This paper investigates the effect of sub-Nyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear time-invariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of right-invertible time-preserving sampling methods which includes irregular nonuniform sampling, and characterize in closed form the channel capacity achievable by this class of sampling methods, under a sampling rate and power constraint. Our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal-to-noise ratio among all spectral sets of measure equal to the sampling rate. This can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates, or through a single branch of modulation and filtering followed by uniform sampling. These results reveal that for a large class of channels, employing irregular nonuniform sampling sets, while are typi- cally complicated to realize in practice, does not provide capacity gain over uniform sampling sets with appropriate preprocessing. Our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario, which is in contrast to the benefits obtained from random mixing in spectrum-blind compressive sampling schemes.
A sequence of binary ( spl plusmn 1) random variables is generated by sampling a hard-limited version of a bandlimited process at n times the Nyquist rate. The information rate spl Iscr carried by these binary samples is investigated. It is shown by constructing a specific nonstationary, bounded, bandlimited process (the real zeros of which are independent and identically distributed, isolated, and lying in different Nyquist intervals) that spl Iscr =log sub 2 (n+1) bits per Nyquist interval is achievable. A more complicated construction in which L distinct zeros are placed in L consecutive Nyquist intervals yields achievable rates that approach (for L spl rarr spl infin ) spl Iscr arbitrarily closely, where spl Iscr =log sub 2 n + (n-1)log sub 2 [n (n-1)], n spl ges 2 (and spl Iscr =1 for n=1 and L=1). By exploiting the constraints imposed on the autocorrelation function of a stationary sign (bilevel) process with a given average transition rate, the latter expression is shown also to yield an upper bound on the achievable values of spl Iscr . The logarithmic behavior with n (n spl Gt 1) is due to the high correlation between the oversampled binary samples, and it is established that this trend is also achievable with stationary sign processes. This model may be used to gain insight into the effect of finite resolution on the information (in Shannon's sense) conveyed by the sign of a bandlimited process, and also to assess the limiting performance of certain oversampling-based communication systems. > A noiseless ideal low-pass filter, followed by a limiter, is normally used as a binary data channel by sampling the output once per Nyquist interval. Detectors that sample more often encounter intersymbol interference, but can be used in ways that increase the information rate. A signaling system that achieves an information rate of 1.072 b Nyquist interval by allowing the detector to sample at times a half Nyquist interval apart is presented. This rate increase is small, but it shows that, in principle, oversampling does permit faster rates. Another signaling system that more closely resembles a digital data system is also presented; it has rate 1.050. By allowing a more irregular pattern of sampling times, the rate is increased to 1.090 (or 1.062 for the 'digital' system). >  We present a model for time-varying communication single-access and multiple-access channels without feedback. We consider the difference between mutual information when the receiver knows the channel perfectly and mutual information when the receiver only has an estimate of the channel. We relate the variance of the channel measurement error at the receiver to upper and lower bounds for this difference in mutual information. We illustrate the use of our bounds on a channel modeled by a Gauss-Markov process, measured by a pilot tone. We relate the rate of time variation of the channel to the loss in mutual information due to imperfect knowledge of the measured channel. We show that very large bandwidths on fading multipath channels cannot be effectively utilized by spread-spectrum systems that (in a particular sense) spread the available power uniformly over both time and frequency. The approach is to express the input process as an expansion in an orthonormal set of functions each localized in time and frequency. The fourth moment of each coefficient in this expansion is then uniformly constrained. We show that such a constraint forces the mutual information to 0 inversely with increasing bandwidth. Simply constraining the second moment of these coefficients does not achieve this effect. The results suggest strongly that conventional direct-sequence code-division multiple-access (CDMA) systems do not scale well to extremely large bandwidths. To illustrate how the interplay between channel estimation and symbol detection affects capacity, we present results for a specific channel and CDMA signaling scheme. Shannon's determination of the capacity of the linear Gaussian channel has posed a magnificent challenge to succeeding generations of researchers. This paper surveys how this challenge has been met during the past half century. Orthogonal minimum-bandwidth modulation techniques and channel capacity are discussed. Binary coding techniques for low-signal-to-noise ratio (SNR) channels and nonbinary coding techniques for high-SNR channels are reviewed. Recent developments, which now allow capacity to be approached on any linear Gaussian channel, are surveyed. These new capacity-approaching techniques include turbo coding and decoding, multilevel coding, and combined coding precoding for intersymbol-interference channels. 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. The discrete-time Gaussian channel with intersymbol interference (ISI) where the inputs are subject to a per symbol average-energy constraint is considered. The capacity of this channel is derived by means of a hypothetical channel model called the N-circular Gaussian channel (NCGC), whose capacity is readily derived using the theory of the discrete Fourier transform. The results obtained for the NCGC are used further to prove that, in the limit of increasing block length N, the capacity of the discrete-time Gaussian channel (DTGC) with ISI using a per block average-energy input constraint (N-block DTGC) is indeed also the capacity when using the per symbol average-energy constraint. > Communication Systems and Information Theory. A Measure of Information. Coding for Discrete Sources. Discrete Memoryless Channels and Capacity. The Noisy-Channel Coding Theorem. Techniques for Coding and Decoding. Memoryless Channels with Discrete Time. Waveform Channels. Source Coding with a Fidelity Criterion. Index.