Outage Prediction for Ultra-Reliable Low-Latency Communications in Fast Fading Channels
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Traßl et al.
RESEARCH
Outage Prediction for Ultra-Reliable Low-Latency
Communications in Fast Fading Channels
Andreas Traßl1,2* , Eva Schmitt1 , Tom Hößler1,3 , Lucas Scheuvens1 , Norman Franchi1 , Nick
Schwarzenberg1 and Gerhard Fettweis1,2,3
Abstract
The addition of redundancy is a promising solution to achieve a certain Quality of Service (QoS) for
ultra-reliable low-latency communications (URLLC) in challenging fast fading scenarios. However, adding more
and more redundancy to the transmission results in severely increased radio resource consumption. Monitoring
and prediction of fast fading channels can serve as the foundation of advanced scheduling. By choosing
suitable resources for transmission, the resource consumption is reduced while maintaining the QoS. In this
article, we present outage prediction approaches for Rayleigh and Rician fading channels. Appropriate
performance metrics are introduced to show the suitability for URLLC radio resource scheduling. Outage
prediction in the Rayleigh fading case can be achieved by adding a threshold comparison to state-of-the-art
fading prediction approaches. A LOS component estimator is introduced that enables outage prediction in line
of sight (LOS) scenarios. Extensive simulations have shown that under realistic conditions, effective outage
probabilities of 10−5 can be achieved while reaching up-state prediction probabilities of more than 90 %. We
show that the predictor can be tuned to satisfy the desired trade-off between prediction reliability and
utilizability of the link. This enables our predictor to be used in future scheduling strategies, which achieve the
challenging QoS of URLLC with fewer required redundancy.
Keywords: channel prediction; URLLC; radio resource scheduling
1 Introduction could cause damage or even human harm. Latency-
One of the main pillars of the fifth generation (5G) mo- critical mobile connectivity is also required when hu-
bile broadband standards is ultra-reliable low-latency mans are involved in the control loop [4, 5]. In indus-
communications (URLLC), which aims to provide ex- try, this is the case, e.g, in teleoperating applications or
tremely high service availabilities paired with latency during installation of machines, where a human could
values of only a few milliseconds. To realize even more train the machine instead of programming it. In the
ambitious quality of service (QoS) requirements com- future, URLLC might also find its way to consumer
pared to 5G, URLLC inevitably has to play a key role products for entertainment, for health or even for hu-
also during research of the sixth generation (6G) mo- man learning.
bile broadband standards [1, 2]. The major challenges for URLLC are the imperfec-
The ongoing development of URLLC is driven by a tions of the wireless link, especially the fast fading of
wide variety of applications. In recent yeas, many of the channel. Due to reflections in the environment,
these applications were industry-focused, where wire- many copies of the transmit signal arrive at the re-
less solutions allow for shorter product cycles, more ceiver simultaneously and interfere which each other.
product individualization and an overall increased flex- When the transmitter or the receiver moves, the chan-
ibility [3]. One major challenge is wireless closed-loop nel conditions continuously change since the waves in-
control as losing packets and the latency of the trans- terfere differently at different locations [6]. In the best
mission might lead to plant instability, which in turn case, all signals constructively add up at the receive
antenna. In the worst case, however, all signals de-
*
Correspondence: andreas.trassl@tu-dresden.de structively cancel each other out, effectively leading to
1
Vodafone Chair Mobile Communications Systems, Technische Universität
Dresden, Germany
zero receive power. Situations where the receive power
2
Centre for Tactile Internet with Human-in-the-Loop (CeTI) is low, so-called outages, have to be avoided for the
Full list of author information is available at the end of the article successful realization of URLLC.
This is a post-peer-review, pre-copyedit version of an article published in EURASIP Journal on Wireless Communications and Networking. The final authenticated version is available online at:
http://doi.org/10.1186/s13638-021-01964-wTraßl et al. Page 2 of 16
Different from well-established conventional ap-
10 0
proaches that cost severe resources either in hard-
ware (spatial diversity through many antennas and
signal processing chains) or at the air interface (band- 10 -1
width), the basic concept of this article is to monitor
the fast fading channel and schedule users only to re-
sources that are operational. Due to the spatial vari- 10 -2
ation of the channel, a resource that is in outage for
one user can have perfect channel conditions for an-
10 -3
other. Therefore, this approach is expected to realize
URLLC’s ambitious QoS targets while keeping the re-
quired additional resource consumption low. This is of 10 -4
utmost importance when considering the scalability of 0 0.1 0.2 0.3 0.4
a URLLC deployment, i.e., when many devices need
to be served simultaneously. The resource consump-
tion becomes even more important when realizing high Figure 1 Performance of outage identification, when relying
purely on the latest channel observation
payload applications with latency requirements, e.g.,
cloud rendered virtual- or augmented reality.
Rapidly varying channel conditions are challenging
when monitoring fast fading channels. In a real deploy- • An analysis of the predictor parameters for Ri-
ment, a monitoring delay τ exists between receiving cian fading is presented. The parameters define
the last channel observation, scheduling and eventu- the number and periodicity of channel observa-
ally transmitting the actual payload. Consider the case tions at the input of the predictor.
that users were scheduled based on the last channel ob- • Performance metrics tailored to the use-case of ra-
servation. In this case, many actual outages would be dio resource scheduling for URLLC are proposed.
missed; i.e., the channel is monitored to be operable • Performance evaluation is conducted by means of
at time t, but non-operable during payload transmis- extensive simulation. Compared to our previous
sion at t + τ . The performance for Rayleigh fading is works [7, 8], generalized results are provided.
visualized in Fig. 1, where the effective outage proba-
bility of such a system Pr(effective outage) is plotted 2 Methods/Experimental
against the Doppler frequency normalized monitoring This article aims to study the performance of a novel
outage prediction scheme in Rayleigh and Rician fad-
delay τ fm for different fading margins F . It is clearly
ing channels. The predictor combines a Wiener filter
visible that most of the monitoring gain disappears al-
with a threshold comparison for the identification of
ready for small monitoring delays. In other words, with
future outages. In the Rician fading case, additional
increasing delay the monitoring quickly becomes point-
line-of-sight (LOS) parameter estimation is employed.
less and the effective outage probability asymptotically
The focus during design and analysis of the prediction
converges towards the average link outage probability. scheme is set on a prospective application to URLLC
A solution to overcome this time delay is to employ radio resource scheduling.
predictive methods. Although the fast fading chan- The performance of the outage prediction scheme is
nel conditions change quickly, they still change con- analyzed using analytical statements and Monte-Carlo
tinuously, which enables predictions. The main con- computer simulation for a practical set of parameters.
tribution of this article is to describe the design and Noisy channel coefficients are generated randomly and
the performance of an outage predictor. This article is fed into the predictor. The predictions are then com-
based on and extends our previous works [7, 8], where pared with the respective true future value. Evaluation
outage predictors for Rayleigh and Rician fading were is conducted using classical binary classification anal-
proposed for the first time. The contributions of this ysis and application-related metrics that are proposed
article are summarized as follows: in this work. The number of repetitions is individu-
• Outage predictors for Rayleigh and Rician fading ally specified during discussion. The underlying data
channels are described. of this study is generated from mathematical models,
• The Rayleigh und Rician fading outage predictors which are completely described in Sec. 4.
are compared. Their differences are highlighted
and their area of application is contrasted. 3 Related Work
• Relevant related work regarding fading prediction Estimating the current state and predicting future be-
is discussed. havior of wireless channels has been a research chal-
This is a post-peer-review, pre-copyedit version of an article published in EURASIP Journal on Wireless Communications and Networking. The final authenticated version is available online at:
http://doi.org/10.1186/s13638-021-01964-wTraßl et al. Page 3 of 16
lenge for many years. As expected, the used estimation to OFDM, [15] also exploits channel prediction for
methods have been evolving along with the wireless adaptive frequency hopping by adapting transmission
communications technologies and standards. parameters to the channel conditions at the next cho-
With the ever-growing demand of higher transmis- sen frequency. Other approaches include the estima-
sion rates without sacrificing transmission quality in tion of time varying fading parameters with the help
terms of bit error rate (BER), the availability of chan- of Kalman filter variants to enhance the prediction
nel state information became a necessity. Adaptive performance [17]. In [18], a Kalman filter is used to
transmission techniques allow a more efficient resource directly predict Rayleigh fading. The used state space
usage, e.g., by choosing the modulation scheme ac- model is based on the SOS model. Recently, opposing
cording to the current fading conditions [9]. Early ap- to statistical methods, machine learning approaches
proaches were based on the sum-of-sinusoids (SOS) have been proposed for fading channel prediction. For
modelling. The deterministic fading modelling is based example, in [19] a back propagation neural network is
on the estimation of Doppler shifts, amplitudes and used to predict I/Q channel coefficients in Rayleigh
phases of the overlaid sinusoids through ESPRIT- and fading. A recurrent neural-network-based approach is
MUSIC-based algorithms [10, 11]. This is done un- used in [20] and analyzed in terms of average pre-
der the assumption that scatterers in the environment diction errors and bit error rates. Those data-driven
remain constant and the process is dominated by a approaches do not require any modelling or parame-
few dominant scatterers. The first advances were fol- terization.
lowed by many variants of auto-regressive (AR) ap- In today’s 5G and future’s 6G mobile broadband
proaches. These model predicted channel samples as a standard, very high data rates are only one of the en-
weighted sum of previous samples using the minimum visioned features. For the realization of URLLC, more
mean square error (MMSE) criterion. This method re- attention needs to be directed towards the reliability
quires an estimate of the correlation function of the of transmissions rather than solely maximizing spec-
channel samples. Promising results were obtained, e.g. tral efficiency. Consequently, research on fast fading
in [9], where a Wiener filter is used for the predic- channel prediction has to perform a paradigm shift
tion of in-phase and quadrature (I/Q) Rayleigh fading as well. To allow the scheduling system to achieve a
channels. The predictor is tested using data generated certain QoS, the predictor needs to be built around
by ray-tracing and real data from vehicular channel suitable reliability measures. For URLLC, the analysis
measurements. Similar investigations were performed of average prediction errors and BERs is not enough
in [12, 13]. In these works, an unbiased power predic- anymore. Under the URLLC premise, only very few
tor is additionally derived based on solely the channel investigations have been conducted. In [21, 22] coop-
power instead of the complex channel gain. The works erative communications schemes, where messages are
[9, 14, 15] extended the predictor by the use of effi- transmitted over multiple relays, are investigated for
cient adaptive filtering techniques to update the pre- URLLC. The authors employ fading monitoring and
dictor coefficients in case of varying long-term channel prediction to choose the most suitable relays. It is
conditions. Motivation for investigating fading predic- shown that the coherence time is an insufficient metric
tion techniques was exclusively the raise of spectral to quantify the reliability of fading prediction methods.
efficiency. This article contributes to fill this gap and provide
When orthogonal frequency-division multiplexing prediction methods and metrics for general URLLC
(OFDM) became part of the fourth generation (4G) architectures.
mobile broadband standard, channel prediction re-
search was directed towards adaptive bit allocation in 4 System Model
OFDM symbols, aiming again to increase the spectral The overall system model considered in this article is
efficiency. In [16], the MMSE predictor, the Wiener shown in Fig. 2. It is assumed that the user equipments
filter predictor, as well as the adaptive methods nor- (UEs) periodically transmit training signals, which are
malised least mean squares (NLMS) and recursive least used to acquire channel information between the base
squares (RLS) are derived for a single input single out- station (BS) and the individual UE. We assume chan-
put (SISO) OFDM system. The predictors are com- nel reciprocity such that by measuring the uplink (UL)
pared in terms of their mean prediction errors. Sim- channel, the downlink (DL) channel can also be rated.
ilarly, for SISO OFDM systems, a simplified MMSE This is practical after a calibration phase to account
predictor and its extension to the adaptive least mean for differences in the circuits of transmitter and re-
squares (LMS) and RLS techniques were proposed ceiver as shown in [23]. Monitoring the uplink channel
in [14]. Evaluations were conducted in terms of av- is preferred over the downlink since thereby the neces-
erage prediction error and spectral efficiency. Similar sary information is directly available to the scheduler
This is a post-peer-review, pre-copyedit version of an article published in EURASIP Journal on Wireless Communications and Networking. The final authenticated version is available online at:
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The variance of the complex channel coefficient 2σ 2 is
Periodic Training Signal
then determined by the mean power of the channel
ΩNLOS according to 2σ 2 = ΩNLOS .
Base Station An underlying assumption for the widely assumed
Fading Channel
URLLC UEs
classical Doppler spectrum is that waves arrive solely
Channel Estimation in the horizontal plane with equally distributed an-
gles of arrival. The UE is considered to move with
Outage Prediction
a constant velocity v in an otherwise static environ-
Scheduler ment, which results in a maximum Doppler shift fm .
It is well-known that this leads to the classical Doppler
spectrum with its autocovariance function
Scheduling Decision
rNLOS (t1 , t2 ) = 2σ 2 J0 2πfm (t2 − t1 ) .
(2)
Figure 2 Overall system model; The outage predictor is
highlighted and topic of this article Thereby, J0 denotes the zeroth order Bessel function
of the first kind.
at the BS. The channel estimations are fed into the 4.1.2 Rician fading
outage predictor, whose design and performance will When additionally allowing for a LOS component, the
be the main topic of this article. For each monitored Rician fading case arises with its I/Q channel coeffi-
carrier frequency, the predictor calculates for the next cient [24]
possible UL and DL scheduling opportunity if the re- √
spective link is operational or not. Based on this infor- h(t) = 2σ · hNLOS (t) + A · hLOS (t) . (3)
mation, the scheduler allocates resources to the UEs. √
As usual, the scheduling decision is transmitted to the The NLOS component 2σ · hNLOS (t) is similar to the
UEs. In the following, the necessary assumptions for Rayleigh fading case described above. The LOS com-
the fading channel and the communications system are ponent A · hLOS (t) is modeled to be purely determin-
explained. istic following
4.1 Fading Channel A·hLOS (t) = A·exp j(2πfD,LOS (t−t0 )+ϕ0 ) . (4)
In this article, we first consider Rayleigh fading with
classical Doppler spectra before extending our find- In this formula, A is the amplitude, fD,LOS is the
ings to the Rician fading model. Rayleigh fading can Doppler frequency, ϕ0 is the initial phase of the LOS
be used to model challenging non-line-of-sight (NLOS) component and t0 is the reference time at which the
conditions and is often used as a starting point for phase of the LOS component equals the initial phase
analysis due to its beneficial mathematical properties. ϕ0 .
In the second part of this paper, we extend our findings In Rician fading the K-factor defines the ratio of
to the Rician fading case by allowing for a LOS com- power in the LOS component ΩLOS over the power in
ponent. By doing so the question how the presence of the NLOS component ΩNLOS
a LOS component affects the prediction performance
is answered. ΩLOS A2
K= = . (5)
ΩNLOS 2σ 2
4.1.1 Rayleigh fading
Rayleigh fading assumes that numerous independent Thus, the standard
√ deviation of the complex NLOS
multi-path components arrive at the receive antenna component 2σ and the amplitude of the LOS com-
simultaneously. In this case, the central limit theorem ponent A can alternatively be expressed over the K-
can be applied and therefore the real and imaginary factor and the average power Ω = ΩLOS +ΩNLOS using
part of the channel coefficient h(t) can be modelled as
√
r r
Gaussian distributed. Thus, the channel coefficient Ω ΩK
2σ = , A= . (6)
√ K +1 K +1
h(t) = 2σ · hNLOS (t) , (1)
For the special case of K = 0, (1) and (3) coincide.
follows a zero mean complex Gaussian distribution Thus, the more general Rician fading model also in-
with variance 2σ 2 , since we define hNLOS (t) ∼ CN (0, 1). cludes the Rayleigh fading case.
This is a post-peer-review, pre-copyedit version of an article published in EURASIP Journal on Wireless Communications and Networking. The final authenticated version is available online at:
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|havg | With knowledge of the sent pilot symbols at the re-
-80 ceiver, the influence of the fading can be observed
|h(t)| [dBm]
up
-90 |hmin | by estimating the complex channel coefficient h(t).
For this purpose, we use the minimum variance un-
-100
outage
biased (MVU) channel estimator [25]
-120
-130 ĥ(t) = (pH p)−1 pH y , (9)
t
Figure 3 Two state fading model
which corresponds the least squares (LS) and maxi-
mum likelihood (ML) channel estimator. Inserting (8)
in (9) yields
4.1.3 Two State Fading Model ĥ(t) = h(t) + n0 (t) . (10)
Our predictor is built upon an abstract fading model
in which the fading is classified in two states up and Thus, under the given assumptions the estimate ĥ(t)
outage depending on the channel gain. This fading is superimposed by CWGN n0 with variance 2σn2 0 =
model is depicted in Fig. 3. The respective fading 2σn2 (pH p)−1 . When only one pilot is used for channel
state is based on the relation of the channel gain to estimation, 2σn0 2 = 2σn2 applies. The relationship be-
a chosen threshold value |hmin |. A different form to tween channel estimation and noise (10) is the starting
characterize the threshold |hmin | is the fading margin point for the derivation of the predictor. The perfor-
F = |havg |2 /|hmin |2 , which relates the threshold |hmin | mance of the predictor will be determined by the SNR
to the average channel gain |havg |. When the chan- of the channel estimation, SNR = 2σΩ0 2 .
n
nel gain is greater than the threshold |h(t)| > |hmin |
the current fading state is denoted as up. In the up- 5 Problem Statement
state the signal/noise ratio (SNR) at the receiver is Prediction of the channel state involves so-called bi-
high enough for an URLLC application to be work- nary classification.Usually, the results of the classifica-
ing satisfactory. Packet errors are still possible in the tion problem are simply called positive and negative.
up-state, but the probability for an error is low and Since the outage predictor aims at predicting outages,
long error bursts are rare. Analogously, an outage oc- this article defines the up-state prediction as the neg-
curs if the channel gain is below the threshold value ative and the outage prediction as the positive classi-
|h(t)| < |hmin |. In outage, the SNR is usually too low fication result. In binary classification, four potential
for successful decoding, leading to high probabilities outcomes exist. Apart from true positive and true neg-
of packet errors and long error bursts. Following these ative (correct classification), also the two error types
considerations, a URLLC service is expected to work false positive and false negative prevail. In the context
satisfactory in the up-state and fail in the outage state. of outage prediction these four outcomes represent the
following:
4.2 Communications System and Channel Estimation • true positive – detection of a future outage
For the communications system we assume that the • true negative – detection that an outage will not
transmission between the UE and the BS is affected occur
only by the fading of the wireless channel, resulting • false positive – miss that an outage will not occur
in the complex channel coefficient h(t), and complex • false negative – miss of a future outage
white Gaussian noise (CWGN) n(t) with variance 2σn2 . The evaluation of such binary classification problems
Hence, the transmit signal x(t) and the receive signal is well known and various metrics for different applica-
y(t) are related by tions are available. Three important metrics are sum-
marized from [26], where a good overview is given. An
intuitive metric for evaluation of a classifier is its ac-
y(t) = x(t) · h(t) + n(t) . (7)
curacy, which is defined as
To acquire information about the wireless channel, a TP + TN
column vector p consisting of P pilot symbols is trans- accuracy = . (11)
TP + FP + TN + FN
mitted for the estimation of h(t). Thus, when taking
(7) into account, this leads to In this formula TP is the number of true positives and
TN is the number of true negatives. Similarly, FP is
y = p · h(t) + n . (8) the number of false positives and FN is the number of
This is a post-peer-review, pre-copyedit version of an article published in EURASIP Journal on Wireless Communications and Networking. The final authenticated version is available online at:
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false negatives. An accuracy of 0.8 means that 20 % of
the prediction results are wrong. However, by investi- Channel Estimation
gating the accuracy metric alone it is impossible to say
if these errors are false positives or false negatives. The I/Q Prediction
metric is of limited benefit for URLLC outage predic-
tion since the error types have a different impact on the Comparison with Prediction Error
reliability of the wireless link. In the context of QoS- Threshold |h0min | Analysis
focused URLLC, false negative classifications have a
much higher impact than false positives as they can be Future
a cause for transmission errors which ultimately lowers Outage Prediction
Outage Probability
the reliability of the wireless communications system.
This emphasizes the importance of choosing the right
Figure 4 Structure of the outage predictor for Rayleigh fading
metric. Investigating the individual probabilities for
correct or wrong classification allows for more explicit
statements. Two common metrics to fully describe the
classifier are the probability of false alarm (also known to assess the utilizability of the resource, i.e., how
as the false positive rate) often the observed link can be used for URLLC
traffic of a specific UE. The more false alarms oc-
FP
Pr(false alarm) = (12) cur, the lower Pr(predicted up) gets. For example,
TN + FP Pr(predicted up) = 0.8 indicates that the observed
and the probabability of detection (also known as the link can be considered for URLLC traffic in 80 % of
true positive rate) the time, whereas in the remaining 20 % the link will
not be assigned to that particular UE for transmission.
TP The ultimate goal for the predictor is to maximize
Pr(detection) = . (13)
FN + TP Pr(predicted up) and minimize Pr(effective outage)
given a certain prediction horizon tp .
The above metrics are well suited to investigate bi-
nary test results. However, they do not provide in-
tuitive interpretation in the context of URLLC radio 6 Outage Prediction
resource scheduling. For example, in the case of very In the following, we describe the structure of the out-
high Rician K-factors even Pr(detection) = 0 (all out- age predictor. We first address the Rayleigh fading case
ages are missed) could be acceptable as outages are and pursue to the more complex Rician fading after-
already very rare. Therefore, Pr(detection) has only wards.
little qualitative meaning if the predictor performs well
enough for scheduling purposes. Here, we propose two 6.1 Rayleigh Fading Prediction
new metrics: the compound probability for an up-state The outage predictor for the Rayleigh fading case is
prediction, but the channel being truly in outage shown in Fig. 4 and its structure is briefly explained
FN here before providing mathematical details.
Pr(effective outage) = (14) The starting point for outage prediction is a history
TP + FP + TN + FN
of channel estimations which is collected at the input
and the average probability to have an up-state pre- of the outage predictor. Afterwards, I/Q channel coef-
diction on the monitored link ficients need to be predicted from the available channel
estimations by an appropriate prediction technique.
FN + TN
Pr(predicted up) = . (15) In order to obtain a binary prediction for the chan-
TP + FP + TN + FN nel state (up or outage), the predicted I/Q channel
First, Pr(effective outage) covers the risk of fatal fail- coefficient is compared with a threshold |h0min |. Sub-
ures due to prediction errors. Thus, this metric en- sequently, an outage prediction is available which can
ables statements about the reliability of the system. be used, e.g., for scheduling purposes. For the Rayleigh
For example, Pr(effective outage) = 10−3 indicates fading case, the exact distribution of the I/Q predic-
that on average 1 in 1000 predictions will result in tion error is known at the Wiener filter output. Thus,
an outage. Here, we assume that a (perfect) sched- additionally to the outage prediction also the probabil-
uler can prevent any predicted outage, since the de- ity for a future outage can be calculated for future time
sign of the scheduler is beyond the scope of this ar- instants. In the next sections a detailed description of
ticle. Second, Pr(predicted up) is defined in a way each block in Fig. 4 is provided.
This is a post-peer-review, pre-copyedit version of an article published in EURASIP Journal on Wireless Communications and Networking. The final authenticated version is available online at:
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6.1.1 I/Q Prediction
Based on the history of channel estimations, predic-
tions of I/Q channel coefficients need to be calcu-
lated. For this purpose, a well investigated Wiener-
filter-based approach is employed [9, 12]. It was shown
that the Wiener filter has a promising performance
not only under the Rayleigh fading assumption, but
also for real fading channels where empirical covari-
ances need to be utilized. In contrast to machine learn-
ing based approaches, analytical statements about the
prediction error can be derived, which allows calcu-
lation of future outage probabilities in the Rayleigh
fading case. For these reasons, the Wiener filter was
preferred over other available fading prediction tech-
niques. Nevertheless, if only the outage prediction is of
interest, other I/Q prediction techniques can be eas-
ily incorporated into the proposed framework as well
and replace the Wiener filter. The key statements to
implement the Wiener filter for the Rayleigh case are
summarized from [12]. Figure 5 Outage prediction concept; A threshold for the
prediction |h0min | different from the outage threshold |hmin | is
For a prediction horizon tp , the prediction of the I/Q introduced to tune the prediction uncertainty
channel coefficient
ĥ(t + tp | t) = ϕθ (16) not in measured fading channels, since the autocovari-
ance is directly estimated in this case anyway. There-
is the output of a finite impulse response (FIR) filter fore, we do not introduce estimators for these parame-
with coefficients θ. The observation vector ters and instead assume that they are known through-
out the article. Using the outage predictor for mea-
sured fading channels is beyond the scope of this arti-
ϕ = ĥ(t) ĥ(t − ∆t) ... ĥ(t − (M − 1)∆t) (17)
cle and instead left for future work.
contains M past channel estimations with a fixed time
6.1.2 Comparison with Threshold
between the observations ∆t. The filter coefficients
Since we are interested if an up-state or an outage will
occur, the predicted I/Q channel coefficient is com-
θ = R−1
NLOS r NLOS (18) pared with the threshold |h0min |. Our idea is to choose
a different threshold value for the predicted channel co-
are calculated from the cross-covariance between chan- efficient |h0min | and not the threshold in the two state
nel coefficient and observations r NLOS and the autoco- fading model |hmin |. This idea is depicted in Fig. 5.
variance matrix RNLOS of the observations according By using the threshold |h0min | for the prediction, we
to are able to adjust the trade-off between the effective
outage probability and the probability for an up-state
[r NLOS ]j = 2σ 2 J0 2πfm (tp + (j − 1)∆t) ,
prediction as discussed in Sec. 5. The objective is to get
a more conservative predictor, such that falsely pre-
(19) dicted up-states are rare, which allows the predictor
2
2σ J0 (2πfm |j − i|∆t), i 6= j to be used for URLLC scheduling. In return, falsely
[RNLOS ]ij = . predicted outages occur more frequently, which the
2σ 2 + 2σn2 0 , i=j
(20) scheduler has to deal with. Numerical evaluation of
this trade-off is presented in Sec. 7.3.
To design the Wiener filter, knowledge about the 6.1.3 Prediction Error Analysis
variance 2σ 2 , the maximum Doppler frequency fm and For the given assumptions in Sec. 4 it can be shown
the noise variance 2σn20 is needed. However, knowledge that the prediction error
of these parameters is only required in a model-based
analysis, which we concentrate on in this article, and e(t) = h(t + tp ) − ĥ(t + tp | t) (21)
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Channel Estimation
Parameter
Estimation
Subtraction of the Â, fˆD,LOS , ϕ̂0
LOS Component
I/Q Prediction
Addition of the Â, fˆD,LOS , ϕ̂0
LOS Component
Figure 6 Illustration of the integration area to calculate the Comparison with
future outage probability; The red circle marks the integration Threshold |h0min |
area and has radius |hmin |
Outage Prediction
follows a zero mean complex Gaussian distribution
Figure 7 Structure of the outage predictor for Rician fading
2σe2
e(t) ∼ CN 0, . (22)
This originates from the fact that both h(t) and Here fe(t) (x, y) is the zero mean bivariate Gaussian
ĥ(t+tp | t) are zero mean complex Gaussian distributed probability density with variance σe2 for both dimen-
and therefore also their difference follows a zero mean sions I and Q. As there is no closed-form solution avail-
complex Gaussian distribution. Similarly, the filter- able for this integral, it must be evaluated numerically.
ing operation in (16) does not change the distribu-
tion type, as scaled and summed zero mean complex 6.2 Rician Fading Prediction
Gaussian random variables are again zero mean com- We now extend the outage predictor for the more gen-
plex Gaussian. Thus, the distribution of (22) is com- eral Rician fading, where not only NLOS fading, but
pletely parameterized by the variance of the prediction also a LOS component is present. The structure of the
error [12] outage predictor for the Rician fading case is presented
in Fig. 7.
2σe2 = IE |e(t)|2 = 1 − r T
−1 Rician fading has a non-zero I/Q mean generated
NLOS RNLOS r NLOS . (23)
by the LOS-component, which is incompatible with a
Knowing the distribution of the prediction error, Wiener filter prediction. Therefore, the strategy when
a predicted channel coefficient value can now be dealing with non-zero mean processes is to subtract
associated with the probability of outage. We de- the mean before filtering and adding the mean back
note the probability for a future outage given a again at the Wiener filter output [27]. As the time
certain predicted channel coefficient ĥ(t + tp | t) as varying LOS-component can hardly be assumed to be
Pr(future outage). As illustrated in Fig. 6, a future known, the outage predictor in the Rician fading case
outage occurs when the prediction error e(t) lies in has to employ estimators for the LOS parameters A,
the complex plane within an area S of a circle around fD,LOS and ϕ0 as first step. The estimated LOS pa-
−ĥ(t + tp | t) with radius |hmin |. This is because the rameters lead to the full description of the LOS com-
ponent at time t. After subtracting it from the history
sum of predicted channel coefficient ĥ(t + tp | t) and
of channel estimations, the filter coefficients are cal-
prediction error e(t) is the true value of the future
culated and the NLOS component can be predicted
fading (rearranged version of (21)). Consequently, for
equal to the Rayleigh fading case. In parallel, the LOS
a prediction error within the area S the true chan-
parameters are used to calculate a prediction of the
nel coefficient lies within the outage region. Therefore,
LOS component at time t + tp which can then be
Pr(future outage) is determined by a double integral
added to the Wiener filter output. This leads to a pre-
over the area S according to
dicted I/Q channel sample, which can be thresholded
Z against |h0min | to obtain an outage prediction. All steps
Pr(future outage) = fe(t) (x, y) dS . (24) from the LOS parameter estimation to the comparison
S
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with the threshold, are repeated when a new prediction consists of N values and is sampled at a discrete sam-
needs to be calculated. pling period ∆t. Furthermore, a vector of exponential
When comparing the outage predictor in Fig. 7 with terms
Fig. 4, it can be seen that for Rician fading no outage
probability is calculated. This is due to the fact, that e = exp −j(2πfD,LOS (N − 1)∆t) ...
it is not possible to analytically calculate the error dis-
exp −j(2πfD,LOS ∆t) 1 (27)
tribution of the introduced LOS parameter estimation.
The result is that also the distribution of the prediction is part of the estimator. Since the true value of the LOS
error is unknown and outage probabilities cannot be Doppler frequency fD,LOS could be located between
calculated. The same problem arises when measured the bins of the periodogram, the frequency estimation
fading is predicted and the assumptions about the dis- can be greatly improved by interpolation as shown in
tributions do not hold anymore. In the rest of this [29]. The authors propose an iterative approach, which
section, we explain the individual predictor elements we also employ in this article to refine the frequency
shown in Fig. 7 in detail. estimate (25). As suggested by the authors, we also
use two iterations.
6.2.1 Parameter Estimation The ML estimates for the remaining parameters can
Different from the Rayleigh fading case, a time vary- then be calculated by inserting the frequency estimate
ing LOS component is present and the parameters A, in (27) (we denote this vector as ê in the following)
fD,NLOS , ϕ0 need to be estimated from a history of and using
channel estimations.
When considering (10) as an estimation problem, ϕ0 êH
 = , (28)
where 2σ 2 , A and fD,LOS are the unknown parameters, êêH
both √the CWGN n0 (t) and the random NLOS compo- ( )
ϕ0 êH
nent 2σ·hNLOS (t) act as noise. We neglect√ the tempo- ϕ̂0 = arg . (29)
ral correlation of the NLOS component 2σ · hNLOS (t) êêH
for parameter estimation, since the complexity of the
derivation is lower and the resulting estimators still The estimator (29) yields a phase estimate of the last
perform very well in our outage prediction use-case. element in (26), which is preferable from a prediction
With both the CWGN and the NLOS component be- point of view. Combining the estimates fD,LOS , Â and
ing complex Gaussian distributed the sum is complex ϕ̂0 gives an estimate of the LOS component
Gaussian, too, and can be combined into a single vari-
· ĥLOS (t) = ·exp j(2π fˆD,LOS (t−t0 )+ ϕ̂0 ) . (30)
able.
This leads to the standard problem of estimating the
6.2.2 I/Q Prediction and Comparison with Threshold
parameters of a complex sinusoid in CWGN, which can
With the available parameter estimations, a prediction
be tackled using a ML estimation approach as shown
of the I/Q channel coefficients in the Rician fading case
in [25]. For the special case of noiseless Rician fading
can be performed. Since we are again using a Wiener
the desired ML estimators were derived in [28]. In the
Filter which relies on the input to be zero mean, a
following, we adapt the estimators from [28] and em-
prediction of future I/Q channel samples
ploy an optimization to the frequency estimation.
A ML estimation of the frequency
ĥ(t + tp | t) = ϕ − Â · ĥLOS θ + Â · ĥLOS (t + tp ) (31)
2
!
ϕ0 eT consists of the estimated LOS component at prediction
fˆD,LOS = −arg max (25)
eeH time  · ĥLOS (t + tp ) added to the FIR filter output
ϕ − Â · ĥLOS θ, with the observation vector of the
is found by maximizing the periodogram with respect Wiener filter ϕ being adjusted for the estimate of the
to fD,LOS . Since the periodogram is the square of a dis- LOS component vector
crete Fourier transform (DFT), a practical implemen-
tation would utilize the fast Fourier transform (FFT) Â · ĥLOS = Â · ĥLOS (t0 ) ĥLOS (t0 − ∆t)
algorithm. The observation vector of the LOS estima-
... ĥLOS (t0 − (M − 1)∆t) (32)
tion
at the input of the filter. Since after subtraction of the
ϕ0 = ĥ(t − (N − 1)∆t)
... ĥ(t − ∆t) ĥ(t) (26) LOS component only the NLOS fading remains, the
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Table 1 Values for numerical evaluation
Parameter Value Throughout the whole section, the normalized predic-
Rician K factor 0, 5, 10 tion horizon tp fm is arbitrary set to 0.1. The results of
mean channel estimation SNR 20 dB, 10 dB this section were generated by means of computer sim-
fading margin F 10 dB
ulation according to the Monte-Carlo approach. For
each point 4 × 105 predicted fading samples were com-
pared with the respective ideal future fading value.
filter coefficient can be calculated in the same way as
in the Rayleigh fading case, shown in Sec. 6.1.1. Also, 7.2.1 Sampling Period
the comparison with the threshold is no different from In Fig. 8, the mean squared error (MSE) of the I/Q
the Rayleigh fading case in Sec. 6.1.2. prediction is plotted against the normalized sampling
period ∆tfm for a fixed history length of the Wiener
7 Results and Discussion filter and the LOS estimator. Although the MSE is
In this section, the performance of the outage predictor not suitable to describe the outage prediction perfor-
for the Rayleigh and the Rician fading case is analyzed mance directly, it can be used as a performance in-
numerically. The scenario and the chosen parameters dicator. Generally speaking, the higher the error of
for numerical evaluation is described in Sec. 7.1. After the I/Q prediction, the worse the outage prediction
investigating the influence of the predictor parameters performance after comparing the predicted I/Q chan-
in Sec. 7.2, the performance evaluation of the outage nel coefficient with the threshold |h0min |. When look-
prediction schemes is conducted in Sec. 7.3. ing at the curves in Fig. 8, it can be seen that very
small as well as very large sampling periods do not
7.1 Scenarios perform well for the investigated SNRs and K factors.
The numerical evaluation is conducted for selected For a fixed history length, very small sampling periods
numerical values. In the following, three different K- ∆t result in the observation not spanning enough to
factors are investigated to reflect the case of a strong capture the continuous variation of the fading. Simi-
LOS component (K = 10), a medium LOS component larly, very large sampling periods ∆t, which are greater
(K = 5) and no LOS component (K = 0, Rayleigh fad- than the coherence time, lead to uncorrelated observa-
ing case). The performance is shown for two different tions. According to the popular rule of thumb from
mean channel estimation SNRs of 20 dB and 10 dB. In [6], the coherence time tcoh can be approximated as
both cases, the fading margin is set to F = 10 dB. All tcoh = 0.423
fm , which is close to the point in Fig. 8 where
plots in the following sections are generalized by using the MSE begins to rise steeply. In all cases, a long
times that are normalized to the maximum Doppler plateau of the MSE can be observed, where a wide
frequency fm . This allows the results to be used for range of sampling periods ∆t perform almost equally
evaluation of various applications without the need of well. In case of a small SNR of 10 dB and especially for
re-simulation. To put the provided normalized plots Rayleigh fading (K = 0), a clear optimum for the sam-
into perspective, the example of a remote-controlled pling period arises. For practical systems, a sampling
automated guided vehicle (AGV) in an industrial cam- period between this optimum and the coherence time
pus network is considered. For this use-case we assume could be chosen. The choice of the sampling period is
a constant relative velocity v = 0.8 m/s and a carrier a trade off: For scheduling purposes, large sampling
frequency fc = 3.75 GHz. According to fm = fcc·v , periods unavoidably lead to large prediction horizons,
where c is the speed of light, this yields a maximum which generally lead to worse prediction performance
Doppler shift of 10 Hz. In the Rician fading case, the than short-term predictions. On the other hand, small
LOS Doppler frequency fD,LOS and the starting phase sampling periods imply that training signals need to
of the LOS component ϕ0 were varied randomly. be sent more frequently. This, however, has a nega-
tive impact on the efficiency of the communications
7.2 Parameter Analysis system and the number of users which can be allo-
Before being able to investigate the performance of cated to send these training signals. For numerical
the outage prediction schemes, the predictor requires evaluation, we settled on a normalized sampling pe-
parameterization. The Wiener filter is parameterized riod of ∆tfm = 0.05. This equals a sampling period of
by its history length M and its sampling period ∆t. ∆t = 5 ms in the AGV use case described in Sec. 7.1.
Furthermore, the history length N of the LOS estima-
tor needs to be specified. In the following, an analy- 7.2.2 Wiener Filter History Length
sis of these parameters is conducted for the scenario In Fig. 9, the Wiener filter history length M is in-
described in Sec. 7.1 to obtain a satisfactory con- vestigated. Throughout all curves, an increase of the
figuration for the following performance evaluation. history length M results in a reduction of the MSE.
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0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40
Figure 8 Influence of the sampling period ∆t on the MSE of Figure 9 Influence of the Wiener filter history length M on
the prediction for F = 10 dB, M = 25, N = 128 and the MSE of the prediction for F = 10 dB, ∆tfm = 0.05 and
tp fm = 0.1 N = 128
0.2
This is intuitive as adding more information to the
estimation will not lead to degradation. However, the
performance gain lowers with increasing M , which is 0.15
also intuitive as recent samples carry more information
about the future channel state compared to outdated
samples. However, in the case of a very low K factor 0.1
of K = 0, high values of M still improve the perfor-
mance. For higher K factors, as the LOS component
dominates the NLOS component, the curves become 0.05
more flat even for small Wiener filter history lengths
M . Since the Wiener filter is responsible for the NLOS
0
prediction, the choice of M becomes less relevant for 0 50 100 150
these high K factors. Throughout our numerical eval-
uation, we settle for a Wiener filter history length of
M = 25. Figure 10 Influence of the LOS estimation history length N
on the MSE of the prediction for F = 10 dB, ∆tfm = 0.05
7.2.3 LOS Estimation History Length and M = 25
Finally, in Fig. 10 the MSE is plotted for different his-
tory lengths of the LOS estimator N . Similar to the
history length of the Wiener filter M , high values of 7.3.1 Rayleigh Fading Prediction
N are beneficial for the performance of the predictor. We first investigate the performance of the outage
Again, the steepness of the curves decreases with rising prediction, which is one of the two predicto outputs.
N , thus, after a certain value the increase of N does
The shown values originate from Monte-Carlo simu-
not lead to a significant performance increase anymore.
lations. In our simulations, 2 · 108 I/Q channel co-
Due to complexity minimization, N should be kept as
efficients were fed into the outage predictor for each
low as possible as the calculation of the FFT is re-
prediction horizon and compared with the true future
quired in (25). For numerical evaluation of the Rician
fading state. To put this into perspective, at a sam-
fading case, we settled on N = 128.
pling period of 5 ms, which is used in the AGV sce-
7.3 Numerical Evaluation nario, this equals 10.6 days of consecutive fading. Two
With the scenario from Sec. 7.1 and the predictor pa- examples for classical metrics to investigate our bi-
rameters from Sec. 7.2, the prediction schemes can be nary classification problem were introduced in Sec. 5
evaluated numerically. In the following, we conduct and are plotted in Fig. 11 for different prediction hori-
the evaluation for the Rayleigh fading outage predic- zons. In Fig. 11(a) the accuracy of the outage predic-
tor and the Rician fading outage predictor separately tor is plotted against different threshold values for the
as they have different feature sets. prediction h0min . One can see that an optimal accu-
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1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0
0
0 0.5 1 1.5 0.2 0.4 0.6 0.8 1
(a) Accuracy (b) receiver operating characteristic (ROC) curve
Figure 11 Analysis of the outage prediction performance using common binary classification metrics for 20 dB SNR, F = 10 dB,
∆tfm = 0.05 and M = 25
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
10-6 10-5 10-4 10-3 10-2 10-1 100 10-6 10-5 10-4 10-3 10-2 10-1 100
(a) 20 dB SNR (b) 10 dB SNR
Figure 12 Analysis of the outage prediction performance for 20 dB SNR, F = 10 dB, ∆tfm = 0.05 and M = 25
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6 to evaluate the percentage of time the observed link
can be utilized for URLLC traffic of a specific UE. The
performance curves with these metrics are plotted in
Fig. 12. Similar to the ROC, each line spans differ-
4 ent operating points, which can be adjusted by vary-
ing the threshold |h0min |. A prominent point in these
curves is |h0min | = 0, where the channel is predicted as
2 up 100 % of the time and the effective outage proba-
bility equals the average outage probability. The av-
erage outage probability for Rayleigh fading can be
calculated using Pr(outage) = 1 − exp(−1/F ) [30]. In
0 Fig. 12(a) the results for a mean channel estimation
-1 -0.5 0 0.5 1 SNR of 20 dB are shown and discussed using the AGV
scenario with fm = 10 Hz. If, for instance, a prediction
horizon of tp = 5 ms is needed to overcome the delay
Figure 13 Validation of the prediction error distribution τ between monitoring and payload and an effective
outage probability Pr(effective outage) = 10−3 is tar-
geted, the link can be used approximately 82 % of the
racy arises for small prediction horizons tp when the time for URLLC traffic. If a higher prediction horizon
threshold for prediction h0min is chosen near the actual of tp = 10 ms is required and the same effective outage
outage threshold hmin . This might appear appealing probability of Pr(effective outage) = 10−3 is targeted,
for the choice of h0min , however, the optimum and the the link can only be utilized approximately 76 % of the
metric as a whole have only little practical relevance time. In Fig. 12(b), a lower mean channel estimation
for the outage predictor. The metric combines both SNR of 10 dB is shown. The lower SNR leads to a worse
error types (false positives and false negatives) within overall performance, e.g., when using our previous ex-
a single number. However, false positives are consid- ample with tp = 5 ms and Pr(effective outage) = 10−3 ,
erably more critical for an URLLC scheduler as they the probability of having a predicted up link is only
are the defining parameter for the QoS. Even worse, 62 % instead of 82 %. An increase of the maximum
as we tune the outage predictor to be more conser- Doppler frequency, resulting for example from a higher
vative by increasing h0min , the number of false posi- carrier frequency fc or an increasing movement speed
tives is orders of magnitude smaller than the number v, will reduce the achievable prediction horizon tp for
of false negatives. Therefore, the accuracy of the pre- a set of Pr(effective outage) and Pr(predicted up). As
dictor almost only reflects false negatives, which is the a result it can be concluded, that the proposed pre-
dominating error type in this case. A more informa- diction approach is unsuitable for realizing URLLC
tive performance evaluation of the predictor can be
services in future mmWave and Terahertz communi-
done by studying a ROC, which is shown in Fig. 11(b).
cations systems.
In this performance figure the probability of detection
Additional to the prediction of outages, the Rayleigh
Pr(detection) is plotted against the probability for a
fading outage predictor is able to calculate future out-
false alarm Pr(false alarm). The threshold value for
age probabilities under the assumptions discussed in
the prediction |h0min | now shifts the operating point
in the ROC. We can learn from the ROC curve that, Sec. 6.1. Basis for the calculation is (22), which states
when increasing |h0min | we get a more conservative pre- that the real and imaginary parts of the prediction er-
dictor so that outages are more likely to be detected ror follow a zero mean Gaussian distribution. The vari-
at the cost of more false alarms. However, we are still ance of the zero mean Gaussian distribution was cal-
not able to make qualitative statements about the ex- culated in (23). These findings are validated in Fig. 13,
pected scheduling performance since we are not able where an empirical estimate (a normalized histogram)
to evaluate which combination of Pr(detection) and of the probability density is compared with the analyt-
Pr(false alarm) is acceptable in the context of URLLC ical calculation for different prediction horizons. One
radio resource scheduling. can see that the empirical histograms fit very well be-
Therefore, following our discussion in Sec. 5, we use neath the calculated probability densities.
Pr(effective outage) instead of Pr(false alarm). This With the known error distribution, the probability
metric shows the effective outage probability of a per- for a future outage can be calculated over the dou-
fect scheduler and thus can be used to qualitatively ble integral (24). An analysis of this integral reveals
evaluate the risk of fatal failures due to prediction er- that the probability for a future outage only depends
rors. Instead of Pr(detection) we use Pr(predicted up) on the amplitude of the predicted fading |ĥ(t + tp |t)|.
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pronounced, however, the curves are still close to their
10 0
ideal counterparts.
10 -1 When comparing the same prediction horizons for
different K factors, one can observe that the out-
10 -2 age predictor performs better at high K factors. For
10 -3 example, when in the AGV scenario a prediction
horizon tp = 10 ms is utilized in case of K = 0
10 -4 and the effective outage target probability is set to
Pr(effective outage) = 10−5 , the observed link is only
10 -5 predicted as up with a 61 % probability. However, for
10 -6 K = 5 the same prediction horizon and effective prob-
ability is achieved while the channel is predicted as
0 0.5 1 1.5 2
up with a much higher probability of 92 % and for
K = 10 even 99 % is reached. A reason for that is the
decrease of randomness in the fading for increasing
Figure 14 Future outage probability depending from the
prediction amplitude for 20 dB SNR, F = 10 dB,
K factors. The randomness originates from the NLOS
∆tfm = 0.05 and M = 25 component, whose impact is reduced when a strong
LOS component is present. Ultimately, the determin-
istic LOS component is easier to predict resulting in a
better outage prediction performance.
The resulting future outage probabilities are plotted
In Fig. 16, an overall worse performance can be ob-
in Fig. 14. Using the AGV scenario with fm = 10 Hz,
served compared to Fig. 15 due to the lower SNR.
if a channel coefficient with an absolute value of 0.6
While high K factors of K = {5, 10} still show a
is predicted at a prediction horizon of tp = 5 ms, the
promising performance, for K = 0 a small prediction
channel state would be in the outage region with a
horizon of 5 ms achieves Pr(effective outage) = 10−5
probability of 10−4 . If, instead, the distance of the pre-
only with a predicted up probability of 26 %. To allevi-
dicted value from the origin is 0.7, the probability is
ate this behaviour to some extent, the number of pilot
2 × 10−7 and therefore approximately three orders of
symbols P can be increased, though leads to a reduced
magnitude smaller. For higher prediction horizons tp ,
spectral efficiency. However, analyzing this trade-off is
the predicted channel coefficient has to be farther away
out of scope of this article and will be left for future
from the origin to achieve the same outage probabil-
work.
ities. This is due to the fact that the variance of the
error distribution is higher.
8 Conclusion
7.3.2 Rician Fading Prediction In view of reducing radio resource consumption for
The performance curves of the outage predictor de- ultra-reliable wireless communication links, monitor-
scribed in Sec. 6.2 for the more general Rician fading ing the fast fading channel and taking measures based
case are shown as solid lines in Fig. 15 for a SNR of on the predicted fading state is a promising strategy.
20 dB and in Fig. 16 for a SNR of 10 dB. Each line is This article provided outage prediction schemes for
based on 2 × 108 predicted I/Q channel coefficients. To Rayleigh and Rician fading and introduced suitable
investigate the influence of the LOS estimation, which metrics that describe their performance.
is the novel component compared to the Rayleigh fad- For the Rayleigh fading case and especially for small
ing case, we also show the case of ideal LOS parameter prediction horizons, the presented predictor features
estimation as dashed lines, where ideal estimates are a low missed outage probability while simultaneously
used for subtraction and prediction of the LOS com- not rigorously denying the current channel. For the Ri-
ponent and only the NLOS fading is realistically pre- cian fading case, i.e., with a LOS component present, a
dicted using the Wiener filter. Therefore, the dashed LOS estimator is utilized. Compared to the case where
lines for K = 0 correspond to the performance curves the LOS component is known, the LOS estimator only
discussed in Fig. 12. The performance loss which can degrades prediction performance minorly. Generally in
be observed between solid and dashed lines originates the presence of a LOS component, the outage predic-
from the imperfections of the introduced LOS estima- tion performance is improved to the Rayleigh fading
tor. Overall, the performance loss is the smallest when case. Evidently, the LOS estimation comes at the cost
tp is small and the SNR is high. For large prediction of increased complexity, mainly due to the calculation
horizons and a small SNR the performance loss is more of the FFT as part of the LOS Doppler estimation.
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