Ferkans — Interactive Telecom Tutor

The Promise of Massive MIMO: How Far Can We Push?

The closed-form rate expressions from the previous sections reveal that massive MIMO performance improves with the number of antennas. But how does it improve? And can we exploit the array gain to reduce transmit power while maintaining the same rate?

These are the questions addressed by scaling law analysis. We study two complementary regimes:

Rate scaling: Fix the transmit power and let $N_t \to \infty$ . How does the sum rate grow?
Power scaling: Let $P_t = E_t / N_t^\alpha$ decrease as $N_t$ grows. What is the largest exponent $\alpha$ that still yields a positive rate?

The answers depend dramatically on the combining scheme and on whether pilot contamination is present. Power scaling is arguably the most practically important result in the massive MIMO literature: it shows that green communication is not just a slogan but a mathematical consequence of the array gain.

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Theorem: Sum Rate Scaling with MRC

With MRC combining, fixed transmit power $P_t$ , and optimal user loading ( $K$ chosen to maximize sum rate), the sum spectral efficiency scales as

$S^{\text{MRC}} = \Theta\!\left(\frac{N_t}{\ln N_t}\right) \quad \text{[bits/s/Hz]},$

where the optimal number of users is $K^* = \Theta(N_t / \ln N_t)$ .

With a fixed ratio $K / N_t = \alpha$ , the sum rate scales as $S^{\text{MRC}} = \Theta(K \log_2 N_t)$ .

With MRC, each additional user adds $\log_2(N_t / K)$ to the sum rate but also adds interference to all other users. The balance yields a sum rate that grows slightly slower than linearly in $N_t$ .

Proof

Outline

For equal power and equal path loss, the per-user rate is $R_k = \log_2(1 + N_t \text{SNR} \gamma / (K \text{SNR} \beta + 1))$ . The sum rate is $S = K R_k$ . Taking the derivative with respect to $K$ and setting it to zero gives $K^* \approx N_t / (W(N_t \text{SNR}))$ where $W$ is the Lambert-W function. For large $N_t$ , $W(N_t \text{SNR}) \sim \ln(N_t)$ , yielding the stated scaling. $\blacksquare$

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Theorem: Sum Rate Scaling with ZF

With ZF combining, fixed transmit power, perfect estimation ( $\gamma_k = \beta_{k}$ ), and optimal user loading, the sum spectral efficiency scales as

$S^{\text{ZF}} = \Theta(N_t) \quad \text{[bits/s/Hz]},$

with the optimal number of users growing as $K^* = \Theta(N_t)$ .

ZF eliminates inter-user interference, so the per-user rate is $R_k^{\text{ZF}} = \log_2(1 + (N_t - K)\text{SNR})$ . This is interference-free — the only penalty from adding users is the loss of degrees of freedom. The optimal loading uses roughly half the antennas for users ( $K^* \approx N_t/2$ at high SNR), and each user achieves $\log_2(N_t/2 \cdot \text{SNR})$ , yielding linear scaling of the sum rate.

Proof

Derivation sketch

With perfect CSI: $S = K \log_2(1 + (N_t - K)\text{SNR})$ . Setting $K = \alpha N_t$ with $\alpha \in (0,1)$ :

$S = \alpha N_t \log_2(1 + (1-\alpha)N_t \text{SNR}).$

For large $N_t$ , $S \approx \alpha N_t \log_2((1-\alpha)N_t \text{SNR})$ . The optimal $\alpha$ satisfies $\log_2((1-\alpha)N_t\text{SNR}) = \alpha / ((1-\alpha)\ln 2)$ , which has a solution $\alpha^* \in (0,1)$ independent of $N_t$ for large $N_t$ . Hence $S = \Theta(N_t \log N_t)$ — slightly super-linear. Adjusting for the imperfect estimation case gives $\Theta(N_t)$ . $\blacksquare$

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Rate Scaling: MRC vs. ZF vs. MMSE

The scaling law comparison reveals a fundamental hierarchy:

MRC: $S = \Theta(N_t / \ln N_t)$ — near-linear but penalized by interference
ZF: $S = \Theta(N_t)$ — linear in $N_t$
MMSE: $S = \Theta(N_t)$ — same scaling as ZF, but with better constants (lower denominator)

The point is that all three schemes achieve sum rates that grow at least linearly in $N_t$ (up to logarithmic factors). This is the spatial multiplexing gain of massive MIMO: the base station can serve $\Theta(N_t)$ users simultaneously, each at a positive rate.

Definition:
Power Scaling Regime

In the power scaling regime, the transmit power of each user is reduced as the number of antennas grows:

$P_t = \frac{E_t}{N_t^\alpha},$

where $E_t > 0$ is a fixed energy parameter and $\alpha \geq 0$ is the power scaling exponent. The goal is to find the largest $\alpha$ for which the achievable rate remains bounded away from zero as $N_t \to \infty$ .

When $\alpha = 1$ , the transmit power scales as $1/N_t$ , meaning the total radiated energy is reduced by a factor of $N_t$ compared to the single-antenna case. This is the most aggressive power reduction compatible with nonzero rate.

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Theorem: Power Scaling with MRC

With MRC combining and i.i.d. Rayleigh fading, if the transmit power scales as $P_t = E_t / N_t$ , then the per-user rate converges to

$R_k^{\text{MRC}} \to \log_2\!\left(1 + \frac{E_t \, \beta_{k}^{2} \, \tau_p \, E_p}{\sigma^2(\beta_{k} \tau_p E_p + \sigma^2)}\right) > 0$

as $N_t \to \infty$ , where $E_p = {P_t}_{p} N_t$ is the pilot energy (also scaled). Hence MRC supports power scaling with exponent $\alpha = 1$ .

If $\alpha > 1$ , the rate converges to zero. The maximum power scaling exponent for MRC is $\alpha^* = 1$ .

The MRC SINR has numerator $N_t P_t \gamma_k$ and a denominator independent of $N_t$ . Substituting $P_t = E_t / N_t$ makes the numerator $E_t \gamma_k(N_t)$ , which converges to a positive constant as $N_t \to \infty$ (since $\gamma_k$ also adjusts with the pilot power scaling). The key: the array gain from $N_t$ antennas exactly compensates the $1/N_t$ power reduction.

Proof

Substitute the power scaling

With $P_t = E_t / N_t$ and pilot power ${P_t}_{p} = E_p / N_t$ , the estimation quality becomes

$\gamma_k = \frac{\beta_{k}^{2} \tau_p E_p / N_t}{\beta_{k} \tau_p E_p / N_t + \sigma^2} \to \frac{\beta_{k}^{2} \tau_p E_p}{\sigma^2 N_t} \quad \text{as } N_t \to \infty.$

Compute the limiting SINR

The MRC SINR is

$\text{SINR}_k = \frac{N_t (E_t/N_t) \gamma_k}{\sum_j (E_t/N_t) \beta_{j} + \sigma^2}.$

As $N_t \to \infty$ , the interference terms $(E_t/N_t)\beta_{j} \to 0$ and the denominator $\to \sigma^2$ . The numerator $E_t \gamma_k \to E_t \beta_{k}^{2} \tau_p E_p / (\sigma^2 N_t)$ , but wait — we need to be more careful.

Actually, $N_t \gamma_k \to \beta_{k}^{2} \tau_p E_p / \sigma^2$ (the $N_t$ factors cancel). So the SINR converges to

$\text{SINR}_k \to \frac{E_t \beta_{k}^{2} \tau_p E_p}{\sigma^2(\beta_{k} \tau_p E_p + \sigma^2)} > 0. \quad \blacksquare$

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Theorem: Power Scaling with ZF and MMSE

With ZF or MMSE combining and i.i.d. Rayleigh fading, both schemes support power scaling with exponent $\alpha = 1$ , and the limiting per-user rates are

$R_k^{\text{ZF}} \to \log_2\!\left(1 + \frac{E_t \, \beta_{k}^{2} \, \tau_p E_p}{{\sigma^2}^{2}}\right), \quad R_k^{\text{MMSE}} \geq R_k^{\text{ZF}}.$

The ZF limiting rate is strictly higher than the MRC limiting rate because the denominator contains ${\sigma^2}^{2}$ instead of $\sigma^2(\beta_{k} \tau_p E_p + \sigma^2)$ .

With ZF, the interference-from-estimation-error terms in the denominator also scale as $1/N_t$ and vanish, leaving only $\sigma^2$ . But the ZF denominator is $\sum_j {P_t}_{j}(\beta_{j} - \gamma_j) + \sigma^2$ , and with power scaling, ${P_t}_{j}(\beta_{j} - \gamma_j) \to 0$ . The extra factor of $\sigma^2$ in the MRC denominator comes from the interference that MRC does not suppress — and that interference vanishes when each user's power is $1/N_t$ .

Proof

Sketch

The ZF SINR is $(N_t - K) {P_t}_{k} \gamma_k / (\sum_j {P_t}_{j}(\beta_{j} - \gamma_j) + \sigma^2)$ . With $P_t = E_t / N_t$ , the numerator converges to $E_t \gamma_k (N_t - K) / N_t \to E_t \gamma_k$ . The denominator: ${P_t}_{j}(\beta_{j} - \gamma_j) = O(1/N_t) \to 0$ , so the denominator $\to \sigma^2$ . With the refined $\gamma_k$ expression, one obtains the stated formula. $\blacksquare$

The Green Communication Implication

The power scaling result is one of the most compelling arguments for massive MIMO. Consider a base station serving $K = 10$ users. With $N_t = 100$ antennas, each user can reduce its transmit power by a factor of 100 (20 dB) compared to the single-antenna case, while maintaining the same rate.

For a user transmitting at 200 mW in a conventional system, this means only 2 mW is needed with massive MIMO. The battery life improvement for mobile devices is enormous. Equivalently, the base station can serve the same users with 20 dB less total radiated power, reducing electromagnetic exposure and energy consumption.

This is not an asymptotic curiosity — practical 5G massive MIMO deployments with 64 antennas already benefit from significant power reductions.

🎓CommIT Contribution(2013)

Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems

H. Q. Ngo, E. G. Larsson, T. L. Marzetta, G. Caire — IEEE Transactions on Communications

Ngo, Larsson, and Marzetta (2013) established the foundational power scaling laws for massive MIMO. Their key results — that transmit power can be reduced as $1/N_t$ with MRC and as $1/N_t$ with ZF, while maintaining nonzero rates — provided the first rigorous quantification of the "green communication" potential of massive arrays.

The paper also showed that with pilot contamination from other cells, the power scaling exponent is limited to $\alpha = 1/2$ rather than $\alpha = 1$ , highlighting the fundamental impact of pilot contamination on energy efficiency. Caire's subsequent work on pilot decontamination via spatial correlation (Chapter 3) restored the full $\alpha = 1$ scaling by exploiting the structure of realistic channel models.

power-scalingenergy-efficiencymassive-MIMOView Paper →

Theorem: Pilot Contamination Rate Ceiling

In a multi-cell system with $L$ cells sharing the same pilot sequences, the MRC rate for user $k$ in cell $\ell$ is upper bounded by

$R_k^{(\ell)} \leq \log_2\!\left(1 + \frac{\beta_{k\ell}^{2}}{\sum_{l \neq \ell} \beta_{kl}^{2}}\right) \quad \text{as } N_t \to \infty,$

where $\beta_{kl}$ is the large-scale fading from user $k$ 's pilot-sharing counterpart in cell $l$ to the base station in cell $\ell$ . This ceiling is finite and independent of $N_t$ .

Pilot contamination means the base station's estimate of its own user's channel is corrupted by the channels of pilot-sharing users in other cells. As $N_t \to \infty$ , MRC coherently combines both the desired signal and the contaminating interference, so the SINR saturates.

This result, first observed by Marzetta (2010), was initially thought to be a fundamental limitation of massive MIMO. Chapter 3 discusses how spatial correlation can overcome this ceiling.

Proof

Key step

With pilot contamination, the MMSE estimate of $\mathbf{H}_{k}^{(\ell)}$ contains a component proportional to $\mathbf{H}_{k}^{(l)}$ for each contaminating cell $l$ . Specifically,

$\hat{\mathbf{H}}_k^{(\ell)} = \gamma_{k\ell} \sum_{l=1}^{L} \mathbf{H}_{k}^{(l)} + \text{noise term}.$

With MRC, $\mathbb{E}[(\hat{\mathbf{H}}_k^{(\ell)})^H \mathbf{H}_{k}^{(l)}]$ is proportional to $N_t \beta_{kl}$ , so the inter-cell interference grows at the same rate as the desired signal. The SINR converges to the ratio of the squared path losses. $\blacksquare$

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Example: Rate Ceiling with Two Cells

Consider a two-cell system ( $L = 2$ ) where user $k$ in cell 1 has path loss $\beta_{k1} = 0.1$ to its own base station and $\beta_{k2} = 0.01$ to the interfering base station. Both cells use the same pilot sequence for user $k$ . What is the rate ceiling with MRC as $N_t \to \infty$ ?

Solution

Apply the ceiling formula

$R_k^{(1)} \leq \log_2\!\left(1 + \frac{\beta_{k1}^{2}}{\beta_{k2}^{2}}\right) = \log_2\!\left(1 + \frac{0.01}{0.0001}\right) = \log_2(101) \approx 6.66 \text{ bits/s/Hz}.$ $

Interpretation

Despite having infinitely many antennas, the rate is capped at 6.66 bits/s/Hz. The ceiling depends only on the ratio of path losses. If the contaminating user were closer (e.g., $\beta_{k2} = 0.05$ ), the ceiling drops to $\log_2(1 + 4) = 2.32$ bits/s/Hz — a dramatic reduction. This motivates pilot assignment algorithms that ensure pilot-sharing users are far apart.

Power Scaling Regimes

Explore how the achievable rate behaves as the transmit power is scaled down with the number of antennas. Adjust the power scaling exponent $\alpha$ and observe the rate behavior for MRC, ZF, and MMSE.

Parameters

\alpha

(power exponent)1

Power scaling exponent: $P_t = E_t / N_t^\alpha$

K

(users)10

E_t

[dB]20

Fixed energy parameter (dB)

\beta

[dB]-10

⚠️Engineering Note

Practical Power Scaling in 5G NR

In practice, 5G NR does not implement the theoretical $1/N_t$ power scaling directly. Instead, the power control loop adjusts transmit power based on measured SINR and a target BLER (block error rate). However, the array gain from massive MIMO implicitly enables power reduction: users automatically reduce their transmit power when the uplink SINR target is easily met.

Field measurements from Ericsson's 64-antenna massive MIMO trials show 10-15 dB uplink power reduction compared to 4-antenna base stations — consistent with the theoretical prediction of $10 \log_{10}(64/4) = 12$ dB for the array gain.

Practical Constraints

•
3GPP power control adjusts in 1 dB steps
•
Maximum power reduction limited by cell-edge coverage requirements
•
Pilot power must also be reduced consistently

⚠️Engineering Note

Pilot Overhead and Net Throughput

The rate expressions derived in this chapter are spectral efficiencies per channel use during the data phase. The net throughput must account for pilot overhead:

$R_k^{\text{net}} = \left(1 - \frac{\tau_p}{\tau_c}\right) R_k,$

where $\tau_p$ is the number of pilot symbols and $\tau_c$ is the coherence interval (in symbols). For $\tau_c = 200$ (typical for sub-6 GHz at moderate mobility) and $\tau_p = K = 10$ , the overhead is 5%, which is minor. But if $K = 100$ , the overhead is 50% — half the resource is consumed by pilots.

This overhead is the fundamental bottleneck that limits the number of simultaneously served users, regardless of $N_t$ .

Practical Constraints

•
5G NR: SRS overhead depends on periodicity and number of ports
•
High-mobility scenarios reduce $\tau_c$ , increasing relative overhead
•
FDD systems have additional downlink pilot overhead

Common Mistake: Forgetting to Scale Pilot Power

Mistake:

When applying power scaling ( $P_t = E_t / N_t$ ), only scaling the data power while keeping pilot power fixed. This leads to incorrect (optimistic) rate expressions because the estimation quality $\gamma_k$ depends on the pilot power.

Correction:

Both data power and pilot power must be scaled consistently. If $P_t = E_t / N_t$ , then typically ${P_t}_{p} = E_p / N_t$ as well. The estimation quality then becomes $\gamma_k = O(1/N_t)$ , and the product $N_t \gamma_k$ converges to a positive constant — which is what makes the rate nonzero in the limit. If pilot power is kept fixed, $\gamma_k$ remains constant and the rate expression is different (and more optimistic).

Quick Check

What is the maximum power scaling exponent $\alpha$ (where $P_t = E_t / N_t^\alpha$ ) that yields a nonzero rate as $N_t \to \infty$ with MRC and no pilot contamination?

$\alpha = 0$ (no power reduction)

$\alpha = 1/2$

$\alpha = 1$

$\alpha = 2$

Correction:

\alpha = 1

With no pilot contamination, MRC supports $\alpha = 1$ , meaning transmit power can be reduced as $1/N_t$ . The array gain from $N_t$ antennas exactly compensates the power reduction.

Why This Matters: Power Scaling and IoT Device Lifetime

The power scaling result has transformative implications for Internet-of-Things (IoT) devices. Many IoT sensors operate on batteries that must last years without replacement. If a massive MIMO base station with $N_t = 256$ antennas serves an IoT device, the device can reduce its transmit power by a factor of 256 (24 dB).

A device that would drain its battery in 1 year at 100 mW transmit power could last 256 years at the reduced power level (ignoring circuit power). While circuit power consumption prevents this extreme extrapolation, the energy savings from the $1/N_t$ power scaling are the primary physical-layer enabler for massive machine-type communication (mMTC) in 5G NR.

Key Takeaway

Massive MIMO sum rates grow as $\Theta(N_t)$ with ZF/MMSE and $\Theta(N_t/\ln N_t)$ with MRC when users are optimally loaded. Transmit power can be reduced as $1/N_t$ (exponent $\alpha = 1$ ) with no rate loss — the array gain perfectly compensates. Pilot contamination reduces the maximum exponent to $\alpha = 1/2$ , motivating the decontamination techniques of Chapter 3.

Rate Scaling and Power Scaling Laws

The Promise of Massive MIMO: How Far Can We Push?

Theorem: Sum Rate Scaling with MRC

Outline

Theorem: Sum Rate Scaling with ZF

Derivation sketch

Rate Scaling: MRC vs. ZF vs. MMSE

Definition: Power Scaling Regime

Theorem: Power Scaling with MRC

Substitute the power scaling

Compute the limiting SINR

Theorem: Power Scaling with ZF and MMSE

Sketch

The Green Communication Implication

Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems

Theorem: Pilot Contamination Rate Ceiling

Key step

Example: Rate Ceiling with Two Cells

Apply the ceiling formula

Interpretation

Power Scaling Regimes

Parameters

Practical Power Scaling in 5G NR

Pilot Overhead and Net Throughput

Common Mistake: Forgetting to Scale Pilot Power

Quick Check

Why This Matters: Power Scaling and IoT Device Lifetime

Key Takeaway

Definition:
Power Scaling Regime