Modeling and Mitigation of Time Delay Error for Non-Timed Arrays in Satellite Communications

A hybrid electromagnetics and communications model is proposed to quantify the performance degradation of non-timed array antennas (e.g. phased arrays or reflectarray antennas) from time delay induced errors. Partial mitigation techniques are investigated. As a non-timed array increases in size, steering angle, frequency, or symbol rate the signal degradation due to lack of true time delay compensation increases from the combined effect of the various per-element fractional time delays. The authors propose that the signal degradation from intersymbol interference due to timing effects can be thought of as an efficiency term and be used to assign a relative energy per symbol to noise power spectral density (EsNo) degradation based on standard satellite communications protocols (specifically DVB-S2X) with an assumed acceptable link margin. From these results, design recommendations are proposed for maximum array size, steering angle, and symbol rate for assumed operating conditions in satellite communications. The work also explores how partial mitigation techniques (e.g. equalization-based, spatial-based or amplification-based approaches) might be applied to the problem as an alternative to true time delay.


I. INTRODUCTION
Signal degradation from physical telecommunications hardware is one of the oldest problems in the communications field, being first studied almost a century ago with respect to signal design for bandwidth constrained telegraph lines [1], [2]. The basic problem is that hardware or transmission lines can degrade a signal through broadening, time delay, attenuation or distortion which results from a transmitted symbol non-uniformly passing through the system with resulting self-interference known as intersymbol interference (ISI) [3], [4]. ISI has been studied for many types of transmission lines and devices, including but not limited to: optical fibers (pulse dispersion) [5], [6], wireless channels (multipath and fading) [7], [8] and power amplifiers (intermodulation and spectral regrowth) [9], [10].
In the antenna array community, the ISI issue limits the instantaneous bandwidth for non-timed (NT) arrays (e.g. phased arrays and reflectarrays), known since at least the The associate editor coordinating the review of this manuscript and approving it for publication was Faissal El Bouanani . 1970's [11]. A classic NT array typically uses phase shifters on each element or group of elements to introduce a fixed phase delay at some frequency that is limited to the range from 0 to 2π radians or 0 • to 360 • . For phases greater than 360 • , phase wrapping occurs, providing the correct phase for the carrier but introducing a timing error of one carrier period for any modulated signals [12]. As the timing delay errors increase, the combination of signals from different elements can introduce an ISI-based signal degradation similar to the ISI degradation on a classical telephone line or optical fiber.
Research in the array community has focused on correcting the ISI problem through the implementation of ''true time delay'' to make the array a ''timed'' array with no associated ISI [13], [14]. Photonics offers an adaptive or reconfigurable method from which to correct the problem through optical coupling of the signal and processing and correcting in that domain [15], [16]. Similar adaptive methods have been proposed in the silicon level [17], [18]. Other implementations of true time delay for reflectarray or phased array applications are passive configurations (e.g. the Rotman lens, coupled lines, etc.) [19], [20], [21], [22].
While there may be relatively few passive or active implementations of true time delay for arrays, a number of adaptive, NT phased array and reflectarrays have been proposed [23], [24], [25]. Schemes for adaptive phasing include coupled delay lines, size variability, feed motion, variable element rotation angle, diodes, MEMs, material tuning, ferrites, integrated-ICs,. . . etc. [26]. These offer a low-cost and simpler alternative to realize adaptive arrays.
Research to model the signal degradation for basic digital communications environments (e.g. phase-shift keying) for reflectarrays and NT phased arrays is typically based on bit error rate degradation [27], [28]. Other research approaches the problem from an even simpler pulsed waveform perspective [29]. Related measurement techniques model the antenna as a filter that can be deconvolved from a measured reference [30]. While these approaches handle the problem from a high-level perspective, they do not consider actual communications frameworks (e.g. Digital Video Broadcasting for Satellite, Second Generation with Extensions or ''DVB-S2X'') or realistic operating conditions which can substantially alter the practical modeling of this phenomenon. For example, DVB-S2X incorporates error-correcting codes and therefore the packet error rate won't degrade for a chosen ''physical layer frame configuration field'' or ''MODCOD'' (the modulation and error correcting code pair) unless the Signal-to-Noise Ratio degrades substantially. In these systems, bit error rate is not an appropriate metric for system performance (as discussed in detail in the next section) [31], [32].
Research in signal degradation has received increased focus recently as many companies are moving towards the non-geostationary (NGSO) marketplace (specifically Low Earth Orbit (LEO) satellites) with many countries meeting at the ITU (the United Nations' telecommunications agency) to prioritize this area [33]. Since the signals from NGSO satellites received by the ground-terminals change as the satellite's position moves across the sky, it is necessary for an adaptive terminal to be used, with the industry focusing on the NT phased array as the subject of most interest as a potential low-cost adaptive array [34]. This work will show illustrative examples using typical performance and operating conditions of these earth stations (based on public filings) as presented in [35], [36], [37], and [38].
Research into signal processing of spatial arrays is relevant but focuses on signal estimation, such as angle of arrival, beamforming, multiple signals, and sparse sampling [39], [40], [41]. While many of these classical models look at the signal from a time-domain perspective, ultimately the problem considered in this work is an extended problem from this field: given perfect sampling, angle of arrival, and beamforming for an NT array, what is the expected performance degradation within a specific digital communications environment and is it possible to mitigate it?
This report builds upon previous work by the authors in [42] and seeks to address unique problems not yet addressed in the literature, by 1) providing a novel framework to model the timing error of digital communications for non-timed arrays which relates to both radiated concepts and also system level concepts in the context of satellite communications, specifically DVB-S2X (Sections II-IV), 2) providing design recommendation guidelines for the use of NT arrays in high symbol rate environments (Section VI-A), and 3) providing a discussion on partial mitigation techniques that could be applied to mitigate the problem rather than using physical time delay units (Section V). The principal motivation of this work is to address these three areas in order to bridge the gap between electromagnetics and communications by directly modeling the array induced time delay degradation and discussing its impact on satellite communications. This work takes a unique approach relative to any other existing literature, with detailed comparisons being expanded upon in Section VII-A.

II. HYBRID SYSTEMS MODEL
A signal-level system model is presented in Figure 1. The model can be considered ''hybrid'' in the sense that it will take an electromagnetics ray-tracing algorithm to compute the antenna time delay and uses it as the ''channel'' block in the digital communications system block diagram to distort the signal; effectively, an electromagnetics model is integrated into the larger digital communications systems model. The overall systems model and algorithm can be described in the following manner: 1) Modulation Mapping: Generate a randomized set of symbols of arbitrarily large size and set a transmission symbol rate of ''R s '' 2) Pulse Shaping (Ideal): A sinc pulse train (assuming infinite neighbor extent in positive and negative time) will be used to eliminate any inherent ISI and discretization will be based on carrier period ''T c '' such that mod(T c , R s ) is zero a) The discretization requirement is based on the fact that the timing error can be approximated by an integer multiple of T c (as described in sec. III) 3) Non-Timed Array Degradation Channel: Pass the pulse shaped symbol train through each element ''ij'' of the NT array model to introduce a per-element time delay and then sum up and normalize the aggregate response of all various timing-delayed symbol trains a) The ''ij'' timing error is computed with ray tracing b) Due to the ''normalization'' at least one element should have zero timing error applied c) Zero filling is used as necessary to have equal length per-element time delay for the summation 4) Timing Synchronization/ Demapper: Sample on the (optimal) peak of the aggregated response with a rate of one symbol period ''R s '' and compare to the original/input data set for an estimate of degradation 5) (Optional) Mitigation: Either true time delay units or partial time delay compensation can be used to mitigate the waveform degradation from time delay. The following non-comprehensive list of idealized assumptions are made in the model as outlined below: 1) Ideal carrier and symbol timing recovery (i.e. sampling at the ideal symbol peak) with practical details omitted 2) Complex AWGN or ''additive white Gaussian noise'' is considered implicitly; only relative EsNo is modeled directly (See Discussion in IV.C) 3) Ideal sinc pulse shaping to eliminate any inherent ISI 4) No inter-carrier frequency-based phase error included (only the phase wrapping phenomenon is considered) 5) Other RF component effects are quantitatively excluded (due to scale); these effects are qualitatively discussed though (See Section IV-D) These assumptions ensure that the degradation modeled by the authors is solely related to the NT array model and not some other factor related to an imperfect communications system. Thus, the conclusions of this paper can be considered an upper bound or an optimistic approach to the degradation modeling and can be used as a guideline for design. However, for a practical implementation of this method into specific terminals for satellite communications, these effects should be addressed. The 1) ideal timing and 5) RF component effects in particular could greatly impact the resulting performance.

A. CONSUMER TERMINAL PARAMETERS FROM LEO SATELLITE CONSTELLATIONS
A summary of typical consumer terminal earth station operating conditions can be found in Table 1 from both publicly available filing information and the literature [35], [36], [37], [38]. These values will be used as realistic inputs to the proposed model to draw conclusions for this application area. Previous research has focused on the bit error rate degradation as the figure of merit [27], [28], but DVB-S2X satellite communications use error correction codes which correct out these bit errors at the Low-Density Parity-Check Decoder [31], [32]. In fact, DVB-S2X is ''Quasi Error Free''. This means it maintains a constant ratio of corrupted received packets to useful packets of no more than 10 −5 . DVB-S2X does this by adapting the MODCOD to match the channel ''energy per symbol to noise spectral density ratio'' or EsNo VOLUME 11, 2023  (analogous to the signal-to-noise ratio ''SNR'' per symbol) with a corresponding MODCOD that has higher or lower spectral efficiency. Essentially, DVB-S2X will transmit with reduced spectral efficiency (i.e. fewer bits per symbol) at lower channel EsNo levels and higher spectral efficiencies at higher channel EsNo levels [31], [32].
It is more appropriate to assess the relative EsNo degradation (rather than the bit error rate) for DVB-S2X. This can be modeled by considering that from a ''link budget'' perspective the EsNo is proportional to the antenna gain. Thus both the time delay degradation considered in this work along with array scan loss can be understood to also degrade the EsNo. However, not all spectral efficiencies may be desirable (i.e. the user may expect a certain data rate) so to maintain a stable operating condition and to account for any degradation, a ''link budget'' is computed such that it has positive margin called the ''link margin'' -where the link margin is defined as the ratio of actual vs needed EsNo. It needs to be positive and finite to allow for these losses, as well as weather events and other random effects.
With this in mind, the authors' proposed metric for time delay degradation is the relative EsNo degradation with a ''link margin relative allowance'' (LMRA), where the LMRA is the fixed, minimum acceptable link margin. Note this differs from the actual link margin (which varies with antenna size, symbol rate, spectral efficiency, etc.), in that it is the desired lower bound to this value and fixed. In adaptive antenna systems, the degradation is not easy to account for since it can change with other parameters. Thus the LMRA may need to be on the order of ∼ 6 dB for adaptive antennas (especially planar, NT antennas) to have a fully closed link given various configurations [38].
As an example, consider a satellite communications terminal using DVB-S2X and operating at the MODCOD 16APSK-3/4 with associated threshold EsNo of 10.21 dB and an LMRA of 6.01 dB ( Figure 2). As long as the ideal MODCOD is maintained, the link can be said to be in the ''link operational region'' but otherwise a lower MODCOD is needed and the link can be said to be in the ''link outage region.'' These concepts are generalized to frame the final results of section VI.

III. RAY TRACE METHOD FOR ELEMENT TIME DELAY ERROR
For general wireless communication systems and especially satellite communications systems, the receive antenna under consideration is very far from the transmitting antenna. Thus, the waves to or from the antenna under consideration can be considered uniform plane waves, and ray tracing can be used to compute the phase or time delay (or error) at each element.
For non-normal incidence (i.e. non-zero angle of incidence), each element in the array will have a varying ray length to the uniform steered wavefront; this is because the reference NT array is flat but reference plane wave front is tilted (with respect to the array). The relative path length per element indexed by ''ij'' (where ''i'' and ''j'' are the position indices in the two-dimensional array) can then be converted into both a time delay ''t ij '' (through the speed of light) or a phase delay ''φ ij '' (through the speed of light and wavenumber), as described by the following equations: In (1)-(5), k = 2πf/c is the wavenumber (with associated frequency ''f'') and c is the speed of light. Note that (1)-(5) are all a function of frequency/wavelength with more nuances being discussed in (6)- (7). The path length L ij is the path length from element ''ij'' to the uniform reference incident plane wave front (or the ''progressive'' path length PL ij ) and includes the ''spatial'' combiner path length (or SL ij ) as a reference. This terminology is similar to that used in [25]. The spatial combiner length SL ij can be thought of as the normalization of the path length from element ''ij'' to combined signal reference point (R ij ); for reflectarrays this is typically the feed horn and for phased arrays this typically is the input of a corporate power divider. We assume the phased array has a uniform path length corporate combiner, thus R ij is constant among all elements and the normalized R ij is zero. The reflectarray, however, has a feed with non-uniform distances to the feed phase center (an assumed spherical wave) and thus the normalized R ij is not zero. The associated geometries for both a 1D and 2D non-timed array are depicted in Figures 3 and 4. The 1D case uses a single index ''i''. For both, the potential reflectarray feed location is suggested by the ''Feed Phase Center.'' The ''collection time'' t ij in equation (2) (from [11]) is frequency independent and purely based on the path length and speed of light, while the phase offset in equation (1) varies with frequency. The practical implications are that perfect phase delay compensation requires ''true time delay'' with no phase wrapping as well as a uniform frequency response.
To model the degradation, the metric that ultimately needs to be considered is the phase and timing error. This can be found from considering an ideal phase shifter limited to  correcting the phase to modulo-2π/360 • phase wrapping. Thus, the error equations on element ''ij'' can be approximated using (1)-(5) as: Equation (6) represents the phase error at element ''ij'' due to phase wrapping from a single frequency, phase limited phase shifter. Equation (7) represents the timing error introduced due to the aforementioned phase shifters effects. Note: the frequency-based effects are ignored in the approximation ''≈ '' of (7) as discussed in Section II, assumption #4 even though in reality the true error deviates from the carrier center frequency by (f o /f).
With the geometrical and error equations now defined, three configuration examples are considered. Each example considers an array length of roughly 0.7 m (or a 51 × 51 element square array) and F/D of 0.8 (the practical industry upper limit due to feed size constraints). Each example considers θ, ϕ as elevation (EL) angle referenced to zenith or azimuth (AZ) angle to suggest directionality or forward and backward half hemispherical scanning. Example #1 ( Figure 5) compares a phased array with a centerfed reflectarray using a steering angle of θ = 60 • (EL) and ϕ = 0 • (AZ). Example #2 ( Figure 6) compares a phased array with an offset reflectarray using a steering angle of θ = 60 • (EL) and ϕ = 0 • (AZ). Example #3 (Figure 7) compares a phased array with an offset reflectarray using a steering angle of θ = -10 • (EL) and ϕ = 0 • (AZ).
It is seen that both antenna type (reflectarray vs. phased array) and steering angle determine the characteristics of the phase and time errors. In particular for the reflectarray, introducing an offset (i.e. placing the feed at one edge) or using a centerfed configuration (i.e. placing the feed in the center of the reflectarray) both with the same associated z-offset or F/D further changes the response. It can be seen in Figure 6 that the offset has a low associated time error for the higher steering angles (due to the spatial path length or SL ij serving to negate the progressive path length or PL ij ); however, this backfires and the terms add together for negative angles as shown in Figure 7. This suggests an offset reflectarray would provide acceptable adaptive performance within a half hemisphere (using a mechanical turn-table to scan the other half, for instance). Conversely, the performance of a phased array vs reflectarray for the centerfed configuration is very similar as shown in Figure 5; suggesting the centerfed configuration of a reflectarray may be best for full hemisphere adaptive applications and have similar timing performance as an equivalent sized phased array.
For all three examples, the carrier center frequency is f o = 11 GHz and the element spacing is dx = dy = λ 0 /2. In all three, the frequency dependent effect is plotted for both time and phase in the left plot of each figure, but is negligible compared to the large modulo wrapping error effect. Thus, it is ignored in the rest of this work (section II, assumption #4) but could be considered in a more complicated receiver model. Also note that in all three reflectarray examples, the included spatial path length is normalized to allow for at least one element to have zero timing error, but in reality, there would be some offset (section II, model step #3,b).

IV. WAVEFORM DEGRADATION FROM TIME DELAY ERROR
Inherent in any digital communication scheme and separate from the physical time delay discussed in the previous section is the symbol timing, consisting of the symbol period/duration ''T sym '' (the time from the start to the end of a single symbol) and the intersymbol separation. Symbols are conveyed in a waveform (t) which has mathematically desirable properties. For example, the ideal waveform for digital communications is the infinite ''sinc pulse'' waveform with sinc pulses spaced by T sym . While not practical due to its infinite nature, it offers simplicity and no inherent ISI. The    infinite waveform can be approximated by a large, finite train of nearest neighbors as described in equation (8), where Q is a sufficiently large number (such as 100). The sinc pulses in (8) are windowed as described in equation (9). The sinc waveform concept is illustrated graphically in Figure 8.
The time delay error can be obtained by evaluating the waveform k at each element ''ij'' with associated timing error t e ij , then summing the responses (with zero padding to equalize domains), and finally normalizing by dividing by the total number of elements (N × M). The result will represent the degraded waveform and is described by equation (10): Equation (10) represents the final model. It is useful to define a ratio of the worst-case element time delay to the symbol rate. This is because we are concerned with high symbol rate satellite communications, where the signal period is typically around 4 ns for consumer terminal applications (see Table 1). Figure 9 shows that if the timing error approaches 2 ns (for example), the symbol smears halfway into the next adjacent symbol. The worst-case smearing ratio can be defined by the ratio: It is apparent that ISI will occur for any finite t w.c. (assuming no time buffer) because now at least one element will contribute a time delayed symbol (which overlaps slightly into the adjacent symbol) to the total combined waveform of (10). However, at lower t w.c. values, the overlap occurs only at the adjacent symbol's edges and thus this ISI will not be sampled (assuming ideal sampling, i.e. section II, step 4) since ideal sampling occurs at the symbol peak not the edges. To avoid confusion, the authors will use the terms ''Sampled ISI'' or (SISI) and ''Non-sampled/edge-only ISI'' or (EISI) going forward. Stated formally, assuming perfect sampling and a linear time delay profile: if t w.c. ≤ 1 then only EISI occurs and not SISI; otherwise for t w.c. > 1, SISI occurs.
In either case, for both SISI and EISI, the normalized waveform response of (10) will always have a sampled peak strictly less than unity. The authors propose that this is an ''efficiency'' which can be expressed as either a magnitude varying between 0 and 1 or in decibels as a dB ''loss'' form from 0 dB to negative infinity dB. This time delay efficiency concept is intuitive because for any finite ISI (SISI or EISI) the waveform now imperfectly combines (or adds together out of sync). This efficiency is used as the basis for the subsequent EsNo degradation modeling with the rationale that EsNo is roughly proportional to gain/loss from a link budget perspective and efficiency can be considered as loss (the same as spillover, illumination, taper, radiation etc. efficiencies are considered as losses in dB form). Note that when SISI occurs, there will be an additional interference term that will add interference gain into an adjacent symbol. To illustrate and further explain these concepts, ''subsection A'' will consider EISI modeling and ''subsection B'' will delve into SISI modeling.
A. EISI MODELING: NON-Sampled/EDGE ISI ( t w .c. ≤ 1) Figure 9 demonstrates the EISI case using a QPSK modulation scheme, 250 MHz symbol rate, with 60 • steering and a 0.68 m length phased array with carrier center frequency of 11 GHz (typical values as outlined in Table 1). The top shows the max/min element delay (i.e. t w.c. ) shifting of the first 5 symbols of the waveform described by (11) depicting a 48% total response of the 5 symbols for all elements after applying (10). Visually, a broadening, attenuation and shifting occurs in the waveform train with spots like ∼ 17 ns clearly showing EISI.
With EISI, there are two waveform degradation metrics to consider: namely, the efficiency (i.e. loss due to the time delay degradation) of the waveform and the waveform sampling time delay. The EISI itself is only considered indirectly since the sampled signal is what is seen at the modem (section II, step 4).
Since equation (10) is normalized, the magnitude at sampling ''η attn '' is already an efficiency with 0 ≤ η attn < 1. The sampling delay ''τ delay '' is defined as the difference between the sampling time ''t samp '' of the time-error normalized sum of equation (10) and the input waveform time ''t 0 '' from equation (9). Thus the following equations can be deduced: Upon inspection, the aggregate response (independent of modulation scheme) for a single pulse (i.e. ''single shot'') can be obtained as: norm total;k (t samp ) ≈ η attn k (t 0 + τ ) This ''single shot'' model is depicted in Figure 10 with the same input parameters as the previous example. Note that the ''single shot'' approach is not accurate for modeling EISI but is accurate for peak modeling. This is acceptable for the model in Figure 1, which ignores EISI (section II, step 4). The efficiency presented in (12) can be converted into an equivalent loss as more appropriate for satellite communications link budget analysis using: While the efficiency of ∼ 0.8 may seem small, it translates to a loss of ∼ 2 dB which greatly reduces the spectral efficiency and DVB-S2X MODCOD operating point.

B. SISI MODELING: SAMPLED ISI (WHERE t w .c. > 1)
Once the aggregate smearing ratio ( t w.c. ) passes the 100% symbol shifting threshold ( t w.c. > 1), the ISI becomes a SISI error. Since the received symbol period is greater than twice the symbol rate (due to the aggregate smearing from the per-element time delay with t w.c. > 1), sampling anywhere  on the symbol (peak or otherwise) leads to self-interference and ISI from the adjacent received symbols.
The previous subsection treated the EISI induced degradation as a scalar efficiency and ignored the actual ISI component of the EISI as it was contained in the symbol edge. The SISI effect, however, cannot ignore the ISI (since it is sampled) and will introduce complex additive energy as an undesirable, directional interference gain into the received symbol towards an adjacent symbol (i.e. shifting the symbol towards the Voronoi region of the adjacent symbol or towards the adjacent amplitude alphabet member in the APSK-case). In all cases, it erodes the useful margin between symbols 96430 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. which is analogous to degrading the EsNo further in addition to the efficiency loss.
This phenomenon is best understood with a constellation diagram which plots all the overlays of the complex magnitude of the sampled signal. An example with a 500symbol-length signal and tw.c. of 1.4 (or symbol rate of 733 Msps given previous assumptions) is plotted (normalized) in Figure 11. Interestingly, the interference produces a diamond-shaped pattern of which the degradation conforms to (or is perpendicular to) the Voronoi regions.
Considering that the relevant loss is the marginal boundary degradation (due to the spectral efficiency analog), and that the degradation is a boundary-conformal diamond shape, the ''Euclidean'' distance metric does not apply and instead the loss topology of the problem lends itself to a ''taxicab'' or ''Manhattan'' distance metric [43]. While the general distance is best expressed by a vector, the authors propose a simpler scalar worst-case efficiency ''α attn '' defined as the minimum distance from normalized symbol with interference to the adjacent boundary of the nearest symbol boundary -see (16).
Equation (16) represents a scalar-generalization of the efficiency concept so far. In the intermediate case where 1 < t w.c. < 2, the efficiency can be understood as the minimum symbol-to-symbol boundary margin across any two distinct regions ''i'' and ''j'' and should be symmetric in d taxi for a linear time delay response. It is seen that in the case t w.c. ≤ 1, equation (16) reduces to (12). The efficiency drops to zero when t w.c. ≥ 2, the point at which the undesirable interference is equal or larger than the desired signal and total destructive interference is possible. Stated differently, this is the point where the margin between Voronoi regions (or the taxicab distance) becomes zero. This is depicted in Figure 12 for t w.c. = 2.
The optimal sampling ''τ '' is the same as the previous section. In these examples (i.e. Figures 9-12) the array delay response is roughly linear (apparent from Figures 5-7) so the optimal sampling point is in the center of the smeared waveform. For a non-linear response (e.g. reflectarray), the optimal point or center of mass may be slightly off the median sample time to locate the apparent ''peak''.
Because ''α attn '' is a worst-case scalar value, equations (14) and (15) are the special case of recasting the worst-case terms. So the generalized time delay efficiency term now yields the following worst-case recasting (denoted by the w.c. in the waveform subscript): norm w.c;k (t samp ) ≈ α k (t 0 + τ ) (17) α ANT dB = 20log 10 (α attn ) (18) While not precisely modeled in this work, the fact that the SISI induces a vectoral spreading means recovering the waveform becomes more difficult. For example, if a low noise amplifier were used to improve the SNR margin, then the ISI would get amplified as well at the same rate meaning the SNR would not improve. Other partial mitigations techniques are discussed in the next section.
To showcase the effect of equations (16)- (17), consider the following parametric analysis: a 0.7 m phased array steered to angles θ = {20 • ,40 • ,60 • }, with ϕ = 0 • , at an 11 GHz carrier center frequency varied by the symbol rate and array size. The resulting efficiency as well as contours of the worst-case time delay are plotted in Figure 13. The time delay induced efficiency degrades to zero at t w.c. = 2 as expected. Note that α attn is affected by a combination of steering angle, symbol rate, frequency, and the array size.

C. NOISE MODELING CONSIDERATIONS
One of the principles behind this approach is that in the DVB-S2X environment, the channel noise is considered additive white gaussian noise or ''AWGN'' and already considered in the MODCOD vs EsNo lookup table [31], [32]. From this observation two fundamental axioms can be developed: 1) given the noise is additive, it can be considered before or after the time delay degradation analysis is conducted (given an appropriate power-scaling factor); 2) fundamentally, the DVB-S2X scheme is robust to noise and already factors it into the acceptable quality and detection assumptions, so only the overall EsNo degradation matters. A corollary can be developed that considering both of these axioms, the time delay degradation does not affect noise itself and so preserves it -and thus noise does not actually need to be modeled to infer the appropriate impact to the DVB-S2X model -only the relative EsNo degradation.
To illustrate this effect, consider the following illustrative example in Figure 14. It uses the same assumptions as Figure 11 ( t w.c. of 1.4) but this time has both appropriate AWGN power levels added before and after the time delay degradation analysis such that the AWGN doesn't affect the decision boundary margin. The only difference from the before/after model is that the output is referenced to a reduced power level α attn from (18) -and the appropriate scaling factor of α attn is used as a reference. The Figure shows that the AWGN effect gives the same results before and after AWGN (in a probabilistic way) and the modeled relative EsNo remains unaffected. Therefore, the authors suggest that the overall noise need not be considered for this DVB-S2X based model as long as the relative EsNo degradation is understood and MODCODs are chosen appropriately.

D. OTHER RF COMPONENT-LEVEL TRANSIENT CONSIDERATIONS
Other non-antenna components could contribute to the time delay transient error effect as well; the phase transient contribution of phase shifters is particularly large [27]. According to a data sheet cited in [27], the scale is multiple orders of magnitude larger for electronic phase shifters; taking on the order of ∼ ms to complete the phase shifting operation vs. the symbol period which is discussed here on the order of ∼ ns. Similar times can be considered for other phase shifters such as servo motors used in mechanically reconfigurable reflectarrays [25]. Given the time scale of these effects is greater than ∼ 10 6 off it really is not appropriate to consider them in this type of analysis. It really just needs to be considered on a macro-level metric, perhaps a black-out time ratio when coherent communications may not be possible due to such a larger timing error vs the symbol rate. At any rate, it is out of scope of this type of analysis where the antenna-level transient is on the same order of magnitude as the symbol rate.
The noise analysis of other components such as local oscillators and amplifiers is very well known. In particular, recent work in high symbol rate devices shows the noise contribution could be additive white, multiplicative white, or primarily phase only [47], [48]. This type of analysis is compatible with the proposed method and could be done in addition to this time delay degradation analysis to form a more holistic picture of the degradation. It is out of the 96432 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
scope of this article to include such analysis (as it is less than modulo 2π phase error so not strictly transient) but the modeling details can be found in multiple sources, e.g. [47] and [48], and could be easily integrated to the analysis proposed here.

V. PARTIAL TIME DELAY MITIGATION TECHNIQUES
With the generalized time delay induced efficiency α attn now defined, consider the relative terminal-level efficiency as ''beta'' described as the logarithmic addition of the generalized time delay efficiency term and the cosine theta ''scan'' loss: β Rel dB = 20log 10 (α ′ attn ) + 10log 10 (cos(θ)) (19) where α ′ attn is α attn after any partial or total/true time delay mitigation or compensation is applied. If no mitigation technique is used α ′ attn is identically α attn . While not explicitly shown, the α ′ attn dependencies (i.e. the input domain) are still the steering angles, symbol rate, frequency, and the array size. The cosine theta term represents the ''scan loss.'' Planar arrays and reflectarrrays have a peak directivity somewhat less than their physical area, and when scanning by an angle ''θ'' off the normal direction, this directivity drops in proportion to cos(θ) (the area projection). As discussed in the introduction, there are methods for ''true'' time delay compensation that can be applied to elements or sub-arrays (e.g. photonics) to eliminate the time delay efficiency term by forcing the per-element t e ij response to be zero, making α' attn zero, and leaving just the cosine theta term. This trivializes the time delay problem and is not considered here.
In a practical sense however, it may not be feasible to have perfect compensation -rather many techniques only provide partial or imperfect compensation. It is worth exploring how the partial techniques could be realized in practice. The following subsections will focus on three of these, namely: equalization-based, spatial delay-based and amplificationbased methods.

A. EQUALIZATION BASED PARTIAL TIME DELAY COMPENSATION
The time delay error modeling approach presented in Sections III and IV is iterative and does not have a closed form solution. However, it is desirable to express the channel as a filter ''h'' in discrete-time closed-form such that the convolution of the filter approximates the time delay effect. Technically this problem is ill-posed, as the response changes iteratively and is influenced by nearest neighbors. The problem can be approximated in the SISI region however by the single shot approximation (noted by superscript ''sing'') for both continuous and discrete time: While a matched filter works perfectly in the single symbol case, the actual system is multi-symbol and may exhibit ISI. However, applying the single shot matched filter to the multi-symbol case (or the actual received/degraded symbol train as described by equation (10) and the term norm total;k ) can still yield some improvements. This can be described by the following approximation: So while the input symbol is not identically recovered, it can be approximately recovered (i.e. the time delay efficiency is mitigated). For more details on matched filtering and other methods containing additional noise terms see [44].
To illustrate this concept, consider the example where t w.c. is 1.4. The inverse filter function is shown in Figure 15 along with the simulated comparison post-lownoise-amplifier with and without the matching filter in Figure 16. The results show a spreading reduction of ∼ 50% which is a significant error improvement.
While this example is simplistic and may not be appropriate for all cases, it illustrates how equalization could be used as signal-processing based alternative to implementing expensive time delay hardware. Many other more effective and complicated equalizer configurations could be examined or perhaps some adaptive equalizer bank could be employed to realize compensation in a physical product.
This formulation does not include noise effects, but more complex approaches can take into account the noise: for example, maximum-a-posteriori equalization is a regularized maximum likelihood optimum trellis-based method that is able to account for the system noise [44]. Future work could consider whether this could be practically implemented as a time delay unit in a similar way as the proposed matched filter.

B. SPATIAL DELAY BASED PARTIAL TIME DELAY COMPENSATION
Another popular technique is based on spatial delay matching (e.g. coupled delay meander lines or offset fed reflectarray compensation). In particular, Figure 6 shows that using an offset reflectarray gives a better response as it is effectively adding an inverse response to the time delay, giving a higher overall delay, but a more normalized one. It is imperfect in the sense that while it works well at high steering angles, it works less well at low angles as depicted by Figure 7: for example, it may need additional features (e.g. a turn-table) to give it full hemispherical scan coverage. Still, it is a viable alternative since a lower per-element time delay will mitigate the overall ISI issues.

C. AMPLIFICATION BASED PARTIAL TIME DELAY COMPENSATION
In the EISI region (when t w.c. ≤ 1) the efficiency is strictly non-vectoral and therefore can be corrected by a low-noise amplifier (LNA). However, any physical amplifier has a finite noise figure (NF), always greater than 0 dB, which serves to necessarily degrade the SNR. Thus, when using an LNA to correct for the EISI, the EsNo cannot be perfectly recovered due to the finite degradation from the LNA. It is therefore important to choose an LNA with a minimal NF to improve the recoverability of the EISI using the LNA. Most practical LNAs have NFs of about 1 dB [38] but better performance can be achieved using techniques such as cryogenic cooling.

VI. RESULTS
The previous sections have provided a framework from which the time delay effects can be modeled for various interrelated, NT array configuration inputs, namely: size, steering angle, frequency (or carrier period), and symbol rate. This section  will contextualize these concepts for the DVB-S2X framework (as described in section II-B with parameters taken from Table 1). This combines the relative terminal-level efficiency given by (19) for an assumed LMRA and EsNo margin to maintain a minimum operational MODCOD(s) (e.g. APSK-3/4) and data rate(s). While we are not able to determine an absolute EsNo without doing an entire link budget for a given system, a relative EsNo degradation and relative margin allowance may be found from the relative terminal efficiency presented earlier. This can be expressed explicitly as a logical classification function δ OP to determine operability given a relative EsNo degradation and a link margin relative allowance (LMRA): A true δ OP is ''operational'' while a false denotes an ''outage''. The α attn dependencies (i.e. the input domain) as expressed in the β Rel dB term include: the steering angles, the symbol rate, frequency, and the size of the array.
It was shown in Section II-B that the absolute EsNo directly relates to the spectral efficiency and thus the MODCOD 96434 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. operating point and data rate, so the relative EsNo degradation can simply be offset to the appropriate absolute EsNo value to characterize any satellite system fully for a given adaptive NT array user terminal. It could take the form of a ''look-up'' loss table which can be incorporated as standard practice in any link budget calculation, albeit an adaptive loss.
There are four dependencies to this adaptive relative loss which suggests there are a few ways to visualize the data. Assuming the frequency point will always be fixed (since from a relative perspective the impact of it is minor), it suggests that the relative EsNo degradation can be visualized by either: 1) fixing array size and varying symbol rate and steering angle, or 2) fixing steering angle and varying symbol rate and array size (the second approach is how Figure 13 was presented).
To help illustrate these concepts, approach 1) is plotted in Figure 17 for a 0.7 m array, and approach 2) is plotted in Figure 18 for ϕ = 0 • , θ = 60 • steering. Both plots assume a 6.01 dB link margin (which is suggested in [38] as an appropriate value) as the quality metric where the minimum desirable operational MODCOD(s) and data rate(s) are used. Both methods are helpful for design purposes in their own right and can be used to understand the fundamental limitations of non-timed arrays. Figures 17 and 18 show the fundamental limitations or upper bounds on non-timed array performance by graphically depicting the relationship between size, steering angle, frequency (or carrier period), and symbol rate and how they contribute in tandem to the relative EsNo degradation. Taking this concept further, a new Figure can be created that is the collection of all possible link operational contours and shows the maximum possible symbol rate a non-timed array can have vs steering angle and size; in particular, the link margin of 6.01 dB and center frequency of 11 GHz is still assumed. Figure 19 presents this and essentially ties all the concepts together so far to give design recommendations on maximum allowable array steering angles and sizes for consumer applications. Now a new contour can be proposed that defines the possible class of antenna: for example, a ''consumer class'' (according to Table 1) would correspond to a contour of 225 Msps. Any steering angle or size combination below this contour would be a fully operational array; anything above would be below the ideal symbol rate. Note that this is a ''recommendation'' rather than a rule because upon violation, throttled transmission is still a viable option as long as the symbol rate is adjusted down.

A. DESIGN RECOMMENDATIONS FOR NON-TIMED ARRAYS
Notice that there is a small sliver between the dashed contour and the red exclusion zone where with typical analysis tools, one might think the phased array would function as intended. While small in this case, at a higher symbol rate (like the second contour for Gsps+ operations) the region is much larger. For example, a 0.5 m array is predicted to only be steerable up to roughly 35 degrees. The plot clearly shows that conventional design approaches fail to properly model non-timed arrays and a more detailed analysis (such as the method proposed here) is needed to fill the gap. Otherwise, improper array sizes or scan angle specifications may be chosen for a given application.

VII. CONCLUSION
With the advent of phased arrays and other NT arrays as consumer terminals in high data rate low-earth orbit satellite communications comes the unique opportunity to understand and characterize time delay waveform degradation in a hybrid, multidiscipline analysis. The authors have presented one method to do this, subject to many simplifications: e.g. optimal sampling, ideal sinc pulse shaping, and no inter-carrier phase error included (only the phase wrapping phenomenon is considered). Thus, the proposed analysis shows the ideal NT array capability. Further work can be done to improve the model with more realistic details. Particularly, to make the method practical for realistic implementation, the major considerations to be added are the timing effects from other RF components and the finite time synchronization error (as discussed in section I).
Partial ISI mitigation techniques are relatively unexplored at the system level. A brief survey of some of these techniques (e.g. equalization, spatial and amplification-based methods) show that they may be advantageous compared to using complicated and expensive true time delay compensation (e.g. photonics). Further modeling or implementation of these or any other mitigation techniques should be topics of future research.

A. IMPACT AND COMPARISON TO EXISTING APPROACHES
The following subsections each consider different aspects of the proposed method compared to existing work. It shows the work in context and the advantages over existing approaches. VOLUME 11, 2023 This work uses a DVB-S2X model whereas other work considers a direct PSK modulation approach [27], [28]. The main advantage with this method is the implicit noise handling and the smooth relationship between EsNo, array time delay degradation efficiency, and max symbol rate.

2) NOISE CONSIDERATION
From a noise perspective, this method implicitly considers noise by looking at the total EsNo degradation output rather than an input as is done in [27] and [28]. The assumption (given AWGN) is shown to hold from the discussion in Section IV-C.

3) WAVEFORM DEGRADATION METRICS
This work considers the time delay degradation directly by considering its normalized result as an efficiency term that can be applied to the relative EsNo degradation. Other methods [27], [28] take a BER degradation approach which isn't directly applicable to frameworks like DVB-S2X that use error correction codes; a similar conclusion is drawn in [27].

4) OTHER RF-COMPONENT CONSIDERATIONS
This paper (Sec. IV-D) reiterates a point from [27] about the timing delays introduced by phase shifters, while can be orders of magnitude greater than the time delays considered here. Although omitted here as not explicitly transient, future work could consider including component noise effects (such as local oscillator and amplifier noise) as it is missing from this but featured in other work such as [47] and [48].

5) MITIGATION TECHNIQUES
A very recent paper [45] proposed a fractional delay filter bank as a form of time delay which is similar to approaches taken in beamforming and data processing in radio astronomy [46]. This is similar to the technique proposed here, but is essentially another quasi-equalization approach. The vast majority of the literature (outside of reflectarray's implicit spatial delay considerations [21], [22]) only proposes true time delay techniques [13].

6) ANTENNA DESIGN RECOMMENDATIONS
References such as [11] and [14] give high level suggestions to not overlap pulses given a simple analytical equation but otherwise no detailed design guidance relating symbol rate, size and steering angle exists outside of this work.