Demystifying the Power Scaling Law of Intelligent Reflecting Surfaces and Metasurfaces

Emil Bj\"ornson; Luca Sanguinetti

Demystifying the Power Scaling Law of Intelligent Reflecting Surfaces and Metasurfaces

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A same-location comparison shows that an N-element intelligent reflecting surface can never exceed the SNR of an N-antenna Massive MIMO receiver, because the quadratic scaling contains an energy-conservation factor that is at most one.

desk verdict A clean, short proof that under co-located free-space assumptions mMIMO always beats IRS in SNR; the N^2 scaling is real but the second N is just pathloss from the surface to the receiver. read the letter →

arxiv 1908.03133 v2 pith:NJRXW35A submitted 2019-08-08 eess.SP cs.ITmath.IT

Emil Björnson , Luca Sanguinetti This is my paper · ORCID

classification eess.SPcs.ITmath.IT

keywords intelligentreflectingsurfacemassiveMIMOpowerscalinglawsignal-to-noiseratioenergyconservationwirelesspropagationreconfigurablepassivebeamforming

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper challenges the widespread interpretation that intelligent reflecting surfaces (IRSs) have a better power scaling law than Massive MIMO because their SNR grows as $N^2$ rather than as $N$. It claims that, when an IRS and an $N$-antenna Massive MIMO receiver occupy the same location and are compared under free-space line-of-sight propagation, the quadratic scaling is the artifact of an incomplete comparison. The paper proves that Massive MIMO always achieves at least as high an SNR, and its numerics show that an IRS needs thousands of reflecting elements to match a 64-antenna array. The practical message is that the advertised $N^2$ scaling does not translate into a power advantage for IRSs.

What carries the argument

The load-bearing identity is the factorization in Eq. (18): $\mathrm{SNR}_{\mathrm{IRS}} = (\mu^2 N \beta_g)\, (N \beta_h P_{\mathrm{tx}}/\sigma^2)$, where the second factor is exactly $\mathrm{SNR}_{\mathrm{mMIMO}}$. The apparent quadratic gain is isolated in the prefactor $\mu^2 N \beta_g$: $\beta_g$ is the free-space channel gain between the IRS and the receiver, $N\beta_g$ is the total channel gain of that hop, and energy conservation bounds it above by one, while $\mu^2$ is the fraction of incident power the surface reflects. Proposition 1 follows because that prefactor cannot exceed unity; equivalently, the IRS behaves as an $N$-element receiver using a fixed combiner $v = \Theta g$ instead of maximum-ratio combining.

What would settle it

Under the paper's assumptions, compute or measure the SNR of an $N$-element IRS and an $N$-antenna mMIMO receiver with optimal phases and perfect channel knowledge. Any instance with $\mathrm{SNR}_{\mathrm{IRS}} > \mathrm{SNR}_{\mathrm{mMIMO}}$ refutes Proposition 1; the analytic check is whether the prefactor $\mu^2 N \beta_g$ can exceed one while respecting energy conservation.

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Extended reading notes

Core claim

The central discovery is Proposition 1: under optimal phase-shift configuration and perfect channel knowledge, the SNR of an IRS-aided link factors as $\mathrm{SNR}_{\mathrm{IRS}} = (\mu^2 N \beta_g)\, \mathrm{SNR}_{\mathrm{mMIMO}}$, where $\mu^2 \le 1$ is the reflection efficiency, $\beta_g$ is the channel gain from the IRS to the single-antenna receiver, and $N \beta_g \le 1$ by energy conservation. Since the prefactor is at most one, $\mathrm{SNR}_{\mathrm{mMIMO}} \ge \mathrm{SNR}_{\mathrm{IRS}}$ for every $N$. The apparent $N^2$ growth is really one factor of $N$ from the same aperture gain that Massive MIMO already enjoys, multiplied by the bounded fraction of reflected power that reaches the receiver.

Load-bearing premise

The result assumes the IRS and the Massive MIMO array are compared at the same location, with the receiver physically separated from the IRS, under free-space line-of-sight propagation and isotropic antennas; outside that deployment the ranking is not proved.

Editorial extensions

If this is right

An IRS-aided link cannot achieve a higher SNR or information rate than an $N$-antenna Massive MIMO receiver in the modeled same-location, line-of-sight setup, regardless of $N$.
The advertised $N^2$ scaling of IRS SNR should not be read as a power advantage; the extra factor is the bounded fraction of reflected power that reaches the receiver.
Numerically, matching a 64-antenna mMIMO array requires more than $10^4$ IRS elements when the receiver is 25 m away and roughly $3\times10^3$ elements when it is 2.5 m away.
For very large $N$, the far-field linear model must give way to the planar-array formula, whose received power saturates at half the transmit power; the IRS comparison inherits that saturation.
Placing the receiver directly behind the surface (holographic beamforming) is the configuration that can theoretically match mMIMO performance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same factorization argument suggests that any passive, power-conserving reradiating structure pays a comparable two-hop loss, so quadratic scaling is not a general route to beating a same-aperture active array.
The paper's comparison fixes the IRS and the mMIMO array at one site; an undeveloped regime is the common deployment where the surface sits near the user while the base station cannot host a large active array there, in which case the relevant baseline is a remote mMIMO array with different pathloss.
A testable extension would allow directive elements or multipath propagation; under those conditions the bound $N \beta_g \le 1$ may loosen or tighten, so the inequality should be re-checked before being used as a design rule.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Emil and Luca have a short conference paper that should be read as a corrective to the IRS hype. The new thing is the factorization in Eq. (18): SNR_IRS = (mu^2 N beta_g) * SNR_mMIMO, with the first factor provably <= 1 by energy conservation. Earlier papers touted the N^2 scaling without noticing that one of the N's is essentially the fraction of power that escapes between the IRS and a physically separated receiver. That is a genuine and clarifying observation, not just a restatement. The proof is transparent: from the channel models and maximum-ratio combining you get SNR_mMIMO = N beta_h Ptx/sigma^2; with optimal phase shifts the IRS gives mu^2 N^2 beta_g beta_h Ptx/sigma^2; and the energy-conservation bound N beta_g <= 1 closes the inequality. The paper is honest about the model: co-located arrays, free-space, isotropic elements, far-field for g. It shows numerically that even with the receiver 2.5 m from the IRS, thousands of elements are needed to match a 64-antenna mMIMO array.

Soft spots, in proportion. The result is exactly as strong as the stated model. The paper says 'always' in the abstract, but that 'always' is conditional on the IRS and mMIMO array being at the same location and the receiver being physically separated from the IRS. That is not the deployment that motivates most IRS work, where the surface is hung close to the user to shorten the second hop. The authors acknowledge this and partially address it with the d_g = 2.5 m case, but they do not analyze the case where the receiver is effectively attached to the surface (which they mention as 'holographic beamforming' in the conclusion). Also, the comparison is single-user; in a multi-user setting mMIMO's spatial multiplexing only widens the gap, so the conclusion would not flip, but the paper does not make that argument. The far-field model for g limits N below ~1/beta_g, and they note it; the exact planar model (4) preserves the inequality via alpha_g <= 1/2, so the stress-test checks out. The citation pattern is fine: [13] and [18] are their own prior work but they are the right sources for the IRS signal model and the relay intuition.

Net: this is a small but useful paper. It deserves a serious referee; it is the kind of thing that should be in the record so that the 'N^2 beats mMIMO' misreading stops being repeated. I would cite it, and I would bring it to a reading group if anyone in the group works on IRSs. Recommend accept, with a request that the abstract's 'always' be scoped to the assumptions.

Referee Report

0 major / 5 minor

Summary. The paper addresses the claim that intelligent reflecting surfaces (IRSs) enjoy a more favorable power scaling law than Massive MIMO because their SNR scales as N^2 instead of N. Under a single-user line-of-sight free-space model, the authors compare an N-antenna mMIMO receiver with an N-element IRS placed at the same location, with the IRS-aided receiver physically separated from the surface. They derive the SNR expressions, factor the IRS SNR as SNR_IRS = mu^2 N beta_g * SNR_mMIMO, and use the energy-conservation bound mu^2 N beta_g <= 1 to prove Proposition 1: SNR_mMIMO >= SNR_IRS. Numerical results quantify the gap, showing that thousands of IRS elements are needed to match a 64-antenna mMIMO array, and the authors provide an exact planar-array correction for the IRS-to-receiver channel when the far-field approximation breaks down.

Significance. The paper makes a clean, self-contained analytical point that corrects an overinterpretation in the IRS literature: observing an N^2 scaling law does not by itself imply that an IRS is more power-efficient than mMIMO, because the complete SNR expressions include a channel-gain factor that is bounded by energy conservation. The proof is transparent, the assumptions are stated, and the numerical comparison gives concrete, falsifiable orders of magnitude. The paper is honest about its scope, restricting the formal result to the co-located, free-space, LOS setup of Figure 3. This is a useful contribution that should help discipline future claims about IRS power scaling.

minor comments (5)

[Abstract and Proposition 1] The abstract and Proposition 1 state that mMIMO always provides higher SNRs, but the comparison is carried out for the specific co-located deployment in Figure 3, where the IRS and the mMIMO array occupy the same location and the receiver is separated from the IRS; please add this qualification to the abstract and to the statement of Proposition 1 so that the claim is not read as applying to other IRS deployment geometries.
[§3.2, Eq. (18)] The proof of Proposition 1 should state explicitly that the factorization in Eq. (18) and the bound mu^2 N beta_g <= 1 are used in the far-field regime; for larger N the same conclusion follows by replacing N beta_g with the exact planar-array gain alpha_g <= 1/2 used in Eq. (21), but this replacement is not mentioned in the proof.
[§3.3, Eq. (20)] The sentence after Eq. (20) would be clearer if it distinguished between the norm loss ||v||^2 and the suboptimality of the combining vector v; the IRS SNR equals ||v||^2 times the SNR of an mMIMO receiver using the same v, and it is further below MRC because v = Theta g is not the maximum-ratio combiner.
[Fig. 4] In Figure 4, please distinguish the approximate IRS curve based on Eq. (17) from the exact curve based on Eq. (21) in the legend, and mark the truncation point N = 1/beta_g where the approximate curve is stopped.
[References and §2.2] Reference [19] has a duplicated year in the bibliographic entry, and the rule-of-thumb condition NA/10 < d^2 in §2.2 would benefit from a short justification or a pointer to the error level it corresponds to.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified; Proposition 1 is proven algebraically from the stated channel model and energy conservation.

full rationale

The derivation is self-contained. The paper defines the mMIMO SNR in Eq. (13) as N beta_h P_tx / sigma^2 and the IRS SNR in Eq. (17) as mu^2 N^2 beta_g beta_h P_tx / sigma^2, both obtained directly from the stated free-space line-of-sight channel model (9) and the IRS phase-shift model (14)-(15). Eq. (18) factors the IRS SNR as (mu^2 N beta_g) times the mMIMO SNR, and the bound mu^2 N beta_g <= 1 follows from mu <= 1 and the energy-conservation bound N beta_g <= 1 derived in Section 2 (Eq. (3)). Proposition 1 is therefore a direct algebraic consequence of the model's assumptions rather than an imported conclusion. No parameter is fitted to data and then 'predicted' as a separate result; the numerical section only instantiates physical parameters such as distance, frequency, transmit power, and noise power. The authors' prior work is cited only for the IRS received-signal model [18] and for the relay-like interpretation [13]; neither citation defines or contains Proposition 1. The paper explicitly restricts the comparison to co-located arrays and the free-space model, which is a scope statement rather than a circular step. When the approximate N beta_g model breaks down, the paper switches to the exact planar-array gain alpha_g from Eq. (4)-(5), with alpha_g <= 1/2, which preserves the inequality without assuming the conclusion. Thus no self-definitional, fitted-input, self-citation load-bearing, or uniqueness-imported circularity is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; numerical examples use fixed scenario values. The paper relies on standard free-space propagation and energy conservation, plus an IRS model from earlier work.

assumptions (6)

domain assumption The total channel gain N beta cannot exceed 1 due to conservation of energy (Eq. (3)).
Used in Section 2.1 and Section 3.2 to bound N beta_g in (18), ensuring SNR_IRS <= SNR_mMIMO.
domain assumption Free-space pathloss model beta = A/(4 pi d^2) for isotropic antennas (Eq. (10)).
Used to compute channel gains for h and g in Fig. 3.
domain assumption IRS received signal model y = g^T Theta h sqrt(Ptx) s + n from [18].
Adopted from prior work; not derived here.
domain assumption The planar array gain formula (5) from [17] for large arrays.
Used for exact numerical comparison in Eq. (21)-(22).
domain assumption Perfect channel knowledge at both mMIMO receiver and IRS phase-shift controller.
Assumed throughout Section 3.
standard math Maximum ratio combining v = h*/||h|| maximizes the SNR in (12).
Standard result from [3], used to derive (13).

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Cite this review

Pith. "Pith review of Demystifying the Power Scaling Law of Intelligent Reflecting Surfaces and Metasurfaces." pith.science (2026). https://pith.science/paper/NJRXW35A

@misc{pith2026190803133,
  author       = {Pith},
  title        = {Pith review of: Demystifying the Power Scaling Law of Intelligent Reflecting Surfaces and Metasurfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NJRXW35A}},
  note         = {Machine review of arXiv:1908.03133}
}

abstract

Intelligent reflecting surfaces (IRSs) have recently attracted the attention of communication theorists as a means to control the wireless propagation channel. It has been shown that the signal-to-noise ratio (SNR) of a single-user IRS-aided transmission increases as $N^2$, with $N$ being the number of passive reflecting elements in the IRS. This has been interpreted as a major potential advantage of using IRSs, instead of conventional Massive MIMO (mMIMO) whose SNR scales only linearly in $N$. This paper shows that this interpretation is incorrect. We first prove analytically that mMIMO always provides higher SNRs, and then show numerically that the gap is substantial; a very large number of reflecting elements is needed for an IRS to obtain SNRs comparable to mMIMO.

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Reference graph

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [13]

Smart radio environments empowered by re- conﬁgurable AI meta-surfaces: an idea whose time has come,

M. D. Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.- S. Alouini, C. Yuen, V . Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceu, G. Lerosey, and M. Fink, “Smart radio environments empowered by re- conﬁgurable AI meta-surfaces: an idea whose time has come,” EURASIP Journal on Wireless Communications and Network- ing, v...

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INTRODUCTION Massive MIMO (mMIMO) is the key physical layer technology in 5G [1]. In a nutshell, mMIMO uses a base station with many an- tennas (e.g.,≥ 64) to deliver large beamforming gains and perform spatial multiplexing of many users on the same time-frequency re- source [2–4]. In this way, the spectral efﬁciency can be increased by, at least, an orde...

work page 2018

[4] [2]

Demystifying the Power Scaling Law of Intelligent Reflecting Surfaces and Metasurfaces

PRELIMINARIES To explain the power scaling laws related to an IRS, we ﬁrst need to review the basics of wireless propagation. We consider the free- space propagation scenario depicted in Fig. 1a. An isotropic antenna transmits a signal with power Ptx and a receive antenna is located at distance d from the transmitter. Suppose the receive antenna is lossle...

work page Pith review arXiv 1908

[5] [3]

more efﬁcient than conventional mMIMO

POWER SCALING LA WS In this section, we will compare the SNR expressions and power scaling laws achieved by mMIMO and an IRS-aided transmission when the arrays are deployed at the same location. The considered mMIMO setup is illustrated in Fig. 3a, where a single-antenna trans- mitter communicates with a planar array ofN antennas. We consider a line-of-si...

work page

[6] [4]

CONCLUSION This paper compared conventional mMIMO with IRS-aided com- munications, when the arrays are deployed at the same location. We proved analytically that the latter can never provide higher informa- tion rate or SNR—even if the SNR scales proportionally toN 2 when using an IRS and only proportionally toN in mMIMO [6]. The rea- son is that oneN-ter...

work page

[7] [5]

NR: The new 5G radio access technology,

S. Parkvall, E. Dahlman, A. Furuskär, and M. Frenne, “NR: The new 5G radio access technology,” IEEE Communications Standards Magazine, vol. 1, no. 4, pp. 24–30, 2017

work page 2017

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T. L. Marzetta, “Noncooperative cellular wireless with unlim- ited numbers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, 2010

work page 2010