Non-orthogonal multiple access (NOMA) is envisioned as a key technology to enhance the spectrum efficiency for 5G cellular networks. Meanwhile, ambient backscatter communication is a promising solution to the Internet of Things (IoT), due to its high spectrum efficiency and power efficiency. In this paper, we are interested in a symbiotic system of cellular and IoT networks and propose a backscatter-NOMA system, which incorporates a downlink NOMA system with a backscatter device (BD). In the proposed system, the base station (BS) transmits information to two cellular users according to the NOMA protocol, while a BD transmits its information over the BS signals to one cellular user using the passive radio technology. In particular, if the BS only serves the cellular user that decodes BD information, the backscatter-NOMA system turns into a symbiotic radio (SR) system. We derive the expressions of the outage probabilities and the ergodic rates and analyze the corresponding diversity orders and slopes for both backscatter-NOMA and SR systems. Finally, we provide the numerical results to verify the theoretical analysis and demonstrate the interrelationship between the cellular networks and the IoT networks. INDEX TERMS Non-orthogonal multiple access (NOMA), Internet-of-Things (IoT), ambient backscatter communication (AmBC), symbiotic radio (SR), outage probability, ergodic rate. I. INTRODUCTION Non-orthogonal multiple access (NOMA) is an effective solution to accommodate the data traffic in 5G networks due to its spectrum-efficiency [1]-[6] and has stimulated the upsurge of interest from both academia and industry [7]-[16]. Different from the conventional orthogonal multiple access (OMA) techniques which allow only one user to access the networks in each orthogonal resource block (e.g., a time slot, a frequency channel, a spreading code, or an orthogonal spatial degree of freedom), NOMA enables more than one user to access the networks in the same resource block and distinguishes them by exploiting the power domain. Thus, NOMA system is more spectrum-efficient than conventional OMA system especially when one of the NOMA users is far away from the base station (BS) [3]. The associate editor coordinating the review of this manuscript and approving it for publication was Zhiyong Chen.

The problem of estimating the frequency and carrier phase of a single sinusoid observed in additive, white, Gaussian noise is addressed. Much of the work in the literature considers maximum likelihood (ML) estimation. However, the ML estimator given by the location of the peak of a periodogram in the frequency domain shown in D.C. Rife and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE TRANSACTIONS ON INFORMATION THEORY, vol. IT-20, pp. 591-598, Sep. 1974, has a very high computational complexity. This paper derives the explicit structure of the ML estimator for data processing in the time domain, assuming only reasonably high signal-to-noise ratio (SNR). The result of this approximate ML estimator shows that both the phase and the magnitude of the noisy signal samples are utilized in the estimator, and the phase data alone as assumed in S. A. Tretter, "Estimating the frequency of a noisy sinusoid by linear regression," IEEE TRANSACTIONS ON INFORMATION THEORY, vol. IT-31, pp. 832-835, Nov. 1985 and S. Kay, "A fast and accurate single frequency estimator," IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, vol. 39, pp. 1203-1205, May 1991, is not a sufficient statistic. The sample-by-sample iterative processing nature of the estimator enables us to propose a novel, recursive phase-unwrapping algorithm that allows the estimator to be implemented efficiently. To facilitate the performance analysis, a new, linearized observation model for the instantaneous signal phase that is more accurate than that of S. A. Tretter, "Estimating the frequency of a noisy sinusoid by linear regression," IEEE TRANSACTIONS ON INFORMATION THEORY, vol. IT-31, pp. 832-835, Nov. 1985 and of S. Kay, "A fast and accurate single frequency estimator," vol. 39, pp. 1203-1205, May 1991, is proposed. This new model explains physically why the phase data are weighted by the magnitude information in the ML estimator. Moreover, by incorporating a priori knowledge via the a priori probability density function of the unknown frequency and the carrier phase, the explicit structure of the approximate maximum a posteriori probability (MAP) estimator is derived, and the Bayesian Cramer-Rao lower bound (BCRLB) on the mean-square error (mse) is obtained. Our analysis shows that the mse performance of the MAP estimator can approach the BCRLB very closely. Index Terms-Cramer-Rao lower bound (CRLB)/Bayesian Cramer-Rao lower bound (BCRLB), frequency, maximum a posteriori probability (MAP)/maximum likelihood (ML) estimation, phase, phase noise model, phase unwrapping.

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The generalized Marcum Q-function, Qm(a, b), is here explained geometrically as the probability of a 2m-dimensional, real, Gaussian random vector, whose mean vector has a Frobenius norm of a, lying outside of a hypersphere of 2m dimensions, with radius b, and centered at the origin. Based on this new geometric interpretation, a new closed-form representation for Qm(a, b) is derived for the case where m is an odd multiple of 0.5. This representation involves only the exponential and the erfc functions, and thus is easy to handle, both numerically and analytically. For the case where m is an even multiple of 0.5, Qm+0.5(a, b) and Qm−0.5(a, b), which can be evaluated using our new representation mentioned above, are shown to be tight upper and lower bounds on Qm(a, b), respectively. They are shown in most cases to be much tighter than the existing bounds in the literature, and are valid for the entire ranges of a and b concerned. Their average is also a good approximation to Qm(a, b).

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