Characterization of Randomly Time-Variant Linear Channels
TL;DR: Several new canonical channel models are derived in this paper, some of which are dual to those of Kailath, and a model called the Quasi-WSSUS channel is presented to model the behavior of such channels.
Abstract: This paper is concerned with various aspects of the characterization of randomly time-variant linear channels. At the outset it is demonstrated that time-varying linear channels (or filters) may be characterized in an interesting symmetrical manner in time and frequency variables by arranging system functions in (timefrequency) dual pairs. Following this a statistical characterization of randomly time-variant linear channels is carried out in terms of correlation functions for the various system functions. These results are specialized by considering three classes of practically interesting channels. These are the wide-sense stationary (WSS) channel, the uncorrelated scattering (US) channel, and the wide-sense stationary uncorrelated scattering (WSSUS) channel. The WSS and US channels are shown to be (time-frequency) duals. Previous discussions of channel correlation functions and their relationships have dealt exclusively with the WSSUS channel. The point of view presented here of dealing with the dually related system functions and starting with the unrestricted linear channels is considerably more general and places in proper perspective previous results on the WSSUS channel. Some attention is given to the problem of characterizing radio channels. A model called the Quasi-WSSUS channel is presented to model the behavior of such channels. All real-life channels and signals have an essentially finite number of degrees of freedom due to restrictions on time duration and bandwidth. This fact may be used to derive useful canonical channel models with the aid of sampling theorems and power series expansions. Several new canonical channel models are derived in this paper, some of which are dual to those of Kailath.
Citations
More filters
TL;DR: This paper describes the statistical models of fading channels which are frequently used in the analysis and design of communication systems, and focuses on the information theory of fading channel, by emphasizing capacity as the most important performance measure.
Abstract: In this paper we review the most peculiar and interesting information-theoretic and communications features of fading channels. We first describe the statistical models of fading channels which are frequently used in the analysis and design of communication systems. Next, we focus on the information theory of fading channels, by emphasizing capacity as the most important performance measure. Both single-user and multiuser transmission are examined. Further, we describe how the structure of fading channels impacts code design, and finally overview equalization of fading multipath channels.
2,017 citations
Cites background from "Characterization of Randomly Time-V..."
...A fading multipath channel is generally characterized as a linear, time-varying system having an (equivalent lowpass) impulse response (or a time-varying frequency response ) which is a wide-sense stationary random process in the -variable....
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TL;DR: JSDM achieves significant savings both in the downlink training and in the CSIT uplink feedback, thus making the use of large antenna arrays at the base station potentially suitable also for frequency division duplexing systems, for which uplink/downlink channel reciprocity cannot be exploited.
Abstract: We propose joint spatial division and multiplexing (JSDM), an approach to multiuser MIMO downlink that exploits the structure of the correlation of the channel vectors in order to allow for a large number of antennas at the base station while requiring reduced-dimensional channel state information at the transmitter (CSIT). JSDM achieves significant savings both in the downlink training and in the CSIT uplink feedback, thus making the use of large antenna arrays at the base station potentially suitable also for frequency division duplexing (FDD) systems, for which uplink/downlink channel reciprocity cannot be exploited. In the proposed scheme, the multiuser MIMO downlink precoder is obtained by concatenating a prebeamforming matrix, which depends only on the channel second-order statistics, with a classical multiuser precoder, based on the instantaneous knowledge of the resulting reduced dimensional “effective” channel matrix. We prove a simple condition under which JSDM incurs no loss of optimality with respect to the full CSIT case. For linear uniformly spaced arrays, we show that such condition is approached in the large number of antennas limit. For this case, we use Szego's asymptotic theory of Toeplitz matrices to show that a DFT-based prebeamforming matrix is near-optimal, requiring only coarse information about the users angles of arrival and angular spread. Finally, we extend these ideas to the case of a 2-D base station antenna array, with 3-D beamforming, including multiple beams in the elevation angle direction. We provide guidelines for the prebeamforming optimization and calculate the system spectral efficiency under proportional fairness and max-min fairness criteria, showing extremely attractive performance. Our numerical results are obtained via asymptotic random matrix theory, avoiding lengthy Monte Carlo simulations and providing accurate results for realistic (finite) number of antennas and users.
1,347 citations
TL;DR: In this article, an adaptive digital filtering algorithm that can compensate for both frequency-selective multipath and interference on constant envelope modulated signals is presented, which exploits the fact that multipath reception and various interference sources generate incidental amplitude modulation on the received signal.
Abstract: An adaptive digital filtering algorithm that can compensate for both frequency-selective multipath and interference on constant envelope modulated signals is presented. The method exploits the fact that multipath reception and various interference sources generate incidental amplitude modulation on the received signal. A class of so-called constant modulus performance functions is developed which sense this AM term but are insensitive to the angle modulation. Simple adaptive algorithms for finite-impulse-response (FIR) digital filters are developed which employ a gradient search of the performance function. One of the resulting algorithms is simulated for the example of an FM signal degraded by specular multipath propagation. Substantial improvements in noise power ratio (NPR) are observed (e.g., 25 dB) with moderately rapid convergence time. These results are then extended to include tonal interference on a FM signal and intersymbol interference on a QPSK data signal.
1,339 citations
TL;DR: A stylized compressed sensing radar is proposed in which the time-frequency plane is discretized into an N times N grid and the techniques of compressed sensing are employed to reconstruct the target scene.
Abstract: A stylized compressed sensing radar is proposed in which the time-frequency plane is discretized into an N times N grid. Assuming the number of targets K is small (i.e., K Lt N2), then we can transmit a sufficiently ldquoincoherentrdquo pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel-compressed sensing approach offers great potential for better resolution over classical radar.
1,113 citations
Cites background from "Characterization of Randomly Time-V..."
...We observe the signal over a duration4 T and for simplicity sample it at the Nyquist rateB....
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References
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01 Mar 1950
TL;DR: It appears that the frequency domain approach using H(jω; t) possesses significant advantages over the conventional approach using the impulsive response of the network.
Abstract: This paper describes an approach to the analysis of linear variable networks which is essentially an extension of the frequency analysis techniques commonly used in connection with fixed networks. It is shown that a function H(jω; t), termed the system function of a variable network, possesses most of the fundamental properties of the system function of a fixed network. Thus, once H(jω; t) has been determined, the response to any given input can be obtained by treating H(jω; t) as if it were the system function of a fixed network. It is further shown that H(jω; t) satisfies a linear differential equation in t, which has complex coefficients and is for the same order as the differential equation relating the input and the output of the network. Two methods of solution of this equation covering most cases of practical interest are given. In addition to H(jω; t), a network function introduced is the bi-frequency system function Γ(jω; ju) which is shown to relate the Fourier transforms of the input and the output through a superposition integral in the frequency domain. On the basis of the results obtained in this paper it appears that the frequency domain approach using H(jω; t) possesses significant advantages over the conventional approach using the impulsive response of the network.
571 citations
TL;DR: By using the pre-envelope, the envelope of the output of a linear filter is easily calculated, and this is used to compute the first probability density for the envelopes of an arbitrary linear filter when the input is an arbitrary signal plus Gaussian noise.
Abstract: Rice's formula ^1 for the "envelope" of a given signal is very cumbersome; in any case where the signal is not a single sine wave, the analytical use and explicit calculation of the envelope is practically prohibitive. A different formula for the envelope is given herein which is much simpler and easier to handle analytically. We show precisely that if \hat{u}(t) is the Hilbert transform of u(t) , then Rice's envelope of u(t) is the absolute value of the complex-valued function u(t) + i \hat{u}(t) . The function u + i\hat{u} is called the pre-envelope of u and is shown to be involved implicitly in some other usual engineering practices. The Hilbert transform \hat{u} is then studied; it is shown that \hat{u} has the same power spectrum as u and is uncorrelated with u at the same time instant. Further, the autocorrelation of the pre-envelope of u is twice the pre-envelope of the autocorrelation of u . By using the pre-envelope, the envelope of the output of a linear filter is easily calculated, and this is used to compute the first probability density for the envelope of the output of an arbitrary linear filter when the input is an arbitrary signal plus Gaussian noise. An application of pre-envelopes to the frequency modulation of an arbitrary waveform by another arbitrary waveform is also given.
239 citations
TL;DR: In this article, the effect of frequency-selective fading on the binary error probabilities of incoherent and differentially coherent matched filter receivers employing postdetection diversity combining was investigated. But the analysis was limited to the case of a Gaussian frequency autocorrelation function.
Abstract: The presence of frequency-selective fading in a communication channel limits the maximum data rate capability of conventional communication systems. Surprisingly little analysis has been carried out to determine the effect of frequency-selective fading on digital (or analog) communication systems. This paper considers the effect of frequency-selective fading on the binary error probabilities of incoherent and differentially coherent matched filter receivers employing postdetection diversity combining. In the analysis it is assumed that the fading is slow enough so that over a few bits at least the channel transfer function remains constant. In addition, it is assumed that the amplitude and phase fluctuations on a received carrier have the same statistical character as those of narrow-band Gaussian noise. The general analytical results are specialized to the cases of frequency-shift keying using incoherent detection, and phase-shift keying using differentially coherent detection for the case of a Gaussian frequency autocorrelation function. For these special cases, signal-to-noise degradation curves are given as a function of the ratio of the binary data rate to the correlation (or coherence) bandwidth. Two types of FSK are considered, phase-continuous and phase-discontinuous. In phase-continuous FSK there is no discontinuity in the phase of the transmitted waveform at the markspace or space-mark epochs. Such an FSK system results when the mark and space frequencies are obtained by frequency modulating an oscillator. In phase-discontinuous FSK phase discontinuities exist at the transition epochs. Such an FSK system results, for example, when the mark and space frequencies are derived by switching between two independent oscillators. An interesting result of the analysis is that the differentially coherent PSK system and the phase-discontinuous FSK system degrade considerably more rapidly with increasing (normalized) data rate than the phasecontinuous FSK system. The existence of an irreducible error probability is demonstrated for the incoherent and differentially coherent matched filter receivers. Thus, in general, an increase in transmitted signal power cannot reduce the error probability below a certain value depending upon the ratio of data rate to correlation bandwidth and order of diversity. Theoretical curves of irreducible error probability are given for the incoherent FSK and differentially coherent PSK systems.
209 citations