OFDM has become a popular choice for modulation and is widely used in many communication systems, especially in wireless systems. For ex-ample, in the latest fourth-generation of wireless systems, it is used in the downlink. It is also a contender for the upcoming fifth-generation wireless systems as well. It facilitates simple implementation of the base-band modulator and demodulator and, most importantly, it trivializes the task of channel equalization. It, however, has its drawbacks and, by far, the most important being its sensitivity to RF-impairments. This thesis contributes to the field of analysis and estimation for OFDM systems im-paired by phase noise which forms one type of RF-impairment.
5.1 Contributions in OFDM under Oscillator Phase Noise
With regard to performance analysis, this thesis contributes by providing new closed-form analytical expressions of capacity for OFDM systems im-paired by phase noise. The capacity analysis is also extended to include the effect of carrier frequency offset. The resulting expressions provide quantitative as well as qualitative insight on the relationship between the phase noise process and capacity. Through these expressions, a clear degradation in capacity of the OFDM system in the presence of phase noise is seen. Fortunately, this degradation can be controlled either by proper choice of oscillator design, or by adjusting the OFDM system pa-rameters, or by performing phase noise estimation and compensation.
This thesis also makes two new contributions to the field of phase noise estimation in OFDM. Specifically, two novel aspects about the desired phase noise parameter are used during the estimation step. In the first contribution, subspace-based information is used, i.e., possible subspaces in which the desired phase noise spectral vector may lie are explored. This
subspace-based approach avoids the need to make assumptions about the nature of the phase noise process which is generally used in the litera-ture. For example, phase noise processes are assumed to be slow-varying in nature which is, no doubt, a reasonable assumption. For moderate or fast-varying phase noise processes, the proposed subspace-based ap-proach will perform better compared to algorithms specifically designed with the slow-varying assumption.
In the second contribution, information on the geometry of the desired phase noise spectral vector is used in the estimation step. The goal is to estimate the spectral vector of the complex exponential of the phase noise process. It is shown in this thesis that this spectral vector is always drawn from a non-convex set which can be expressed using a set of quadratic forms that involve permutation matrices. This geometry is nothing but a frequency domain manifestation of the constant magnitude property of the complex exponential function. The constant magnitude property is a well-known fact but its equivalent frequency domain manifestation has not been observed and utilized in the research community.
5.2 Contributions in Applied Statistics and Optimization Theory
This thesis also presents some new fundamental results in the fields of applied statistics and optimization theory. These results are application independent and can be applied wherever suitable. As an example, per-taining to the field of statistics, the PDF of a sum of correlated gamma random variables with a normalized covariance matrix of any rank is rived. The state-of-the-art result was limited to the full-rank case as de-rived in [96]. The framework, based on the work done by Moschopoulos in [95], naturally allows to extend the result to deriving the PDF of a sum of correlated gamma and Gaussian distributed random variables.
This thesis also contributes to the field of optimization theory and pro-vides some new results on the losslessness of the S-procedure that involve equality constraints. The S-procedure is a method of replacing a set of quadratic equalities or inequalities with a linear-matrix-inequality. Con-ditions for the S-procedure to be lossless for the case of quadratic inequal-ities is well-established and used extensively. This thesis fills the void by providing conditions for the S-procedure to be lossless for any number of quadratic equality constraints.
5.3 Directions of Future Work
We end this chapter with a brief treatise on possible research directions related to OFDM and phase noise. The next-generation of wireless sys-tems, namely 5G, have set forth gigantic goals on data rate, latency and spectral efficiency to name a few. For example, peak data rates are ex-pected to deliver three orders of magnitude more than the current 4G sys-tems [138]. The key physical layer technologies that promise to deliver such massive data rates are primarily: massive MIMO, millimeter wave communications and heterogeneous networks [4]. In the context of phase noise, there is abundant research that shows a performance degradation for MIMO systems corrupted by phase noise, for example, when perform-ing beamformperform-ing [139]. The effect of phase noise on MIMO systems serves as an indicator of what to expect with regard to massive MIMO systems that employ hundreds of antennas at the base station. This indeed marks the beginning of a new research area dedicated to analyzing and address-ing the effects of phase noise and RF-impairments, in general, on massive MIMO systems. Some new studies on the topic can be found in [140]
and [141].
An important aspect to delivering high data rates is the notion of spec-tral efficiency which is intrinsically linked to the underlying waveform and symbol constellation. For example, in LTE, the OFDM waveform is used. It is a popular contender for the upcoming 5G systems, however, alternatives are being sought after [142]. Two main reasons for seeking alternatives are: low spectral efficiency of OFDM; and stringent synchro-nization requirements [143]. By synchrosynchro-nization, we refer to timing and frequency synchronization. For example, in a multi-user uplink scenar-ios, the base station needs to estimate the timing and frequency offset of all the users it services in order to avoid block and multi-user inter-ference. This stringent requirement arises fundamentally because of the susceptibility of OFDM to timing and frequency offset. Frequency offset can be viewed as a deterministic version of phase noise. Alternative multi-carrier waveforms, such as the filter bank multi-multi-carrier, are more robust to synchronization errors compared to OFDM and also have higher spec-tral efficiency [144]. The impact of phase noise on these waveforms is not well-known, and we envisage a new research field dedicated to this area.