In this paper, the capacity and energy efficiency of training-based communication schemes employed for transmission over a-priori unknown Rayleigh block fading channels are studied. In these schemes, periodically transmitted training symbols are used at the receiver to obtain the minimum mean-square-error (MMSE) estimate of the channel fading coefficients. Initially, the case in which the product of the estimate error and transmitted signal is assumed to be Gaussian noise is considered. In this case, it is shown that bit energy requirements grow without bound as the signal-to-noise ratio (SNR) goes to zero, and the minimum bit energy is achieved at a nonzero SNR value below which one should not operate. The effect of the block length on both the minimum bit energy and the SNR value at which the minimum is achieved is investigated. Flash training and transmission schemes are analyzed and shown to improve the energy efficiency in the low-SNR regime. In the second part of the paper, the capacity and energy efficiency of training-based schemes are investigated when the channel input is subject to peak power constraints. The capacity-achieving input structure is characterized and the magnitude distribution of the optimal input is shown to be discrete with a finite number of mass points. The capacity, bit energy requirements, and optimal resource allocation strategies are obtained through numerical analysis. The bit energy is again shown to grow without bound as SNR decreases to zero due to the presence of peakedness constraints. The improvements in energy efficiency when on-off keying with fixed peak power and vanishing duty cycle is employed are studied. Comparisons of the performances of training-based and noncoherent transmission schemes are provided.
Deep Dive into On the Capacity and Energy Efficiency of Training-Based Transmissions over Fading Channels.
In this paper, the capacity and energy efficiency of training-based communication schemes employed for transmission over a-priori unknown Rayleigh block fading channels are studied. In these schemes, periodically transmitted training symbols are used at the receiver to obtain the minimum mean-square-error (MMSE) estimate of the channel fading coefficients. Initially, the case in which the product of the estimate error and transmitted signal is assumed to be Gaussian noise is considered. In this case, it is shown that bit energy requirements grow without bound as the signal-to-noise ratio (SNR) goes to zero, and the minimum bit energy is achieved at a nonzero SNR value below which one should not operate. The effect of the block length on both the minimum bit energy and the SNR value at which the minimum is achieved is investigated. Flash training and transmission schemes are analyzed and shown to improve the energy efficiency in the low-SNR regime. In the second part of the paper, the
In wireless communications, channel conditions vary randomly over time due to mobility and changing environment, and the degree of channel side information (CSI) assumed to be available at the receiver and transmitter is a key assumption in the study of wireless fading channels. The case in which the channel is assumed to be perfectly known at the receiver and/or transmitter has been extensively studied. In an early work, Ericsson [1] obtained the capacity of flat fading channels with perfect receiver CSI. More recently, Ozarow et al. [2] studied the average and outage capacity values in the cellular mobile radio setting assuming perfect channel knowledge at the receiver. Goldsmith and Varaiya [3] analyzed the capacity of flat fading channels with perfect CSI at the transmitter and/or receiver.
The assumption of having perfect channel knowledge is unwarranted when communication is trying to be established in a highly mobile environment. This consideration has led to another line of work where both the receiver and transmitter are assumed to be completely uninformed of the channel conditions. Abou-Faycal et al. [4] studied the capacity of the unknown Rayleigh fading channel and showed that the optimal input amplitude has a discrete structure. This is in stark contrast to the optimality of a continuous Gaussian input in known channels.
In [16] and [18], the discreteness of the capacity-achieving amplitude distribution is proven for noncoherent Rician fading channels under input peakedness constraints. When the input is subject to peak power constraints, the discrete nature of the optimal input is shown for a general class of single-input single-output channels in [7]. Marzetta and Hochwald [5] gave a characterization of the optimal input structure for unknown multipleantenna Rayleigh fading channels. This analysis subsequently led to the proposal of unitary space-time modulation techniques [6]. Chan et al [8] considered conditionally Gaussian multiple-input multiple-output (MIMO) channels with bounded inputs and proved the discreteness of the optimal input under certain conditions. Zheng and Tse [10] analyzed the multiple-antenna Rayleigh channels and identified the high signal-to-noise ratio (SNR) behavior of the channel capacity.
Heretofore, the two extreme assumptions of having either perfect CSI or no CSI have been discussed. Practical wireless systems live in between these two extremes. Unless there is very high mobility, wireless systems generally employ estimation techniques to learn the channel conditions, albeit with errors. Hence, it is of utmost interest to analyze fading channels with imperfect CSI. Médard [13] investigated the effect upon channel capacity of imperfect channel knowledge and obtained upper and lower bounds on the input-output mutual information.
Lapidoth and Shamai [12] analyzed the effects of channel estimation errors on the performance if Gaussian codebooks are used and nearest neighbor decoding is employed. The capacity of imperfectly-known fading channels is characterized in the low-SNR regime in [14] and in the high-SNR regime in [9].
The aforementioned studies have not considered explicit training and estimation techniques, and resources allocated to them. Recently, Hassibi and Hochwald [23] studied training schemes to learn the multiple-antenna channels. In this work, power and time dedicated to training is optimized by maximizing a lower bound on the capacity. Similar training techniques are also discussed in [10]. Due to its practical significance, the informationtheoretic analysis of training schemes has attracted much interest (see e.g., [24]- [35]). Since exact capacity expressions are difficult to find, these studies have optimized the training signal power, duration, and placement using capacity bounds. Since Gaussian noise is the worst-case uncorrelated additive noise in a Gaussian setting [23], a capacity lower bound is generally obtained by assuming the product of the estimate error and the transmitted signal as another source of Gaussian noise. In the above cited work, training symbols are employed to solely facilitate channel estimation. However, we note that training symbols can also be used for timing-and frequencyoffset synchronization, and channel equalization [36]- [38]. Tong et al. in [22] present an overview of pilotassisted wireless transmissions and discuss design issues from both information-theoretic and signal processing perspectives.
Another important concern in wireless communications is the efficient use of limited energy resources. In systems where energy is at a premium, minimizing the energy cost per unit transmitted information will improve the efficiency. Hence, the energy required to reliably send one bit is a metric that can be adopted to measure the performance. Generally, energy-per-bit requirement is minimized, and hence the energy efficiency is maximized, if the system operates in the low-SNR regime. In [14], Verdú has analyzed the
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