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Rate Analysis of Two-Receiver MISO Broadcast Channel with Finite Rate Feedback: A Rate-Splitting Approach

Chenxi Hao, Yueping Wu, Bruno Clerckx

arXiv:1506.09023v1cs.IT

TL;DR

Imperfect or quantized CSIT limits interference mitigation in two-receiver MISO broadcast channels, motivating analysis of rate-splitting schemes. The paper derives sum-rate loss bounds and feedback-scaling laws for RS-S and RS-ST, showing feedback reductions and significant high-SNR gains over conventional SU/MU switching.

  • Problem

    The sum-rate performance and feedback benefits of RS-S and RS-ST under quantized CSIT remain insufficiently studied despite the practical difficulty of obtaining accurate CSIT.

  • Method

    The paper analyzes ergodic sum rates for RS-S and RS-ST, deriving loss bounds and feedback-scaling laws relative to ZFBF with perfect or quantized CSIT.

  • Results

    RS-S reduces feedback overhead relative to quantized-CSIT ZFBF, RS-ST provides further reduction with alternating feedback qualities, and both yield significant high-SNR gains over SU/MU switching.

  • Takeaways & Limitations

    Rate splitting provides feedback-efficient multiuser transmission under imperfect CSIT, with space-time common-message transmission benefiting alternating receiver-specific feedback qualities.

  • Takeaways & Limitations

    The analysis does not cover RS-S and RS-ST with the alternative precoders specified for other conventional multiuser transmissions.

Abstract

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To enhance the multiplexing gain of two-receiver Multiple-Input-Single-Output Broadcast Channel with imperfect channel state information at the transmitter (CSIT), a class of Rate-Splitting (RS) approaches has been proposed recently, which divides one receiver's message into a common and a private part, and superposes the common message on top of Zero-Forcing precoded private messages. In this paper, with quantized CSIT, we study the ergodic sum rate of two schemes, namely RS-S and RS-ST, where the common message(s) are transmitted via a space and space-time design, respectively. Firstly, we upper-bound the sum rate loss incurred by each scheme relative to Zero-Forcing Beamforming (ZFBF) with perfect CSIT. Secondly, we show that, to maintain a constant sum rate loss, RS-S scheme enables a feedback overhead reduction over ZFBF with quantized CSIT. Such reduction scales logarithmically with the constant rate loss at high Signal-to-Noise-Ratio (SNR). We also find that, compared to RS-S scheme, RS-ST scheme offers a further feedback overhead reduction that scales with the discrepancy between the feedback overhead employed by the two receivers when there are alternating receiver-specific feedback qualities. Finally, simulation results show that both schemes offer a significant SNR gain over conventional single-user/multiuser mode switching when the feedback overhead is fixed.

I. INTRODUCTION · II. SYSTEM MODEL · A. Random Vector Quantization

The paper analyzes finite-rate-feedback rate-splitting in a two-receiver MISO broadcast channel, establishing rate-loss and feedback-scaling results for RS-S and RS-ST and validating their SNR gains. It models quantized channel directions using RVQ under equal and receiver-specific feedback qualities.

  • I. INTRODUCTION: Multiple antennas can increase multiplexing gain and throughput, but interference mitigation depends strongly on accurate CSIT, which is difficult to obtain practically.
  • I. INTRODUCTION: With CSIT error scaling as SNR^−α, conventional ZFBF achieves sum DoF 2α, while rate-splitting superposes a common message on ZF-precoded private messages.The RS receiver first decodes the common message and then its desired private message via successive interference cancellation.
  • I. INTRODUCTION: The paper addresses the previously uninvestigated ergodic sum-rate benefits of RS-S and RS-ST with quantized CSIT, including message splitting and space-time transmission under alternating qualities.
  • I. INTRODUCTION: RS-S rate-loss upper bounds are derived relative to ZFBF with perfect CSIT for equal and alternating receiver-specific feedback qualities.For equal feedback qualities, the analysis also shows that a b-bit feedback increase yields a 3b/(M−1) dB high-SNR sum-rate improvement when feedback bits are SNR-invariant.
  • I. INTRODUCTION: (M−1)log2δ−1 feedback bits are saved asymptotically by RS-S versus ZFBF with RVQ for maximum rate loss log2δ bps/Hz and equal feedback qualities.
  • I. INTRODUCTION: τ feedback bits are asymptotically saved by RS-ST over RS-S for alternating receiver-specific CSIT qualities, with the reduction scaling as τ^2−2(M−1) for large τ.
  • I. INTRODUCTION: At high SNR with a fixed feedback-bit count, simulations show that RS-S and RS-ST provide significant SNR gains over conventional SU/MU mode switching.
  • II. SYSTEM MODEL · A. Random Vector Quantization: The system is a two-receiver MISO BC with M≥2 transmit antennas and SNR P, using FDD finite-bit feedback and RVQ that quantizes only channel direction.Each receiver uses a receiver-specific isotropic codebook; the quantization error is the minimum of 2^Bkl independent beta (1,M−1) random variables.

B. Rate-Splitting Approach · C. Rate-Splitting Approach with Space-Time design

The paper introduces RS-S, which splits one receiver’s message into common and private parts and superposes the common message on ZF-precoded private messages. It then extends this idea to RS-ST, transmitting an additional common message across two channel uses to exploit alternating feedback qualities.

  • B. Rate-Splitting Approach: RS-S splits one receiver’s message into common and private parts, while the other receiver’s message remains private-only.The common message is decoded by both receivers; private messages are decoded only by their intended receivers.
  • B. Rate-Splitting Approach: RS-S allocates Pc=P(1−t) to the common message and P1=P2=Pt to private messages, with t∈(0,1].Equal private-message powers simplify the rate-loss analysis relative to perfect-CSIT ZFBF, although they are not sum-rate optimal.
  • B. Rate-Splitting Approach: 1+α sum DoF is achieved by RS-S, exceeding the 2α achieved by ZFBF when feedback scales as B=α(M−1)log2P+o(log2P).The private-message power is scaled according to Pα, while the common message uses the remaining power.
  • B. Rate-Splitting Approach: Alternating receiver-specific feedback qualities motivate RS-ST because applying RS-S independently does not exploit the differing feedback qualities across channel uses.RS-ST is designed for Bα=α(M−1)log2P+o(log2P) and Bβ=β(M−1)log2P+o(log2P), with 0≤α<β≤1.
  • C. Rate-Splitting Approach with Space-Time design: RS-ST transmits an additional common message c0 across two channel uses, alongside per-use common messages and ZF-precoded private messages.The c0 power equals the difference between the private-message powers associated with tβ and tα.
  • C. Rate-Splitting Approach with Space-Time design: RS-ST decodes common messages sequentially with SIC before removing them and decoding private messages in both channel uses.Each receiver decodes its relevant per-use common messages and c0 before recovering the private symbols.
  • C. Rate-Splitting Approach with Space-Time design: β−α DoF is achieved by c0, raising RS-ST’s total sum DoF above the 1+α achieved by RS-S.The quantization errors associated with the weaker and stronger feedback qualities decay as P−α and P−β, respectively.

III. RS-S WITH EQUAL FEEDBACK QUALITIES · A. Preliminary Calculations

For equal feedback qualities, RS-S analysis develops tractable distributions and bounds for common-message SINR before studying rate loss relative to perfect-CSIT ZFBF. The derivation uses RVQ error properties, independence approximations, and CDF-based bounds under a good-feedback assumption.

  • III. RS-S WITH EQUAL FEEDBACK QUALITIES: The analysis assumes equal feedback qualities, B_11=B_21=B, and derives preliminary results used throughout the RS-S development.These preliminaries support the later rate-loss and feedback-scaling analysis.
  • A. Preliminary Calculations: Lemma 1 decomposes the RVQ projection term into an independent quantization-error factor and a beta (1,M−2) random variable.The lemma is used to upper-bound ZF-precoder rate loss in RS-S and RS-ST.
  • A. Preliminary Calculations: Under Assumption 1, common-message SINRs are approximated using P(1−t)Y_k and P(1−t)Y, with Y=min(Y_1,Y_2).The assumption applies only to common-message rate derivations; residual ZFBF interference remains in private-message rates and exact SINRs are used in simulations.
  • A. Preliminary Calculations: The correlated exponential variables X_k1 and X_k2 have a joint CDF characterized in Lemma 2 because both depend on the same channel realization.Each variable is exponential with parameter 1, while their dependence motivates the joint-CDF formulation.
  • A. Preliminary Calculations: To simplify analysis, X_k1 and X_k2 are approximated as independent; Figure 2 indicates this is sufficiently accurate for large M and already adequate at M=4.The independent approximation is then used to obtain the CDF of the resulting variable through exponential distributions and Monte Carlo comparison.
  • A. Preliminary Calculations: Because Y_1 and Y_2 are correlated, the paper uses an upper bound on the CDF of Y=min(Y_1,Y_2) rather than deriving its exact distribution.The bound is subsequently approximated to make the analysis tractable.
  • A. Preliminary Calculations: Lemma 4 converts a CDF ordering into an expectation bound, enabling a lower bound on E[ln Y] used for common-message rate analysis.This bound feeds the proofs of Propositions 1, 3, and 5.
  • A. Preliminary Calculations: The section then evaluates RS-S sum-rate loss relative to perfect-CSIT ZFBF and the feedback scaling required to meet a maximum allowable loss.This establishes the transition from preliminary distributional calculations to the main rate-scaling results.

B. Sum Rate Loss

This section upper-bounds the RS-S sum-rate loss relative to perfect-CSIT ZFBF, then derives a high-SNR power-splitting choice and examines feedback-quality effects. Simulations show the proposed choice nearly matches exhaustive optimization, while RS-S avoids the high-SNR saturation observed for quantized-CSIT ZFBF.

  • Sum-rate-loss definition: The RS-S loss ΔR_eq^S(t) is defined as the difference between perfect-CSIT ZFBF and RS-S sum rates with power-splitting ratio t∈(0,1].The reference private rates use ZFBF with perfect CSIT, while RS-S includes private and common-message rates.
  • Upper bound: For equal feedback qualities, Proposition 1 upper-bounds the RS-S sum-rate loss with RVQ relative to perfect-CSIT ZFBF.The bound decomposes into private-power reduction, quantized-CSIT ZF-precoder loss, and common-message rate contributions.
  • Power splitting: At high SNR, the analysis replaces the generally intractable optimal ratio t* with a closed-form equivalent choice teq^S.The threshold in the resulting expression switches from ZFBF with RVQ to RS-S when feedback quality is insufficient.
  • Simulation validation: The teq^S allocation yields almost the same performance as exhaustive search in simulations.Figure 3(a) compares analytical upper bounds and Monte Carlo simulations for M=4 and B=10.
  • Feedback-quality effect: 5dB SNR gain is observed for RS-S with B=15 over B=10 at high SNR, whereas quantized-CSIT ZFBF saturates when B is fixed.For M=4, Figure 3(b) compares B=10 and B=15 and verifies the feedback-quality SNR gain.

C. A New Scaling Law of B

With equal receiver feedback qualities, RS-S specifies a feedback-bit scaling law that maintains a bounded sum-rate loss relative to perfect-CSIT ZFBF. At 15 dB, it requires 5 fewer feedback bits than ZFBF with RVQ for a 6 bps/Hz allowable loss, while achieving nearly the same sum rate.

  • Scaling law: RS-S enables feedback reduction while maintaining a constant rate offset relative to ZFBF with perfect CSIT.This scaling law applies when ZFBF with RVQ achieves full DoF through feedback scaling with M and SNR.
  • Scaling law: Under equal feedback qualities, RS-S specifies the required feedback bits for a maximum sum-rate loss of log2δ bps/Hz relative to perfect-CSIT ZFBF.The result is stated as Proposition 2 for the equal-feedback-quality scenario.
  • Simulation results: Both schemes incur less than 6 bps/Hz rate loss relative to ZFBF with perfect CSIT under their respective feedback-bit scaling laws.The comparison uses log2δ=6 bps/Hz in the simulation.
  • Simulation results: RS-S and ZFBF with RVQ achieve almost the same sum-rate performance under the respective scaling laws.The result supports the characterization of RS-S feedback-overhead reduction for matching ZFBF with RVQ performance.

IV. RS-S AND RS-ST WITH ALTERNATING RECEIVER-SPECIFIC FEEDBACK QUALITIES · A. Performing the RS-S scheme · 1) Sum rate loss:

This section analyzes RS-S under alternating receiver-specific feedback qualities, where the receivers exchange feedback-bit allocations across channel uses. It bounds RS-S’s loss relative to perfect-CSIT ZFBF and characterizes how feedback asymmetry affects the threshold and degradation.

  • IV. RS-S AND RS-ST WITH ALTERNATING RECEIVER-SPECIFIC FEEDBACK QUALITIES: Alternating feedback assigns Rx1 Bβ and Rx2 Bα bits in one channel use, then reverses the assignments in the next, with τ=Bβ−Bα and Bα<Bβ.The average feedback overhead is denoted by B̄.
  • A. Performing the RS-S scheme: RS-S is studied first in this alternating-feedback scenario, while RS-ST is later assessed by comparison with RS-S’s sum rate.The supplied passage frames the section as an extension of the preceding RS-S analysis and a comparison of the two schemes.
  • A. Performing the RS-S scheme: RS-S achieves statistically equivalent sum rates in the two channel uses, so the analysis focuses on a single channel use.The loss is denoted by ΔR_rs-S(t).
  • 1) Sum rate loss:: Proposition 3 upper-bounds the RS-S sum-rate loss with RVQ relative to ZFBF with perfect CSIT under alternating receiver-specific feedback qualities.The bound reuses the proof structure of Proposition 1 and depends on the scheme’s power-splitting parameter t.
  • 1) Sum rate loss:: The optimal power-splitting ratio t* generally lacks a closed-form solution at arbitrary SNR because the loss expression is complicated, so a high-SNR ratio is derived instead.The high-SNR optimization minimizes the upper-bounded loss.
  • 1) Sum rate loss:: The feedback threshold that switches between RS-S and ZFBF with RVQ decreases monotonically with Θ, equivalently with τ.When Bα=Bβ, the threshold reduces to the equal-feedback case.
  • 1) Sum rate loss:: For τ>0, RS-S incurs sum-rate degradation relative to τ=0; the paper characterizes this degradation through a logarithmic term and interprets sufficiently large τ as an SNR loss.The cited remarks compare the asymmetric-feedback case with Θ=4 when τ=0.
  • 1) Sum rate loss:: The analysis then inverts the loss bound with respect to average feedback overhead B̄ to obtain a feedback-overhead characterization.This inversion follows the RS-S loss expressions developed for the alternating-feedback setting.

2) Scaling law of ¯B:

The section characterizes the RS-S feedback overhead needed for a prescribed rate loss under alternating receiver-specific feedback qualities and analyzes its high-SNR power split. It also shows that RS-S feedback savings over ZFBF with RVQ diminish as feedback-quality discrepancy τ increases, with a threshold governed by Θ (or τ).

  • Scaling law of ¯B:: For a maximum rate loss of log2δ bps/Hz relative to perfect-CSIT ZFBF, Proposition 4 gives the RS-S average feedback-bit requirement under alternating receiver-specific feedback qualities.The expression depends on Θ and the parameters introduced in Proposition 1.
  • Scaling law of ¯B:: At arbitrary SNR, the optimal RS-S power-splitting ratio is difficult to calculate, but its high-SNR limit is characterized by a quadratic equation.When Bα=Bβ, the quadratic equation reduces to a linear one.
  • Scaling law of ¯B:: The threshold δ0(Θ) increases monotonically with Θ or τ, so the threshold for S-JMB to reduce feedback over ZFBF with RVQ grows with Θ or τ.The monotonicity is stated for Θ≥4.
  • Scaling law of ¯B:: RS-S reduces average feedback overhead relative to ZFBF with RVQ while maintaining the same prescribed rate-loss target under alternating receiver-specific feedback qualities.The comparison is made against the ZFBF-with-RVQ overhead at t=1.
  • Scaling law of ¯B:: When τ increases, the feedback-overhead gap between RS-S and ZFBF with RVQ decreases, indicating that RS-S savings diminish with τ.Figure 5 reports this behavior for M=4 and P=30dB.

B. Benefit of the Space-Time Transmission · 1) Sum rate loss:

RS-ST uses an additional common message, c0, transmitted in space-time to mitigate the sum-rate degradation that RS-S experiences with alternating receiver-specific feedback qualities. At high SNR, RS-ST can provide substantial SNR and feedback-overhead gains over RS-S while maintaining bounded rate loss relative to perfect-CSIT ZFBF.

  • B. Benefit of the Space-Time Transmission: RS-ST transmits an additional common message, c0, using space-time transmission to improve performance under alternating receiver-specific CSIT qualities.The analysis compares RS-ST with RS-S to characterize the benefit of space-time transmission.
  • 1) Sum rate loss:: Proposition 5 upper-bounds the RS-ST sum-rate loss relative to ZFBF with perfect CSIT for arbitrary SNR under alternating receiver-specific feedback qualities.The bound decomposes losses from private-message power reduction, imperfect-CSIT ZF precoding, and common-message transmission.
  • 1) Sum rate loss:: The analytical upper-bounds for both RS-S and RS-ST agree with simulations, while the RS-S allocation t_rs,S performs almost identically to exhaustive-search optimization.This supports t_rs,S as a proper RS-S power allocation in the alternating-quality scenario.
  • 1) Sum rate loss:: At high SNR with fixed B̄, RS-ST sum-rate loss is independent of Θ, so its performance is similar across choices of Bα and Bβ.This contrasts with RS-S, whose sum rate degrades dramatically with τ.
  • 1) Sum rate loss:: 3(τ/(2(M−1))−2) dB is the high-τ SNR gain RS-ST offers over RS-S, avoiding RS-S degradation through space-time transmission of c0.The gain is derived from the high-SNR rate-loss expressions for the two schemes.
  • 1) Sum rate loss:: 2∼3dB and 8∼9dB are the simulated RS-ST SNR gains over RS-S for τ=6 and τ=10, respectively, with M=2 and B̄=20.RS-ST with τ=6 and τ=10 has the same high-SNR performance in the reported comparison.
  • 1) Sum rate loss:: Less than 6bps/Hz rate loss is achieved by all schemes under their respective feedback-scaling laws, with nearly identical sum-rate performance.This confirms the feedback-overhead reduction benefits of the proposed scaling laws.
  • 1) Sum rate loss:: 1∼2 bits is the feedback-overhead reduction RS-ST achieves over RS-S when M=4, τ=14, and the maximum allowable rate loss is log2δ=6 bps/Hz.RS-S itself provides roughly 1 bit reduction over ZFBF with RVQ at high SNR in this setting.

V. PERFORMANCE COMPARISON

With fixed feedback, RS-S and RS-ST outperform SU/MU switching in sum rate and avoid high-SNR saturation by allocating most power to common messages. RS-S provides SNR gains over SU/MU for equal feedback qualities, while RS-ST provides an additional gain under alternating receiver-specific feedback qualities.

  • Fixed-feedback comparison: At fixed feedback, RS-S and RS-ST perform similarly to ZFBF with RVQ at low and medium SNR but increase rather than saturate at high SNR.At high SNR, both schemes transmit common messages with most of the power.
  • Equal feedback qualities: For equal feedback qualities, RS-S achieves an SNR gain over SU/MU for B=10 and 15 with M=4, despite offering no DoF gain.The comparison is reported for both listed feedback budgets.
  • Equal feedback qualities: The RS-S rate gap over SU/MU increases with B when feedback quality is sufficiently good.The common and private message rates together produce the higher sum rate, although the common-message rate is limited by the weakest effective channel.
  • Alternating feedback qualities: 3dB SNR gain is offered by RS-ST over RS-S when alternating receiver-specific feedback qualities satisfy τ=18.Both RS schemes also yield a significant SNR gain over SU/MU in this scenario.

VI. CONCLUSION

The paper derives feedback-overhead scaling laws for RS-S and RS-ST in a two-receiver MISO broadcast channel with quantized CSIT, showing reductions relative to ZFBF and additional gains for alternating feedback qualities. Simulations show significant high-SNR gains over SU/MU switching.

  • VI. CONCLUSION: RS-S reduces feedback overhead relative to ZFBF with quantized CSIT while achieving a specified maximum allowable rate loss.The paper derives a scaling law for the required feedback overhead under this rate-loss constraint.
  • VI. CONCLUSION: RS-ST provides a further feedback-overhead reduction over RS-S when receiver-specific feedback qualities alternate.Both schemes are based on rate splitting, with one receiver’s message divided into common and private parts.
  • VI. CONCLUSION: At high SNR, both RS schemes achieve significant SNR gains over SU/MU switching.The schemes transmit private messages through ZFBF using part of the total power and common messages through the rate-splitting design.

APPENDIX … C. Proof of Proposition 1

The appendix derives the distributions and conditional-independence relations needed for Lemma 2, evaluates the expectation used in Lemma 4, and proves Proposition 1 through upper and lower rate bounds. It also notes that Proposition 3 follows similarly with receiver-specific feedback quality replacing the final bound term.

  • A. Proof of Lemma 2: A. Proof of Lemma 2 reduces the analysis using statistical equivalence and defines X1=β1a and X2=β2a.Here β1 and β2 are beta (1,M−1) random variables, while a=∥h∥2.
  • A. Proof of Lemma 2: A. Proof of Lemma 2 establishes that a d∼χ2(M) is independent of β1 and β2, whose CDFs and a’s PDF are then used.The distributional setup supplies the ingredients for the subsequent joint-CDF calculation.
  • A. Proof of Lemma 2: A. Proof of Lemma 2 derives the joint CDF of X1 and X2 by conditioning on A=a and using conditional independence.The conditional probabilities are replaced by the beta-variable CDFs in the integral expression.
  • A. Proof of Lemma 2: A. Proof of Lemma 2 completes the derivation by shifting a and evaluating the remaining term, yielding (11) and (10).The proof identifies the first term as 1 before evaluating the final contribution.
  • B. Proof of Lemma 4: B. Proof of Lemma 4 evaluates E[˜Z] for ˜Z supported on (−∞,∞).The passage states that the expectation is obtained for this support.
  • C. Proof of Proposition 1: C. Proof of Proposition 1 uses statistical equivalence to upper-bound Rp1 and Rp2 through the corresponding private-rate terms.The proof states that R1(t) and R2(t) are statistically equivalent with Rp1 and Rp2, respectively.
  • C. Proof of Proposition 1: C. Proof of Proposition 1 models the zero-forcing residual projection as exponential with parameter 1, applies Jensen’s inequality, and uses Lemma 1 with ∥hk∥2 d∼χ2(M).These steps produce the private-rate upper-bound terms and the common-rate lower bound involving E[lnY].
  • C. Proof of Proposition 1: C. Proof of Proposition 1 combines the resulting bounds to obtain (16), while Proposition 3 follows similarly with Bkl as the final upper bound.The distinction for Proposition 3 is confined to the last term in (52d).

D. Proof of Corollary 4 and Derivation of (35)

The section derives the relevant stationary point by solving a quadratic under τ>0 (or Θ>4), then shows that r∗ minimizes ¯Brs and yields a closed form for trs,2.

  • Proof of Corollary 4 and Derivation of (35): The derivation also obtains a solution by solving the quadratic formula.This intermediate algebraic step precedes the stationary-point evaluation.
  • Proof of Corollary 4 and Derivation of (35): Under τ>0 (or Θ>4), solving quadratic formula (56) gives a stationary point.The derivation identifies one stationary point in closed form.
  • Proof of Corollary 4 and Derivation of (35): The stationary point r∗ in (57) minimizes ¯Brs.This establishes the minimizing choice used in the subsequent derivation.
  • Proof of Corollary 4 and Derivation of (35): Evaluating r∗ in (57) yields the closed form of trs,2.The closed-form expression follows after substituting the minimizing stationary point.

E. Proof of Proposition 5

The proof of Proposition 5 bounds the RS-ST private rates from above and the common rates from below. It applies the same derivation as Proposition 1, uses the feedback-bit relation and signal-to-interference-plus-noise ratio bounds, and treats the two receiver-specific common rates symmetrically before bounding the remaining common rate.

  • Rate bounds: The proof upper-bounds the two private-rate terms and lower-bounds the three common-rate terms in the RS-ST scheme.The targeted bounds are 1−R_u11(tβ,tα), R_p2−R_u21(tβ,tα), R_c1(tβ,tα), R_c2(tβ,tα), and R_c0(tβ,tα).
  • Rate bounds: The private-rate upper bound depends on B_11=B_β, while the common-rate lower bound uses Assumption 1 and the equality w_01=w_11.The derivation follows Proposition 1 and incorporates the corresponding SINR bound.
  • Common-rate bounds: R_c2(tβ,tα) is derived analogously to R_c1(tβ,tα), and R_c0(tβ,tα) is bounded using monotonicity of log2( (1+bx)/(1+ax) ) and the minimum of two independent exponential variables.The minimum has CDF F(x)=1−e^−2x when the two variables are independent and identically distributed.
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