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On Peak versus Average Interference Power Constraints for Protecting Primary Users in Cognitive Radio Networks

Rui Zhang

arXiv:0806.0676v2cs.IT

TL;DR

The paper examines the tradeoff between CR transmission throughput and PR protection under average and peak interference constraints. It reports that AIP-based control outperforms PIP-based control for CR transmission and yields larger PR ergodic and outage capacities, attributed to interference diversity.

  • Problem

    Cognitive radio transmission must balance maximizing its own throughput with protecting the primary radio, while PIP initially appears preferable for PR protection.

  • Method

    The paper compares AIP-based and PIP-based power control using available channel-state information at the CR transmitter.

  • Results

    AIP-based optimal power control outperforms PIP-based control for the CR and produces larger PR ergodic and outage capacities across various PR power-control policies.

  • Takeaways & Limitations

    Interference diversity gives AIP an advantage over PIP for the PR, while AIP remains more flexible for exploiting channel information at the CR transmitter.

  • Takeaways & Limitations

    The concluding remarks assume availability of only statistical channel knowledge.

Abstract

from arXiv · show

This paper considers spectrum sharing for wireless communication between a cognitive radio (CR) link and a primary radio (PR) link. It is assumed that the CR protects the PR transmission by applying the so-called interference-temperature constraint, whereby the CR is allowed to transmit regardless of the PR's on/off status provided that the resultant interference power level at the PR receiver is kept below some predefined threshold. For the fading PR and CR channels, the interference-power constraint at the PR receiver is usually one of the following two types: One is to regulate the average interference power (AIP) over all the fading states, while the other is to limit the peak interference power (PIP) at each fading state. From the CR's perspective, given the same average and peak power threshold, the AIP constraint is more favorable than the PIP counterpart because of its more flexibility for dynamically allocating transmit powers over the fading states. On the contrary, from the perspective of protecting the PR, the more restrictive PIP constraint appears at a first glance to be a better option than the AIP. Some surprisingly, this paper shows that in terms of various forms of capacity limits achievable for the PR fading channel, e.g., the ergodic and outage capacities, the AIP constraint is also superior over the PIP. This result is based upon an interesting interference diversity phenomenon, i.e., randomized interference powers over the fading states in the AIP case are more advantageous over deterministic ones in the PIP case for minimizing the resultant PR capacity losses. Therefore, the AIP constraint results in larger fading channel capacities than the PIP for both the CR and PR transmissions.

I. INTRODUCTION

The paper studies spectrum sharing in which a cognitive radio transmits over a primary radio’s bandwidth while limiting interference at the primary receiver. It compares average and peak interference constraints and shows that AIP is superior for both links’ fading-channel capacities.

  • Spectrum-sharing setting: The CR shares the PR’s spectrum and uses an interference-temperature constraint to transmit while keeping received interference below a predefined threshold.Dynamic resource allocation adapts CR transmission parameters using available channel-state information.
  • Constraint comparison: AIP regulates interference averaged over fading states, whereas PIP limits interference separately at every fading state.With equal thresholds, PIP is more restrictive because it implies the corresponding average constraint, but not vice versa.
  • Research question: The previously unstudied effect of AIP- and PIP-based CR power control on PR transmission motivates the paper’s comparison.Although PIP initially appears better for protecting the PR, the paper examines this assumption using PR capacity limits.
  • Main result: AIP is rigorously shown to outperform PIP for PR ergodic and outage capacities under the same interference-power threshold.The paper attributes this result to interference diversity: randomized interference powers over fading states reduce PR capacity losses more effectively than deterministic interference.
  • Implication: AIP provides larger fading-channel capacities than PIP for both CR and PR transmissions.The paper presents this as a practical design rule involving the tradeoff between maximizing CR throughput and minimizing interference to PR transmission.

II. SYSTEM MODEL

The system model describes a single-antenna, slow block-fading PR–CR network in which the CR controls its transmit power using channel knowledge and an AIP or PIP constraint at the PR receiver.

  • Network and channel model: The network contains a PR link and a CR link sharing bandwidth, with single antennas and slow block-fading channels.The model uses coherent communication and focuses on channel power gains.
  • Channel variables: The fading power gains h, g, and f describe the CR-Tx–CR-Rx, CR-Tx–PR-Rx, and PR-Tx–PR-Rx channels, respectively.These gains are assumed independent and continuously distributed.
  • Interference model: CR transmission creates PR-side interference I_i = g_i p_i at fading state i, where p_i is the CR transmit power.The PR treats CR interference as additional Gaussian noise and uses a power-control policy subject to E[q_i] ≤ Q.
  • CR power control: The CR power-control policy depends on h_i and g_i and satisfies its average transmit-power constraint E[p_i] ≤ P.The CR transmitter is assumed to know these channel power gains for each fading state.
  • Interference constraints: AIP constrains average received interference across fading states, whereas PIP constrains peak received interference at every state.For equal thresholds, PIP is generally more restrictive, while AIP gives the CR greater flexibility to adapt transmit powers.

III. CR CAPACITIES UNDER AIP VERSUS PIP CONSTRAINT

For the CR fading channel, the paper compares capacity-achieving power control under AIP and PIP constraints and finds AIP more flexible and superior in capacity limits.

  • Ergodic capacity: AIP ergodic-capacity control resembles water filling, but its water level depends on the interference-channel gain g_i.The policy is evaluated by substituting it into the per-state CR mutual information and averaging over fading states.
  • PIP policy: Under PIP, maximizing ergodic and outage capacities requires using the maximum permitted CR transmit power in each fading state.The permitted power is determined by the peak interference threshold and the CR-to-PR channel gain.
  • Power-control comparison: Under AIP, CR power control for ergodic and outage capacity can use both h_i and g_i, whereas PIP power control depends only on g_i.This gives AIP greater flexibility across fading states.
  • Capacity comparison: With equal AIP and PIP thresholds, AIP is superior to PIP in the achievable CR fading-channel capacity limits.The comparison covers ergodic, outage, and delay-limited capacity results described for the two constraints.

IV. PR CAPACITIES UNDER AIP VERSUS PIP CONSTRAINT

The PR-capacity analysis compares AIP and PIP under equal interference thresholds, with both constraints active at the PR receiver, and evaluates ergodic and outage capacities.

  • Comparison setup: The PR analysis uses the same average and peak interference threshold, Γ_a = Γ_p = Γ, for a fair AIP–PIP comparison.Under the considered CR power-control policies, both constraints are met with equality at the PR receiver.
  • PR capacity analysis: The section studies ergodic and outage capacities of the fading PR channel under interference generated by AIP- or PIP-based CR power control.The analysis begins with the constant-power policy.

1) Constant-Power Policy:

Across constant-power and water-filling policies, randomized interference under AIP is generally at least as favorable as constant PIP interference for PR ergodic capacity, with the comparison driven by convexity and interference diversity.

  • Constant-Power Policy: AIP randomizes interference across fading states, whereas PIP produces constant interference levels at all states under the same threshold Γ.The resulting advantage is attributed mainly to the convexity of capacity with respect to noise/interference power.
  • Constant-Power Policy: Interference diversity names the advantage of randomized AIP interference over constant PIP interference for PR transmission.The phenomenon is explicitly identified in the paper as arising in a CR network.
  • Water-Filling Power Control: The water-filling comparison is nontrivial because a direct comparison based on the relative water levels μ_a and μ_p does not establish the capacity ordering.The paper proves by contradiction that the intuitive inequality between these water levels is reversed.
  • Water-Filling Power Control: For water-filling comparison, the paper uses Lagrange duality and min-max optimization to establish the AIP-versus-PIP capacity ordering.The proof compares optimal water levels and power allocations while invoking E[I_i] = Γ, convexity, and Jensen’s inequality.
  • Water-Filling Power Control: Under water-filling power control, AIP randomized interference is superior to PIP constant interference for maximum achievable PR ergodic capacity.The interference-diversity gain is less obvious than under constant-power control because water-filling PR power control is more complex.

B. Outage Capacity

With constant-power control, the paper introduces the PR outage-capacity analysis for a given outage probability ε_0.

  • Outage Capacity: For a given outage probability, ε_0, the analysis considers the maximum achievable constant SNR at PR-Rx under the interference constraints.The corresponding AIP and PIP outage capacities are developed from these constant-SNR quantities.

1) Constant-Power Policy:

Under constant-power control, the AIP-versus-PIP outage-capacity comparison depends on the convexity or concavity of the PR fading power-gain CDF.

  • Constant-Power Policy: The AIP and PIP outage capacities are obtained from maximum constant SNRs γ_a and γ_p at PR-Rx for the same outage probability ε_0.The analysis compares equivalent outage-probability conditions rather than comparing the capacities directly.
  • Constant-Power Policy: If the fading-gain CDF G_f(x) is convex, Jensen’s inequality gives ε_a ≤ ε_p; if it is concave, ε_a ≥ ε_p.The direction of the outage-probability comparison determines the corresponding AIP-versus-PIP outage-capacity comparison.
  • Constant-Power Policy: With constant-power control, AIP outage capacity is at least PIP outage capacity when G_f(x) is convex.This is stated as Theorem 4.3 for the same average and peak threshold Γ.
  • Constant-Power Policy: For standard Rayleigh fading, the fading-gain CDF is convex, so the theorem’s convex-CDF condition applies.The paper notes that the interference-diversity gain generally depends on the PR fading-channel distribution.

2) Channel-Inversion Power Control:

Under channel-inversion power control, the AIP and PIP cases yield identical PR delay-limited capacities, while AIP is no worse for zero-delay constant-rate traffic.

  • Channel-Inversion Power Control: For channel inversion, the optimal PR power allocations are formulated separately for AIP and PIP using constant SNRs γ_a and γ_p.The comparison assumes the same average and peak power threshold Γ.
  • Channel-Inversion Power Control: Since f_i is independent of I_i and E[q_i] = Q, the analysis obtains γ_a = γ_p under AIP and PIP.This equality leads directly to equal PR delay-limited capacities.
  • Channel-Inversion Power Control: The PR delay-limited capacities are identical under channel-inversion power control: C_DL,a^PR = C_DL,p^PR.This result is stated as Theorem 4.4.
  • Channel-Inversion Power Control: For zero-delay and constant-rate data traffic, AIP is at least no worse than PIP from the PR’s perspective.The result concerns the PR’s delay-limited capacity under channel-inversion control.

3) Truncated-Channel-Inversion Power Control:

The PR’s truncated-channel-inversion power control compares AIP and PIP constraints by selecting transmission thresholds to meet a target outage probability. The AIP constraint provides an interference-diversity advantage, yielding at least as large PR outage capacity as PIP under the same threshold.

  • Truncated-channel-inversion power control: TCI maintains a constant receiver SNR when the effective channel gain exceeds its threshold and disables transmission otherwise.For AIP, the effective gain is fi/(1 + Ii); for PIP, interference is fixed at Γ.
  • Truncated-channel-inversion power control: The AIP and PIP cases choose truncation thresholds θa and θp to satisfy the same target outage probability ϵ0.The thresholds are determined from the respective effective-channel outage probabilities.
  • Capacity comparison: With the same average and peak threshold Γ, AIP-based TCI achieves PR outage capacity at least as large as PIP-based TCI.The comparison is stated as COUT,a^PR,TCI(ϵ0) ≥ COUT,p^PR,TCI(ϵ0).
  • Capacity comparison: The proof establishes ϵa ≤ ϵp for every target SNR γ0 by comparing the corresponding max-min optimization problems.The argument uses Lagrange duality and concavity with respect to average interference power.
  • Capacity comparison: Theorem 4.5 attributes the AIP advantage over PIP to interference diversity, which persists regardless of outage probability.For zero outage probability, however, the delay-limited capacities are equal under AIP and PIP.

V. SIMULATION RESULTS AND DISCUSSIONS

Simulations compare AIP- and PIP-constrained CR power control under shared fading-channel assumptions. Across ergodic and outage settings, AIP generally improves CR and PR capacities through flexible allocations and interference diversity, with gains diminishing when interference becomes effectively constant.

  • Simulation setup: The simulations use Rayleigh fading for h, g, and f, with Γa = Γp = 1 and independent squared-norm CSCG channel gains.The average power for g is set 10 dB above that for h or f to emphasize the CR-to-PR interference channel.
  • CR capacities: As g-channel attenuation increases, CR ergodic and outage capacities increase under both AIP and PIP constraints.Lower average power for g permits higher CR transmit power under the fixed interference threshold.
  • CR capacities: AIP-based optimal power control outperforms PIP-based control for CR ergodic and outage capacities by exploiting all available CSI at CR-Tx.The AIP allocation is more flexible across fading states, whereas PIP-based control depends only on g_i.
  • Interference diversity: In the AIP case, randomized interference powers create interference diversity, whereas PIP fixes interference at Γ in every fading state.This randomness explains the PR capacity advantage of AIP; for the CR, the AIP and PIP ergodic capacities eventually converge as g_i decreases.
  • PR capacities: For PR transmissions, AIP yields larger ergodic and outage capacities than PIP under various PR power-control policies.The simulations agree with the analytical results for PR capacity under AIP and PIP.
  • Interference diversity: As g_i approaches zero, AIP interference also approaches Γ in every state, so the interference-diversity gain and the AIP–PIP capacity difference diminish.For CR outage capacity, the gap converges to log(ζa/ζp), equal to 2.6791 bits/complex dimension when ϵ0 = 0.1.

VI. CONCLUDING REMARKS

This paper compares AIP and PIP interference constraints for spectrum sharing and finds that AIP can better protect PR capacity while also supporting CR transmission. The result is attributed to interference diversity and motivates AIP as a practical design rule under the paper’s channel-knowledge assumptions.

  • Conclusions: AIP can be more advantageous than PIP for minimizing capacity losses of the PR fading channel at the same threshold.This challenges the traditional view that PIP protects PR transmission better than AIP.
  • Conclusions: Interference diversity explains why randomized interference powers across fading states can outperform deterministic PIP interference for PR capacity.The paper identifies this phenomenon as the main reason for AIP’s advantage in many cases.
  • Conclusions: The paper recommends AIP for both protecting PR transmission and maximizing CR throughput.This design rule is stated for practical CR networks.
  • Scope and assumptions: The analysis assumes perfect interference-channel state information at the CR transmitter for each fading state.In practice, statistical channel knowledge may be more realistic, and the AIP definition can be extended to such cases.
  • Scope and assumptions: The results extend beyond fading PR and CR channels to parallel Gaussian channels with average and peak power constraints, including OFDM-decomposable broadband channels.The paper identifies OFDM as an example of a time-dispersive broadband channel represented by parallel narrow-band channels.
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