min is equal to d min = m i n i min z2C zi=+1 w H (z) = m i n i Estimation of the minimum distance as the minimum error impulse We assume that d min is in the range d 0 d 1 ] where d 0 and d 1 are two i n tegers. Then d min can be determined as follows. set A min = d 1 + 0 :5. for i=1 to n do { A = d 0 ; 0:5 { set (x = x) = T R UE] { while (x = x) = T R UE] and (A A min ; 1:0) do A = A + 1 :0 y = (;1;1 ;1 + A ;1 ;1) where ;1 + A is in position i ML decoding of y ) x if (x 6 = x) t h e n ( x = x) = F ALSE] end while { A min = A end for d min is the integer part of A min .

Figure 1 :

 1 Figure 1: Frame Error Rate of the DVB-RCS turbo code for k = 1504 and rates 2=3 and 4=5. Both simulated FER and estimated FER (UB, eq. (2))) are given.

Figure 4 .

 4 1 shows the performance in FER measured on FPGA hardware, for MPEG packets (188 bytes) and for rates 2=3 and 4=5. The asymptotic performance given by -Eq.-(2) is also displayed. The parameters obtained by the error impulse method are: d min = 1 3 a n d n(d min ) = 752 for R = 2 =3 d min = 8 a n d n(d min ) = 9 4 0 f o r R = 4 =5 Measured and estimated curves are in good agreement.

Figure

  Figure 2: Frame Error Rate of the UMTS/3GPP turbo code for k = 640 and rate 1=3. Both simulated FER and estimated FER (UB, eq. (17)) are given.


Sao Mai Nguyen
email: nguyensmai@gmail.com

Philippe Tanguy

Olivier Remy-Neris

André Thépaut

A humanoid robot for coaching patients for physical rehabilitation exercises

Robot Coach for Physical Rehabilitation

1.1 Motivation 50 to 80% of the world population suffers at a given moment from back pain, which makes it in the lead in terms of health problems occurrence frequency [START_REF] Mounce | Back pain[END_REF][START_REF] Who | The burden of musculoskeletal conditions at the start of the new millenium[END_REF]. It is the third common cause of disabling condition in the 45-65 years old population, whose proportion in European societies keeps rising. This incurs growing concern, as medical professionals will not be able to face this increasing demand for healthcare.

A Robot Coach

Active rehabilitation is considered as more effective than usual care [START_REF] Kent | The efficacy of targeted interventions for modifiable psychosocial risk factors of persistent nonspecific low back pain -a systematic review[END_REF]. To enable rehabilitation to an increasing number of elderly patients, we propose a robot coach for physical rehabilitation exercises. The long-term goal of this project is to provide patients with a e-Health solution for personalised follow-up of long-term rehabilitation at home. We develop a robot coach capable of demonstrating physical exercises to patients, watch a patient carry out the exercises and give him feedback so as to improve his performance. Our system aims to be easily used by medical experts. Fig. 1. Overview of the computational architecture for the robot to learn a model of the exercise from the doctor's gestures, then to make demonstrations from the model, and to give feedback to the patient

State of the art and challenges

Our approach for robotic systems coaching physical exercises is related to those presented in [START_REF] Fasola | A socially assistive robot exercise coach for the elderly[END_REF][START_REF] Görer | A robotic fitness coach for the elderly[END_REF] and [START_REF] Takenori | Imitation learning for daily exercise support with robot partner[END_REF]. Takenori et al [START_REF] Takenori | Imitation learning for daily exercise support with robot partner[END_REF] developed a system of imitation learning for daily physical exercises. The robot could learn new exercises from the therapists and be an exercise demonstrator. However, it could not provide feedback or active guidance to the patient. Fasola et al. in [START_REF] Fasola | A socially assistive robot exercise coach for the elderly[END_REF] and Goerer et al. [START_REF] Görer | A robotic fitness coach for the elderly[END_REF] presented a socially assistive robot (SAR) to engage elderly users based on their intrinsic motivation or through HRI. They proposed an automatic evaluation of physical exercises by computing the distance between the user's current arm angles and the specified goal arm angles. However, it does not take into account the speed or the dynamics of the movement. It is ill-adapted to complex full-body exercises that can involve several parts of the body but not necessarily all parts. We would like to focus on the intelligent tutoring system (ITS), and more specifically an algorithm capable of assessing which parts of the body are important to the exercise, and what are the acceptable ranges of freedom. More specifically we use probabilistic imitation learning algorithms to model each movement and detect errors.

Proposed system 3.1 The Hardware

The system is composed with a Microsoft Kinect v2 and an open source humanoid robot Poppy [START_REF] Lapeyre | Poppy : open-source, 3D printed and fully-modular robotic platform for science, art and education[END_REF]. A demonstration video can be watched at http ://nguyensmai.free.fr/roman2016.html.

The Computational Architecture

Figure 1 depicts the computational architecture with a humanoid robot during physical rehabilitation exercises. The schema combines three different phases : learning, demonstration and feedback. In the first step, the therapist records in front of the camera several movements of the same exercise. These therapist's gestures enable the robot to learn a Gaussian Mixture Model (GMM) with Expectation Maximization, as described in [START_REF] Calinon | On learning, representing and generalizing a task in a humanoid robot[END_REF]. In the demonstration phase, the "ideal movement" can be generated in order to be reproduced by the robot, using Gaussian Mixture Regression (GMR).

Finally, during the exercise all movements done by the patient are recorded and analysed for vocal feedback by the robot. This part is realized with the Feedback Engine (section 3.3) and Analyse component.

For the Analyse component, a metric of imitation allows the robot to assess the movement of a patient for each joint j automatically:

H j =δ j T W δ j (1)
where W represents parameters of the GMM and δ = x -xˆ is the difference between the observed attempt x and the generalised motion xˆ (obtained by GMR). The lower Hj, the better the imitation is. We arbitrarily set a threshold ∆ to get the list of outstanding errors Ej = {(j, δj), if Hj > ∆}.

The Patient-Robot Interaction

This list of errors is sent to the Feedback Engine for audio feedback to translate this mathematical analysis into recommendation sentences for the Human-Patient Interaction. The system currently allows a one-way interaction with the patient, using a text to speech server (MaryTTS) to provide vocal feedbacks.

Results

Dataset

We set up with the Kinect a database for 2 different physical exercises, including 6 different recordings for each exercise. We obtained a model of exercise 1 using the 5 first recordings of exercise 1.

Modelling and Generating

The GMM model obtained is represented in fig. 2. This probabilistic representation is then used to generate an ideal movement, to be played by the robot coach to patients.

Assessing

We tested with the following data: -d1 : a recording of exercise 1 (in the training set) -d2 : d1 where we added an offset to the x position of the left shoulder -d3 : a recording of exercise 1 (not in the training set) -d4 : a recording of exercise 2

The Analysis results are shown in Fig 3 . Our analysis outputs a general evaluation of the attempted imitation as H. As expected, we can observe that the error is acceptable for d1 and d3, which are movements of exercise 1. On the contrary the error is very high for d2 where an error has been artificially introduced, and d4, which is a recording of a different exercise.

Moreover, our analysis finds out which joint positions are accountable for the errors. It shows that the robot was satisfied for d1 and d2. When the patient repeats correctly exercise 1. In the case of d2, where we added an offset on the x position of the joint named "rShoul", we can verify that it reported correctly this error. In the case of d4, as the patient was executing another exercise, our system reported several errors. Thus, our ITS reports correctly the errors seen in the attempted imitations. This output is then used by the Feedback Engine to translate this analysis into sentences for audio feedback.

Discussions

In this paper, we have presented a general software architecture for a robot coach for physical exercises of rehabilitation. Using programming by demonstration as a probabilistic imitation learning algorithm, our robot coach is capable of learning new exercises from demonstrations by the therapists, then demonstrate exercises to the patient, and assess the performance of the patient and provide him with corrective feedback. However, these are only preliminary results, as this system does not yet run online. In particular, we plan on developing an interface to be able to switch seamlessly from the 'learning from the therapist' mode to the 'assessing the patient mode'. Second, we plan to adapt the threshold ∆ to the patient. If it's too high, the robot will not correct movements that may be ineffective or even harmful, if too low, the patient will easily become frustrated. Furthermore, in our tests the same person has acted as the therapist and the patient. We will need to address the correspondence problems when we face several users with different sizes and kinematic models. We approach this problem by using joint angles instead of Cartesian positions.

Fig. 2 .

 2 Fig. 2. Modelling the demonstrations with GMM and generating an ideal movement with GMR. Is represented a recorded joint position with respect to time (normalised values). The green ellipses plot a representation of the components of the GMM. The ideal trajectory (GMR) is represented with the thick blue line.

Fig. 3 .

 3 Fig. 3. Output of the analysis of test movements compared to the learned movement model (no units as the positions are normalised). The analysis outputs both a global evaluation of the test movements, and a list of the outstanding errors for each joint position. These figures are then interpreted by the robot to give an audio feedback.

Acknowledgment:

The research work presented in this paper is partially supported by the EU FP7 grant ECHORD++ KERAAL and by the European Regional Fund (FEDER) via the VITAAL Contrat Plan Etat Region.


Ricardo M Ribeiro

Andrés P L Barbero

Vinicius N H Silva

Frédéric Lucarz
email: frederic.lucarz@telecom-bretagne.eu

Bruno Fracasso

Electrical Modulation of a Semiconductor Optical Amplifier Inserted into a Sagnac Loop by Injecting Microwave Carrier

Keywords: optical modulator, optical signal processing, Microwave-Photonics, semiconductor optical amplifier, radio-overfibre I S,O,A, , ,, , Delay,2,, PC, , Delay,1,, RF, generator, , DC,bias, source, , Bias&T, !

This paper describes for the first time, the design and numerical simulations of an optical modulator from a semiconductor optical amplifier (SOA) inserted into a fibre-optic Sagnac loop electrically modulated by a microwave carrier at 20 GHz frequency and higher. Even though the adopted SOA model presents > 1.4 ns gain recovery time, the device is optically modulated up to 30 GHz because a completely gain recovery is not needed. The interferometric device was shown to present better performance than the solely SOA. It is potentially useful to simultaneously carry out optical modulation, sampling and mixing or to be used simultaneously as a modulator (uplink) and amplifier (downlink) in analogue links.

INTRODUCTION

Radio-over-Fibre (RoF) technology enables the fusion of fibre-optic and wireless communications networks [START_REF] Al-Raweshidy | Radio over Fiber Technologies for Mobile Communications Networks[END_REF][START_REF] Abdollahi | Fully-Photonic Analogue-to-Digital Conversion Technique for Super-Broadband Digitized-Radio over Fibre Link[END_REF][START_REF] Williamson | RF Photonics[END_REF]. Classical RoF systems are relatively simple and very powerful. In the transmission end, signals at microwave carrier frequencies analogically modulate an optical source (ARoF) and the generated light signal is transmitted through an optical fibre. Reciprocally, in the receiver end, a photo-detector recovers the original analogue signal that modulates the optical carrier [START_REF] Al-Raweshidy | Radio over Fiber Technologies for Mobile Communications Networks[END_REF][START_REF] Abdollahi | Fully-Photonic Analogue-to-Digital Conversion Technique for Super-Broadband Digitized-Radio over Fibre Link[END_REF][START_REF] Williamson | RF Photonics[END_REF]. An arbitrary wireless signal in either standard is seamless despite the use of optical fibres. Because of the intrinsic nature of analogue transmission systems, ARoF links and networks feature limited dynamic range [START_REF] Al-Raweshidy | Radio over Fiber Technologies for Mobile Communications Networks[END_REF][START_REF] Abdollahi | Fully-Photonic Analogue-to-Digital Conversion Technique for Super-Broadband Digitized-Radio over Fibre Link[END_REF][START_REF] Williamson | RF Photonics[END_REF]. Furthermore, ARoF systems are more difficult to integrate with the conventional digital networks [START_REF] Abdollahi | Fully-Photonic Analogue-to-Digital Conversion Technique for Super-Broadband Digitized-Radio over Fibre Link[END_REF].

Although the high-frequency electrical modulation of SOAs is not new [START_REF] Mork | The Modulation Response of a Semiconductor Laser Amplifier[END_REF], it is not as usual when compared with the SOA used as an amplifier or optical signal processor fed only by a DC bias current.

The electronic digitisation of ARoF links and networks using fast AD and DA converters [START_REF] Nirmalathas | Digitized Radio-Over-Fiber Technologies for Converged Optical Wireless Access Network[END_REF] have being gaining interest from Telecommunications carriers [START_REF] Saadani | Digital Radio over Fiber for LTE-Advanced: Opportunities and Challenges[END_REF]. The digitised ARoF (DRoF system) may extend the coverage, increase the system bandwidth together with a higher degree of capillarisation and enables a higher number of access points, improves the signal quality and energy saving when compared with the classical ARoF [START_REF] Abdollahi | Fully-Photonic Analogue-to-Digital Conversion Technique for Super-Broadband Digitized-Radio over Fibre Link[END_REF][START_REF] Nirmalathas | Digitized Radio-Over-Fiber Technologies for Converged Optical Wireless Access Network[END_REF][START_REF] Saadani | Digital Radio over Fiber for LTE-Advanced: Opportunities and Challenges[END_REF]].

An advanced, interesting and underexplored issue is the alloptical digitising of ARoF links and networks (o-DRoF), i.e. all-optical ADC (transmitter) and DAC (receiver) both intended to process RoF signals, which the authors has been working [START_REF] Abdollahi | Fully-Photonic Analogue-to-Digital Conversion Technique for Super-Broadband Digitized-Radio over Fibre Link[END_REF][START_REF] Ribeiro | A Thresholder for All-Optical Digitising Radio-over-Fibre Transceivers[END_REF][START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF][START_REF] Ribeiro | Proposal and Design of an All-Optical Encoder for Digitising Radio-over-Fibre Transceivers[END_REF][START_REF] Ribeiro | An Optical Pulse Generator from a Sinusoidal Optical Signal Using Sagnac Loop -Self-Sampling[END_REF][START_REF] Ribeiro | Design of an Analogiser for Optically Digitised Radio-over-Fibre Signals[END_REF]. Furthermore, in the best of our acknowledgement, there are none commercially available o-DRoF systems. Such all-optical approach does not require signal demodulation. Since multi-GHz sampling rates are offered by pulse train produced by stable mode-locked lasers, it may eliminate or reduce the number of mixer-stages (intermediary frequencies -IFs) that are needed to downconvert high frequency signals before digitisation and enables optical sampling with ≤10 fs timing jitter [START_REF] Kärtner | Electronic Photonic Integrated Circuits for High Speed, High Resolution, Analog to Digital Conversion[END_REF].

Previously [START_REF] Ribeiro | A Thresholder for All-Optical Digitising Radio-over-Fibre Transceivers[END_REF][START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF][START_REF] Ribeiro | Proposal and Design of an All-Optical Encoder for Digitising Radio-over-Fibre Transceivers[END_REF][START_REF] Ribeiro | An Optical Pulse Generator from a Sinusoidal Optical Signal Using Sagnac Loop -Self-Sampling[END_REF][START_REF] Ribeiro | Design of an Analogiser for Optically Digitised Radio-over-Fibre Signals[END_REF], the authors described an o-DRoF system able to process analogue signals already in the optical domain. This paper proposes, designs and carry out numerical simulations using the VPI Transmission Maker TM platform of an interferometric front-end device able to directly receive microwave carriers. It is intended to be simultaneously an amplifier and broadband optical modulator regarding an ARoF network. Furthermore, the device may be simultaneously an optical modulator, optical sampling [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF] & mixing [START_REF] Rampone | Conversion gain characterisation of optical radiofrequency signal mixing by all-optical sampling on SOA-MZI[END_REF][START_REF] Capamany | Optical mixing of microwave signals in a nonlinear semiconductor laser amplifier modulator[END_REF] as to be the transmitter device of a future complete o-DRoF link. However, only the proof-of-principle operation of the ARoF generator (optical modulator) is here described. It is based on the optical switching provided by a Sagnac loop with an in-line inserted SOA [START_REF] Eiselt | Optical Loop Mirror with Semiconductor Laser Amplifier[END_REF] with a long recovery-time. Comparison of performance is carried out with a solely SOA-based device. Interestingly, the electrical modulation of SOAs may be classified as an intermediary kind between the "direct modulation" of LDs and LEDs and "external modulation" of electro-optic and electro-absorption modulators.

II. THE OPTICAL CIRCUITS & PRINCIPLE

Two optical circuits were designed and numerically simulated using the bulk model of SOA with L SOA = 500 µm and alpha-factor α = 6 [START_REF] Van Der Poel | Alpha parameter in quantum-dot amplifier under optical and electrical carrier modulation[END_REF].

The Circuit #1 is schematically shown in Fig. 1. It uses a single SOA and as result the modulation is achieved by means of the gain modulation driven by the injected microwave carrier in a non-interferometric configuration. The Circuit #2 is schematically shown in Fig. 2. It uses a SOA inserted into a Sagnac loop and as a result the optical modulation is achieved by means of the gain and phase modulation simultaneously also driven by the injected microwave carrier now in an interferometric configuration. Comparisons between the performances of both circuits are outlined ahead. In both schemes, a microwave carrier generator replaces a microwave receiver antenna. The 1550 nm wavelength CW (unmodulated) light signal P in = 20 mW power to be modulated is launched into the single SOA or the Sagnac loop + SOA. In both circuits, the microwave carrier at f RF frequency and I RF current amplitude to be transmitted along the optical fibre is launched into the SOA superimposed with a bias current I bias by means a bias-T. The composite signal is injected through the electrical port of the SOA. The SOA works as a one-travelling optical amplifier enhancing the energy of the switched light. In this way, the launched optical signal is gain-modulated in the Circuit #1 and gain + phase-modulated in the Circuit #2 both by the I bias + I RF input. Because the SOA is asymmetrically inserted in the Sagnac loop (Fig. 2), a differential and periodic phase-shift Φ 0 senωt is modulated between the clockwise and anticlockwise interfering signals when I RF is launched, where ω = 2πf RF . Because of the long recovery-time of the SOA, the negative cycle of the input microwave current is likely to be not clipped. The constant Φ 0 depends on the nonlinearity strength of the SOA, the recovery-time, the bias current and the asymmetry (38 ps/62 ps) of the inserted SOA into the Sagnac loop. By setting the Sagnac loop to operate in the linear region of the transfer function, the P OUT (transmitted power) for small phase-shift is given by [START_REF] Al-Raweshidy | Radio over Fiber Technologies for Mobile Communications Networks[END_REF] where the parameter B depends on losses and amplifier gain of P IN .

P !"# = 1 2 BP !" 1 -cos π 2 + Φ ! senωt P !"# ≈ ! ! BP !" 1 + Φ ! senωt (1)
The numerical simulations were first carried out using a single SOA in a travelling-wave configuration (noninterferometric) subject only to gain modulation. The second set of simulations was carried out using a SOA in an interferometric Sagnac based device subject to gain and phase modulation simultaneously. The SOA and electrical/optical supply parameters were the same for both configurations and none band-pass filter is used despite the setting of I bias > 0 mA, but limited to 50 mA.

III. RESULTS AND DISCUSSIONS

A. Single SOA in travelling-wave configuration

Figure 3 shows three plots of the optical output from the Circuit #1 at f RF = 20 GHz carrier frequency, I RF = 80 mA microwave current amplitude, I bias = 0, 10 and 50 mA bias current, respectively. All plots are assigned with the optical offset level P DC and amplitude P OUT of the output waveform. When the SOA is unbiased, the output amplitude is very low (0.24 µW) with a high optical offset (129 µW). The origin of the latter comes from the excitation of SOA with a highfrequency I RF microwave carrier that interacts with long-lived charge carriers featuring recovery time > 1.4 ns, since 1.4 ns was previously characterised for I bias = 100 mA (not shown here).

When I bias = 10 mA is applied, both offset P DC and output amplitude P OUT dramatically increases because much more charge carriers are available in the conduction band. Therefore, the ratio P OUT /P DC is increased from 0.24/124 = 0.002 to 0.43/1.3 = 0.331 (x 165.5 times). When I bias = 50 mA is applied, it leads to 4.2/15.9 = 0.264 comparable with the ratio achieved when I bias = 10 mA. For all injected I bias , the SOA gain is not fully recovered because P DC > 0 mW. As expected, for polarisation bias at 26° and 50° the output signals are distorted by clipping since the device operates in a non-linear region of the transfer curve. For 90° polarisation bias, a non-distorted signal very faithful with the input sinusoidal signal is calculated because the interferometer operates in the region of linear response with maximum derivative (quadrature). Nevertheless, for polarisation bias at 0°, the output waveform is little distorted because the interferometer operates in a maximum transmission around Φ 0 = 0 rad (I bias = 0 mA) since the light carrier is not suppressed and the 2 nd harmonic contribution is not very high. An advantage by using 0° polarisation bias at I bias = 0 mA for small input signal (at least I RF < 80 mA), is to generate an output optical signal accompanied with an almost null optical DC [START_REF] Ribeiro | Sagnac Loop Incorporating a Semiconductor Optical Amplifier Without Bias Current to Receive/Modulate Weak RF Signals[END_REF], but with a reduced amplitude when compared with the 90° (maximum linearity).

B. Sagnac interferometer with a SOA inserted into the loop

Figure 5 shows three plots of the optical output from the Circuit #2 by injecting unbiased and biased (I bias = 0, 10 and 50 mA) drive signal of I RF = 80 mA amplitude and f RF = 20 GHz microwave carrier frequency. Different output signals are achieved by using the same polarisation bias of 90°. When the I bias is increased, the P DC and the amplitude of optical output signal P OUT are also increased. When I bias = 10 mA is applied, both P DC and P OUT dramatically increases because much more carriers are available. However, the ratio P OUT /P DC is increased from 0.080/0.045 = 1.78 to 2.4/0.7 = 3.43 (x 1.93 times). When I bias = 50 mA is applied, it leads to 8.9/4.6 = 1.93, less than the ratio when I bias = 10 mA. For all applied I bias , the SOA gain is not fully recovered. At least in the range 0-50 mA, the P OUT increases with I bias , but not the P OUT /P DC ratio.

C. General discussions

From ~ 300 MHz to 30 GHz even with unbiased SOA, none negative cycle clipping of the sinusoidal current input was already calculated for the optical output [START_REF] Ribeiro | Sagnac Loop Incorporating a Semiconductor Optical Amplifier Without Bias Current to Receive/Modulate Weak RF Signals[END_REF]. The clipping is expected since a SOA is a diode as LDs and LEDs are. A possible explanation is because the SOA presents a recoverytime greater than 1.4 ns when it is biased with I bias < 50 mA. Therefore, for f RF > 300 MHz the period of the input signal becomes lesser than the recovery-time, e.g. 1/20 GHz = 50 ps is at least ~ 28 times less. The "memory effect" of SOAs seems to be useful for continuous analogue signals whereas the resulting pattern effect is likely to appear for digital streams [START_REF] Zoiros | Theoretical analysis of pattern effect suppression in semiconductor optical amplifier utilizing optical delay interferometer[END_REF]. When the SOA is biased, carriers are injected in the conduction band by the I bias and the contribution of I RF is less critical. However, the devices become noisier due the ASE generation. SOAs biased below the transparency current should be further investigated since it is still almost noiseless.

It was already reported [START_REF] Ribeiro | Sagnac Loop Incorporating a Semiconductor Optical Amplifier Without Bias Current to Receive/Modulate Weak RF Signals[END_REF] for low microwave current amplitude (I RF = 1 mA and f RF = 5 GHz) a flat response from 0.3 to 10 GHz and f CUT = 26 GHz cut-off frequency for the interferometric device whereas the single SOA was not able to respond in either input frequency. Now, from the calculated plots shown in Figs. 3 and5, the Table I show the comparison between the P OUT and P DC from the single SOA and Sagnac with an inserted SOA (polarisation bias = 90°) modulators electrically driven by I RF = 80 mA and f RF = 20 GHz.

Table I shows that when both devices are unbiased a dramatic increase around x188 times is observed for the P OUT whereas a modest decrease of P DC is calculated when the SOA is inserted into a Sagnac loop. When the devices are biased, both P OUT and P DC increase as I bias is augmented. However, the increase of P OUT is less dramatic when I bias > 0 mA and is increased when the SOA is again inserted into the Sagnac loop. Moreover, it is achieved a greater decrease of P DC . As a result of present simulations and those shown on [START_REF] Ribeiro | Sagnac Loop Incorporating a Semiconductor Optical Amplifier Without Bias Current to Receive/Modulate Weak RF Signals[END_REF] for small microwave current amplitude, a significant improvement of performance is achieved when the electrically driven SOA is merely inserted into a Sagnac loop despite the long recovery-time of the amplifier when compared with the period of the microwave carrier to be modulated.

As modulation frequency increases, traveling-wave effects can degrade the bandwidth of directly modulated laser diodes (LDs) [START_REF] Iezekiel | Travelling-Wave Effect in Microwave Photonics[END_REF]. Furthermore, pulses are generated with optical chirp that in general negatively interacts with the fibre chromatic dispersion thus limiting the bandwidth. In order to circumvent the limitations of directly modulated light sources, some devices have being used and are firmly established as external optical modulators as Mach-Zehnder electro-optic (MZ-EOM) and electro-absorption (EAM) modulators in analogue [START_REF] Cox | Analog Optical Links: Theory and Practice[END_REF] and digital modulations.

In principle, the SOA itself exhibits a reduced intrinsic capacitance because of its low length typically of 0.5 mm whereas a typical MZ-EOM presents electrode length in the few mm to few tenths mm range [START_REF] Wooten | A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems[END_REF]. However, the package parasitics of SOAs should not be neglected [START_REF] Toazza | Experimental and Theoretical Analysis of Packaging Inductances and Stray Capacitances of a Semiconductor Optical Amplifier[END_REF]. By using an unpackaged device [START_REF] Figueiredo | Chip-on-Carrier Microwave Mount for Semiconductor Optical Amplifier Measurement[END_REF] the parasitics of SOA may be reduced and f RF > 10 GHz electrical modulation was already achieved. It has been shown that at microwave frequencies, typically above 10 GHz, usually the electrode geometry of SOAs yields a large propagation loss, which means that the details of the way the modulation signals are fed to the device becomes important [START_REF] Mork | The Modulation Response of a Semiconductor Laser Amplifier[END_REF].

A more detailed comparative analysis still needs to be carried out amongst the MZ-EOMs, EAMs and SOAs-based modulators. However, SOAs can automatically amplify the signals, has a very high and fast nonlinearity (e.g. quantumdot SOAs), when inserted in an interferometric device it can perform several functions simultaneously and due it reduced size the resulting circuit is prone to optical integration.

IV. CONCLUSIONS

In this work, were reported two electrically driven SOAbased devices able to receive a guided or free-propagating microwave carrier thus creating a sub-carrier onto the light wave.

The first device is the well-known single SOA operating in travelling-wave regime where the microwave carrier modulates the gain of the amplifier. In the second device, the same SOA is merely inserted into an interferometric Sagnac loop where the microwave carrier simultaneously modulates the gain of the amplifier and the differential phase between the counterpropagating signals. As a result, the modulation efficiency was significantly improved for very low microwave current amplitude (< 1 mA) at 5 GHz when the device is used as a probe [START_REF] Ribeiro | Sagnac Loop Incorporating a Semiconductor Optical Amplifier Without Bias Current to Receive/Modulate Weak RF Signals[END_REF] and for electrically amplified input (80 mA) at 20 GHz.

Even though the adopted SOA model presents a long recovery-time > 1.4 ns, both configuration and mainly the interferometric one, could be modulated at 20 GHz carrier frequency and higher.

The main aims of this paper were to show the improvements when an interferometric configuration is used instead of a single SOA and to take the advantage of the partial recovery gain for high-frequency modulation. In practice, it is not easy to directly modulate a SOA in carrier frequencies higher than 10 GHz [START_REF] Toazza | Experimental and Theoretical Analysis of Packaging Inductances and Stray Capacitances of a Semiconductor Optical Amplifier[END_REF][START_REF] Figueiredo | Chip-on-Carrier Microwave Mount for Semiconductor Optical Amplifier Measurement[END_REF], but improvements have being achieved.

Further numerical simulations regarding device sensitivity under variations of many physical parameters and also including the electrical model of SOA/package are on going. Experimental validations are also scheduled to be held in our laboratory.

Fig. 1 .

 1 Fig. 1. Optical circuit diagram #1 of an optical modulator based only on an electrically driven SOA.

Fig. 2 .

 2 Fig. 2. Optical circuit diagram #2 of an interferometric optical modulator based on a Sagnac loop incorporating an electrically driven SOA.

Fig. 3 .

 3 Fig. 3. Outputs from the single SOA optical modulator at 20 GHz for 80 mA microwave current, biased with Ibias = 0, 10 and 50 mA.

Figure 4

 4 shows four plots of the optical outputs from the Circuit #2 by injecting drive signal of I RF = 80 mA amplitude and f RF = 20 GHz microwave carrier frequency into the unbiased (I bias = 0 mA) SOA. Different output signals are achieved by varying the polarisation bias (phase angle) of the Sagnac loop.

Fig. 4 .

 4 Fig. 4. Optical outputs from the Sagnac loop interferometer incorporating an unbiased (Ibias = 0 mA) SOA driven with 20 GHz frequency and 80 mA microwave current amplitude. The polarisation bias of the loop are set at 0°, 26°, 50° and 90°.

Fig. 5 .

 5 Fig. 5. Optical outputs from the Sagnac loop interferometer incorporating an unbiased (Ibias = 0 mA) and biased at Ibias = 10 and 50 mA SOA electrically driven with 20 GHz frequency and 80 mA microwave current amplitude. The polarisation bias is set at 90° for all bias current.

TABLE I

 I

	.	COMPARISON OF OUTPUT POWERS FROM SINGLE SOA AND
		SAGNAC INTERFEROMETER + SOA MODULATORS
	Ibias (mA)	Power	SOA	Sagnac with	Increase
				SOA	(dB)
		POUT	0.24 µW	45 µW	+ 22.7
	0	PDC	129 µW	80 µW	-2.1
		POUT	0.43 mW	2.4 mW	+ 7.5
	10	PDC	1.3 mW	0.7 mW	-2.7
		POUT	4.2 mW	8.9 mW	+ 3.3
	50	PDC	15.9 mW	4.6 mW	-5.4

Ricardo M. Ribeiro thanks Foundation Capes/MEC-Brasil for the financial support through a Post-Doctoral fellowship under the grant BEX 9096/11-6. The authors thank INCT-Fotonicom/Campinas, SP-Brasil.

ACKNOWLEDGMENT Ricardo M. Ribeiro thanks the team at Département d'Optique of Telecom Bretagne and the CapilR TM platform for hosting his Post-Doctoral study during which the present work was carried out. The authors also thank Cristina Arellano of VPI Photonics GmbH for her technical help.


W J Ellison

J M Moreau

Keywords: Error analysis, Open-ended coaxial probe, Permittivity model

We propose a procedure to determine the validity range of any empirical model of the open-ended coaxial probe transition that is used to measure the permittivity of materials. The procedure is illustrated by an application of the method to a standard coaxial cable probe, a permittivity range of interest in the food industry and, as an empirical model, the well-known "lumped capacitor model".

I. INTRODUCTION

The long history of the open-ended probe technique for the dielectric characterization of materials is reviewed in [START_REF] Stuchly | Coaxial line reflection methods for measuring dielectric properties of biological substances at radio and microwave frequencies -A Review[END_REF], [START_REF] Gardiol | Open-ended waveguides: Principles and applications[END_REF], [START_REF] Gex-Fabry | Reflection and radiation from an open-ended circular waveguide: Applications to nondestructive measurement of materials[END_REF], [5], [START_REF] Ellison | Theory and practice of open-ended dielectric probes[END_REF]. The method is widely used because it is simple to implement. It suffices to cut a wave-guide perpendicular to its axis, place the open end flush against the substance to be characterized and measure the admittance of the waveguide -material transition. The determination of the permittivity from this measurement is more delicate. The admittance, Y(f ,ε), is a relatively complicated function of: the frequency f, the permittivity ε of the substance under investigation and the physical characteristics of the probe. However, at "low" frequencies the transition behaves like two simple capacitors in parallel and it has become customary to approximate the function Y(f, ε) by the "lumped capacitor model": Y(f, ε) = j2πf(C 1 + C 2 ε), where C 1 , C 2 are assumed to represent respectively the "line capacity" and the capacity of the transition "without the sample". A common procedure is to assume that this model is valid for frequencies less than a few GHz, that C 1 and C 2 do not depend upon ε or f, and determine the constants C 1 , C 2 by measuring the admittances of substances with known permittivities and then use the calibrated probe to measure substances with unknown ε.

Open-Ended Coaxial Probe: Model Limitations William Ellison and Jacques-Marie Moreau > IM 06-8408 < 2 Surprisingly, the question of the validity of the model does not seem to have been systematically explored from the theoretical point of view. It is recognised that for a given probe this model is certainly not valid for "high" frequencies, but precisely what does "high" mean ? Is it 20 GHz, 10 GHz, 5 GHz, 1 GHz or 100 MHz ? The supposed precision of the values of ε' and ε" cannot be seriously estimated without information about the precision of the model itself to represent the admittance function. The object of this article is to propose a procedure to determine, for a given probe and admittance model, frequency and permittivity ranges for which the model is valid to specified precision. As an example, we shall consider the case of the "lumped capacitor" model. The general idea of our method is to compare the contours induced in the complex admittance plane by the model with the ideal contours obtained by an exact calculation and then determine frequency and permittivity intervals for which the two reticulations differ by less than a specified amount (say 1%).

II. ANALYTIC EXPRESSIONS

In certain idealised cases it is possible to calculate an exact analytical expression for Y(f,ε) and so compute the corresponding contour grid. These cases are: a circular waveguide with an infinite ground-plane [5], [START_REF] Ellison | Theory and practice of open-ended dielectric probes[END_REF], [START_REF] Fray | TM01 admittance and radiation from a flanged open-ended waveguide into layered absorbing media[END_REF], a rectangular waveguide with an infinite ground-plane [START_REF] Decreton | Nondestructive measurement of complex permittivity for dielectric slabs[END_REF], [START_REF] Levine | Theory of the circular diffraction antenna[END_REF], [START_REF] Mosig | Reflection of an open-ended coaxial line and application to nondestructive measurement of materials[END_REF] and a coaxial waveguide with an infinite groundplane [START_REF] Bois | Multimode solution for the reflection properties of an open-ended rectangular waveguide radiating into a dielectric halfspace[END_REF], [START_REF] Chang | Nondestructive determination of electromagnetic parameters of dielectric materials at X-band frequencies using a waveguide probe system[END_REF], [START_REF] Gex-Fabry | Reflection and radiation from an open-ended circular waveguide: Applications to nondestructive measurement of materials[END_REF], [START_REF] Gajda | An equivalent circuit of an open-ended coaxial line[END_REF], [START_REF] Li | Determination of electromagnetic properties of materials using flanged open ended coaxial probe -Full wave analysis[END_REF], [START_REF] Blackham | An improved technique for permittivity measurements using a coaxial probe[END_REF].

We consider a perfectly conducting coaxial waveguide of internal radius a, external radius b, filled with a dielectric of permittivity ε d and possessing an infinite ground plane placed flush against a semi-infinite substance of permittivity ε and permeability μ = 1. The expression for the admittance is:

0,0 0, . ( / ) 1 ( , ) d n n c ln b a n Y f I I νε ε ε ∞ = ⎛ ⎞ = + Λ ⎜ ⎟ ⎝ ⎠ ∑ , (1)
where the sequence {Λ n : n = 1, 2,…} is the solution of the infinite system of linear equations:

∑ ∞ = Λ = - 1 , 0 , n n n m m A I m = 1, 2, 3,… (2)
The coefficients A m,n of this system are given by ,

2 2 2 2 2 , , 4
d mn n d c m n m n c k f A I πε δ ε π ε - = + , (3)
where: the sequence { k n : n = 1, 2, 3,… } are the consecutive positive roots of the equation J 0 (ax)N 0 (bx) = N 0 (ax)J 0 (bx), δ m,n = 1 if m = n and zero otherwise,

∫ ∫ ∫ - = π ρ ρ φ φ ρ ρ ρ ρ ε 2 0 ) cos( ) ( ) ( , 2 1 0 2 1 2 1 r d d d e R R b a b a n m d r jk n m I , (4)
and

r 2 = ρ 1 2 + ρ 2 2 -ρ 1 ρ 2 cos(φ)
The functions R n (ρ) in the integrals (4) are defined by

1 0 ( / ) ( ) ln b a R ρ ρ = (5)
( )

0 1 0 1 0 2 2 0 0 ( ) ( ) ( ) ( ) ( ) 2 ( ( ) ( ))
( )

n n n n n n n n k J bk J k N ak N k J ak n J ak J bk R π ρ ρ ρ - - = (6)
for n = 1, 2, … Thus, in order to calculate the admittance we must solve an infinite system of linear equations to determine the coefficients {Λ n }. The infinite system (2) has to be truncated to a finite system. How many terms should be taken ? Various suggestions have been made, zero terms ( [START_REF] Noh | Radiation from a flanged coaxial line into a dielectric slab[END_REF], [START_REF] Misra | A quasistatic analysis of open ended coaxial lines[END_REF], [START_REF] Misra | Noninvasive electrical characterization of materials at microwave frequencies using an open-ended coaxial line: Test of an improved calibration technique[END_REF]), three terms [START_REF] Blackham | An improved technique for permittivity measurements using a coaxial probe[END_REF], six terms [START_REF] Li | Flanged coaxial microwave probes for measuring thin moisture layers[END_REF], twelve terms [START_REF] Baker-Jarvis | Analysis of an open ended coaxial probe with lift-off for non-destructive testing[END_REF]. In reports from the National Physical

Laboratory ( [START_REF] Baker-Jarvis | Shielded open-circuited sample holder for dielectric measurements of solids and liquids[END_REF], [START_REF] Warham | Annular slot antenna radiating into lossy material[END_REF], [START_REF] Jenkins | Comparison of three numerical treatments for the openended coaxial line sensor[END_REF]) comparative tests between various methods of calculation were made and the authors concluded that to have a satisfactory value of Y(f,ε) one must use up to 50 terms! The number of terms that must be used depends upon many factors (probe characteristics, frequency and permittivity) and one cannot assign a fixed universal constant number. The computer programme should be arranged so as to increase this number automatically until the resulting calculated admittance has a predetermined tolerance.

The coefficients in the system of linear equations must be calculated very precisely. Errors in the values of A m,n , together with an inappropriate truncation can lead to very inaccurate values for Λ n and hence for Y(f,ε). The accurate numerical evaluation of the integrals I m,n posed a certain number of technical problems for a long time [START_REF] Bois | Multimode solution for the reflection properties of an open-ended rectangular waveguide radiating into a dielectric halfspace[END_REF], [START_REF] Baker-Jarvis | Shielded open-circuited sample holder for dielectric measurements of solids and liquids[END_REF], [START_REF] Jenkins | Comparison of three numerical treatments for the openended coaxial line sensor[END_REF]. The triple integrals I m,n have lines of singularities in the integrands for ρ 1 = ρ 2 and ϕ = 0 or 2π and routine quadrature methods either gave incorrect results or consumed an inordinate amount of computer time.

One can transform the integrals (4) using Hankel transforms (see the Appendix for a sketch of the proofs and [START_REF] Ellison | Theory and practice of open-ended dielectric probes[END_REF] for the details) into a more manageable form. It is convenient to introduce the following normalised variables:

β = b/a, Ω = 2πfa√ε d /c, q = (ε/ε d ) ½ (7)
In terms of these variables, the integrals I m,n are: [START_REF] Decreton | Nondestructive measurement of complex permittivity for dielectric slabs[END_REF] where m ≥ 1 and n ≥ 1 and F n (ξ) = J 0 (βξ)J 0 (k n ) -J 0 (ξ)J 0 (βk n ).

ξ π ξ ξ βξ ξ d a I q J J ∫ ∞ Ω - - = 0 ) ( )) ( ) ( ( 0 , 0 2 2 2 0 0 2 (8) 0 0 2 2 2 2 ( ( ) ( )) ( ) 0, ( ) ( ) 0 2 n n J J F n k q I a d ξ ξ βξ ξ ξ ξ π ξ ∞ - - -Ω = ∫ (9) ∫ ∞ Ω - - - = 0 ) ( ) )( ( ) ( ) ( , 2 2 2 2 2 2 3 2 ξ π ξ ξ ξ ξ ξ ξ d I q k k F F n m m n n m
The integrands have removable singularities at ξ = 0, λ n and λ m respectively (If ε is real, there is an an integrable singularity at ξ = Ωq.). We write each of the infinite integrals as the sum of an intergral over the range [0, X] plus an integral over the range [X, ∞], where X is ''large'', say, X = max{100, 2|Ωq|, 2λ m , 2λ n }. The integrals over the range [0, X] contain the removable singularities, which now present no particular problems for the numerical integration. One must simply take care to correctly evaluate the integrands in the neighbourhood of the singular points.

The integrals can be very quickly calculated to double precision by using a Gauss-Legendre or a Gauss-Kronrod Quadrature method [START_REF] Press | Numerical recipes in Fortran: The art of scientific computing[END_REF], chapter 4.

For the integrals over the range [X, ∞] the problem lies in the oscillatory nature of the integrands and the fact that they do not tend to zero very quickly. The basic idea is to replace the Bessel functions which occur in the integrands by asymptotic expansions with specific estimates for the error term. Watson [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], chapter 7, gives the following result:

For x > 0:

) ( ) sin( ) ( ) (cos( ) ( 4 4 2 0 x Q x x P x x J n n x π π π - - - = (11)
where for any n ≥ 1

) , ( 1 ) ( 1 1 ) 1 ( 2
2 x n E x P n m x C n m m m + + = ∑ = - (12)
∑ = - + = + + n m x C n x n E x Q m m m 0 2 ) 1 ( ) , ( ) ( 1 2 1 2 . ( 13
)
The sequence {C r } is given for r = 0, 1, 2, 3,… by: ) , ( [START_REF] Li | Determination of electromagnetic properties of materials using flanged open ended coaxial probe -Full wave analysis[END_REF] with an explicit error term. The integrals (15) can be quickly and accurately evaluated to double precision using a standard continued fraction algorithm, [START_REF] Press | Numerical recipes in Fortran: The art of scientific computing[END_REF] The departure from a square grid pattern is beginning to be visible at 2 GHz and it is manifest at 5 GHz.

C 0 = 1, C 1 = 1 2 /(1!8 2 ), C r+1 = C r (2r+1) 2 /8(r+1)
+ + < n n x C x n E , 3 2 3 2 2 ) , ( + + < n n x C x n E . ( 14
For any frequency f less than 1 GHz we have a square grid characterized by C 1 (f) and C 2 (f). To see whether C 1 and C 2 vary with frequency we calculated the grids for f = 0.1, 0.2, …, 1.0 GHz. The results are shown in Table I. We note that C 2 is practically constant and that C 1 varies by a factor of 3 over the frequency range 0.1 to 1.0 GHz The variation of C 1 can be very closely represented by a quadratic function of f: C 1 (f) = a 1 + a 2 4π 2 f 2 , where a 1 = 0.60102 and a 2 = -0.010670, with a correlation coefficient of 0.999. Thus, even in conditions where the simple lumped capacitor model can be fitted to the exact admittance grid, the coefficients C 1 and C 2 vary with frequency. We note that several authors ( [START_REF] Kraszewski | Capacitance of open-ended dielectric filled coaxial lines: Experimental results[END_REF], [START_REF] Athey | Measurement of radio frequency permittivity of biological tissues with an open-ended coaxial line, part 1[END_REF], [START_REF] Gabriel | Admittance models for open ended coaxial[END_REF]) in the course of careful experimental measurements have remarked that the parameters they found for the capacitor model varied with frequency and that this variation seemed to be quadratic. Thus, the variations which our theoretical calculations have revealed are perfectly detectable and should be taken into account. This means that instead of using the model Y(f,ε) = j2πf(C 1 + C 2 ε), where C 1 and C 2 do not depend upon f, one should use:

Y(f, ε) = j2πf(A 1 + A 2 f 2 +A 3 ε), (16)
or Re(Y) = -2πfA 3 ε" Im(Y) = 2πf(A 1 + A 2 (2πf) 2 + A 3 ε'), [START_REF] Misra | A quasistatic analysis of open ended coaxial lines[END_REF] where the constants A 1 , A 2 and A 3 do not depend upon ν.

The A i can be determined by the calibration procedure discussed below.

One way of estimating the validity range of the model is to compute the percentage differences between the results of the exact calculation and the values of the admittance given by the model. The open squares correspond to those pairs (ε',ε") for which the difference is less than 1% and the solid squares to the pairs where the difference is greater than 1%. If we use the interpolation for C 1 and C 2 given above, then the same precision holds for the same permittivity values over the frequency range 0 to 1 GHz. Thus, for permittivities (ε',ε") within the area defined by the solid line in Fig. 5 and for frequencies between 0 and 1 GHz the admittance model ( 16) can be calibrated so as to represent the theoretical admittance model with a difference of less than 1%.

IV. PROBE CALIBRATION

Why use the lumped capacity model at all? The exact calculation of the impedance will correspond to the measured impedance of a "real" probe if the hypotheses and numerical values assumed in the calculation correspond to the "real" probe. One could, in principle, use precalculated grids, such as Fig. 4, to read directly the permittivity from the measured admittance at any given frequency.

Unfortunately real probes do not have infinite groundplanes, they are not perfect conductors, their linear dimensions are known with a relative error, perfect cylindricity and axiality are true up to a point, the central conductor is not exactly flush with the ground-plane etc.

Variations in the calculated admittace due to an uncertainty in the physical charaaacteristics of the probe (such as the internal and external radii, r and R, the permittivity of the coaxial cable filling etc.) are best estimated statistically by taking sequences of random values of the parameters r, R, ε d within realistic error intervals and hen calculating the probbility distribution of the resulting values of the admittance. In precisely controlled laboratory conditions it is possible to use the exact calculations to determine absolute values of permittivity, but even so, the effect of the finite ground-plane is observed at certain resonant frequencies when measuring pure water [START_REF] Gabriel | Admittance models for open ended coaxial[END_REF]. Our concern is primarily with quality control in the food industry where the field conditions are far from being perfect and it is preferable to calibrate a given probe in terms of a specific empirical admittance model.

The admittance will usually be measured with a vector network analyser (VNA). The defects in the cables up to the connection with the coaxial probe will be taken into account by the manufactuers calibration kit and the measured admittance will be relative to this connection plane. However, what is required is the admittance at the probe/material interface at the end of the probe. There are three possibilities:

-a. If the probe is rigid, relatively short in length and in pristine condition, then one can suppose that there are no defects and the admittance at the probe/material interface can be calculated from the measured value at the VNA/probe connection using the electrical length of the probe to determine the phase change in the admittance at the end of the probe.

-b. Certain VNA's have a built-in time domain gating procedure that determines the admittance at the probe/material interface. The technique is attractive, but poses a certain number of subtle problems which are not easy to resolve [START_REF] Jenkins | Use of open-ended coaxial line sensor with a liminar or liquid dielectric backed by a conducting plane[END_REF], [START_REF] Porrata-Doria | Analysis and reduction of the distortion induced by time domain filtering techniques in network analysers[END_REF].

-c. If the time domain gating method is not available or the probe cannot be considered perfect, then it is usual to model the defects along the line with a two port transfer matrix. The transfer matrix combined with the lumped capacity model yields: Y(f,ε) = j2πf(D 1 + εD 2 ), where D 1 and D 2 are complex numbers which depend upon f.

In order to calibrate the probe we must work in frequency and permittivity ranges for which the lumped capacity model differs by say 1% from the theoretical calculation and use reference media with permittivities within the valid range to determine either the coefficients a 1 , a 2 , a 3 or the coefficients D 1 (f) and D 2 (f).

If we are in cases (a) or (b) then a 1 , a 2 , a 3 can be determined uniquely from the imaginary part of the admittance and a knowledge of ε' of the reference media. This observation is important because the number of references substances is rather limited and for calibration purposes one should cover the desired permittivity range. It is easy to prepare water-alcohol solutions or saline solutions that have ε' at any given value in the range 10 to 80 by an appropriate mixture. Precise permittivity data for these solutions over a wide temperature range are readily available ( [START_REF] Broersen | Facts and fiction in spectral analysis[END_REF], [START_REF] Evans | Intercomparison of dielectric reference materials available for the calibration of an open-ended probe at different temperatures[END_REF], [START_REF] Mashimo | The dielectric relaxation of mixtures of water and primary alcohol[END_REF], [START_REF] Bertolini | The dielectric properties of alcohol-water solutions. I The alcohol rich region[END_REF], [START_REF] Evans | Intercomparison of dielectric reference materials available for the calibration of an open-ended probe at different temperatures[END_REF]).

V CONCLUSION

The open-ended coaxial probe is frequently used to chrcterise the dielectric properties of maaterials via admittance measurements. The deduction of the permittiviy of the material from the admittance data is made by assuming that the admittance of the probe/materiaal interface is described by some simple empirical model. Such models are not universally valid. We propose a method to determine the frequence range an permittivity intervals for which a given model has at most a specified deviation from an exact theoretical calculation of the interface admittance. As an illustration of the general method we apply it to the commly used open-ended coaxial probe and the "lumped capacity" model.

APPENDIX

The transformation of the integrals (4) to the integrals [START_REF] Decreton | Nondestructive measurement of complex permittivity for dielectric slabs[END_REF] is not new, but we have not found an easily accessible selfcontained exposition and the derivation is not obvious without some guidelines. We refer to a standard treatise for specific theorems and explicitly indicate the more routine mathematical operations. Proof See [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], page 384 and page 405.

Lemma 2 If Re(a) > 0, R > 0 then λ λ λ λ d a R J R e aR . ) . ( . 0 2 2 0 ∫ ∞ - + = (18)
Proof Use Hänkel's double integral theorem [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], chapter 14:

If the integral 0 ( ) F r rdr ∞ ∫ exists and is absolutely convergent, then ∫ ∫ ∞ ∞ = 0 0 0 0 ) ( ) ( ) ( ) ( λ λ λ λ drd r J r rF R J R F with F(R) =e -aR /R and use lemma 1. Lemma 3 If Re(a) > 0, R 2 = ρ 1 2 + ρ 2 2 -ρ 1 ρ 2 cos(φ) then, λ λ λρ λρ λ π ϕ ϕ π d a J J d R e aR ∫ ∫ ∞ - + = 0 2 2 2 1 1 1 2 0 ) ( ) ( 2 ) cos(
Proof From Lemma 2 we have:

∫ ∫ ∫ ∫ ∫ ∞ ∞ - + = + = = π π π λ ϕ ϕ λ λ λ ϕ λ ϕ λ λ λ ϕ ϕ 2 0 0 0 2 2 0 2 2 0 2 0 2 0 ) cos( ) ( ) cos( ) ( 2 ) cos( d d R J a d d a R J d R e aR I
We now apply Neumann's addition theorem [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF] , Chapter 11:

"If R 2 = ρ 1 2 + ρ 2 2 -ρ 1 ρ 2 cos(φ) then, ( ) ∑ ∞ = - = 0 2 1 , 0 0 ) cos( ) ( ) ( 2 )
( n n n n n J J R J ϕ λρ λρ δ λ "
Substitute the above expression for J 0 (Rλ) in the final integral to obtain:

( ) λ ϕ ϕ ϕ λρ λρ δ λ λ π d d n J J a n n n n ) cos( ) cos( ) ( ) ( 2 0 2 0 0 2 1 , 0 2 2 ∫ ∫ ∑ ∞ ∞ = - + = ( ) ∑ ∫ ∫ ∞ = ∞ - + 0 2 0 2 1 , 0 0 2 2 ) cos( ) cos( ) ( ) ( 2 n n n n d d n J J a π λ ϕ ϕ ϕ λρ λρ δ λ λ Since ∫ π ϕ ϕ ϕ 2 0 d ) cos( ) n cos(
is equal to π if n = 1 and is zero otherwise we obtain the stated result.

Lemma 4 If Re(κ) > 0 and R 2 = ρ 1 2 + ρ 2 2 -ρ 1 ρ 2 cos(φ) then { } 2 2 0 0 0,0 1 2 2 2 0 0 ( ) ( ) cos( ) 2 R b b a a J a J b e d d d d R κ π λ λ ϕ ϕ ρ ρ π λ λ λ κ - ∞ - = = + ∫ ∫ ∫ ∫ I Proof 2 1 2 0 R b a b a 0 , 0 d d d ) cos( R e ρ ρ ϕ ϕ π κ ∫ ∫ ∫ - = I 2 1 0 2 2 2 1 1 1 b a b a d d d ) ( J ) ( J 2 ρ ρ λ κ λ λρ λρ λ π ∫ ∫ ∫ ∞ + = ∫ ∫ ∫ ∞ + = b a b a 2 2 1 1 1 1 0 2 2 d d ) ( J d ) ( J 2 λ ρ λρ ρ λρ κ λ λ π Recall that ) ( ) ( 0 0 x J dx x dJ - =
and so

1 0 0 ( ) ( ) ( ) b a J x dx J a J b = - ∫ . Thus { } 0 0 1 1 1 ( ) ( ) ( ) b a J a J b J d λ λ λρ ρ λ - = ∫ and { } 0 0 1 2 2 ( ) ( ) ( )
{ } 1 1 1 0 0 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) n n nn n J J d J J J J ρ ρ λρ λ ρ ρ λ λ ρ λρ λ λ ρ λρ λ λ = - - ∫ { } 1 1 1 0 0 1 2 2 ( ) ( ) ( ) ( ) ( ) ( )
{ } 1 1 2 2 ( ) ( ). ( ) ( ) ( ) ( ) z z z z d z k k z z k z z k μ μ μ μ μ μ + + = - - ∫ C k C l C C l lC C l l
From [START_REF] Press | Numerical recipes in Fortran: The art of scientific computing[END_REF], page 82, we have, for all Cylinder functions, the relation:

1 1 2 ( ) ( ) ( ) z z z z μ μ μ μ - + + = C C C
If we use this relation to eliminate

) ( 1 z + μ C and C μ +1 ( ) z
from the above expression we obtain, after algebraic manipulation, the following expression for the indefinite integral:

{ } 1 1 2 2 ( ) ( ). ( ) ( ) ( ) ( ) z z z z d z k z z k k z z k μ μ μ μ μ μ - - = - - ∫ C k C l lC C l C C l l To obtain (1), take z = ρ, k = λ, l = λ n , C μ = J 1 , C μ = J 1 .
To obtain [START_REF] Gardiol | Open-ended waveguides: Principles and applications[END_REF]

take z = ρ, k = λ, l = λ n , C μ = J 1 , C μ = N 1 . Lemma 6 If n ≥ 1 and R 2 = ρ 1 2 + ρ 2 2 -ρ 1 ρ 2 cos(φ), then 2 1 2 0 2 2 , 0 ) cos( ) ( ρ ρ ϕ ϕ ρ ρ π κ d d d R e R b a R n b a n ∫ ∫ ∫ - = I = { } ( ) λ λ κ λ λ λ λ λ λ λ λ d F b J a J b J n n n n ) ( ) ( ) ( ) ( 4 0 2 2 2 2 0 0 0 ∫ ∞ + - - where F n (λ) = C n .{ J 0 (λb)J 0 (λ n .a) -J 0 (λ n b)J 0 (λ.a) } Proof 1 2 2 0 2 2 , 0 ) cos( ) ( ρ ρ ϕ ϕ ρ ρ π κ d d d R e R R b a n b a n ∫ ∫ ∫ - = I = 1 2 0 2 2 2 1 1 1 2 2 ) ( ) ( ) ( 2 ρ ρ λ κ λ λρ λρ λ ρ ρ π d d d J J R b a n b a ∫ ∫ ∫ ∞ + = ∫ ∫ ∫ ∞ + b a b a n d d J d J R a λ ρ λρ ρ λρ ρ ρ λ λ π . ). ( . ). ( ). ( . 2 1 1 1 2 2 1 2 2 0 2 2 = { } 0 0 2 2 1 2 2 2 2 0 ( ) ( ) 2 ( ) () b n a J a J b R J d d a λ λ λ π ρ ρ λρ ρ λ λ λ ∞ - + ∫ ∫ = { } λ ρ λρ ρ ρ λ λ λ π d d J R a b J a J b a n ∫ ∫ ∞ + - . ) ( ) ( ) ( ) ( 2 2 2 1 2 2 0 2 2 0 0 { } { } λ ρ ρ λ λ λ ρ λ λρ ρ λ λ λ π d . d ) ( N ) a ( J ) a ( N ) ( J ) ( J a ) b ( J ) a ( J C . 2 b a 2 2 n 1 n 0 n 0 2 n 1 2 1 2 0 2 2 0 0 n ∫ ∫ - + - = ∞ We now have to evaluate the integral: { } ∫ - b a n n n n d N a J a N J J 2 2 1 0 0 2 1 2 1 2 ) ( ) ( ) ( ) ( ) ( ρ ρ λ λ λ ρ λ λρ ρ . It is equal to: ∫ ∫ - b a n n b a n n n d N J a J d J J a N 2 2 1 2 1 2 0 2 2 1 2 1 2 0 ) ( ) ( ) ( ) ( ) ( ) ( ρ ρ λ ρ λ ρ λ ρ ρ λ λρ ρ λ
The integrals:

∫ b a n d J J 2 2 1 2 1 2 ) ( ) ( ρ ρ λ λρ ρ and ∫ b a n d N J 2 2 1 2 1 2 ) ( ) ( ρ ρ λ λρ ρ
have been evaluated in Lemma 5. If we substitute the expressions we find that the integral is equal to:

[ ] ) ( ) ( ) ( ) ( ) ( ) ( 0 0 1 0 0 1 2 2 a J b J b N a N b J b J b n n n n n λ λ λ λ λ λ λ λ λ - - [ ] ) ( ) ( ) ( ) ( ) ( ) ( 0 1 0 0 1 0 2 2 a J b J b N a N b J b J b n n n n n n λ λ λ λ λ λ λ λ λ - - - [ ] ) ( ) ( ) ( ) ( ) ( ) ( 0 0 1 0 0 1 2 2 a J b J a N a N a J a J a n n n n n n λ λ λ λ λ λ λ λ λ - - -
The term in the second bracket is equal to zero because the quantities λ n are defined so that J 0 (λ n b).N 0 (λ n a) = N 0 (λ n b).J 0 (λ n a). We now use the relation: J 1 (z)N 0 (z) -J 0 (z)N 1 (z) = 2/πz, taking in turn z = λ n b and z = λ n a , to eliminate N 1 (λ n b) and N 1 (λ n a) from the first and third brackets. After a simple algebraic manipulation we arrive at the following expression for the integral:

{ } ) ( ) ( ) ( ) ( . ) ( 2 0 0 0 0 2 2 0 a J b J b J a J b J n n n n n λ λ λ λ λ λ λ λ πλ - -
Substituting this expression in the integral we obtain the stated result.

Lemma 7 If m > 0, n > 0, Re(κ) > 0, R 2 = ρ 1 2 +ρ 2 2ρ 1 ρ 2 cos(φ) then

  If we write ε = ε' -jε'' and Y(f,ε) = j2πf(C 1 + εC 2 ), thenRe(Y) = 2πfC 2 ε'' and Im(Y) = 2πf(C 1 + C 2 ε').For a fixed f and for fixed values of C 1 and C 2 the model induces a reticular parameterisation of the complex Y plane as a function of ε' and ε'', i.e. the contours for fixed ε' and variable ε" are parallel horizontal straight lines and the contours for fixed ε" and variable ε' are parallel vertical stright lines. (See Fig.1). Conversely, it follows that if the true contours of ε' and ε" in the Y plane do not form a reticulation, then the simple capacitor model is certainly not valid for the given frequency and permittivity range.

)

  Using the above expansion the three types of integrals can be written as linear combinations of integrals of the kind

  chapter 6. The explicit nature of the error term[START_REF] Delecki | Analysis of the open-ended coaxial line sensors[END_REF] guarentees the precision of the calculated values of the intergrals.We have implemented the above ideas in a PC program (Fortran + Visual Basic user interface). The program and its source code are freely available from the authors, together with the mathematical details of the algorithms that are used.III. A SPECIFIC EXAMPLEAs an illustration of the procedure we consider the common 3.6 mm rigid coaxial line excited in the TEM mode. The relevant physical parameters of the line are: a= 0.45925 mm, b = 1.4925 mm, ε d = 2.15. A permittivity range 5 ≤ ε' ≤ 100, 5 ≤ ε" ≤ 100 was used, since we are interested in humidity control testing. We calculated nomograms for Y(f, ε) at 100, 1000 2000 and 5000 MHz. They are shown in Figs. 1, 2, 3, and 4. The admittances were calculated with a precision of ±0.001. This involved trucating the infinite linear system (2) at up to 60 terms for some of the calculations. The horizontal lines correspond to constant values of ε'. The bottom line is ε' = 5, the top line is ε' = 100, the increment is 5 units. The vertical lines correspond to constant values of ε". The far left line is ε" = 5, the far right line is ε" = 100, the increment is 5 units.

Fig. 5

 5 is such a representation for f = 1 GHz.

Lemma 1

 1 If z ∈ C, b ≥ 0 and if either Re(z) > 0 and b ≥ 0 or Re(z) = 0 and b ≠ Im(z)

Lemma 5

 5 If λ ≠ λ n then the following indefinite integrals hold:

  indefinite integrals are special cases of a general formula for Cylinder functions,[START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], Chapter 5, equation[START_REF] Bois | Multimode solution for the reflection properties of an open-ended rectangular waveguide radiating into a dielectric halfspace[END_REF]. The formula in question is: Cylinder functions of order μ and k ≠ l, then

  In the proof of Lemma 6 we showed that:

∫

  Replacing the integrals with respect to ρ 1 and ρ 2 by the corresponding expressions gives the stated result.

TABLE I VARIATION

 I OF C 1 AND C 2 WITH FREQUENCY FOR A 3.6 COAXIAL PROBE FOR THE PERMITTIVITY RANGE 5 ≤ ε' ≤ 100 AND 5 ≤ ε'' ≤ 100 FIG. 1 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 100MHZ. FIG. 2 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 1000MHZ. FIG.3 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 2000MHZ. FIG. 4 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 5000MHZ FIG. 5 SCATTER DIAGRAM (ε' IS ABSCIISSA, ε" IS ORDINATE) FOR THE VALIDITY OF THE LUMPED CAPACITY MODEL AT 1 GHZ. (OPEN SQUARES CORRESPOND TO PAIRS (ε',ε'') FOR WHICH THE MODEL ERROR IS < 1%)

	> IM 06-8408 <		14
	ONE TABLE
	GHz	C 1 pF	C 2 pF
	0.1	0.597943	0.899251
	0.2	0.586221	0.899513
	0.3	0.562271	0.899963
	0.4	0.535866	0.900591
	0.5	0.498545	0.901392
	0.6	0.453240	0.990236
	0.7	0.400015	0.903505
	0.8	0.338888	0.904828
	0.9	0.270034	0.906322
	1.0	0.184453	0.909966

TABLE I VARIATION

 I OF C1 AND C2 WITH FREQUENCY FOR A 3.6 COAXIAL PROBE FOR THE PERMITTIVITY RANGE 5 ≤ ε' ≤ 100 AND 5 ≤ ε'' ≤ 100

FIVE FIGURES, ONE PER PAGE

FIG. 1 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 100MHZ.

FIG. 2 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 1000MHZ.

FIG.3 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 2000MHZ.

FIG. 4 PARAMETRIZATION OF THE COMPLEX ADMITTANCE PLANE AT 5000MHZ