Complex Noise Analysis of DMT

Abstract

In this paper, we consider discrete multitone (DMT) or baseband orthogonal frequency-division multiplexing (OFDM) modulation and perform a detailed noise analysis which takes into account dependencies and power (variance) differences of real and imaginary part after the complex-valued discrete Fourier transform (DFT). The derivation is based on the so-called pseudocovariance matrix of a complex random vector, which was introduced by Neeser and Massey (1993). We show that the relevant pseudocovariance matrix is not the zero matrix in general, in contrast to passband OFDM, for which it can be proven (under certain assumptions) that all occurring pseudocovariance matrices are vanishing. We show that for colored noise rotated rectangular symbol constellations are more appropriate than the common quadratic quadrature amplitude modulation (QAM) symbol constellations with respect to capacity and symbol error probability, and we derive formulas for the rotation angles and constellation sizes/densities. Finally, we extend the results to a multitransceiver [multiple-input-multipleoutput (MIMO)] scenario, for which we assume a very general noise model at the inputs of the receivers, allowing correlations between the noise signals of different receivers. This requires the introduction of pseudocross-covariance matrices of complex random vectors, which are the important objects (together with cross-covariance matrices) in the MIMO situation.

Complex-Valued Random Vectors and Channels: Entropy, Divergence, and Capacity

“…Whereas in many real-world scenarios the noise vector happens to be circular, so that the results of Section 5.1 apply, there are also practically relevant scenarios, where the noise vector is known to be improper. More specifically, DMT modulation, which is widely used in xDSL applications [18], yields an equivalent system channel that exactly matches the situation considered here [8,9]. We also note that capacity results for improper Gaussian distributed noise vectors could be alternatively derived by making use of an equivalent real-valued channel of dimension 2n × 2m that is obtained by appropriate stacking of real and imaginary parts.…”

“…In recent research-for an comprehensive overview see [3]-it has been shown that exploiting circularity/properness of complex-valued signals or lack of it (non-circularity/improperness) is able to significantly enhance the performance of the applied signal processing techniques. More specifically, for the field of communications, it has been observed that important digital modulation schemes including binary phase shift keying (BPSK), pulse amplitude modulation (PAM), Gaussian minimum shift keying (GMSK), offset quaternary phase shift keying (OQPSK), and baseband (but not passband) orthogonal frequency division multiplexing (OFDM), which is commonly called discrete multitone (DMT), (potentially) produce non-circular/improper complex baseband signals, see e.g., [4][5][6][7][8][9]. Non-circular/improper baseband communication signals can also arise due to imbalance between their in-phase and quadrature (I/Q) components, and several techniques for compensating for I/Q imbalance have been proposed [10][11][12].Information theory, on the other hand, addresses fundamental performance limits of communication systems and also has large impact on many other scientific areas, where stochastic models are used.…”

A Journey From Improper Gaussian Signaling to Asymmetric Signaling

“…Power Systems Modulation DoA Estimation [7], [32], [159] Multiple Access Coding Radio Communications [5], [28] Communications BPSK/PAM [4], [5], [22] GMSK OQPSK [23]- [25] Baseband OFDM [27] Transceiver Impairments [ Rotationally Variant Channel [6] Feature Extraction Improved Estimation Biomedical Engineering [102] Medicine ECG [145] EEG [147] EOG [147] fMRI [6], [20], [21] , [124], [146] SSB/VSB [26] DS-CDMA [45] STBC [28] Analysis Treatment Squeezed States of Light [153] Biphoton States in Q-OCT [154] Anisotropic Noise Imbalanced Data Channels Reflections…”

“…Similarly, Subramaniam et al introduce asymmetry in 8-PSK to increase the minimum product distance reducing error events in TCM [268]. For colored noise compensation, Taubock prefers optimally rotated rectangular symbol constellations over common quadratic QAM in order to minimize capacity loss and SEP [27].…”

“…Besides covering the broader aspects of various disciplines, this survey is focused on the communication systems with intentional/unintentional improper signatures. Various discrete modulation schemes that realize the asymmetric complex characteristics include Binary Phase Shift Keying (BPSK) [4], [22], Pulse Amplitude Modulation (PAM) [5], Gaussian Minimum Shift Keying (GMSK) [23], [24], Offset Quadrature Phase Shift Keying (OQPSK) or staggered QPSK [25], single sideband (SSB), vestigial sideband (VSB) [26] and baseband (but not passband) Orthogonal Frequency Division Multiplexing (OFDM) or discrete multitone (DMT) modulation [27]. On the other hand, coding schemes like space-time block coding (STBC) also results in asymmetric signaling [28].…”