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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 9, NO. 10, OCTOBER 2020
Long Term Fade Margin for 90% Availability
in Fixed Wireless Links With Diversity
Reinaldo A. Valenzuela, Fellow, IEEE, Rodolfo Feick , Life Senior Member, IEEE, Patricio Alegre,
Mauricio Rodríguez , Senior Member, IEEE, Luciano Ahumada , Senior Member, IEEE,
and Dmitry Chizhik , Fellow, IEEE
Abstract—Fixed wireless access is becoming a major 5G application. We measured received power for 72 urban fixed wireless
links at 5.8 GHz over a period of 48 hours each. We find that
5-minute power averages show temporal variations requiring
additional fade margins close to 6 dB at the 90th percentile. The
average power temporal variation was found to increase with
the excess path loss of the link. We propose a model that accurately characterizes first and second order link statistics. Spatial
and frequency diversity measurements allowed us to also assess
the effectiveness of the corresponding fade mitigation techniques.
From them we conclude that the long-term fades appear to be
consistent with time-varying multipath propagation, as opposed
to shadowing.
Index Terms—Channel models, propagation, fading, diversity.
I. I NTRODUCTION
IXED wireless links (FWL) are experiencing intense
interest for applications such as small cell, relay backhaul, local access, etc. [1], [2]. For such links, temporal fades
have been characterized with respect to the average power
calculated over time intervals in the order of minutes, typically in a range of 5 to 15 minutes [2]–[5] to exclude the
non-stationarities associated with changes in average power.
We refer to these fades as “short-term fades”. In urban environments such fades have been reported to be quite shallow,
characterized by Ricean K-factors that exceed 13 dB under
virtually all conditions [3]. Somewhat lower values for K
were observed in a suburban fixed wireless link with foliage
F
Manuscript received April 18, 2020; revised May 22, 2020; accepted
May 28, 2020. Date of publication June 3, 2020; date of current version
October 7, 2020. This work was supported in part by the Chilean Research
Agency Agencia Nacional de Investigación y Desarrollo (ANID), through
Research under Grant ANID PIA/APOYO AFB180002, Grant ANID BASAL
AC3E FB0008, Grant ANID FONDECYT/INICACION 11171159, and Grant
ANID REDES 180144, and in part by the Pontificia Universidad Católica de
Valparaíso under Project VRIEA-PUCV 039.430/2020 and Project VRIEAPUCV 039.437/2020. The associate editor coordinating the review of this
article and approving it for publication was V. Raghavan. (Corresponding
author: Mauricio Rodríguez.)
Reinaldo A. Valenzuela and Dmitry Chizhik are with Bell Laboratories,
Nokia, Holmdel, NJ 07974 USA.
Rodolfo Feick and Patricio Alegre are with the Electronics Engineering
Department, Universidad Técnica Federico Santa María, Valparaíso 1680,
Chile.
Mauricio Rodríguez is with the Escuela de Ingeniería Eléctrica, Pontificia
Universidad Católica de Valparaíso, Valparaíso 2950, Chile (e-mail:
mauricio.rodriguez.g@pucv.cl).
Luciano Ahumada is with the Escuela de Informática y
Telecomunicaciones, Universidad Diego Portales, Santiago 8370190,
Chile.
Digital Object Identifier 10.1109/LWC.2020.2999564
obstructions subject to wind-induced fluctuation [4], [6]. In
contrast, little is known about the temporal variation of the
average powers used as reference for the short-term fades. We
refer to these variations as the “long-term fades”. Intuition
suggests that such long-term fades will exhibit the effect of
cyclic daily patterns such as temperature or human activity,
but the significance of this has not been reported for urban
environments. Such fades are an important consideration for
FWLs because a terminal placement will typically be based
on a short-term assessment of link quality. Thus, a position
found to be favorable, will only remain so over longer time
intervals if the short-term average power remains higher than
that of alternative positions. Statistical data on the variation
of average power will provide information on whether or not
this variation is of significance in planning actual FWLs. It
is often assumed that long-term power variations are dealt
with through link adaptation techniques, for example by power
control and/or by changing modulation and error control. As
pointed out in [6], being able to characterize the slow-fading
phenomena will be useful to the designer in selecting adequate fade mitigation techniques and appropriate adaptation
rates. Equally important is the assessment of the effectiveness
of diversity techniques, a topic well understood for short-term
fades but not discussed to date for time scales of several days.
We present the first measurements of fading in urban FW
channels over days, in scenarios with very scarce vegetation
and moderate wind. This letter includes single-input singleoutput (SISO) as well as diversity non-line-of-sight (NLOS)
FWLs. The network topography corresponds to a base servicing multiple users as described in the ITU document “Fixed
service use and future trends” [7]. We propose a statistical
fading model for an ensemble of users of such a service.
Previous work with similar time scales was focused on the
effect of wind on trees [6] and on factory scenarios [8], [9].
In contrast to this letter, the results reported in [6] specifically
exclude all effects other than vegetation by using narrow beam
antennas with the first Fresnel zone obstructed by treetops.
Long-term measurements were also included in an outdoorto-indoor path-loss study at 28 GHz [10]. However, this
considered only a single scenario with short-range line-of-sight
(LOS) links.
The large volume of empirical data we collected allowed
us to relate the severity of fades to the excess path loss characterizing the link and to propose a very simple model for
the fading dynamics of the Mean Received Signal Strength
(MRSS). We also evaluated the performance improvement
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VALENZUELA et al.: LONG TERM FADE MARGIN FOR 90% AVAILABILITY IN FIXED WIRELESS LINKS WITH DIVERSITY
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Fig. 1. Measurement region: a) Viña del Mar and b) Valparaíso. (Source:
Google Earth).
from spatial and frequency diversity. Interestingly, we find
that slow fades can be effectively mitigated with spatial diversity, using antenna separations similar to those required for
fast fades, suggesting that fading is caused by multipath
propagation in both cases.
II. M EASUREMENT S CENARIOS AND S ETUP
A. Measured Scenarios
We measured received power for narrowband urban fixed
wireless links over uninterrupted periods of 48 hours each
at a carrier frequency of 5.8 GHz for over 70 links in residential urban environments in Valparaíso and Viña del Mar,
Chile. These are shown in Fig. 1(a) and 1(b) respectively.
We measured a wide variety of links, with varying degrees
of obstruction and with path lengths from 150 m to 2 km.
The test area contains nearby hills and flat regions with residential houses and buildings up to 20 stories high. Moderate
vehicular traffic occurred during the daytime measurements,
with almost no traffic at night. Pedestrian presence was sparse.
There are few trees in the area, with heights not exceeding 5 m
and no significant wind induced movement.
Our links emulate the connection between a base station
with a sector antenna and a remote terminal with an omnidirectional antenna located within the sector to be covered.
Arbitrarily, we placed the transmitter at the base station. Its
location was chosen at an unobstructed position overlooking
the receiving terminal positioning area on an 180 m high hill
in Viña del Mar, and on a radio mast 15 m above nearby
rooftops in Valparaíso. Transmitter output was 30 dBm, fed to
a vertically polarized 15 dBi gain antenna with 90◦ -azimuth
and 8◦ -elevation beamwidth between half power angles. The
remote terminal used 1 or 2 vertically polarized dipoles
for frequency diversity or spatial diversity respectively. This
terminal was located outdoors below the clutter, attached to an
exterior facing wall or window located well within the sector
covered by the transmitting antenna.
B. Measurement Setup
We used two narrowband sounding systems.
1) For spatial diversity measurements we used a twoantenna data acquisition system, where a single receiver
was switched between 2 antennas every 10 ms. The
receiver samples power at intervals less than 2 ms, which
allows eliminating the values affected by antenna switching. For each antenna, a power measurement is obtained
every 20 ms averaging 3 samples, which we found only
differed by fractions of 1 dB. We note that 20 ms is
much shorter than the reported FW coherence times [3].
Fig. 2.
Typical MRSS time series with different values of EPL.
2) For frequency diversity measurements, we used a similar arrangement with a two-tone CW transmitter and
a corresponding 2-channel receiver. Tone frequency
separations of 0.5, 2, 4, 10 and 20 MHz were used.
To verify long-term stability of the measurement system, we
calibrated often, using back-to-back connections. We observed
power variations within +/−1 dB of average over several
days. We also measured absolute transmit power output,
receiver gains and receiver noise floor to compute path losses.
The receiver has a noise figure of 11 dB and allows up to
145 dB path loss with 5 dB minimum SNR.
III. E MPIRICAL R ESULTS AND A NALYSIS
A. Characterization of Slow Fades and First Order Statistics
As mentioned in [8], slow fades may have characteristics
that differ significantly from those of short time scales. For
the latter we found that amplitude variations with respect
to 5-minute averages were well characterized by Rician fade
statistics with K-factors that even for the most obstructed links
exceed 9 dB in 90% of cases, matching previous results [3]. In
what follows, we propose a model for the power averaged over
the 5-minute intervals, i.e., the MRSS of the long-term fades.
As shown in Fig. 2, greater power variation was observed for
links with larger excess path loss (EPL), defined as the dB
difference between the full-record average and the free space
received power [3]. First-order statistics for the individual
voltage magnitude time series did not fit well with any of
the classical distributions, i.e., Rice/Rayleigh, Nakagami, etc.
and in many cases the probability density functions (PDFs)
were bi-modal. Alternatively, we define for each time series
Δ90% [dB] as the range containing 90% of the power differences (up-and down-fades) of the MRSS around the mean.
Fig. 3 shows a scatter plot of Δ90% [dB] vs. EPL for all tested
positions. Also shown are points generated with the simulation model to be presented in Section III-D. A simple linear
regression fit given by (1)
Δ90% [dB] = 1.3 + 0.29(EPL[dB])
(1)
resulted in a small RMS error of 2.3 dB. Higher order models
did not improve the fit significantly.
Although first-order statistics for individual time-series were
often not Rician, our measurements included a sufficiently
large number of locations (i.e., individual time-series) to allow
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Fig. 3.
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 9, NO. 10, OCTOBER 2020
Fig. 4.
Autocovariance example and CDF of the decorrelation interval.
Fig. 5.
Power spectrum magnitude grouped in EPL ranges.
Signal variation Δ90% [dB] vs. EPL [dB].
grouping them into different EPL ranges and to estimate
ensemble statistics for each range. Such statistics are relevant
when considering fade margins to assure a degree of coverage for a group of users rather than a single individual. We
chose the ranges 0 to 10 dB, 10 to 20 dB and > 20 dB.
This separation assured a comparable amount of empirical
data for each range. We find that the resulting ensembles are
now well described by Rician distributions. The empirical fade
Cumulative Distribution Functions (CDFs) in dB fit those of
Rician voltage envelopes to within 0.6 dB. The K-factors for
each range were estimated using the two-moment method [4]
and were found to be 19 dB for EPL < 10 dB, 13 dB for
10 dB < EPL < 20 dB and 6 dB for EPL > 20 dB. The
observed slow-fades are thus characterized by lower K-factors
(i.e., deeper fades) than those reported when subtracting the
short-term average [3].
B. 2nd Order Statistics: Characterization of Fade Dynamics
Fade dynamics were found to be best characterized based
on the voltage
magnitude time series {v (ti )}i=1,2...N with
v (ti ) =
P (ti ), where {P (ti )}i=1,2...N are the MRSS
samples for each measurement record.
1) Slow-Fade Decorrelation Interval: We define the “slowfade decorrelation interval” for the time series v (ti ) by
considering a “slow-fade time-scale” corresponding to the 5minute sampling interval. This decorrelation interval is the
time separation where the autocovariance of the process v (ti )
drops to 0.5. We note that this definition excludes the effect
of the averaged-out short-term fades. The threshold value 0.5
has been used in the literature before, as have been values of
0.9 and exp(−1) [11]. While this choice affects the numerical values of the slow-fade decorrelation interval, their order
of magnitude remains the same. The results indicate that the
long-term fades are characterized by time constants in the
range of hours. A typical normalized autocovariance is shown
in Fig. 4(a). Fig. 4(b) shows the CDF for the decorrelation
interval we found in all our measurements. The large MRSS
decorrelation interval suggests that its origin is attributable
to slow environmental changes in the propagation path (for
example ambient temperature), as will be discussed later.
2) Periodograms: Further insight into the fluctuation
dynamics and the basis for a simulation model can be obtained
from the periodograms for the voltage time series. These have
a constant (DC) part and a fluctuating component, the latter
corresponding to the temporal variations. In order to combine
empirical data from different links we first normalized each
series for unit mean voltage to eliminate the effect of different large-scale path losses. The normalized voltage vn (ti ) is
N
1 thus calculated through vn (ti ) = v (ti )/
v (ti ), where
N i=1
N = 576 is the number of 5-minute segments in each record.
We then obtain the modified periodograms of vn (ti ) using
a Hanning-windowed FFT to yield N equi-spaced frequency
samples [12]. Their average allows estimating the Power
Spectral Density (PSD). The choice of windowing was based
on the need to minimize spectral masking due to the sidelobes of the strong DC component. Using this windowing
function, masking was only observed to affect the first nonzero frequency component (f1 ), which was therefore deleted.
The reduced resolution that results from windowing is of no
consequence here, since no purely periodic temporal components are expected to occur in these random processes. The
periodograms and the PSD estimate are found to be monotonically decreasing, as reported before in [13] for short-term
fades. We note that the value of fp is two orders of magnitude lower than the reciprocal of the 5-minute averaging time.
Thus, the low-pass characteristic of the observed spectra is not
a consequence of this averaging. We grouped periodograms
corresponding to different values of EPL into three ranges as
before: 0 to 10 dB, 10 to 20 dB and > 20 dB and plotted them
in Fig. 5. We observed that the three groups have the same
general shape. The results confirm the observation that follows
from Fig. 3, in that higher values of EPL correspond to higher
variability, as seen by the higher total power of the spectral
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VALENZUELA et al.: LONG TERM FADE MARGIN FOR 90% AVAILABILITY IN FIXED WIRELESS LINKS WITH DIVERSITY
TABLE I
M ODEL F ITTING PARAMETERS G ROUPED IN EPL R ANGES
chosen as independent uniformly distributed random variables
over (0, 2π). To obtain |Vi |, we first use (2) to obtain the average PSD and add to it a zero-mean Gaussian random variable
with a standard deviation as per Table I.
The corresponding power time series is the square of the
voltage, scaled to the desired average power P (t):
P (t) = P (t)g(t)2 .
components with respect to the (0 dB) DC component. As seen
in Fig. 5 the strongest components in the power spectrum correspond to frequencies of around 10 µHz, corresponding to a
period of about 24 hours. This suggests that daily variations
are a possible mechanism driving these fluctuations. This may
include traffic density patterns and temperature induced effects
as further discussed in Section III-D.
C. Slow-Fading Model
The average PSD of the fluctuation Va (fi ) is accurately
described by a single pole model given by (2)
Va (fi )[dB] = 20log10 |Va (fi )|
⎛
⎞
A
⎠, i = 1, 2 · · · N . (2)
= 20log10 ⎝ 2
1 + (fi /fp )
where for our 2-day long time series consisting of N = 576
samples, the frequency spacing fi − fi−1 is 5.79 µHz.
We find A, and fp to minimize the weighted RMS error,
N
i=2
Wi {20log10 |Va (fi )| − 20log10 |SVn (fi )|}2
(3)
where {SVn (fi )}i=2,3,...,N . is the (complex) discrete Fourier
transform of the Hanning-windowed normalized voltage
time-series.
The error and model parameters are listed in Table I. Higher
order models were also tested but did not reduce the error by
more than 0.5 dB. We chose the weighting factors Wi such
that they were inversely proportional to the number of data
points in the respective octave, starting at 20 µHz. This compensates the unequal number of points per octave. We used
a single value for fp and found the value of A that minimizes the sum of RMS errors for the time series in each EPL
range. Optimizing fp separately per EPL range resulted in less
than 0.5 dB decrease in the error. The modeling error (in dB)
matched a zero-mean Gaussian distribution and the correlation of the error between contiguous frequency samples was
typically less than 0.4. We also evaluated the statistics of the
phases of the frequency components of {SVn (fi )}i=2,3,...,N .
which we found to be uniformly distributed over (0, 2π) with
a cross-correlation of less than 0.1. Based on the above, we
propose that the normalized measured time series be modeled
as g(t):
N
|Vi |sin(2πfi t + φi ),
g(t) = 1 +
1651
(4)
i=1
where the constant value 1 corresponds to the DC component
and the time-varying part is the sum of sinusoids with the
amplitudes |Vi | generated by our model. The phases φi are
(5)
As seen in Fig. 3, the model-based simulation is consistent
with our empirical data and results in a regression line that
differs by less than 1 dB from that based on measurements.
D. Fade Mitigation Through Spatial and Frequency Diversity
For spatial diversity we employed the 2-branch CW receiver
with 2 omnidirectional antennas separated between 0.2 and
18 wavelengths. For frequency diversity we used the dual
frequency receiver with a single dipole antenna and frequency
separation in the range of 500 kHz and 30 MHz.
1) Spatial Correlation of Slow Fades: To investigate the
origin of the slow-fade process we performed a total of 32
spatial diversity measurements, covering a distance range from
155 m to 2 km. Only 1 pair of the slow-fade time series
exhibited a cross-covariance exceeding 0.6, but we found no
evidence of higher values occurring at shorter antenna separations. The cross-covariance of the fast fades, i.e., the residual
fades after subtracting the 5-minute averages were as expected
found to be very low, less than 0.35 in all cases. The lack of
dependence on antenna separation of up to 18 wavelengths
suggests that time-varying multipath is also the main contributor to long-term fades, since shadowing would exhibit a
much larger spatial correlation than we observed. The origin
of this slow time-varying multipath in an urban environment
may be attributable to displacements of the scattering, diffracting or reflecting surfaces from nearby construction or from
elements on the street such as parked cars. In particular highrise buildings in the NLOS propagation paths are affected by
temperature-induced deformations in the order of a cm over
periods of hours, as has been reported for building heights similar to our environment [14]. Such a displacement is enough to
cause a significant phase change at the considered frequency
band. During the period in which we carried out our measurements weather conditions remained very stable with a
recurring pattern of day/night temperature variation of 10 ◦ C
to 15 ◦ C. Traffic patterns, for example the density of streetparked cars, also exhibits a slow and daily recurring variation
that will affect multipath components.
2) Spatial Diversity: To assess possible diversity gains we
evaluated the fade reduction that would result from being able
to choose every 20 ms the stronger of the two channels. Note
that this includes fast and slow fades, although as already
observed the latter will be dominant. As a reference for comparison we considered the statistics resulting from a single
channel receiver using the antenna placement that during the
first 5-minute segment had the highest average power. In the
absence of any previous long-term measurements, this is a
logical choice for an installation. We normalized each link
by its average power and then combined the data from all
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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 9, NO. 10, OCTOBER 2020
IV. C ONCLUSION
Fig. 6.
Selection combiner for EPL > 20 dB.
to obtain ensemble fade statistics for the no-diversity case.
We then considered two-branch selection diversity. To this
effect we normalized the second branch by the same factor
of the first. This preserves possible branch imbalance of the
average powers for the full time-series. The diversity gain
will be the result of several factors. Low cross-covariance
implies possible gains, but this may be reduced by imbalance [15]. On the other hand, our results show that a short-term
power measurement is not a reliable indicator of long-term
advantage of a specific antenna placement and thus spatial
diversity can counter a poor initial choice. For our spatial
diversity channels, the power difference (in [dB]) between the
2 slow-fade processes exhibited zero-crossings, i.e., a reversal of the channel with the strongest 5-minute average power.
In less than 10% of cases the interval between zero crossings exceeded 3 hours, longer times being associated with
larger average imbalances. Only 2 series, which had imbalances of 9 dB, exhibited no reversal of the best channel. To
evaluate diversity gains, we again combined data from all
antenna separations since this had no observable effect on
cross-covariance.
Fig. 6 shows the CDFs of fades with and without selection diversity. As a reference we also show Rician fading
with K = 5.2 dB, which matches the data for EPL >
20 dB considering only the initially strongest branch. We
note that this K-factor is only slightly less than that of the
corresponding slow processes (6 dB for EPL > 20 dB),
confirming that the latter are the dominant source of fades.
We also show as reference the output of a selection combiner with 2 balanced and independently fading links with
K = 5.2 dB, which as expected exhibits a larger diversity
gain. We observe in Fig. 6 that the simple two-branch selection combiner reduced the 90% fade from about 6 dB to
around 3 dB, i.e., a 3 dB reduction of fade margin at the 90th
percentile.
3) Frequency Diversity: We tested 2 links with path lengths
of 195 and 250 m and found that for a frequency separation of around 10 MHz the cross-covariance drops below
0.7. This is consistent with results reported in [16] and references quoted therein for small-size urban cells. A frequency
coherence scale of 10 MHz is generally consistent with
multipath fading, rather than shadowing, suggesting again
that time-varying multipath is the origin of the observed
phenomenon.
Narrowband fixed wireless links exhibit significant longterm fades in the average power. The severity of the fades
is related to the link’s excess path loss. We have proposed
a model that accurately captures first order statistics and
dynamics of these fades. First order ensemble statistics of
the channel magnitude were found to be well-described by
a Rician distribution whose K-factor decreased with increasing excess loss. The decorrelation interval of the fluctuations
exceeded 30 min for 90% of locations. Spatial and frequency
diversity measurements show that the long-term fades can
be effectively countered by the same techniques that prove
to be effective for the fast fades, suggesting that their origin also is multipath propagation. We observed that simple
2-branch selection diversity with antenna separations exceeding a quarter of a wavelength can reduce the 90% fade margin
by 3 dB.
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