Massive MIMO with Optimal Power and Training Duration Allocation

We consider the uplink of massive multicell multiple-input multiple-output systems, where the base stations (BSs), equipped with massive arrays, serve simultaneously several terminals in the same frequency band. We assume that the BS estimates the channel from uplink training, and then uses the maximum ratio combining technique to detect the signals transmitted from all terminals in its own cell. We propose an optimal resource allocation scheme which jointly selects the training duration, training signal power, and data signal power in order to maximize the sum spectral efficiency, for a given total energy budget spent in a coherence interval. Numerical results verify the benefits of the optimal resource allocation scheme. Furthermore, we show that more training signal power should be used at low signal-to-noise ratio (SNRs), and vice versa at high SNRs. Interestingly, for the entire SNR regime, the optimal training duration is equal to the number of terminals.

Power Scaling of Uplink Massive MIMO Systems with Arbitrary-Rank Channel Means

This paper investigates the uplink achievable rates of massive multiple-input multiple-output (MIMO) antenna systems in Ricean fading channels, using maximal-ratio combining (MRC) and zero-forcing (ZF) receivers, assuming perfect and imperfect channel state information (CSI). In contrast to previous relevant works, the fast fading MIMO channel matrix is assumed to have an arbitrary-rank deterministic component as well as a Rayleigh-distributed random component. We derive tractable expressions for the achievable uplink rate in the large-antenna limit, along with approximating results that hold for any finite number of antennas. Based on these analytical results, we obtain the scaling law that the users' transmit power should satisfy, while maintaining a desirable quality of service. In particular, it is found that regardless of the Ricean K-factor, in the case of perfect CSI, the approximations converge to the same constant value as the exact results, as the number of base station antennas, M, grows large, while the transmit power of each user can be scaled down proportionally to 1/M. If CSI is estimated with uncertainty, the same result holds true but only when the Ricean K-factor is non-zero. Otherwise, if the channel experiences Rayleigh fading, we can only cut the transmit power of each user proportionally to 1/√M. In addition, we show that with an increasing Ricean K-factor, the uplink rates will converge to fixed values for both MRC and ZF receivers.

Massive MIMO with Non-Ideal Arbitrary Arrays: Hardware Scaling Laws and Circuit-Aware Design

Massive multiple-input multiple-output (MIMO) systems are cellular networks where the base stations (BSs) are equipped with unconventionally many antennas, deployed on colocated or distributed arrays. Huge spatial degrees-of-freedom are achieved by coherent processing over these massive arrays, which provide strong signal gains, resilience to imperfect channel knowledge, and low interference. This comes at the price of more infrastructure; the hardware cost and circuit power consumption scale linearly/affinely with the number of BS antennas N. Hence, the key to cost-efficient deployment of large arrays is low-cost antenna branches with low circuit power, in contrast to today’s conventional expensive and power-hungry BS antenna branches. Such low-cost transceivers are prone to hardware imperfections, but it has been conjectured that the huge degrees-of-freedom would bring robustness to such imperfections. We prove this claim for a generalized uplink system with multiplicative phasedrifts, additive distortion noise, and noise amplification. Specifically, we derive closed-form expressions for the user rates and a scaling law that shows how fast the hardware imperfections can increase with N while maintaining high rates. The connection between this scaling law and the power consumption of different transceiver circuits is rigorously exemplified. This reveals that one can make the circuit power increase as p N, instead of linearly, by careful circuit-aware system design.

Multipair Full-Duplex Relaying with Massive Arrays and Linear Processing

We consider a multipair decode-and-forward relay channel, where multiple sources transmit simultaneously their signals to multiple destinations with the help of a full-duplex relay station. We assume that the relay station is equipped with massive arrays, while all sources and destinations have a single antenna. The relay station uses channel estimates obtained from received pilots and zero-forcing (ZF) or maximum-ratio combining/maximum-ratio transmission (MRC/MRT) to process the signals. To reduce significantly the loop interference effect, we propose two techniques: i) using a massive receive antenna array; or ii) using a massive transmit antenna array together with very low transmit power at the relay station. We derive an exact achievable rate in closed-form for MRC/MRT processing and an analytical approximation of the achievable rate for ZF processing. This approximation is very tight, especially for large number of relay station antennas. These closed-form expressions enable us to determine the regions where the full-duplex mode outperforms the half-duplex mode, as well as, to design an optimal power allocation scheme. This optimal power allocation scheme aims to maximize the energy efficiency for a given sum spectral efficiency and under peak power constraints at the relay station and sources. Numerical results verify the effectiveness of the optimal power allocation scheme. Furthermore, we show that, by doubling the number of transmit/receive antennas at the relay station, the transmit power of each source and of the relay station can be reduced by 1.5dB if the pilot power is equal to the signal power, and by 3dB if the pilot power is kept fixed, while maintaining a given quality-of-service.

On the MIMO Capacity with Residual Transceiver Hardware Impairments

Radio-frequency (RF) impairments in the transceiver hardware of communication systems (e.g., phase noise (PN), high power amplifier (HPA) nonlinearities, or in-phase/quadrature-phase (I/Q) imbalance) can severely degrade the performance of traditional multiple-input multiple-output (MIMO) systems. Although calibration algorithms can partially compensate these impairments, the remaining distortion still has substantial impact. Despite this, most prior works have not analyzed this type of distortion. In this paper, we investigate the impact of residual transceiver hardware impairments on the MIMO system performance. In particular, we consider a transceiver impairment model, which has been experimentally validated, and derive analytical ergodic capacity expressions for both exact and high signal-to-noise ratios (SNRs). We demonstrate that the capacity saturates in the high-SNR regime, thereby creating a finite capacity ceiling. We also present a linear approximation for the ergodic capacity in the low-SNR regime, and show that impairments have only a second-order impact on the capacity. Furthermore, we analyze the effect of transceiver impairments on large-scale MIMO systems; interestingly, we prove that if one increases the number of antennas at one side only, the capacity behaves similar to the finite-dimensional case. On the contrary, if the number of antennas on both sides increases with a fixed ratio, the capacity ceiling vanishes; thus, impairments cause only a bounded offset in the capacity compared to the ideal transceiver hardware case.

Massive MIMO Systems with Hardware-Constrained Base Stations

Massive multiple-input multiple-output (MIMO) systems are cellular networks where the base stations (BSs) are equipped with unconventionally many antennas. Such large antenna arrays offer huge spatial degrees-of-freedom for transmission optimization; in particular, great signal gains, resilience to imperfect channel knowledge, and small inter-user interference are all achievable without extensive inter-cell coordination. The key to cost-efficient deployment of large arrays is the use of hardware-constrained base stations with low-cost antenna elements, as compared to today's expensive and power-hungry BSs. Low-cost transceivers are prone to hardware imperfections, but it has been conjectured that the excessive degrees-of-freedom of massive MIMO would bring robustness to such imperfections. We herein prove this claim for an uplink channel with multiplicative phase-drift, additive distortion noise, and noise amplification. Specifically, we derive a closed-form scaling law that shows how fast the imperfections increase with the number of antennas.

Impact of Residual Transmit RF Impairments on Training-Based MIMO Systems

Radio-frequency (RF) impairments, which intimately exist in wireless communication systems, can severely limit the performance of multiple-input-multiple-output (MIMO) systems. Although we can resort to compensation schemes to mitigate some of these impairments, a certain amount of residual impairments always persists. In this paper, we consider a training-based point-to-point MIMO system with residual transmit RF impairments (RTRI) using spatial multiplexing transmission. Specifically, we derive a new linear channel estimator for the proposed model, and show that RTRI create an estimation error floor in the high signal-to-noise ratio (SNR) regime. Moreover, we derive closed-form expressions for the signal-to-noise-plus-interference ratio (SINR) distributions, along with analytical expressions for the ergodic achievable rates of zero-forcing, maximum ratio combining, and minimum mean-squared error receivers, respectively. In addition, we optimize the ergodic achievable rates with respect to the training sequence length and demonstrate that finite dimensional systems with RTRI generally require more training at high SNRs than those with ideal hardware. Finally, we extend our analysis to large-scale MIMO configurations, and derive deterministic equivalents of the ergodic achievable rates. It is shown that, by deploying large receive antenna arrays, the extra training requirements due to RTRI can be eliminated. In fact, with a sufficiently large number of receive antennas, systems with RTRI may even need less training than systems with ideal hardware.

The modern math movement(s): an essay on how elementary school teachers in Brazil gave meaning to it

The main goal of this paper is to discuss the production of meaning of the Modern Math Movement. The main sources were data available in school archives and interviews with former teachers that we use in order to focus on the diversity of perspectives -that complement it or oppose it-, which comes up when teachers refer to the Movement. Using this process of signification, teachers whether accept it, invalidate it or adapt it to guidelines imposed to them in their teaching activities. We establish a methodology by following the premises of Oral History to gather oral testimonies. The theoretical foundations in which this article is written are the guidelines of Paul Ricoeur's Hermeneutics, John Thompson's Depth Hermeneutics and Bolivar's narrative analysis.

Mining Unit Tests for Discovery and Migration of Math APIs

Today's programming languages are supported by powerful third-party APIs. For a given application domain, it is common to have many competing APIs that provide similar functionality. Programmer productivity therefore depends heavily on the programmer's ability to discover suitable APIs both during an initial coding phase, as well as during software maintenance. The aim of this work is to support the discovery and migration of math APIs. Math APIs are at the heart of many application domains ranging from machine learning to scientific computations. Our approach, called MATHFINDER, combines executable specifications of mathematical computations with unit tests (operational specifications) of API methods. Given a math expression, MATHFINDER synthesizes pseudo-code comprised of API methods to compute the expression by mining unit tests of the API methods. We present a sequential version of our unit test mining algorithm and also design a more scalable data-parallel version. We perform extensive evaluation of MATHFINDER (1) for API discovery, where math algorithms are to be implemented from scratch and (2) for API migration, where client programs utilizing a math API are to be migrated to another API. We evaluated the precision and recall of MATHFINDER on a diverse collection of math expressions, culled from algorithms used in a wide range of application areas such as control systems and structural dynamics. In a user study to evaluate the productivity gains obtained by using MATHFINDER for API discovery, the programmers who used MATHFINDER finished their programming tasks twice as fast as their counterparts who used the usual techniques like web and code search, IDE code completion, and manual inspection of library documentation. For the problem of API migration, as a case study, we used MATHFINDER to migrate Weka, a popular machine learning library. Overall, our evaluation shows that MATHFINDER is easy to use, provides highly precise results across several math APIs and application domains even with a small number of unit tests per method, and scales to large collections of unit tests.

Measuring math anxiety in Italian college and high school students: Validity, reliability and gender invariance of the Abbreviated Math Anxiety Scale (AMAS)

Given that the ability to manage numbers is essential in a modern society, mathematics anxiety – which has been demonstrated to have unfortunate consequences in terms of mastery of math – has become a subject of increasing interest, and the need to accurately measure it has arisen. One of the widely employed scales to measure math anxiety is the Abbreviated Math Anxiety Scale (AMAS) (Hopko, Mahadevan, Bare & Hunt, 2003). The first aim of the present paper was to confirm the factor structure of the AMAS when administered to Italian high school and college students, and to test the invariance of the scale across educational levels. Additionally, we assessed the reliability and validity of the Italian version of the scale. Finally, we tested the invariance of the AMAS across genders. The overall findings provide evidence for the validity and reliability of the AMAS when administered to Italian students.

How to feel "in love" with Math - a MOOC experience

In the past few years we have witnessed the fast development of distance learning tools such as Open Educational Resources (OER) and Massive Open Online Courses (MOOCs). This paper presents the “Mathematics without STRESS” MOOC Project, which is a cooperation between four schools from the Polytechnic Institute of Oporto (IPP). The concepts of MOOC and their quickly growing popularity are presented and complemented by a discussion of some MOOC definitions. The process of the project development is demonstrated by focusing on used MOOC structure, as well as the several types of course materials produced. At last, is presented a short discussion about problems and challenges met throughout the project. It is also our goal to contribute for a change in the way as teaching and learning Mathematics is seen and practiced nowadays.

Innovations in technology: a friendly math project and the learning analytics “challenge”

Currently the world around us "reboots" every minute and “staying at the forefront” seems to be a very arduous task. The continuous and “speeded” progress of society requires, from all the actors, a dynamic and efficient attitude both in terms progress monitoring and moving adaptation. With regard to education, no matter how updated we are in relation to the contents, the didactic strategies and technological resources, we are inevitably compelled to adapt to new paradigms and rethink the traditional teaching methods. It is in this context that the contribution of e-learning platforms arises. Here teachers and students have at their disposal new ways to enhance the teaching and learning process, and these platforms are seen, at the present time, as significant virtual teaching and learning supporting environments. This paper presents a Project and attempts to illustrate the potential that new technologies present as a “backing” tool in different stages of teaching and learning at different levels and areas of knowledge, particularly in Mathematics. We intend to promote a constructive discussion moment, exposing our actual perception - that the use of the Learning Management System Moodle, by Higher Education teachers, as supplementary teaching-learning environment for virtual classroom sessions can contribute for greater efficiency and effectiveness of teaching practice and to improve student achievement. Regarding the Learning analytics experience we will present a few results obtained with some assessment Learning Analytics tools, where we profoundly felt that the assessment of students’ performance in online learning environments is a challenging and demanding task.

The development and implementation of math projects in a HEI: expectations, objectives, experiences and analysis

An overwhelming problem in Math Curriculums in Higher Education Institutions (HEI), we are daily facing in the last decade, is the substantial differences in Math background of our students. When you try to transmit, engage and teach subjects/contents that your “audience” is unable to respond to and/or even understand what we are trying to convey, it is somehow frustrating. In this sense, the Math projects and other didactic strategies, developed through Learning Management System Moodle, which include an array of activities that combine higher order thinking skills with math subjects and technology, for students of HE, appear as remedial but important, proactive and innovative measures in order to face and try to overcome these considerable problems. In this paper we will present some of these strategies, developed in some organic units of the Polytechnic Institute of Porto (IPP). But, how “fruitful” are the endless number of hours teachers spent in developing and implementing these platforms? Do students react to them as we would expect? Do they embrace this opportunity to overcome their difficulties? How do they use/interact individually with LMS platforms? Can this environment that provides the teacher with many interesting tools to improve the teaching – learning process, encourages students to reinforce their abilities and knowledge? In what way do they use each available material – videos, interactive tasks, texts, among others? What is the best way to assess student’s performance in these online learning environments? Learning Analytics tools provides us a huge amount of data, but how can we extract “good” and helpful information from them? These and many other questions still remain unanswered but we look forward to get some help in, at least, “get some drafts” for them because we feel that this “learning analysis”, that tackles the path from the objectives to the actual results, is perhaps the only way we have to move forward in the “best” learning and teaching direction.

Viva@Matemática project: a different way to see math

Viva@Mat is a project developed by four Math teachers from the School of Industrial Studies and Management (ESEIG) that was born with the fundamental objective of engaging ESEIG students with different math backgrounds in Math challenging activities. Some of these activities were transformed into real palpable materials and others into small interactive ones, being the great majority of them proposed by ESEIG’ students themselves. This small project rapidly grew into something we didn’t expect – it did flow over the walls of our institution to the general involving community – specifically to pre-university schools through the Viva@Math Exhibits – Orange, Blue and Green (the fourth, the Purple one is still in development). Nowadays, Viva@Math Exhibits – the public face of the Project – are itinerant and have been travelling between several, and different institutions (pre-university schools, preparatory schools, libraries, among others), around ESEIG and IPP area of influence and having registered visitors/participants of all ages. In this article we will describe the Viva@Math Project, its different activities that are categorized in some “great groups” like Numerical Trivia, Logic Activities and Mental Calculation, Puzzles, Geometric Curiosities, Magic Tricks, among others, designed to challenge students to use the underlying logical-mathematical reasoning to any ordinary and everyday activity. We will give specific and concrete examples of some of the activities developed and, also, reproduce of the general stimulating feedback the Project receives from the enrolled “actors” (teachers, students and their relatives, institutions, among others). We feel that this Project has become a small “bridge” between the pre-university schools and Higher Education Institutions (HEI), in trying to shorten the “gap” between the institutions of different levels of education and bring them to work together.

Using Effective Teaching Strategies and Personality Type to Enhance the Mathematics Classroom: A Handbook for Intermediate Math Teachers

This project addressed the need for more insightful, current, and applicable resources for intermediate math teachers in Canadian classrooms. A need for a handbook in this division seemed warranted by a lack of government resource support. Throughout an extensive review of the literature, themes and topics for the handbook emerged. The handbook was designed to not only provide educators with examples of effective teaching strategies within the mathematics classroom but to also inform them about the ways in which their personal characteristics and personality type could affect their students and their own pedagogical practices. Three teaching professionals who had each taught in an intermediate math class within the past year evaluated the handbook. The feedback received from these educators was directly applied to the first draft of the handbook in order to make it more accessible and applicable to other math teachers. Although the handbook was written with teachers in mind, the language and format used throughout the manual also make it accessible to parents, tutors, preservice education students, and educational administrators. Essentially, any individual who is hoping to inspire and educate intermediate math students could make use of the content within the handbook.