EML-AirComp: Layered Over-the-Air Computation from a Single Nomographic Gate Onur Günlü1,2 1
arXiv:2607.16360v1 [cs.IT] 17 Jul 2026
2
Lehrstuhl für Nachrichtentechnik, Technische Universität Dortmund, Germany Information Theory and Security Laboratory (ITSL), Linköping University, Sweden [email protected]
Abstract—Over-the-air computation (AirComp) exploits multiple-access superposition to compute functions of distributed data without separately decoding all terminal messages. We study a reusable two-input AirComp gate for the exp-minus-log (EML) operation eml(u, v) = exp(u) − log(v), v > 0. Thus, all internal nodes of a prescribed real-admissible EML tree reuse one gate type, avoiding node-specific nonlinear gate designs. Given an explicit EML tree whose intermediate logarithm arguments remain positive on a given compact domain, we derive additive white Gaussian noise (AWGN) and coherent flat fading implementations under peak-power constraints. We then characterize the number of gate evaluations, the dependency depth, evaluation latency, node-wise feasibility, deterministic error propagation, positivity preservation, and a high-probability AWGN error bound for the complete tree. A four-terminal two-hop example gives explicit positivity and end-to-end error conditions, and a digital interface propagates quantization and gate errors across the tree.
I. I NTRODUCTION Over-the-air computation (AirComp) uses the physical superposition of simultaneous wireless transmissions to compute a function of distributed data directly over the channel. This idea appears in computation over multiple-access channels (MACs) [1], uncoded transmission over a Gaussian sensor network [2], compute-and-forward [3], analog function computation over wireless MACs [4], [5], and nomographic function computation in clustered Gaussian sensor networks [6]. It has also become a central primitive in wireless learning, including federated edge learning and collaborative inference [7]–[10]. A standard AirComp-compatible class is the class of nomographic functions. Such a function can be written as ! K X f (s1 , . . . , sK ) = ψ ϕk (sk ) , (1) k=1
where terminal k ∈ [1 : K] applies the local pre-processing function ϕk (·), the MAC provides the sum, and the receiver applies the post-processing function ψ(·). This model directly covers sums, weighted averages, and other functions with a post-processed-sum representation. For multivariate inputs, however, continuous functions admitting a representation of the form (1) form a nowhere-dense subset of the continuous functions under the uniform norm [11]. A broader approach is to compute a finite sequence of nomographic functions. In wireless computation, this finite-succession viewpoint appears in clustered Gaussian sensor networks [6]. This motivates the
question studied here: can nonlinear scalar operations around AirComp sums be evaluated by repeatedly using the same AirComp gate? We consider the exp-minus-log (EML) operation, defined as eml(u, v) = exp(u) − log(v) for v > 0. A recent result gives constructive EML representations of the operations in a specified scientific-calculator basis using the constant 1 and repeated applications of the EML operation [12]. Some of those representations require complex intermediate values, whereas the AirComp model considered here is real-valued. We therefore use only EML identities whose complete trees are real-valued and whose right-child values remain strictly positive on the prescribed operating domain. To this end, we map explicit real-admissible EML trees to layered AirComp evaluation schedules. Each internal node of the tree is evaluated by the same two-input EML-AirComp gate. For any fixed tree on a compact operating domain, the construction gives the number of gate evaluations, the dependency depth, per-node transmit-amplitude bounds, deterministic error propagation, and positivity margins for all logarithm inputs. We also specialize the same gate to additive white Gaussian noise (AWGN) and coherent flat fading MACs under peak-power constraints. Here, reuse refers to the functional gate type: every internal node uses the same nonlinear pre-processing maps exp(·) and − log(·) and the same receiver normalization structure, while the operating intervals, scaling factors, and channel coefficients may be node dependent. The main contributions are as follows. First, we give peakpower-feasible AWGN and coherent flat-fading gate implementations with explicit scaling, mean-squared error (MSE), Gaussian-tail, and Rayleigh channel-inversion formulas. Second, we show that a fixed real-admissible EML tree T requires one gate evaluation per internal node, has criticalpath length depth(T ), and admits explicit resource-latency bounds. Third, we derive deterministic path-product and highprobability AWGN bounds with sufficient positivity conditions, and illustrate them through a three-gate hierarchical example and a generic digital-error interface. II. S INGLE EML G ATE OVER AWGN AND FADING C HANNELS This section provides AWGN and coherent flat-fading realizations of one EML-AirComp gate under peak-power constraints.
A. Nomographic Form and Bounded Operating Intervals Let u ∈ I = [a, b] and v ∈ J = [c, d], where c > 0. As the right AirComp input contributes a scaled version of − log(v), we define BJ = max{| log c|, | log d|},
(2)
which is used to ensure that the transmitted signal satisfies the peak-power constraint. If this bound is zero, then log(v) = 0 over the considered interval, so the corresponding transmitted waveform is zero. In that case, the associated peak-power constraint does√not restrict the scaling parameter and we use the convention P /0 = ∞ for this special case. The following fact provides the exact nomographic representation of the EML gate and the channel-input bounds used in the AWGN and fading implementations below.
then (6) satisfies the peak-power constraints for all (u, v) ∈ I × J. Moreover, we have Z σ2 b = Y = G(u, v) + ξ, G (11) ξ= ∼ N 0, 2 A A A so that we obtain 2 b − G)2 = σ E (G A2 and, for every ϵ > 0, we have Aϵ b P |G − G| > ϵ = 2Q . σ
Fact 1. For (u, v) ∈ I × J, the function (3)
(13)
Among all feasible A, the minimum local MSE is achieved by A = AAWGN and is equal to max MSEAWGN = min
G(u, v) = exp(u) − log(v)
(12)
σ2 (AAWGN )2 max
.
(14)
Proof: The peak-power constraints follow from Fact 1. Substituting (6) into (8) gives
is nomographic with G(u, v) = ψ ϕL (u) + ϕR (v) ,
Y = A(exp(u) − log(v)) + Z,
where ϕL (u) = exp(u),
ϕR (v) = − log(v),
ψ(y) = y.
(5)
With scaling A > 0, the two real channel inputs used by the EML-AirComp gate are XL = A exp(u),
which proves the Gaussian error law, tail probability, and local MSE. Since σ 2 /A2 is decreasing in A > 0, the MSEminimizing feasible scaling is the largest feasible scaling. C. Coherent Flat-Fading EML-AirComp Gate
XR = −A log(v)
(6)
We next consider a coherent flat-fading MAC. Each transmitter applies channel inversion so that the two pre-processed EML inputs add coherently at the receiver. Consider the complex flat-fading MAC
|XR | ≤ ABJ .
(7)
Y = hL XL + hR XR + Z,
which satisfy |XL | ≤ A exp(b),
(15)
(4)
These bounds are used below to choose the scaling A so that the corresponding peak-power constraints are satisfied. B. AWGN EML-AirComp Gate We now specialize the EML gate to a real two-user AWGN MAC. The scaling parameter is chosen subject to the peakpower constraints. Consider the real two-user AWGN multiple-access gate Y = XL + XR + Z,
Z ∼ N (0, σ 2 ),
(8)
with peak-power constraints |XL |2 ≤ PL and |XR |2 ≤ PR . For scaling A > 0, set (6) and form the estimate b = Y /A. G
(9)
(16)
where hL , hR ∈ C \ {0} are independent fading coefficients known at the corresponding transmitters, and Z is independent 2 = E[(ℜ{Z})2 ]. Use of (hL , hR ). Let σR XL = A
h∗L exp(u), |hL |2
h∗R log(v), |hR |2
(17)
ℜ{Z} . A
(18)
XR = −A
and form the estimate b = G(u, v) + ξ, G
ξ=
2 If ℜ{Z} ∼ N (0, σR ), then we have 2 σR ξ ∼ N 0, 2 . A
(19)
The next result gives the exact AWGN error law and the MSE-minimizing scaling under peak-power constraints, where Q(x) denotes the Gaussian tail function. Throughout the AWGN and fading models below, assume PL , PR > 0.
The next result analyzes the coherent flat fading gate under channel inversion. It gives the conditional MSE and tail probability for fixed fading coefficients, and the feasibility probability that a fixed scaling satisfies the peak-power constraints under independent Rayleigh fading.
Fact 2. Consider the AWGN channel. If we have √ √ PL PR , , 0 < A ≤ AAWGN ≜ min max exp(b) BJ
Fact 3. Consider the coherent flat-fading MAC. If we have √ √ PL |hL | PR |hR | 0 < A ≤ Afad ≜ min , , (20) max exp(b) BJ
(10)
then (17) satisfies the peak-power constraints for all (u, v) ∈ I × J. The conditional MSE is 2 b − G)2 | hL , hR = σR . E (G A2
(21)
2 If ℜ{Z} ∼ N (0, σR ), then we have
Aϵ b P |G − G| > ϵ | hL , hR = 2Q . σR
(22)
Among all feasible A for the realized channel, the conditional 2 2 MSE is minimized by A = Afad max . Since |hL | and |hR | are independent exponential random variables with some means ΩL and ΩR , respectively, then a fixed scaling A is feasible with probability A2 BJ2 A2 exp(2b) − . (23) pfeas (A) = exp − PL ΩL PR Ω R
a right child r(q). The value computed at node q is defined recursively as Fq (s) = G Fℓ(q) (s), Fr(q) (s) . (27) The function represented by the whole tree is the value at the root FT (s) = Fρ (s). The number of EML gates in the tree is W (T ), and the depth depth(T ) is the largest number of EML gates on any path from a leaf to the root. ♢ The next definition gives the real-admissibility condition, needed as the second argument of G enters a logarithm. Definition 2. Let D ⊂ RK . An EML tree T is real-admissible on D if all node functions are recursively well-defined and real-valued on D, and if we have Fr(q) (s) > 0,
∀s ∈ D,
∀q ∈ Vint (T ).
♢
Proof: Substituting (17) into (16) gives b = ℜ{Y } = G(u, v) + ℜ{Z} , G A A
(24)
which yields (21) and (22). Moreover, using u ≤ b and | log(v)| ≤ BJ , we obtain |XL |2 ≤
A2 exp(2b) ≤ PL , |hL |2
|XR |2 ≤
A2 BJ2 ≤ PR , (25) |hR |2
2 where the last inequalities follow from (20). Since σR /A2 decreases with A > 0, the conditional MSE is minimized by A = Afad max . Moreover, for a fixed A, feasibility for all (u, v) ∈ I × J is equivalent to
A2 exp(2b) |hL |2 ≥ , PL
A2 BJ2 |hR |2 ≥ . PR
(28)
(26)
Since the channel power gains are independent exponential random variables, their tail probabilities give (23). III. E XPLICIT EML T REES AND L AYERED A IR C OMP S CHEDULES Having established the implementation of one EMLAirComp gate, we now extend it to a fixed EML tree, where each internal node requires one gate evaluation and the longest leaf-to-root path determines the dependency depth. The following definition fixes the tree notation used in all later bounds. Definition 1. An EML computation tree T over inputs s = (s1 , . . . , sK ) is a finite binary computation tree whose internal nodes are EML gates. The leaves provide the starting values of the computation, and the root node gives the final output. Let Vleaf (T ) and Vint (T ) denote the sets of leaves and internal nodes, respectively, and let ρ denote the root. Each leaf is either one of the input variables sk or a copy of the constant 1. Thus, for a leaf i, the associated value is either Fi (s) = sk for some k ∈ [1 : K], or Fi (s) = 1. Each internal node q ∈ Vint (T ) has exactly two children: a left child ℓ(q) and
For each internal node q, we need bounds on the values that can appear at its two inputs. As s ranges over the operating domain D, the left child of node q can take values in Iq = Fℓ(q) (D) and the right child can take values in Jq = Fr(q) (D). If D is compact and T is real-admissible on D, then all node functions are continuous. Thus, the sets Iq and Jq are compact. Moreover, real-admissibility guarantees that the right input of every EML gate is strictly positive on D. In the analysis, we use intervals that contain the following value ranges: Iq ⊆ [aq , bq ],
Jq ⊆ [cq , dq ] ⊂ R++ .
(29)
Thus, Bq = max{| log cq |, | log dq |} is an upper bound on | log v| at node q. An evaluation order is said to be admissible if each internal node is evaluated only after both child values are available. Moreover, at each internal node, the two child values are assumed to be available at the transmitters feeding one EMLAirComp gate. Routing, storage, forwarding, synchronization, and waveform generation between gate evaluations are not modeled. Hence, W (T ) counts only EML-AirComp gate evaluations. The next result counts the required EML-AirComp gate evaluations, identifies the tree depth as the critical-path latency, and gives node-wise peak-power feasibility conditions. Proposition 1. Let D ⊂ RK be compact and let T be realadmissible on D. Consider any evaluation order in which an internal node is evaluated only after the values of its two children have already been computed. Then, the function FT is evaluated by one EML-AirComp gate evaluation for each internal node of T . Hence, the total number of EML-AirComp gate evaluations is W (T ) = |Vint (T )|. A critical-path lower bound on the number of sequential gate-evaluation intervals is depth(T ), achieved when all nodes in each dependency layer can be evaluated in parallel. For each internal node q, let the possible left- and rightchild values be enclosed by the intervals in (29). If node q is
implemented over the real AWGN gate with scaling Aq and peak-power limits Pq,L and Pq,R , then it is sufficient to choose ) (p p Pq,L Pq,R . (30) 0 < Aq ≤ min , exp(bq ) Bq Similarly, if node q is implemented over the coherent flatfading gate with channel coefficients hq,L and hq,R , then it is sufficient to choose ) (p p Pq,L |hq,L | Pq,R |hq,R | . (31) 0 < Aq ≤ min , exp(bq ) Bq These guarantee the peak-power constraints. Proof: Each internal node computes exactly one value of the form G(u, v). Evaluating the nodes in any topological order, therefore, evaluates the root after all internal nodes have been evaluated. The number of gate invocations is |Vint (T )| = W (T ), and the longest dependency chain is the maximum number of internal nodes on a root-to-leaf path. The nodewise AWGN and fading constraints follow from Facts 2 and 3 applied to the enclosing intervals of node q. Remark 1. Assume that each EML-AirComp gate evaluation takes one interval and that at most P ≥ 1 gates can be evaluated in parallel. Let Wℓ be the number of internal nodes whose longest leaf-to-node path contains ℓ gates, and let DP⋆ (T ) be the minimum number of intervals required by an admissible schedule. Then, we have depth(T ) X W (T ) Wℓ ⋆ max depth(T ), ≤ DP (T ) ≤ P P ℓ=1 (32) where the lower bounds follow from the longest dependency path and the total number of gates, while evaluating one layer at a time gives the upper bound. Thus, D1⋆ (T ) = W (T ), and DP⋆ (T ) = depth(T ) if P ≥ maxℓ Wℓ . For a noisy sequential evaluation, the same node-wise conditions apply on any bounded-error event for which the perturbed operands remain in the prescribed intervals. Outside this event, no downstream peak-power guarantee is made, and the evaluation may be declared to be in outage, as discussed in the next section. IV. T REE -L EVEL E RROR P ROPAGATION , P OSITIVITY, AND R ELIABILITY This section derives deterministic and high-probability AWGN tree-error bounds while ensuring positive logarithm arguments. Let Fbq denote the noisy computed value at node q. For each internal node q, define the analysis rectangle Rq = [aq , bq ] × [cq , dq ] ⊂ R × R++ ,
(33)
over which the local sensitivity bounds are evaluated. Define 1 LLq = exp(bq ), LR (34) q = cq
where LLq bounds how sensitive the output of node q is to an error in its left input, and LR q in its right input. For a connection p → q, where p is a child of q, set the sensitivity factor as ( LLq , p = ℓ(q), (35) Lp→q = LR q , p = r(q). Moreover, for a node p, let P (p → ρ) denote the unique path from p to the root, and define Y M (p → ρ) = Le , (36) e∈P (p→ρ)
with the empty product equal to 1, such that it is the product of the sensitivity factors along the path from p to the root, and therefore bounds how much an error introduced at node p can be amplified before it reaches the final output. The next theorem gives a deterministic bound on how local errors propagate through the tree. It also provides checkable range conditions ensuring that every perturbed logarithm argument remains positive. To initialize a common error recursion over all tree nodes, set the error budget of each leaf occurrence i to its given error bound, i.e., δi = εi , and define the error budget of every internal node q recursively as δq = LLq δℓ(q) + LR q δr(q) + ηq .
(37)
K
Theorem 1. Let D ⊂ R be compact, and let T be real-admissible on D. Suppose that every leaf occurrence i ∈ Vleaf (T ) satisfies |Fbi (s) − Fi (s)| ≤ εi ,
s ∈ D,
(38)
and that the noisy evaluation is performed in an admissible order, with every internal node q ∈ Vint (T ) satisfying Fbq (s) = G Fbℓ(q) (s), Fbr(q) (s) + ξq (s), |ξq (s)| ≤ ηq . (39) For every internal node q, suppose Fℓ(q) (D) ⊆ [aq + δℓ(q) , bq − δℓ(q) ],
(40)
Fr(q) (D) ⊆ [cq + δr(q) , dq − δr(q) ].
(41)
Then, every noisy gate evaluation is well defined, the exact and computed child-value pairs remain in Rq , and |Fbq (s) − Fq (s)| ≤ δq ,
q ∈ Vint (T ),
s ∈ D.
(42)
In particular, we have X
|FbT (s) − FT (s)| ≤
M (i → ρ)εi
i∈Vleaf (T )
+
X
M (q → ρ)ηq .
(43)
q∈Vint (T )
Moreover, every computed logarithm argument satisfies Fbr(q) (s) ∈ [cq , dq ] ⊂ R++ ,
q ∈ Vint (T ),
s ∈ D. (44)
Proof: Fix s ∈ D and consider any admissible evaluation order. For every leaf occurrence i, (38), and the definition δi = εi give |Fbi (s) − Fi (s)| ≤ δi .
(45)
Consider an internal node q and assume that the bound (42) has already been established for its two children. From (40), we have Fℓ(q) (s) ∈ [aq + δℓ(q) , bq − δℓ(q) ]. Using |Fbℓ(q) (s) − Fℓ(q) (s)| ≤ δℓ(q) , this implies Fbℓ(q) (s) ∈ [aq , bq ].
(46)
Similarly, (41) and |Fbr(q) (s) − Fr(q) (s)| ≤ δr(q) give Fbr(q) (s) ∈ [cq , dq ] ⊂ R++ .
(47)
Thus, both the exact and computed child-value pairs belong to the rectangle Rq = [aq , bq ] × [cq , dq ], and the logarithm in the noisy evaluation of node q is well defined. Consider arbitrary (u, v), (u′ , v ′ ) ∈ Rq . Since Rq is convex, we have 1 |G(u, v) − G(u′ , v ′ )| ≤ exp(bq )|u − u′ | + |v − v ′ |. (48) cq Applying (48) to the exact and computed child-value pairs at node q, and using (39), gives |Fbq (s) − Fq (s)| ≤ LLq |Fbℓ(q) (s) − Fℓ(q) (s)| + LR |Fbr(q) (s) − Fr(q) (s)| + ηq q
≤ LLq δℓ(q) + LR q δr(q) + ηq = δq .
(49)
This proves (42) by induction over the admissible evaluation order. Equation (47) simultaneously proves (44). Recursively expanding (37) from the root toward the leaves shows that every leaf error εi is multiplied by the product of the sensitivity factors on the unique path from leaf occurrence i to the root. Similarly, every local gate error ηq is multiplied by the product of the sensitivity factors on the path from node q to the root. By the definition of M (· → ρ), giving (43). We next specialize the deterministic tree bound to independent AWGN-induced errors at the EML-AirComp nodes, and give a lower bound on the probability that the final root error satisfies the deterministic path-product bound. Corollary 1. Fix s ∈ D and assume the setting of Theorem 1, except that the internal-node errors are independent AWGNinduced errors such that ! σq2 ξq ∼ N 0, 2 , q ∈ Vint (T ). (50) Aq Choose local tolerances ηq > 0. Assume that the buffered range conditions (40) and (41) hold for the selected local tolerances ηq . Then, for this fixed input s, the bound X |FbT (s) − FT (s)| ≤ M (i → ρ)εi i∈Vleaf (T )
+
X
M (q → ρ)ηq
(51)
q∈Vint (T )
holds with probability at least Y Aq η q 1 − 2Q . σq q∈Vint (T )
(52)
On the same event, every logarithm argument remains in its positive interval, and no interval-outage event occurs. Proof: For every internal node q, Fact 2 gives Aq η q P[|ξq | ≤ ηq ] = 1 − 2Q . σq
(53)
Since the node errors are independent, the probability that all events {|ξq | ≤ ηq } occur is the product in (52). On this event, the local error bounds required by Theorem 1 hold simultaneously. The buffered range conditions then ensure that all gate evaluations are well defined and that all computed child-value pairs remain in their assigned rectangles. A. Finite-Alphabet and Digital Gate Errors The preceding deterministic bound also applies when the leaf values are quantized and the individual EML gates are evaluated by finite-alphabet methods. Let Q : D → DQ ⊂ D be a quantizer with finite image DQ , and write sQ = Q(s). For each coordinate-leaf occurrence i, assume |sk(i) − sQ k(i) | ≤ ∆k(i) .
(54)
Suppose also that a finite-alphabet or digital AirComp method is used for every EML node q, and let Eq be an event on which its local computation error satisfies |ξq | ≤ ηq . Let also FbT (sQ ) denote the output obtained by evaluating the perturbed digital EML tree from the quantized leaf values. The following corollary combines the quantization errors at the leaves with method-dependent errors at the digital EML gates in one end-to-end path-product bound. Corollary 2. Let D ⊂ RK be compact, let T be realadmissible on D, and fix s ∈ D. Suppose that the buffered ¯ i and with range conditions of Theorem 1 hold with εi = ∆ the selected gate-error tolerances ηq . Define ( ¯ i = ∆k(i) , i is a coordinate-leaf occurrence, ∆ (55) 0, i is a constant-1 leaf. T Then, on q∈Vint (T ) Eq , we have X ¯i |FbT (sQ ) − FT (s)| ≤ M (i → ρ)∆ i∈Vleaf (T )
+
X
M (q → ρ)ηq .
(56)
q∈Vint (T )
If P[Eqc ] ≤ pq , (56) holds with probability at least X 1− pq .
(57)
q∈Vint (T )
T ¯ i, Proof: Apply Theorem 1 on q∈Vint (T ) Eq with εi = ∆ and use the union bound. Note that Corollary 2 does not provide a finite-alphabet coding method. If applied to the selected finite input alphabets, a method such as ChannelComp or SumComp [13], [14] can separately provide the method-dependent local error tolerances ηq or failure probabilities pq . The corresponding quantization, coding, modulation, and decoding analyses remain separate from the EML-tree error propagation in (56).
V. H IERARCHICAL EML-A IR C OMP E XAMPLE
VI. C ONCLUSION
Consider four terminals, two relays, and one fusion center. Terminals 1 and 2 transmit to relay 1, terminals 3 and 4 to relay 2, and both relays transmit to the fusion center. The first-hop MACs use noninterfering resources and may operate simultaneously. At each EML gate, the two operands are held by different transmitters and combined over a MAC. Let s1 ∈ [a1 , b1 ], s2 ∈ [c2 , d2 ] ⊂ R++ , s3 ∈ [a3 , b3 ], and s4 ∈ [c4 , d4 ] ⊂ R++ . Define
We derived peak-power-feasible AWGN and coherent flatfading EML-AirComp gates, resource-latency bounds for prescribed real-admissible EML trees, and a deterministic error recursion resulting in a path-product bound, positivity conditions, and a high-probability AWGN guarantee. A two-hop four-terminal example illustrated the hierarchical analysis, and the digital interface propagates leaf-quantization and local gate errors through the tree. The model assumes that the operands of each gate are available at the corresponding transmitters, and routing, storage, and the existence of real-admissible EML representations for arbitrary functions are left for future work.
U1 = G(s1 , s2 ), U2 = G(s3 , s4 ), FT (s) = G(U1 , U2 ). (58) Since G(u, v) is increasing in u and decreasing in v, the exact first-level output intervals are determined by u1 = exp(a1 ) − log d2 ,
u1 = exp(b1 ) − log c2 ,
u2 = exp(a3 ) − log d4 ,
u2 = exp(b3 ) − log c4 .
(59)
If u2 > 0, then the tree is real-admissible on the stated domain. b1 = U1 + ξ1 , U b2 = For AWGN gate evaluations, consider U b1 , U b2 ) + ξ3 , where ξq ∼ N (0, σq2 /A2q ), U2 + ξ2 , and FbT = G(U q ∈ {1, 2, 3}, are mutually independent. The two first-level gates have W1 = 2 and the root layer has W2 = 1, so Remark 1 gives D1⋆ (T ) = 3 and DP⋆ (T ) = 2 for P ≥ 2. The following result gives the end-to-end error and a sufficient positivity condition for the root logarithm. Proposition 2. Assume |ξq | ≤ ηq for q ∈ {1, 2, 3} and η2 < b2 > 0 and u2 . Then, we have U |FbT − FT (s)| ≤ exp(u1 + η1 )η1 +
η2 + η3 . u2 − η2
(60)
b2 ≥ u − η2 > 0, the root logarithm is Proof: Since U 2 well defined. The exact and perturbed root inputs lie in [u1 − η1 , u1 + η1 ] × [u2 − η2 , u2 + η2 ].
(61)
Hence, (48) gives b1 , U b2 ) − G(U1 , U2 )| ≤ exp(u1 + η1 )η1 + |G(U
η2 . u2 − η2 (62)
Combining this bound with |ξ3 | ≤ η3 proves (60). Moreover, for κq ∈ (0, 1), choose σq −1 κq ηq = Q , q ∈ {1, 2, 3}. Aq 2
(63)
Then, we have P[|ξq | ≤ ηq ] = 1−κq . If η2 < u2 , independence of Q3the three node errors implies P3 that (60) holds with probability q=1 κq . On the same boundedq=1 (1 − κq ) ≥ 1 − error event, the node-wise peak-power constraints follow from Proposition 1 using the corresponding error-enlarged operand intervals. Such guarantees are therefore conditioned on the bounded-error event. Under coherent flat fading, the same deterministic error bound holds conditionally on both the nodewise channel-inversion feasibility events and the boundederror events, as used above.
ACKNOWLEDGMENT This work was partially supported by German Federal Ministry of Research, Technology and Space (BMFTR) 6GEM+ Transfer Hub under Grants 16KIS2412 and 16KISS005. The author used OpenAI’s ChatGPT 5.5 for writing improvements and reviewed and verified all manuscript content. R EFERENCES [1] B. Nazer and M. Gastpar, “Computation over multiple-access channels,” IEEE Trans. Inf. Theory (T-IT), vol. 53, no. 10, pp. 3498–3516, 2007. [2] M. Gastpar, “Uncoded transmission is exactly optimal for a simple Gaussian “sensor” network,” IEEE Trans. Inf. Theory (T-IT), vol. 54, no. 11, pp. 5247–5251, 2008. [3] B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interference through structured codes,” IEEE Trans. Inf. Theory (T-IT), vol. 57, no. 10, pp. 6463–6486, 2011. [4] M. Goldenbaum, H. Boche, and S. Stańczak, “Harnessing interference for analog function computation in wireless sensor networks,” IEEE Trans. Signal Process. (TSP), vol. 61, no. 20, pp. 4893–4906, 2013. [5] M. Goldenbaum and S. Stanczak, “Robust analog function computation via wireless multiple-access channels,” IEEE Trans. Commun. (TCOM), vol. 61, no. 9, pp. 3863–3877, 2013. [6] M. Goldenbaum, H. Boche, and S. Stańczak, “Nomographic functions: Efficient computation in clustered Gaussian sensor networks,” IEEE Trans. Wireless Commun. (TWC), vol. 14, no. 4, pp. 2093–2105, 2015. [7] G. Zhu, Y. Du, D. Gündüz, and K. Huang, “One-bit over-the-air aggregation for communication-efficient federated edge learning: Design and convergence analysis,” IEEE Trans. Wireless Commun. (TWC), vol. 20, no. 3, pp. 2120–2135, 2021. [8] A. Şahin and R. Yang, “A survey on over-the-air computation,” IEEE Commun. Surveys & Tutorials (COMST), vol. 25, no. 3, pp. 1877–1908, 2023. [9] A. Pérez-Neira, M. Martinez-Gost, A. Şahin, S. Razavikia, C. Fischione, and K. Huang, “Waveforms for computing over the air: A groundbreaking approach that redefines data aggregation,” IEEE Signal Process. Mag. (SPM), vol. 42, no. 2, pp. 57–77, 2025. [10] S. F. Yilmaz, B. Hasircioğlu, L. Qiao, and D. Gündüz, “Private collaborative edge inference via over-the-air computation,” IEEE Trans. Mach. Learn. Commun. Netw. (TMLCN), vol. 3, pp. 215–231, 2025. [11] R. C. Buck, “Nomographic functions are nowhere dense,” Proc. Amer. Math. Soc., vol. 85, no. 2, pp. 195–199, 1982. [12] A. Odrzywołek, “All elementary functions from a single binary operator,” arXiv preprint arXiv:2603.21852, 2026. [13] S. Razavikia, J. M. Barros da Silva, and C. Fischione, “ChannelComp: A general method for computation by communications,” IEEE Trans. Commun. (TCOM), vol. 72, no. 2, pp. 692–706, 2024. [14] S. Razavikia, J. M. B. da Silva, and C. Fischione, “SumComp: Coding for digital over-the-air computation via the ring of integers,” IEEE Trans. Commun. (TCOM), vol. 73, no. 2, pp. 752–767, 2025.