Adaptation Metrics for Low-Order Nonstationary Line-of-Sight-MIMO Links

We derive metrics for adaptation of antenna-element spacing in nonstationary pure line-of-sight MIMO (LOS-MIMO) links. An example of such a deployment is a high-frequency and very high-speed wireless link between an indoor base-station and an untethered mixed-reality headset worn by a user whose distance from the base-station is time-variant. Due to the narrow antenna beams required to close the link budget and highly directional propagation at high frequencies, achieving high transmission spectral efficiency must rely on LOS-MIMO transmission—realization of which depends critically on the distance between transmitter and receiver, as well as on the antenna element spacing at the transmitter and receiver. Therefore, metrics that facilitate quick adaptation of the antenna element spacings in response to changes in the LOS-MIMO link length are required. We derive these metrics for MIMO orders two, three and four (i.e., number of transceiver antennas) and show how the transmission rate of the LOS-MIMO link depends on them. We also show how they can be quickly computed from simple measurements of path phase.

Adaptation Metrics for Low-Order Nonstationary Line-of-Sight-MIMO Links Kofi D. Anim-Appiah , Senior Member, IEEE, and Tanbir Haque, Senior Member, IEEE Abstract-We derive metrics for adaptation of antenna-element spacing in nonstationary pure line-of-sight MIMO (LOS-MIMO) links.An example of such a deployment is a high-frequency and very high-speed wireless link between an indoor base-station and an untethered mixed-reality headset worn by a user whose distance from the base-station is time-variant.Due to the narrow antenna beams required to close the link budget and highly directional propagation at high frequencies, achieving high transmission spectral efficiency must rely on LOS-MIMO transmission-realization of which depends critically on the distance between transmitter and receiver, as well as on the antenna element spacing at the transmitter and receiver.Therefore, metrics that facilitate quick adaptation of the antenna element spacings in response to changes in the LOS-MIMO link length are required.We derive these metrics for MIMO orders two, three and four (i.e., number of transceiver antennas) and show how the transmission rate of the LOS-MIMO link depends on them.We also show how they can be quickly computed from simple measurements of path phase.

I. INTRODUCTION
A T FREQUENCIES in the terahertz band (0.1 -10 THz)   where signal propagation in wireless channels becomes highly directional due to loss of multipath, Line-of-Sight Multiple-Input Multiple-Output (LOS-MIMO) communication may be one of only a few available techniques for increasing the spectral efficiency η (in b/s/Hz) of wireless links.The increase in η due to LOS-MIMO transmission is realized by taking advantage of propagation path-length differences between pairs of transmitter (TX) and receiver (RX) antenna array elements.With specific antenna placement that depends on the TX-RX separation distance, the pure LOS-MIMO complex baseband channel is independent of the propagation environment and is deterministic, depending only on those propagation path-length differences, [1], [5], [12].Measurements of pure LOS-MIMO links using both off-the-shelf transceivers appear in [4], [8], [9], [10], and [11], and related S-parameter network analysis with high-gain antennas are discussed in [6] and [7].
We consider LOS-MIMO links for which the TX-RX separation distance (link length) varies as a result of movement of one relative to the other.The challenge of realizing such nonstationary links has been highlighted by measurements in [6], [7], and [9].These letters document the degradation in η that occurs when links optimally set up for a particular link length are subsequently operated at a different length.
The key variable for optimally setting up such links is the antenna-element spacing which depends critically on the link length.Related work in [8] proposes the use of an optimization over antenna ports to effectively minimize a drop in η when fixed antenna-element spacings are required to support different link lengths.Also, [10] and [13] rely on the fact that fixed antenna separations are optimal at many discrete values of the link length, and [12] proposes the use of nonuniform antenna array spacings which are suboptimal for some reference link length but which minimize variation of η over some prescribed range of the link length.
Our viewpoint of this challenge is prompted by the prospect of pure LOS-MIMO deployments that make use of very flexible antenna arrays at a base station serving some user equipment (gNB and UE, respectively, in 3GPP language) [14].The gNB flexibility accommodates changing link length by a corresponding change in its antenna array element spacing.We envision such a link deployment for an indoor high-speed pure LOS-MIMO wireless link at subterahertz frequencies between the gNB and an untethered headset worn by a user whose distance from the gNB is time-variant.It will become evident in Sect.II that the primary LOS-MIMO link degradation due to UE motion is a reduction in η due to UE displacement, and therefore the primary effect of UE velocity is the time rate of change of η.Practically, this is far more deleterious than the effect of, say, the equivalent Doppler shift.
The main contribution of this letter is a derivation of metrics that provide a sufficiently accurate measure of the instantaneous η of the pure LOS-MIMO complex baseband channel.These metrics can be easily and rapidly computed to facilitate adaptation of non-fixed antenna array element spacings (typically at the gNB.)Related work in [3] describes measurement of a phase-based metric for antenna set up of a 60 GHz 2 × 2 LOS-MIMO link.Due to very high reflection losses [14] and narrow antenna beams at these high frequencies, the metrics comprehend only a direct LOS path.Nevertheless, they are also applicable where MIMO communication is established by a first-order specular reflection from a single boundary (no direct LOS path between TX and RX) [15].This is because the complex baseband channel MIMO matrix is defined by only propagation path lengths.
The remainder of the letter is organized as follows; Section II covers preliminaries, and metric derivation is given in Section III.Section IV describes measurement of propagation path phase from which the metrics can be computed, and metric numerical performance is discussed in Section V. We provide concluding remarks in Section VI.
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II. PRELIMINARIES Notation: matrices M are in bold uppercase, vectors x are in bold lowercase, M H is the conjugate transpose of M , and scalars M , ϵ are non-bold in either uppercase or lowercase.R and C are the real and complex field, respectively, Z is the set of integers, and j = √ −1.I is an identity matrix.For parallel TX and RX antenna arrays with uniform element spacing in a pure LOS scenario, the N × N complex baseband MIMO channel matrix H = [h mn ] has entries with D the link length, d T and d R the TX and RX antenna element spacing, respectively, λ the carrier wavelength, and assuming D ≫ d T , d R [1].For links operated at and beyond subterahertz frequencies with highly directional antennas, use of H with entries ( 1) is justified both theoretically [1], [2], [13], [14] and empirically [3], [4], [7], [8], [10].In addition to a deterministic H for such a pure LOS-MIMO deployment, practical values of d T and d R at these frequencies are at least an order of magnitude as great as λ for relevant D thus obviating any need for concern about antenna-element correlations.
The capacity of the LOS-MIMO link with channel matrix entries ( 1) is maximized when all N singular values of H are equal, i.e., when H is a scaled unitary matrix.Such a channel matrix characterizes a transmission system with N parallel streams with equal capacity.It is well known [1], [2], [3], [4], [13] that H becomes so when p an integer dependent on N .We restrict attention to p = 1.

A. Antenna Deployment for LOS-MIMO
Due to its relative simplicity and small size, the UE will typically have a small fixed number of antenna elements with spacing d R (for the remainder of this letter, the gNB is the TX and the UE is the RX.)These will support MIMO transmission of modest order N likely two or three.The extent of UE antenna array flexibility then is limited to several fixed-size sub-arrays and to beam steering.The gNB, on the other hand, will have a much larger and very flexible antenna array to enable support of LOS-MIMO links with a large pool of sub-arrays that can be dynamically configured to support a wider range of values of N and link lengths.After a LOS-MIMO link has been optimally set up according to (2), a change in the link length due to UE movement will result in loss of link optimality and, unless restored by reconfiguration of the antenna arrays, will result in a reduction in η [6], [7], [9].Thus, readily-computed metrics which provide a quick indication of the deviation of link properties from (2) are indispensable.

B. Channel Matrix Condition Number Vs. MIMO Rate
Here, we describe how the conditioning of the LOS-MIMO channel matrix relates to the mean-square error (MSE) of the received signal vector resulting from LOS-MIMO transmission.We are interested in such a relationship because we later derive error metrics that estimate the condition number of the LOS-MIMO channel matrix and therefore the MSE.We therefore connect the MIMO rate (equivalently η) and the condition number through the MSE.The MSE, resulting from stream interference due to imperfect spatial equalization of the LOS-MIMO channel (from use of a stale estimate of H), sets an upper bound for the signal-to-interference-plusnoise ratio and thus also for η.Solely for the purpose of showing a relationship between the channel matrix condition number and the MSE, we assume perfect channel knowledge at both link ends.In practice, channel-state information (CSI) could be periodically communicated from the gNB to the UE.Due to UE movement, this CSI will be prone to staleness, and the ensuing analysis provides a heuristic quantitative assessment of this condition.Consider baseband transmission model y = Hx + n of zero-mean complex signal vector x ∈ C N ×1 over H resulting in received signal y ∈ C N ×1 corrupted with complex noise vector n with E nn H = σ 2 n I. Let H = U SV H be a singular value decomposition of H, with U and V ∈ C N ×N unitary, and diagonal S ∈ R N ×N comprised of the ordered singular values {σ 1 > • • • > σ N } of H. Thus, the TX transmits precoded x = V x over the channel and the RX recovers estimate from channel output y = H x + n.Now assume that the LOS-MIMO link at a fixed carrier frequency is optimally set up for a known distance D according to (2) and then, due to UE movement, a different channel H ∆ results, of which the gNB has not yet informed the UE.If antenna-element spacings d T , d R , the precoding V , spatial equalization U and scaling S matrices all remain unchanged, an error vector e will result from use of these matrices to recover x from H ∆ , with e = I − S with ∥M ∥ F the Frobenius norm of M .The matrix condition number κ (H) is defined as κ (H) ≜ σ 1 /σ N and optimal MIMO transmission occurs for κ (H) = 1 so that S in (4) takes the form σI with σ = λ √ N / (4πD) for H with entries given by (1).A general closed-form expression relating the MSE to κ (H) does not seem readily available but we can numerically examine the dependence of the MSE (4) on κ (H) for the case where antenna spacings d  1 shows this numerically-determined characteristic for the dependence of the MSE on κ (H) for d T = d R , and we observe that the MSE degrades very rapidly as H deviates moderately from a scaled unitary matrix for which κ (H) = 1.For a practical receiver, the MSE will be much higher than thermal

III. LINK PHASE-DEPENDENT ERROR METRICS
Here we derive error metrics that can be used for setup and subsequent adaptation of the antenna array element spacing for an order-N LOS-MIMO link using channel phase.The LOS-MIMO channel matrix H depends only on the carrier frequency and link length, therefore properties of the modulating signal (e.g., bandwidth) need not be considered in metric development and application.We derive metrics which are simple combinations of the phase difference between pairs of component MIMO paths.These error metrics result as a direct consequence of requiring a scaled unitary H to maximize the channel capacity, and they have the following desirable properties: (1) they can be easily and rapidly computed from simple channel phase estimates; (2) they provide an indication of κ (H) without explicit estimation of H and subsequent computation of κ (H); (3) the magnitude of the primary metric (defined below) for given N is monotonic in η and its sign indicates how antenna element spacings should be increased or decreased to approach link optimality.While other heuristic adaptation metrics may be contemplated (e.g., the sum rate or MSE), none of them possesses all these characteristics.Estimating H [to subsequently compute κ (H)] by direct measurement of the individual path phases is impractical since it requires very precise coherent path-phase measurement; measurement errors of less than one degree readily compound across the entries of H to produce order-of-magnitude estimation errors of κ (H).For a normalized complex baseband channel, H = e −jϕmn , m, n ∈ {1, . . .N }, with path phase ϕ mn = 2πd mn /λ, and d mn the distance between TX antenna element m and RX antenna element n.When H is a scaled unitary matrix, HH H = H H H = N I, thus, with Maximizing the MIMO transmission rate requires driving all the offdiagonal entries s mn to zero.Because HH H is Hermitian, s mn = s * nm , and thus s mn = 0 ⇒ s * nm = 0.A general solution for s mn = 0 (m ̸ = n) for arbitrary N is not evident, but we show solutions for N = 2, 3, and 4 which lead directly to error metrics.

A. Error Metric Derivation-N = 2
For N = 2, setting s 12 = 0 gives the single constraint for H to be a scaled unitary matrix, e −j(ϕ21−ϕ11) + e −j(ϕ22−ϕ12) = 0, (6) and by setting real and imaginary parts to zero we obtain with A = (ϕ 21 − ϕ 11 ), B = (ϕ 22 − ϕ 12 ), and general solution The one-dimensional error metric ϵ is then which should be driven towards zero during antenna adaptation.The LHS of ( 7) could be viewed as a pair of error metrics but are impractical due to the oscillatory nature of the circular functions over small perturbations of the pathphase differences.Reference [3] uses essentially (9) without derivation.

B. Error Metric Derivation-N = 3
For N = 3, the 3 2 constraints corresponding to the upper off-diagonal terms s 12 = s 13 = s 23 = 0 result in and This results in six equations with solution independent of B, 1 and subsequently implies an error metric ϵ corresponding to each sine-cosine pair from (10) as Equation ( 12) applies to each of the three sine-cosine pairs (10), giving the components of a 3-D error-metric ϵ with ϵ = [ϵ 1 , ϵ 2 , ϵ 3 ], and {k 1 , k 2 , k 3 } ∈ Z. 1 We have used identities (11).
C. Error Metric Derivation-N = 4 For N = 4, constraining to zero the upper off-diagonal terms s mn in ( 5 with l and m odd integers.This has as a solution and the corresponding error metric implied by ( 16) is therefore which applies to each of the six sine-cosine pairs ( 14) giving a six-dimensional error metric ϵ = [ϵ n ], n = 1, . . ., 6.
Minimization of each expression in ( 9), (13), and ( 17) is achieved by a search over k or k i ∈ {0, ±1, ±2}.These metrics can be frequently monitored by the gNB and the antenna-element spacing changed when metric values fall outside a range corresponding to a lower limit for acceptable link transmission rate.

IV. MEASUREMENT OF PATH PHASE DIFFERENCES
We generalize a method described in [3] for N = 2 that makes use of a pair of phase-locked sinusoidal tones with slightly different frequencies close to that of the carrier.Thus, the TX sequentially transmits a pair of phase-locked tones from pair {i, j} of its antenna elements, and the RX estimates the difference (ϕ ik − ϕ jk ) in path phase between that pair and RX antenna element k.Thus, all of the error signals ϵ in (13) [for N = 3] or in (17) [for N = 4] can be measured for all TX antenna pairs.Fig. 2 shows how these measurements can be made for N = 3 using three envelope detector/phase detector (ED/PD) pairs and a pair of coordinated TX switches; three switch-pair states provide all measurements required for metric computation.For general N , the functionality in Fig. 2 can be realized with a pair of N -throw TX switches and N RX ED/PD pairs.

V. NUMERICAL RESULTS AND DISCUSSION
Here we show how these adaptation metrics respond to perturbations of the link length D. We first show how κ (H) varies with a change in D. We do this by assuming that an optimal LOS-MIMO link of length D ref has been set up for given N according to (2).Then D is varied resulting in varying nonoptimal H. Plots of the MSE and κ (H) are shown in Fig. 3. Observe that an MSE better than -20 dB, requires κ (H) ≲ 1.1 for N = 2.This translates into a requirement for array element spacing adaptation when ∆D/D extends  beyond a fixed window; if d T /d R = α when H is optimal, that window is maximized for α = 1.
We next examine the response of the metrics to changes in link length D. In general, each component of the error metrics is a different function of ∆D/D. 2 Recall from ( 9), (13), and (17) that ϵ has dimension N 2 ; we plot in Fig. 4 the first ('primary') component ϵ 1 of ϵ against ∆D/D.A serendipitous benefit of such a monotonic and near-linear characteristic over the operating range is its suitability for also estimating the distance change between UE and gNB if the nominal link length (when ϵ 1 = 0) is known.Fig. 4 also shows [due to (2)] that a positive value of ϵ 1 indicates that d T or d R is larger than required for optimal H-and vice versa.Indeed, the required relative change of d T or d R is directly indicated by ϵ 1 up to a proportionality constant.Shown in Fig. 5 is a plot of κ (H) against the magnitude of ϵ 1 , and the monotonic characteristic makes evident the facility of estimation of κ (H) from |ϵ 1 |.The ordinate in Fig. 5 can be transformed into the MSE by use of Fig. 1; this is shown in Fig. 6.Thus for N = 2, in order to maintain the MSE at better than -20 dB, the antenna element spacing should be changed when |ϵ 1 | ≳ 0.2 (the characteristic is only approximately symmetric about zero.)The element spacing should be increased if the sign of ϵ 1 is positive, and vice versa.And the ideal factor by which it should be changed can be read directly from Fig. 4.
A change in the relative elevation angle ∆θ between the TX and RX antenna arrays will also result in a degradation of the MSE-all the forgoing results have been for ∆θ = 0 • .But we have examined, numerically, the MSE response to a change in ∆θ after optimal link set up at ∆θ = 0 • .At 300 GHz, with D = 5 m, and N = 2, the operating window (MSE< −20 dB) is only about ±7.5 • from vertical.The MSE-∆θ characteristic is dependent on operating frequency and link length, but the operating window is again maximized for d T = d R .

VI. CONCLUSION
A single component of the multidimensional error metric (for N = 3 and 4) cannot distinguish between increased MSE from UE translational motion and that due to changes in antenna elevation angle.However, it may be possible to discriminate between these two modes of UE motion by using more than one component of the metric.The high-order dimensionality of the metrics for N = 3 and 4 also holds promise for distinguishing between limited radial and rotational changes to one end of a LOS-MIMO link.Lastly, it is evident from (2) that carrier-frequency adaptation could be used to supplement that of antenna element spacing to improve link adaptability to UE movement.This may be necessary due to a limitation in the gNB antenna array flexibility that precludes restoration of optimal H at all link lengths.
T and d R optimally set for a given distance D ref are subsequently used at distance D ∆ ̸ = D ref .To this end, for each value of D ∆ in a range centered at D ref we compute both the MSE and κ (H) with fixed d T and d R (both optimally set based on D ref .)The ratios MSE/κ (H) [D ∆ ] are then ordered and plotted.Fig.

Fig. 3 .
Fig. 3. MSE (top) and channel matrix condition number (bottom) vs. normalized distance change for d T = d R .