Full-Duplex Cooperative Cognitive Radio with Transmit Imperfections

I. INTRODUCTION

• The paper proposes full-duplex CBS relaying to avoid the throughput loss of half-duplex cooperation, which uses at least two orthogonal phases.In full-duplex mode, the CBS simultaneously receives the primary message and transmits its processed signal and cognitive signal on the same channel phase.Source: §I · p. 1

A. Related Work

• Prior full-duplex cognitive-radio work either relied on dirty-paper coding with non-causal primary-message information or assumed perfect transmit-receive isolation and no self-interference [19].The paper identifies these limitations as motivation for studying full-duplex cooperation with self-interference.Source: A. Related Work · p. 2

B. Summary of Contributions

• For half-duplex CBS operation, the paper formulates cognitive-rate maximization subject to CBS power and primary-user rate constraints and derives closed-form solutions.Source: B. Summary of Contributions · p. 2
• For full-duplex CBS operation, the paper models residual self-interference from transmit noise and gives closed-form solutions for fixed noise power and an efficient algorithm for scalable noise power.The scalable-noise algorithm is developed by establishing a link between the fixed- and scalable-noise cases.Source: B. Summary of Contributions · p. 2
• The hybrid scheme selects half- or full-duplex operation using simplified zero-forcing solutions that null interference between the primary and secondary systems.The paper also examines how system parameters affect performance.Source: B. Summary of Contributions · p. 2
• Simulations show that the proposed full-duplex and hybrid schemes enlarge the rate region and substantially outperform half-duplex, while the hybrid scheme performs nearly as well as best-mode selection.The approach also applies to cognitive-radio scenarios beyond cellular networks, including ad hoc networks with multiantenna full-duplex secondary transmitters [23].Source: B. Summary of Contributions · p. 2

C. Notations

• The paper uses bold lowercase and uppercase for vectors and matrices, respectively, † for the Hermitian operation, and A ⪰ 0 for positive semidefiniteness.Its complex Gaussian notation is x ∼ CN(m, Θ), with mean m and covariance Θ.Source: C. Notations · p. 2

A. System Model

• The modeled system has a single-antenna primary base station and user, an N-antenna CBS with N > 1, and a single-antenna cognitive user; the CBS is time-synchronized and cooperation is needed when the primary link misses its rate target.In half-duplex operation, the primary broadcasts in Phase I and the CBS forwards using amplify-and-forward or decode-and-forward in Phase II while superimposing its cognitive signal; transmit noise models hardware impairments and may be fixed or scale with relay power.Source: A. System Model · pp. 2–3

B. Achievable Rate Region and Problem Formulation

• The objective is to maximize the CU rate subject to the PU rate constraint r0 and CBS transmit-power constraint PC, jointly optimizing the protocol-specific variables.The variables are (wc, A) for AF and (wc, w0) for DF.Source: B. Achievable Rate Region and Problem… · p. 3
• Received-SINR and achievable-rate monotonicity enables simplified equivalent optimization formulations for both AF and DF.Source: B. Achievable Rate Region and Problem… · p. 3

C. The Optimal Structure of A in P-AF-HD and Physical Interpretation

• The optimal AF relay matrix is rank one, A = w_a g†, combining primary signals with MRC before forwarding them through transmit beamforming.This structure gives a physical interpretation of the relay processing and reduces the matrix optimization to a vector representation.Source: C. The Optimal Structure of A in… · pp. 3–4

D. Simplified AF Problem

• The rank-one structure replaces the general N × N AF matrix with a vector w_a and simplifies P-AF-HD.The simplified problem is given in (14).Source: D. Simplified AF Problem · p. 4
• P-AF-HD is feasible only when ∥g∥2 exceeds γ′0AF; a weak PBS–CBS link cannot support the required primary rate.The AF end-to-end performance is bounded by the PBS–CBS channel quality.Source: D. Simplified AF Problem · p. 4
• For P-DF-HD, the first constraint can be removed when satisfied; otherwise, the required primary rate is infeasible even with CBS assistance.When feasible, the remaining problem has the same structure as (14).Source: D. Simplified AF Problem · p. 4

A. System Model

• In FD mode, the CBS receives and transmits simultaneously using N_r receive and N_t transmit antennas, with loop-interference channel H remaining after self-interference mitigation [11].The residual channel strength depends on mitigation quality, including antenna separation and analog-domain cancellation.Source: A. System Model · p. 4
• The FD–HD comparison assumes equal energy consumption and equal total antenna counts, while also considering a higher-cost HD configuration with N antennas for each phase.The equal-configuration HD case uses N_r receive antennas in Phase I and N_t transmit antennas in Phase II; the higher-cost case uses all N antennas in each phase.Source: A. System Model · pp. 4–5
• Although the CBS can subtract its known transmitted signal, transmit noise passing through H remains as residual interference in its input.The preprocessed signal contains HΔt[i] alongside the relayed primary signal and receiver noise.Source: A. System Model · p. 5
• FD removes the CU rate’s HD prelog loss of 1/2, but AF incurs an additional residual-self-interference term, P_t∥h†_cFD A H∥2.The additional term arises from CBS transmit imperfections.Source: A. System Model · p. 5

B. Problem Formulation

• For FD-AF, the optimal relay processing matrix has the same structure as in HD and enables reformulation of the problem as (28).The AF and DF problems are formulated as (26) and (27), respectively.Source: B. Problem Formulation · p. 5

C. Fixed Transmit Noise Pt

• With fixed Pt, the AF and DF formulations retain the structures of earlier problems, so their solutions follow the Appendix B approach.Source: C. Fixed Transmit Noise Pt · p. 5

D. Scalable Transmit Noise Pt

• Scalable transmit noise is modeled as Pt = ǫ2pR in HD and Pt = ǫ2pR^FD in FD; ǫ2 represents the transmit-noise fraction and may limit useful transmit power.More transmit power also brings more receiver noise in both modes and more self-interference in FD.Source: D. Scalable Transmit Noise Pt · p. 5
• In HD, scalable noise leaves the objective non-decreasing in pR, so the relay uses maximum transmit power; the solution mechanism is unchanged for HD and FD-DF.Source: D. Scalable Transmit Noise Pt · pp. 5, 7
• Unlike HD and FD-DF, scalable noise may change the FD-AF solution because AF amplifies received noise and greater transmit power increases self-interference.The resulting P1 is complicated: its power constraint may be inactive, and its objective involves a product of two quadratic terms.Source: D. Scalable Transmit Noise Pt · p. 7
• For FD-AF, Theorem 2 reduces P1 to maximizing Φ(P) over 0 ≤ P ≤ PC^FD, so P2 can be solved by a one-dimensional search.This reduction applies when P1 is feasible.Source: D. Scalable Transmit Noise Pt · p. 7
• Theorem 2 characterizes P1's objective value as the maximum of Φ(P) over the available FD power range.This lets the difficult P1 be solved through P2's one-dimensional search over P.Source: D. Scalable Transmit Noise Pt · p. 7

E. Implementation Issues

• The CBS must track the CBS-CU, CBS-PU, and PBS-CU links, as well as the self-interference channel, using appropriate periodic pilot signals.Source: E. Implementation Issues · p. 7
• Residual self-interference can be estimated from periodic CBS pilot sequences; [24] implements pilot-based estimation with analogue and digital self-interference mitigation.Source: E. Implementation Issues · p. 7
• The PBS-CU channel is estimated at the CU by overhearing and synchronizing to the primary pilot, then fed back to the CBS; the CBS-CU channel uses the cognitive pilot.The passage states that CBS-CU channel information is then fed back to the CU; [19, Sec. V. D] proposes the estimation approach.Source: E. Implementation Issues · p. 7
• The CBS-PU channel can be obtained using periodic PU pilots or blind estimation; assuming perfect channel knowledge yields performance bounds for practical implementations.Channel-estimation imperfections degrade performance; the paper assumes perfect channel knowledge as in.Source: E. Implementation Issues · p. 7

IV. HYBRID HD/FD MODE SELECTION FOR THE CBS

• The proposed hybrid selector switches between HD and FD using a suboptimal zero-forcing criterion, avoiding the complex closed-form comparison that gives little insight into when each mode is preferable.The motivation is that FD adds self-interference whereas HD reduces spectral efficiency; neither mode is always better.Source: §IV · p. 7
• When the primary direct link is weak (h0 ≈ 0), DF achieves a higher cognitive-user rate than AF in both HD and FD, but this result is not general without that assumption.The weak-link assumption supports a simple switching criterion and is motivated by cases where the primary link is weaker than the primary-to-secondary link or unavailable.Source: §IV · pp. 7–8
• Because AF performance is asymptotically close to DF, the paper derives simpler DF solutions for HD and FD and uses their comparison as a unified mode-selection criterion for both protocols.The derivation assumes transmit-noise power scales with signal power in each mode.Source: §IV · p. 8

A. DF in the HD mode

• The cognitive beamformer satisfies w†_c h_c h_c0 = 0 while maximizing |w†_c h_c| under the zero-forcing criterion.The beamforming vectors are expressed using transmit powers and unit-norm vectors.Source: A. DF in the HD mode · pp. 8–9
• The DF problem formulation is simplified to a power-allocation problem, which yields a CU rate expression.Source: A. DF in the HD mode · p. 9
• An approximation ignores the PBS-PU and PBS-CU links when P_0∥g∥^2 ≥ γ.Source: A. DF in the HD mode · p. 9

B. DF in the FD mode

• The FD decode-and-forward problem yields the CU rate in (43) and approximation (44), which ignores the PBS–PU and PBS–CU links.Source: B. DF in the FD mode · p. 9
• For DF relaying, mode selection compares the achievable rates in (40) and (43).Source: B. DF in the FD mode · p. 9
• The rate analysis assumes sufficiently strong PBS–CBS links to support the required PU rate and focuses on CBS–PU and CBS–CU links.Source: B. DF in the FD mode · p. 9

C. Same RF chains for HD and FD

• With the same RF-chain count and transmit/receive antenna sets, both modes share corresponding channel matrices, and their achievable CU rates are defined.Source: C. Same RF chains for HD and FD · p. 9
• Setting both rates to zero, when possible, gives the zero points for ϵ_2, representing maximum tolerable transmit-noise factors.Source: C. Same RF chains for HD and FD · p. 9
• As P_C increases or ρ_2 decreases, FD tends to outperform HD, but its relative improvement shrinks; as r_0 decreases, HD tends to perform better.Source: C. Same RF chains for HD and FD · p. 9

D. High CBS power PC

• The high-CBS-power analysis approximates the achievable rates for the case where HD and FD may use different RF chains and antenna sets.Source: D. High CBS power PC · p. 9
• FD outperforms HD when N_t is close to N or both antenna counts are large in the high-CBS-power case.This analysis allows HD and FD to have different RF chains and transmit/receive antenna sets.Source: D. High CBS power PC · p. 9

• With fixed transmit noise, FD rates increase with CBS power, while DPC and HD rates saturate from 25 dB; AF approaches DF at high SNR.The saturation indicates that CBS should reserve power to suppress self-interference.Source: §V · pp. 10–11
• In Figs. 4–5, FD achieves approximately 50% higher rates for both PU and CU than HD using the same RF chains.DPC provides an outer bound based on non-causal primary-message knowledge at the CBS,,.Source: §V · p. 10, Figure 4 · p. 11, Figure 5 · p. 11
• At 20 dB CBS power, FD generally outperforms HD as transmit-noise scaling grows, especially for DF; AF-HD can outperform AF-FD because AF amplifies self-interference.Proposed mode selection is close to best selection for AF and almost identical for DF.Source: §V · p. 11
• At 25 dB CBS power, AF-HD outage is about 60%, while hybrid schemes reduce it below 40%; DF-FD reaches a 32% outage above 20 dB.Without CBS assistance, outage is nearly 100%; at equal energy, FD and hybrid schemes can reduce outage to 60%.Source: §V · p. 11
• In Fig. 6, FD and hybrid achieve almost three times the CU rate of HD with the same RF chains; hybrid performs nearly as well as best mode selection.HD may outperform FD at low SNR, but FD's gain widens as CBS power increases; FD and hybrid have the same CU-rate slope as DPC.Source: §V · p. 10, Figure 6 · p. 11
• For FD, (Nt, Nr) = (3, 3) performs best at medium-to-high SNR, while (5, 1) beats (2, 4) at low-to-medium SNR and (2, 4) beats (5, 1) at high SNR.At high SNR, (4, 2) saturates near 20 dB CBS power, while (2, 4) continues improving through 25 dB; the trends hold for AF and DF.Source: §V · p. 11

VI. CONCLUSION

• Transmit imperfections are modeled in both HD and FD modes, with the FD operation’s residual self-interference included in the analysis.The study focuses on CBS operation in a cooperative cognitive network.Source: §VI · pp. 11–12
• The proposed FD and hybrid schemes greatly enlarge the primary-cognitive rate region compared with HD, increasing CU spectrum-access opportunities and overall spectral efficiency.The conclusion presents these schemes as efficient solutions for active cooperation between primary and cognitive systems.Source: §VI · p. 12
• Closed-form solutions or efficient one-dimensional search algorithms provide optimal AF and DF beamforming vectors to characterize the achievable primary-cognitive rate regions.Source: §VI · p. 12
• A hybrid scheme switches between HD and FD using a simplified zero-forcing beamforming design.Source: §VI · p. 12
• With a strong primary direct link, direct and relayed copies create an artificial multipath effect at the PU; equalization to combat it is proposed as future work.The paper and most prior studies discard the direct link and decode only relaying information.Source: §VI · p. 12

A. PROOF OF THEOREM 1

• Theorem 1’s proof sets C, d, and E to zero and obtains the optimal structure A = wa g†, where wa = H̄b.The proof’s partitioned parameterization shows that C, d, and E do not affect the term being maximized and that setting them to zero reduces terms being minimized.Source: A. PROOF OF THEOREM 1 · p. 12

B. CLOSED-FORM SOLUTION TO A GENERAL RATE MAXIMIZATION PROBLEM

• The problem models maximizing user 2’s SINR in a two-user MISO broadcast system, subject to total BS power PC and user 1’s SINR constraint γ1.The BS has N antennas, each user has one antenna, and user noise powers are normalized to one.Source: B. CLOSED-FORM SOLUTION TO A GENERAL… · pp. 12–13
• The solution approach focuses on an equivalent weighted sum-power minimization problem, whose dual LMI constraints determine the dual variables λ1 and λ2.The optimal objective value of the original problem is denoted γ∗2; the passage relates it to the power-minimization formulation’s objective value.Source: B. CLOSED-FORM SOLUTION TO A GENERAL… · p. 13
• Because A > 0, C > 0, B < 0, and f(PC) < 0, f(λ) = 0 has two positive roots and the optimal λ1 is the minimum root.After finding λ1, λ2 and γ2 are derived from equations (59–60); A, B, and C are defined in terms of the channel norms, ρ, γ1, and PC.Source: B. CLOSED-FORM SOLUTION TO A GENERAL… · p. 13