There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.

Of Theory and Practice

Quadrature amplitude modulation (QAM) is an attractive technique to achieve high rate transmission without increasing the bandwidth. A great deal of attention has been devoted to the study of bit error rate (BER) performance of QAM, and approximate expressions for the bit error probability of QAM have been developed in many places in the literature. However, the exact and general BER expression of QAM with an arbitrary constellation size has not been derived yet. We provide an exact and general closed-form expression of the BER for one-dimensional and two-dimensional amplitude modulations, i.e., PAM and QAM, under an additive white Gaussian noise (AWGN) channel when Gray code bit mapping is employed. The provided BER expressions offer a convenient way to evaluate the performance of PAM and QAM systems for various cases of practical interest. Moreover, simple approximations can be found from our expressions, which are the same as the well-known approximations, if only the dominant terms are considered.

On the general BER expression of one- and two-dimensional amplitude modulations

We developed a cross-layer design which combines adaptive modulation and coding at the physical layer with a truncated automatic repeat request protocol at the data link layer, in order to maximize spectral efficiency under prescribed delay and error performance constraints. We derive the achieved spectral efficiency in closed-form for transmissions over Nakagami-m block fading channels. Numerical results reveal that retransmissions at the data link layer relieve stringent error control requirements at the physical layer, and thereby enable considerable spectral efficiency gain. This gain is comparable with that offered by diversity, provided that the maximum number of transmissions per packet equals the diversity order. Diminishing returns on spectral efficiency, that result when increasing the maximum number of retransmissions, suggest that a small number of retransmissions offers a desirable delay-throughput tradeoff, in practice.

Cross-Layer combining of adaptive Modulation and coding with truncated ARQ over wireless links

So far the general expression of bit error rate (BER) has not been derived for M-ary square quadrature amplitude modulation (QAM). In this paper, a generalized closed-form BER expression of M-ary square QAM with Gray code bit mapping is derived and analyzed in the presence of additive white Gaussian noise (AWGN) channel.

Bit error probability of M-ary quadrature amplitude modulation

Deep neural network (DNN) has recently received much attention due to its superior performance in classifying data with complex structure. In this paper, we investigate application of DNN technique to automatic classification of modulation classes for digitally modulated signals. First, we select twenty one statistical features which exhibit good separation in empirical distributions for all modulation formats considered (i.e., BPSK, QPSK, 8PSK, 16QAM, and 64QAM). These features are extracted from the received signal samples and used as the input to the fully connected DNN with three hidden layer. The training data containing 25,000 feature vectors is generated by the computer simulation under both additive Gaussian white noise (AWGN) and Rician fading channels. Our test results show that the proposed method brings dramatic performance improvement over the existing classifier especially for high Doppler fading channels.

Deep neural network-based automatic modulation classification technique

A simple and general bit metric generation method by using the log likelihood ratio (LLR) for a Gray coded QAM signal over an additive white Gaussian noise channel is proposed. The mathematical max (min) functions of the conventional LLR expression are replaced with basic arithmetic functions. Since the derived LLR expression can be used as a soft bit metric for input to a conventional binary turbo decoder, it is suitable for practical applications with a turbo decoder or other related applications as an alternative to conventional methods.

Bit metric generation for Gray coded QAM signals

M-ary quadrature amplitude modulation (M-ary QAM) is an attractive technique to achieve high rate transmission without increasing the bandwidth in wireless communication systems. A great deal of attention has been devoted to the study of bit error rate (BER) performance of M-ary square QAM constellations. Approximate expressions for bit error probabilities of M-ary square QAM have recently been developed by J.J. Lu et al. (1999) and L. Yang and L. Hanzo (2000) based on signal-space concepts and recursive algorithms, respectively. However, it is only recently that an exact generalized closed form expression for the BER of M-ary square QAM constellation has been developed (see Yoon, D. et al., IEEE VTC Fall 2000, vol.5, p.2422-7, 2000). We extend that previous work of ours to an arbitrary I/spl times/J rectangular QAM (R-QAM) scheme. A general closed form expression for the exact BER performance of an arbitrary I/spl times/J R-QAM with Gray code mapping is derived and analyzed under an additive white Gaussian noise (AWGN) channel. The BER performance is also analyzed under fading channels. It is shown that there is a noticeable deviation in the BER performance between the approximation and exact expression over low SNR values.

Dongweon Yoon

Papers

On the general BER expression of one- and two-dimensional amplitude modulations

Bit error probability of M-ary quadrature amplitude modulation

Deep neural network-based automatic modulation classification technique

Bit metric generation for Gray coded QAM signals

BER analysis of arbitrary rectangular QAM