The Topology of Wireless Communication

In this paper we study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. To describe the behavior of a multi-station network, we use the convenient representation of a \emph{reception map}. In the SINR model, the resulting \emph{SINR diagram} partitions the plane into reception zones, one per station, and the complementary region of the plane where no station can be heard. We consider the general case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected. This poses the algorithmic challenge of designing efficient point location techniques as well as the theoretical challenge of understanding the geometry of SINR diagrams. We achieve several results in both directions. We establish a form of weaker convexity in the case where stations are aligned on a line. In addition, one of our key results concerns the behavior of a $(d+1)$-dimensional map. Specifically, although the $d$-dimensional map might be highly fractured, drawing the map in one dimension higher "heals" the zones, which become connected. In addition, as a step toward establishing a weaker form of convexity for the $d$-dimensional map, we study the interference function and show that it satisfies the maximum principle. Finally, we turn to consider algorithmic applications, and propose a new variant of approximate point location.Comment: 64 pages, appeared in STOC'1

On Wireless Scheduling Using the Mean Power Assignment

In this paper the problem of scheduling with power control in wireless networks is studied: given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of concurrently transmitting nodes. The signal interference is modeled by SINR constraints. Approximation algorithms are given for this problem, which use the mean power assignment. The problem of schduling with fixed mean power assignment is also considered, and approximation guarantees are proven

Braess's Paradox in Wireless Networks: The Danger of Improved Technology

When comparing new wireless technologies, it is common to consider the effect that they have on the capacity of the network (defined as the maximum number of simultaneously satisfiable links). For example, it has been shown that giving receivers the ability to do interference cancellation, or allowing transmitters to use power control, never decreases the capacity and can in certain cases increase it by $\Omega(\log (\Delta \cdot P_{\max}))$, where $\Delta$ is the ratio of the longest link length to the smallest transmitter-receiver distance and $P_{\max}$ is the maximum transmission power. But there is no reason to expect the optimal capacity to be realized in practice, particularly since maximizing the capacity is known to be NP-hard. In reality, we would expect links to behave as self-interested agents, and thus when introducing a new technology it makes more sense to compare the values reached at game-theoretic equilibria than the optimum values. In this paper we initiate this line of work by comparing various notions of equilibria (particularly Nash equilibria and no-regret behavior) when using a supposedly "better" technology. We show a version of Braess's Paradox for all of them: in certain networks, upgrading technology can actually make the equilibria \emph{worse}, despite an increase in the capacity. We construct instances where this decrease is a constant factor for power control, interference cancellation, and improvements in the SINR threshold ($\beta$), and is $\Omega(\log \Delta)$ when power control is combined with interference cancellation. However, we show that these examples are basically tight: the decrease is at most O(1) for power control, interference cancellation, and improved $\beta$, and is at most $O(\log \Delta)$ when power control is combined with interference cancellation

Generalized Perron--Frobenius Theorem for Nonsquare Matrices

The celebrated Perron--Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. However, many real-life scenarios give rise to nonsquare matrices. A natural question is whether the PF Theorem (along with its applications) can be generalized to a nonsquare setting. Our paper provides a generalization of the PF Theorem to nonsquare matrices. The extension can be interpreted as representing client-server systems with additional degrees of freedom, where each client may choose between multiple servers that can cooperate in serving it (while potentially interfering with other clients). This formulation is motivated by applications to power control in wireless networks, economics and others, all of which extend known examples for the use of the original PF Theorem. We show that the option of cooperation between servers does not improve the situation, in the sense that in the optimal solution no cooperation is needed, and only one server needs to serve each client. Hence, the additional power of having several potential servers per client translates into \emph{choosing} the best single server and not into \emph{sharing} the load between the servers in some way, as one might have expected. The two main contributions of the paper are (i) a generalized PF Theorem that characterizes the optimal solution for a non-convex nonsquare problem, and (ii) an algorithm for finding the optimal solution in polynomial time

Algorithms for Efficient Communication in Wireless Sensor Networks - Distributed Node Coloring and its Application in the SINR Model

In this thesis we consider algorithms that enable efficient communication in wireless ad-hoc- and sensornetworks using the so-called Signal-to-interference-and-noise-ratio (SINR) model of interference. We propose and experimentally evaluate several distributed node coloring algorithms and show how to use a computed node coloring to establish efficient medium access schedules

Fuzzy FES controller using cycle-to-cycle control for repetitive movement training in motor rehabilitation. Experimental tests with wireless system

A prototype of wireless surface electrical stimulation system combined with the fuzzy FES controller was developed for rehabilitation training with functional electrical stimulation (FES). The developed FES system has three features for rehabilitation training: small-sized electrical stimulator for surface FES, wireless connection between controller and stimulators, and between controller and sensors, and the fuzzy FES controller based on the cycle-to-cycle control for repetitive training. The developed stimulator could generate monophasic or biphasic high voltage stimulus pulse and could output stimulation pulses continuously more than 20 hours with 4 AAA batteries. The developed system was examined with neurologically intact subjects and hemiplegic subjects in knee joint control. The maximum knee joint angle was controlled by regulating burst duration of stimulation pulses by the fuzzy controller. In the results of two experiments of knee extension angle control and knee flexion and extension angle control, the maximum angles reached their targets within small number of cycles and were controlled stably in the stimulation cycles after reaching the target. The fuzzy FES controller based on the cycle-to-cycle control worked effectively to reach the target angle and to compensate difference in muscle properties between subjects. The developed wireless surface FES system would be practical in clinical applications of repetitive execution of similar movements of the limbs for motor rehabilitation with FES. Keywords: Cycle-to-Cycle Control, Functional Electrical Stimulation, Rehabilitation, Surface Electrical Stimulation, Wireless System Introduction Functional electrical stimulation (FES) can be an effective method of assisting or restoring paralyzed motor functions caused by spinal cord injury or cerebrovascular disease. FES has been utilized as an orthotic and therapeutic aid in the rehabilitation of the upper and lower limb motor functions. The therapeutic effects during rehabilitation with FES have been shown to improve muscle strength In motor rehabilitation, goal-oriented repetitive movement training of the paralyzed limbs has been applied. One of the therapeutic effects is motor relearning, which is reacquisition of previously learned motor skills after central nervous system injury. In general, assistance provided by therapists is required to perform repetitive execution of identical or similar movements of the limbs in the rehabilitation training. On the other hand, several large-scale robotic systems have been developed to reduce the workloads for the therapists and improve repeated training for patients For motor rehabilitation with FES, surface electrical stimulation would be useful because of its noninvasive nature. However, the electrical stimulator for surface FES is usually required to generate high stimulation intensity pulses, which leads to an increase of size and power consumption of the stimulator. In addition, wired connection between controller and stimulators and between controller and sensors are sometimes cumbersome and can obstruct the movement of the limb. Therefore, this study focused on miniaturizing the electrical stimulator and on removing the connection code using wireless technology. In training with FES for rehabilitation, repetitive movements of limbs have to be controlled appropriately by stimulating the relevant muscles. Closed-loop FES control is required to suppress variations of initial position and muscle response, and muscle fatigue in the exercise and to derive benefit from InnovatIon Fuzzy FES controller using cycle-to-cycle control for repetitive movement training in motor rehabilitation. Experimental tests with wireless system Fuzzy FES controller for motor rehabilitation 315 Copyright © 2010 Informa UK Ltd. the rehabilitation. Tracking control of joint angles of the lower limb is a difficult problem because of nonlinearity and significant time delay, both affecting the responses of the musculoskeletal system to electrical stimulation. For this purpose, the fuzzy FES controller based on the cycle-to-cycle control The purpose of this study is to show the effectiveness of the wireless FES system in which the fuzzy cycle-to-cycle control was implemented for repetitive movement control through control tests with neurologically intact and hemiplegic subjects. In this paper, the wireless surface electrical stimulation system combined with the fuzzy FES controller based on the cycle-to-cycle control was developed. The developed wireless feedback FES system was examined in knee joint control. First, the maximum knee extension angle control stimulating one muscle was performed to find the basic performance of the closed-loop control with the wireless system with neurologically intact subjects and hemiplegic subjects. Then, the maximum knee flexion and extension angle control was performed as a preliminary test of controlling a sequence of movements stimulating two muscles with neurologically intact subjects. Wireless surface electrical stimulation system The wireless surface electrical stimulation system consists of three parts: the fuzzy FES controller implemented on the PC, surface electrical stimulator and sensor. For wireless communication between the controller and the stimulator and between the controller and the sensor, a 2.4 GHz wireless transceiver module (WCU-241, K2-denshi) was used. The stimulus data determined by the fuzzy FES controller is transmitted to the stimulator through the wireless transceiver modules. The stimulator generates electrical stimulation pulses immediately after receiving the stimulus data. The data was composed of stimulus voltage, stimulus pulse width and monophasic/biphasic pulse type. The current system can send the stimulus data of up to 4 channels together. The sensor data are digitized by a 10 bit A/D converter with 40 Hz of sampling frequency in the wireless transceiver module and transmitted to the fuzzy FES controller through wireless transceiver module as feedback signal. It is possible to receive the sensor data of up to 4 channels simultaneously. The topology of wireless communication is the point-to-point connection between the controller and the stimulator and between the controller and the sensor, in which the original protocol (the packet consisting of the data of Preamble, Address, Payload and Cyclic Redundancy Check) is used. The bit rate and the latency of the transceiver module are up to 250 kbit/s and about 2 ms, respectively. Electrical stimulator consists of the wireless transceiver module, the boost converter and the stimulation pulse generator. The boost converter stores electric charges in a tank capacitor and generates high voltage pulse required for the surface electrical stimulation (maximum output voltage: 128 V). The stimulator generates monophasic or biphasic pulse. The maximum stimulation frequency was 520 Hz. In usual FES control, stimulation pulses with a constant stimulation frequency smaller than about 100 Hz are used. High frequency stimulation pulses are sometimes used in research work as a doublet or a triplet Outline of fuzzy FES controller based on cycleto-cycle control The block diagram of the fuzzy FES control for repetitive movement is shown in where TB[n-1] is the stimulation burst duration for the cycle just before the current one and ΔTB[n] is the output of the fuzzy controller adjusted by the 2 factors. The fuzzy controller was designed as multi-input singleoutput (MISO) controller with two inputs of 'error' and 316 Naoto Miura et al. Journal of Medical Engineering & Technology 'desired range' . The 'error' was defined as the difference between the target angle and the maximum angle elicited by the burst stimulation pulses. The 'desired range' was defined as the difference between the target angle and the angle at the stimulation onset. The E-OAF is determined by the error of the cycle just before the current cycle, which increases the output value of the controller if the error is large, and decreases if the error is small. The S-OAF is determined by joint angle production ratio that is defined as the ratio of joint angle change to stimulation burst duration, θ ch /TB, which means sensitivity of the muscle to electrical stimulation. figure 2 shows an example of input and output membership functions of the fuzzy controller for knee extension angle control. Input membership functions were expressed by triangular and trapezoidal fuzzy sets. The membership functions of the 'error' and 'desired range' comprised 7 and 3 linguistic terms, respectively, and that of the output variable was expressed as 11 fuzzy singletons. The membership functions of the E-OAF comprised 5 linguistic terms, and the output variable was expressed as 5 fuzzy singletons. The membership functions of the S-OAF comprised 3 linguistic terms, and the output variable was expressed as 3 fuzzy singletons. The fuzzy inference was accomplished by using the Mamdani method. Center of gravity (COG) was used in the defuzzification process. Parameter values of the fuzzy controller were determined based on control results and values obtained in our previous studies Knee extension angle control with neurologically intact subjects and hemiplegic subjects Experimental methods The vastus muscles were stimulated through surface electrodes (SRH5080, SEKISUI PLASTICS), and maximum knee extension angle was controlled by the surface electrical stimulation system (figure 3). Two neurologically intact subjects (subject A, B) and two hemiplegic subjects caused by cerebral apoplexy (subject C: 76-year-old right sided hemiplegic male patient, subject D: 38-year-old left sided hemiplegic male patient) participated in the experiments. Subjects' consent to participate in the experiment was obtained. The subject seated in the chair (GT-30, OG Giken) and relaxed his legs during experiments. To maintain the sitting position, the trunk of hemiplegic subject was fixed to the chair with band. The sitting position of the neurologically intact subjects was determined by themselves and that of the hemiplegic subject was determined by adjusting the back of the chair in the forward and backward direction for appropriate knee joint movement. Consequently, the initial joint angle was about 65° in neurologically intact subjects and about 80° in hemiplegic subjects (0° means full knee extension). The target angle was 30° (range of knee extension angle was about 35°) for neurologically intact subjects and was 70° or 65° (range of knee extension angle was about 10° or 15°) for hemiplegic subjects. The target angle was determined based on the maximum knee extension angle developed by electrical stimulation. The Fuzzy FES controller for motor rehabilitation 317 Copyright © 2010 Informa UK Ltd. values of fuzzy membership functions for neurologically intact subjects were determined in previous experiments with other subjects In one experimental session, 100 cycles were performed with neurologically intact subjects, and 35 cycles were performed with hemiplegic subjects to reduce physical load. The knee joint angles were measured with an electric goniometer (M180, Penny & Giles). The output signal of the goniomenter was digitized by a 10 bit A/D converter with 40 Hz of sampling frequency. Pulse width and pulse frequency was fixed at 0.3 ms and 20 Hz, respectively. Stimulus pulse amplitude was determined so as to develop target joint angle without pain before the experiment. Initial value of TB was 0 s. Results An example of control results with a hemiplegic subject (Subject C) was shown in figure 4. Stimulation burst duration TB increased as the number of cycles increased, and then the maximum extension angle was reached to the target angle at the 4th control cycle. In the first few cycles, the value of E-OAF was large, which shows the E-OAF worked effectively in early cycles in order to reach the targets with small number of cycles. The value of S-OAF was large at the most cycles, which shows the S-OAF compensated for the weak muscle response of this subject. For evaluating control results, settling index (SI), mean error (ME) and mean variation (MV) were calculated (table 1). SI was defined as the number of cycles that were required to reach the target joint angle with absolute error that was less than or equal to 3°. ME was mean value of the absolute error between the target angle and the produced maximum extension angle in cycles after reaching the target. MV was mean of the difference in controlled joint angles between two consecutive cycles after reaching the target. The number in parentheses in table 1 shows the result for the first 35 cycles that is the same evaluation condition as the hemiplegic subjects. As seen in table 1, SI was 3-5 cycles, ME was less than 1° for all trials and MV was approximately 1°. The evaluation indices for the hemiplegic subjects showed similar values as those for the neurologically intact subjects. The developed system performed well in the knee extension control with all subjects. However, there were some cases that the value of S-OAF did not change dynamically to the change in the sensitivity. Journal of Medical Engineering & Technology neurologically intact subjects because of safety for hemiplegic subjects in positioning during control. Experimental methods The maximum knee flexion and extension angles were controlled in one cycle stimulating the hamstrings and the vastus muscles by the surface electrical stimulation system with 7 neurologically intact subjects (figure 6). Subject's consent to participate in the experiment was obtained. The subject sat on the equipment keeping his position by his upper limbs. The initial knee joint angle (neutral position) was approximately 30° and target angles were 45-70° for knee flexion and 10° for extension (the maximum angle of knee extension is defined as 0°). Starting condition for each control cycle was when the difference of knee joint angle between two consecutive cycles is less than 0.3° for the 20 consecutive samples after 6 seconds from the time when the maximum extension angle was detected in the previous control cycle. The hamstrings were stimulated first and then the vastus muscles were stimulated after detecting the maximum flexion angle. Pulse width was fixed at 0.2 ms. Other electrical stimulation condition and the measurement method of knee joint angle were same as the experiment of knee extension control. Initial value of TB is 0 s and 35 cycles were performed in each control session. Three control sessions were performed for each subject with the time interval between 20 min and 30 min. The number in parentheses shows the result for the first 35 cycles that is the same evaluation condition as the hemiplegic subjects. the change in sensitivity, the modification of the S-OAF will be necessary. Knee flexion and extension control with neurologically intact subjects Based on the results of the previous section, the range of values of input and output membership functions of the S-OAF were expanded, and the number of terms were increased in order to adapt to changes in these muscle responses. The maximum knee flexion and extension angle control was examined as a sequence of movements stimulating two muscles with For personal use only. Fuzzy FES controller for motor rehabilitation 319 Copyright © 2010 Informa UK Ltd. For the hamstrings, the fuzzy model of the S-OAF was changed to have 7 linguistic terms for input membership function and 7 singletons for the output variable. For the vastus muscles, the fuzzy model of the S-OAF was changed to have 4 linguistic terms for input membership function and 4 singletons for the output variable. Results Both maximum joint angles were controlled with five subjects, but sufficient knee flexion angle was not produced by the electrical stimulation with two subjects. One example of control results is shown in For evaluating control results, SI, ME and MV were calculated (table 2). SI was 3-5 cycles, ME was less than 4° for flexion control and less than 2° for extension control. MV was less than 5° for flexion and less than 2.5° for extension. Discussions The developed wireless surface electrical stimulation system combined with the fuzzy controller performed well in the knee angle controls. The system realized reaching the target within about 5 cycles. In most of trials, the mean error after reaching the target was less than about 3°, and the mean variation after reaching the target was less than about 3°. These control results were similar to the results in our previous reports The developed system worked well with hemiplegic subjects in the knee extension control. Although the muscle response produced by the electrical stimulation with the hemiplegic subjects was weak compared to those of neurologically intact subjects, the ability of fuzzy FES controller based on the cycle-to-cycle control is considered to be appropriate to reach the target angle and to compensate difference in muscle properties between subjects. Therefore, the developed system is expected to be practical in clinical applications. Stimulation burst duration (TB) was adjusted appropriately, and the knee joint angle was controlled stably by the fuzzy controller with two parameters of the E-OAF and the S-OAF in both control tests. For large error between the control angle and the target angle in early cycles, the E-OAF worked effectively to reach the target with small number of cycles in all subjects. After reaching the target angle, the E-OAF was small because the error of the obtained knee joint angle was small. In contrast, the S-OAF worked to compensate for the different muscle responses during all the stimulus cycles automatically based on the value of sensitivity. These results showed that both of the E-OAF and S-OAF would be effective in controlling the repetitive execution of similar movements for rehabilitation. In the knee extension angle control, the value of S-OAF did not change dynamically to the change in sensitivity with some subjects. Therefore, the range of values of input and output membership functions of the S-OAF was expanded and the number of terms was increased, and then the knee flexion and extension angle control was examined. Journal of Medical Engineering & Technology input and output membership functions of the S-OAF for the vastus muscles because of the slight saturation of the S-OAF of the vastus muscles in both the lower and higher sensitivity, if necessary. The S-OAF using the sensitivity, which is the ratio of joint angle change to stimulation burst duration, worked effectively in both hemiplegic subjects and neurologically intact subjects. However, the sensitivity obtained by this method may contain other effects such as movements of other parts of the body or gravitational effect. For optimal movement control, it is required to modify by using directly measured muscle activity such as surface electromyogram elicited by electrical stimulation (M-wave). The developed wireless surface FES system by using the wireless transceiver module is expected to improve ease of use. Max. knee extension control SI ± SD (cycle) ME ± SD (deg) MV ± SD (deg) SI (cycle) ± SD ME ± SD (deg) MV ± SD (deg) E 4.0 ± 1.0 2.3 ± 0.1 2.9 ± 0.2 3.7 ± 0.6 1.3 ± 0.2 1.2 ± 0.3 F 2.7 ± 0.6 3.3 ± 0.5 3.7 ± 1.1 3.3 ± 0.6 0.9 ± 0.1 1.1 ± 0.2 H 3.0 ± 0.0 1.3 ± 0.2 1.6 ± 0.4 3.0 ± 0.0 0.9 ± 0.2 1.3 ± 0.4 J 4.0 ± 1.0 2.3 ± 1.3 3.4 ± 1.6 2.7 ± 1.6 1.4 ± 0.5 1.7 ± 0.8 K 3.7 ± 0.6 2.1 ± 1.2 2.9 ± 1.7 3.0 ± 0.0 1.5 ± 0.1 2.0 ± 0.1 Average 3.5 ± 0.8 2.3 ± 0.9 2.9 ± 1.2 3.1 ± 0.5 1.2 ± 0.3 1.5 ± 0.5 Fuzzy FES controller for motor rehabilitation 321 Copyright © 2010 Informa UK Ltd. However, the wireless communication has problems such as delay and interference. The delay of wireless communication between the wireless modules used in the developed system was small (2 ms) enough compared to the sampling period of ADC (25 ms) and the stimulus period (50 ms). Therefore, it is considered that the influence of the delay of the wireless communication on the performance of the cycle-to-cycle controller was small in this system. The interference in wireless communication was not caused in the experiments using the wireless transceiver module for the 2.4 GHz (Industrial, Scientific and Medical: ISM) band. However, the interference problem does not always cause in the wireless communication. It is necessary to deal with the problem of the wireless communication by modifying the communication software including time management within each module, the retransmission processing and so on. The setting of the goniometer for movement measurements is not so easy for rehabilitation training because of limited attachment position and requirement of complicated calibration process. The measurement of movement using wearable sensor such as a gyroscope and acceleromete