This correspondence corrects an error in our paper titled, "On the Optimality of Beamforming with Quantized Feedback", published in the IEEE Transactions on Communications, vol. 55, no. 12, pp. 2288-2302, Dec. 2007.

We consider a cognitive radio system where the secondary transmitter varies its transmit power based on the information available from the spectrum sensor. The operation of the secondary user is governed by its peak transmit power constraint and an average interference constraint at the primary receiver. Without restricting the sensing scheme (total received energy, or correlation etc), we characterize the power adaptation strategies that maximize the secondary user's SNR and capacity. We show that, in general, the capacity optimal power adaptation requires decreasing the secondary transmit power from the peak power to zero in a continuous fashion as the probability of the primary user being present increases. In contrast, we find that that power control that maximizes the SNR is binary, i.e., if there is any transmission, it takes place only at the peak power level. Numerical results for common spectrum sensing schemes show that the SNR and capacity maximizing schemes can be very different. With an average transmit power constraint at the secondary radio, both the SNR and capacity optimal power control schemes are observed to be non-binary. Further, we find that with secondary channel knowledge at the cognitive transmitter, the optimal SNR with an average transmit power constraint is unbounded.

Cognitive radios hold tremendous promise for increasing spectral efficiency in wireless systems. This paper surveys the fundamental capacity limits and associated transmission techniques for different wireless network design paradigms based on this promising technology. These paradigms are unified by the definition of a cognitive radio as an intelligent wireless communication device that exploits side information about its environment to improve spectrum utilization. This side information typically comprises knowledge about the activity, channels, codebooks, and/or messages of other nodes with which the cognitive node shares the spectrum. Based on the nature of the available side information as well asa priorirules about spectrum usage, cognitive radio systems seek to underlay, overlay, or interweave the cognitive radios' signals with the transmissions of noncognitive nodes. We provide a comprehensive summary of the known capacity characterizations in terms of upper and lower bounds for each of these three approaches. The increase in system degrees of freedom obtained through cognitive radios is also illuminated. This information-theoretic survey provides guidelines for the spectral efficiency gains possible through cognitive radios, as well as practical design ideas to mitigate the coexistence challenges in today's crowded spectrum.

We explore the performance tradeoff between opportunistic and regulated access inherent in the design of multiuser cognitive radio networks. We consider a multichannel cognitive radio system with sensing limits at the secondary users and interference tolerance limits at the primary and secondary users. Our objective is to determine the optimal amount of spectrum sharing, i.e., the number of secondary users that maximizes the total deliverable throughput in the network.We begin with the case of perfect primary user detection and zero interference tolerance at each of the primary and secondary nodes. With identical primary and secondary traffic statistics, we find that the optimal fraction of licensed users lies between the two extremes of fully opportunistic and fully licensed operation and is equal to the traffic duty cycle. When the secondary users can vary their transmission probabilities based on the number of active primary users, we find that the optimal number of opportunistic users is equal to the average number of unoccupied channels. We then consider the more involved case of imperfect sensing and non-zero interference tolerance constraints. We provide numerical simulation results to study the tradeoff between licensing and autonomy and the impact of primary user sensing and interference tolerance on the deliverable throughput for two different subchannel selection strategies at the secondary users.

We consider a cognitive radio system where the secondary transmitter varies its transmit power based on all the information available from the spectrum sensor. The operation of the secondary user is governed by its peak transmit power constraint and an average interference constraint at the primary receiver. Without restricting the sensing scheme (total received energy, or correlation etc), we characterize the power adaptation strategies that maximize the secondary user's SNR and capacity. We show that, in general, the capacity optimal power adaptation requires decreasing the secondary transmit power from the peak power to zero in a continuous fashion as the probability of the primary user being present increases. We find that power control that maximizes the SNR is binary, i.e., if there is any transmission, it takes place only at the peak power level. Numerical results for common spectrum sensing schemes show that the SNR and capacity maximizing schemes can be significantly different.

We explore the performance tradeoff between opportunistic and regulated access inherent in the design of multiuser cognitive radio networks. We consider a cognitive radio system with sensing limits at the secondary users and interference tolerance limits at the primary and secondary users. Our objective is to determine the optimal amount of spectrum sharing, i.e., the number of secondary users that maximizes the total deliverable throughput in the system. We begin with the case of perfect primary user detection and zero interference tolerance at each of the primary and secondary nodes. We find that the optimal fraction of licensed users lies between the two extremes of fully opportunistic and fully licensed operation and is equal to the traffic duty cycle. For the more involved case of imperfect sensing and non-zero interference tolerance constraints, we provide numerical simulation results to study the tradeoff between licensing and autonomy and the impact of primary user sensing and interference tolerance on the deliverable throughput.

We investigate the capacity of opportunistic communication in the presence of dynamic and distributed spectral activity, i.e., when the time varying spectral holes sensed by the cognitive transmitter are correlated but not identical to those sensed by the cognitive receiver. We develop a two switch model that captures the localized spectral activity estimates at the transmitter and receiver. The information theoretic framework of communication with side information is employed to characterize the capacity of the cognitive link with both causal and non-causal side information at the transmitter and/or the receiver. These capacity results are used to determine the benefits of any feedforward and feedback information. We find that cognitive radio capacity is robust to the uncertainties arising out of distributed and dynamic spectral environments, even when the communication occurs in bursts of only 3-5 symbols. The capacity depends strongly on the correlation of the local spectral environment at the cognitive transmitter and receiver.

The ergodic capacity of a fading vector channel with multiple transmit antennas and a single receive antenna is explored. Perfect channel information is assumed to be available at the receiver while the transmitter has only partial knowledge of the direction of the user's channel vector based on quantized feedback. We present necessary and sufficient conditions for the optimality of beamforming in such systems. The conditions are applicable to all quantized feedback scenarios regardless of the channel distribution, number of transmit antennas, number of quantization vectors or transmit power. The optimality conditions are closely related to the iteration conditions of the Lloyd algorithm, revealing an interesting link between the optimality of beamforming and the optimality of the vector quantizers. Using the conditions, we prove the capacity optimality of beamforming for several quantized feedback scenarios such as the antenna-selection scheme. We also point out examples of quantized feedback scenarios where beamforming is not optimal. We find that for the independent identically distributed Rayleigh fading channel with more than a single bit of quantized feedback, there is no capacity benefit from increasing the number of antennas beyond the number of quantization vectors. Extensions of the necessary and sufficient optimality condition to the multiple-input multiple-output case are also provided.

We investigate the capacity of opportunistic communication in the presence of dynamic and distributed spectral activity, i.e. when the time varying spectral holes sensed by the cognitive transmitter are correlated but not identical to those sensed by the cognitive receiver. Using the information theoretic framework of communication with causal and non-causal side information at the transmitter and/or the receiver, we obtain analytical capacity expressions and the corresponding numerical results. We find that cognitive radio communication is robust to dynamic spectral environments even when the communication occurs in bursts of only 3 - 5 symbols. The value of handshake overhead is investigated for both lightly loaded and heavily loaded systems. We find that the capacity benefits of overhead information flow from the transmitter to the receiver is negligible while feedback information overhead in the opposite direction significantly improves capacity.

We explore the tradeoff between hopping and spreading in a wideband spread spectrum transmitter-receiver link in an ad-hoc network with multi-user interference. The receiver is assumed to track the user's channel, spreading signature and the overall SINR. Using expressions for the ergodic capacity of the link, we determine the amount of spreading required that maximizes the link capacity. As a comparison we also discuss optimal spreads in centralized architectures where the receiver has interference knowledge. While we find that in general a combination of spreading and hopping may achieve higher throughputs than either pure spreading or pure hopping, we prove that pure hopping and pure spreading are optimal strategies for ad-hoc networks at low SINR and for centralized networks at all SINR, respectively. Numerical results suggest that in an ad-hoc network, spreading is beneficial at high SNR.

We explore the capacity of opportunistic secondary (cognitive) communication over a spectral pool of two independent channels. Due to the distributed nature of the primary user's spectral activity, the cognitive receiver does not have full knowledge of the channel used at the transmitter for secondary communication. Tracking the transmitter state at the receiver is therefore a primary issue in such channels. The problem is further complicated as the channel availability changes with time. The tracking uncertainty also makes decoding at the receiver non-trivial. Using genie based outer bounds and training based lower bounds, we estimate the capacity of the secondary link. The capacity analysis shows that the benefits of spectral pooling are lost in dynamic spectral environments

We explore the optimality of transmit beamforming for a vector channel with multiple transmit antennas and a single receive antenna. Perfect channel information is assumed to be available at the receiver while the transmitter only has partial/limited knowledge of the user's channel vector based on feedback. Without limiting the kind of partial/limited feedback or the type of channel distribution, we derive a general necessary and sufficient condition for the optimality of beamforming in such channels. The condition we obtain is universal - applicable to all partial/limited feedback scenarios in all ergodic fading channel distributions regardless of the number of transmit antennas or transmit power. Considering different types of partial/limited feedback, we show how our conditions can be employed to obtain previous results on the optimality of beamforming. With Monte Carlo simulations, we provide numerical results comparing different partial/limited feedback schemes.

We explore the capacity of an isotropic fading vector channel with multiple transmit antennas at the base station and a single antenna at the mobile receiver. Perfect channel knowledge is assumed to he available at the receiver while the transmitter has only partial knowledge of the direction of the user's channel vector based on quantized feedback. We determine a necessary and sufficient condition for optimality of beamforming, which turns out to be a precise relationship between the optimality of beamforming and the symmetry of the quantizer. For the cases where beamforming is not optimal, we develop upper and lower bounds. Numerical results are provided to help estimate the capacity.