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Uplink Performance Analysis of Multicell MU-MIMO Systems with ZF Receivers
Hien Quoc Ngo, Michail Matthaiou, Trung Q. Duong, Erik G. Larsson
TL;DR
Earlier ZF studies considered only single-cell scenarios, leaving intercell interference neglected in multicell uplinks. The paper derives exact closed-form uplink rate, SER, and outage expressions for a multicell MU-MIMO system using ZF detection. The results enable each user's transmitted power to be reduced proportionally to 1/N.
Problem
Earlier ZF studies considered only single-cell scenarios, leaving intercell interference neglected in multicell uplinks.
Source: §I · p. 2Method
The paper derives exact closed-form uplink rate, SER, and outage expressions for a multicell MU-MIMO system using ZF detection.
Source: §VI · p. 18Results
The results enable each user's transmitted power to be reduced proportionally to 1/N.
Source: §VI · p. 18Takeaways & Limitations
Very large BS arrays allow reduced per-user transmit power while preserving system performance.
Source: §VI · p. 18
Abstract
from arXiv · showhide
We consider the uplink of a multicell multiuser multiple-input multiple-output system where the channel experiences both small and large-scale fading. The data detection is done by using the linear zero-forcing technique, assuming the base station (BS) has perfect channel state information. We derive new, exact closed-form expressions for the uplink rate, symbol error rate, and outage probability per user, as well as a lower bound on the achievable rate. This bound is very tight and becomes exact in the large-number-of-antennas limit. We further study the asymptotic system performance in the regimes of high signal-to-noise ratio (SNR), large number of antennas, and large number of users per cell. We show that at high SNRs, the system is interference-limited and hence, we cannot improve the system performance by increasing the transmit power of each user. Instead, by increasing the number of BS antennas, the effects of interference and noise can be reduced, thereby improving the system performance. We demonstrate that, with very large antenna arrays at the BS, the transmit power of each user can be made inversely proportional to the number of BS antennas while maintaining a desired quality-of-service. Numerical results are presented to verify our analysis.
I. INTRODUCTION
The paper analyzes multicell MU-MIMO uplinks with ZF, where intercell interference was missing from prior single-cell studies. It derives finite-antenna performance expressions and studies how large arrays affect interference and user power.
I. INTRODUCTION
- Earlier ZF work considered single-cell scenarios and neglected intercell interference, an important impairment in frequency-reuse cellular systems.The cited prior work is described as single-cell; no bibliography marker is attached to that claim in these passages.Source: §I · p. 2
- The analysis focuses on ZF because exact multicell MMSE analysis is challenging and MMSE implementation needs noise and interference statistics.The paper also states that ZF performance provides a lower bound on MMSE performance.Source: §I · p. 3
- The paper derives exact finite-antenna closed forms for uplink rate, SER, and outage probability, plus a tractable lower bound on achievable rate.These results concern the multicell uplink with ZF detection.Source: §I · p. 3
- Large-array asymptotics are used to study transmit power, intercell interference, and antenna count, with distinct interference behavior across user-loading regimes.When users per cell remain fixed, interference and noise are averaged out; when the antenna-to-user ratio is fixed, intercell interference persists.Source: §I · p. 3
- With a fixed number of users per cell, intercell interference and noise are averaged out as the number of BS antennas grows; at a fixed antenna-to-user ratio, intercell interference does not vanish.The power-scaling result holds in both cases.Source: §I · pp. 3–4
- With very large arrays, each user’s transmit power can scale as 1/N with no performance degradation.The paper also distinguishes fixed-user and fixed antenna-to-user-ratio regimes, in which intercell interference behaves differently.Source: §I · p. 4
II. MULTICELL MU-MIMO SYSTEM
The section models uplink transmission in a multicell MU-MIMO system with shared frequency resources and ZF detection. It gives a per-user SINR representation in terms of independent random variables.
II. MULTICELL MU-MIMO SYSTEM
- The system has L cells, each with a BS using N antennas to serve K single-antenna users, with N ≥ K.Users transmit simultaneously in the shared frequency band.Source: §II · p. 4
- Channel matrices model fast fading, path loss, and log-normal shadow fading, while the BS is assumed to have perfect CSI.The model assumes independent fast fading across sufficiently separated antennas.Source: §II · pp. 4–5
- ZF processing is applied to the received vector, and Proposition 1 represents each user's SINR using independent random variables with specified PDFs.The supplied passages do not include the full SINR or PDF expressions.Source: §II · pp. 5–6
III. FINITE-N ANALYSIS
The finite-N analysis derives exact expressions for uplink rate, SER, and outage probability for any N ≥ K. For rate, it also supplies an analytical lower bound and shows that increasing user power cannot overcome the high-SNR interference limit.
III. FINITE-N ANALYSIS
- Exact analytical expressions for ergodic uplink rate, SER, and outage probability apply for any N ≥ K.The section's general finite-N results cover arbitrary numbers of BS antennas satisfying this condition.Source: §III · p. 7
A. Uplink Rate Analysis
- Theorem 1 gives the uplink ergodic rate, and Proposition 2 provides an analytical lower bound that is easier to evaluate.The bound is presented in addition to the exact rate result.Source: A. Uplink Rate Analysis · pp. 7–8
- At high SNR, the SINR is bounded, so increasing each user's transmit power alone cannot improve system performance.Desired-signal and interference powers both increase with transmit power.Source: A. Uplink Rate Analysis · pp. 8–9
B. SER Analysis
The SER analysis uses the SINR moment generating function to derive an exact average SER expression for M-PSK. It also examines high-SNR behavior, where interference makes SER converge to a constant.
B. SER Analysis
- Theorem 2 gives the exact average SER for uplink M-ary phase-shift keying using the SINR moment generating function.The MGF-based approach is used to evaluate SER.Source: B. SER Analysis · p. 9
- At high SNR, SER converges to a constant independent of log(pu), reflecting the presence of interference.The passage identifies the high-SNR diversity gain as zero.Source: B. SER Analysis · p. 9
- A tight approximation avoids the finite integration over θ required by the SER expression and is easier to evaluate.The approximation is applied to the expression in (19).Source: B. SER Analysis · p. 10
C. Outage Probability Analysis
The outage analysis provides an exact probability expression and a small-threshold asymptotic result. The large-N discussion attributes interference reduction to channel-vector orthogonality and assumes sufficiently separated antennas to maintain channel independence.
C. Outage Probability Analysis
- Theorem 3 gives an exact outage probability for a user's transmission, defined by whether instantaneous SINR falls below a threshold.Outage is particularly appropriate for non-ergodic channels such as quasi-static or block-fading channels.Source: C. Outage Probability Analysis · pp. 10–11
- As the threshold γth approaches zero, the outage result is independent of SNR, reflecting interference's deleterious impact.The asymptotic result is obtained by retaining the dominant term and letting γth approach zero.Source: C. Outage Probability Analysis · p. 11
IV. ASYMPTOTIC (N →∞) ANALYSIS
- For large N, channel-vector orthogonality can significantly reduce interference even with ZF, under an assumption of independent channel elements.The analysis requires antennas to be sufficiently well separated to guarantee channel independence.Source: §IV · p. 12
1) Fixed pu, K, and N
The asymptotic behavior depends on how the user count scales with the BS antenna count. With fixed users, interference and noise vanish as antennas grow; with a finite antenna-to-user ratio, intercell interference remains.
1) Fixed pu, K, and N
- As N grows with K fixed, interference and noise disappear and the uplink SINR grows without limit.The random vectors and BS noise become deterministic in this regime.Source: 1) Fixed pu, K, and N · p. 12
- When N and K grow at a finite ratio, user-channel vectors are not pairwise orthogonal and intercell interference cannot be cancelled.The Gram matrix does not converge pointwise to an identity matrix in this regime.Source: 1) Fixed pu, K, and N · pp. 12–13
- At finite κ, the limiting SIR is independent of transmit power and increases with κ, tending to infinity as κ grows.Thus, reducing every user's power by the same factor leaves the limiting SIR unchanged.Source: 1) Fixed pu, K, and N · p. 13
3) Fixed Npu, N
The section studies large-array asymptotics and numerical behavior under fixed and practical channel settings. Large arrays support power reduction, while the numerical results show that antenna count and intercell interference strongly shape rates and error performance.
3) Fixed Npu, N
- With perfect CSI, each user's transmit power can fall proportionally to 1/N while maintaining a desired quality of service.The result is generalized to N≫1; the proposed rate lower bound also becomes exact in the large-antenna regime.Source: 3) Fixed Npu, N · pp. 14–15
- At fixed finite κ, intercell interference remains as N grows; increasing κ reduces its relative importance toward the massive-MIMO regime.The required κ rises with the ultimate rate and with the interference factor.Source: 3) Fixed Npu, N · p. 14, A. Scenario I · p. 16
A. Scenario I
- For N⪆20, the rate lower bound is almost exact across the entire SNR range.Its tightness depends on antenna count rather than interference level.Source: A. Scenario I · p. 15
- At cross gain 0.1, sum rates are 3.76, 38.35, and 73.20 for N=10, 50, and 500; at 0.5, they are 0.93, 19.10, and 50.80.Raising cross gain from 0.1 to 0.5 reduces the rates by 75.27%, 50.20%, and 30.60%, respectively.Source: A. Scenario I · p. 16
- With pu=10/N, sum rate converges to a constant as N increases, whereas with pu=10 it grows logarithmically without bound.These trends are reported irrespective of interference for the power-scaled case.Source: A. Scenario I · p. 16
- At moderately high SNR, SER approaches an interference-driven error floor, but increasing BS antennas improves performance.The analytical approximations match simulations and remain accurate across the reported cases.Source: A. Scenario I · p. 16
B. Scenario II
In the practical fading scenario, BS antenna count strongly affects users' net-rate distribution. Frequency reuse has a rate-dependent tradeoff: lower reuse is preferable at high rates, while higher reuse is preferable at low rates.
B. Scenario II
- The evaluated simulations use 20 MHz bandwidth, place K = 10 users uniformly at random in each cell, enforce a 100-meter minimum user-to-BS distance, and model fading with path-loss exponent 3.8 and 8 dB log-normal shadowing.Source: B. Scenario II · p. 17
- Larger N substantially lowers the probability that a user's net uplink rate falls below a given rate.The random channel also becomes more concentrated around its mean as N grows.Source: B. Scenario II · p. 17
- Increasing N from 20 to 100 with reuse factor 1 raises the 95%-likely net rate eightfold, from 0.170 Mbits/sec to 1.375 Mbits/sec.The 95%-likely rate is the rate met or exceeded with probability 0.95.Source: B. Scenario II · p. 17
- At high rates smaller frequency-reuse factors are preferable, while at low rates larger factors are preferable.The performance gap between reuse factors widens with N as intercell interference is reduced and bandwidth matters more.Source: B. Scenario II · pp. 17–18
VI. CONCLUSION
The paper derives exact per-user uplink performance expressions for multicell MU-MIMO with ZF detection. Its analysis and results identify large antenna arrays as a route to improved performance and lower user transmit power under perfect CSI.
VI. CONCLUSION
- The paper derives exact closed-form expressions for uplink rate, SER, and outage probability under Rayleigh fading and shadowing.The analysis considers single-antenna users, multicell interference, an N-antenna BS, and ZF detection.Source: §VI · p. 18
- As N grows, pairwise asymptotic orthogonality of channel vectors allows interference cancellation, improving multicell performance.Array and diversity gains affect both desired and interference signals.Source: §VI · p. 18
- With perfect CSI, large BS arrays permit each user's transmit power to decrease proportionally to 1/N without performance degradation.The conclusion also discusses the massive-MIMO effect and frequency-reuse factors.Source: §VI · p. 18
APPENDIX
The appendix proves Theorem 1 and Proposition 2 by reducing the rate derivation to an integral identity and evaluating the expectations in the rate expression.
A. Proof of Theorem 1
- Lemma 1 evaluates an exponential-integral expression by changing variables, integrating by parts, and using recursion.The result is then used with [33, Eq. (39)] to obtain (14).Source: A. Proof of Theorem 1 · pp. 18–19
B. Proof of Proposition 2
- Jensen’s inequality rewrites the achievable ergodic rate as a lower-bound expression involving expectations of ln(Xk) and ln(puZl + 1).The derivation uses the convexity of log2(1 + α exp(x)) for α > 0.Source: B. Proof of Proposition 2 · pp. 19–20
- The proof evaluates the rate expression’s integrals using Meijer’s-G representations and identities, then combines the results to obtain the proposition.The integrand ln(1+αz) is expressed as a Meijer’s-G function before simplification.Source: B. Proof of Proposition 2 · p. 20
C. Proof of Theorem 2
The appendix derives Theorem 2 using the distribution of Xk and integral identities, then proves Theorem 3 by expressing the outage probability through its CDF.
C. Proof of Theorem 2
- Theorem 2’s proof starts from the PDF of Xk, applies a binomial expansion, and rewrites an integral using a Meijer’s-G function.The resulting integral manipulations lead to the desired result (20).Source: C. Proof of Theorem 2 · p. 21
- Using the PDF of Xk given by (7), the derivation obtains the last equality via [30, Eq. (3.326.2)] and then substitutes (59) into (58).Source: C. Proof of Theorem 2 · p. 21
D. Proof of Theorem 3
- Theorem 3’s proof expresses the outage probability as an integral, derives the CDF of Xk, and combines the expressions to obtain (28).The derivation uses a binomial expansion and an integral identity.Source: D. Proof of Theorem 3 · pp. 21–22