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                          Víctor M Hinostroza, José Mireles and Humberto Ochoa
                     Institute of Engineering and Technology, University of Ciudad Juárez
                     Valle del tigris # 3247,Ciudad Juárez Chihuahua México C. P. 32306
                            vhinostr@uacj.mx, jmireles@uacj.mx, hochoa@uacj.mx

This work is a study of the effects of frequency selectivity on multi-carrier wideband signals in three different
environments; indoors, outdoor to indoor and outdoors. The investigation was made using measurements carried
out with a sounder with a 300 MHz bandwidth. The main part of this work is related to evaluate the contribution
of several parameters; frequency selective fading, coherence bandwidth and delay spread on the frequency
selectivity of the channel. A description of the sounder parameters and the sounded environments are given. The
300 MHz bandwidth is divided in segments of 60 kHz to perform the evaluation of frequency selective fading.
Sub channels of 20 MHz for OFDM systems and 5 MHz for WCDMA were evaluated. Figures are provided for
a number of bands, parameters and locations in the three environments. It is also shown the variation of the
signal level due to frequency selective fading. The practical assumptions about the coherence bandwidth and
delay spread are reviewed and a comparison is made with actual measurements. Statistical analysis was
performed over some of the results.

Keywords.     Coherence bandwidth, frequency correlation, frequency selective fading and multi-carrier
                                                            characteristics of the wireless communications
I. INTRODUCTION.                                            channel. The dispersive channel characteristics
                                                            arise from the different propagation paths, i.e.
To simulate and evaluate the performance of a               multipath, between the receiver and the transmitter.
wireless mobile system a good channel model is              This dispersion could be measured, if we could
needed. Mobile communication systems are using              measure the channel impulse response (CIR). As a
larger bandwidths and higher frequencies and these          general rule the effects of ISI on the transmission
characteristics impose new challenges on channel            errors is negligible if the delay spread is
estimation. The channel models that have been               significantly shorter than the duration of the
developed for the mobile systems in use may not be          transmitted symbol. Due to the expected increase in
applicable anymore. To validate that the old                demand of higher data rates, wideband multi-
models can be used for future systems or to design          carrier systems such as; OFDM and WCDMA are
new models, it is necessary to answer the question          expected to be technologies of choice [1], [12] and
about how the same parameters performs at higher            [14]. This is because these two technologies can
bandwidths? Also, we have to be able to measure             provide both; high data rates and an acceptable
and validate some parameters and compare them to            level of quality of service. However, these systems
well known practical assumptions. Measurements              need first to address better the problem regarding
for analysis of the fading statistics at common             channel prediction or estimation, because this
frequencies have been performed before, but they            condition is the main boundary for higher data
have been performed at small bandwidths, it is              rates. The study of correlation of the mobile radio
necessary to update the models with higher                  channel in frequency and time domains has helped
bandwidths.                                                 to understand the problem of channel estimation.
                                                            One of this work objectives is to evaluate
As the data rate (the bandwidth) increases the              frequency selective fading (FSF) in several
communication limitations come from the Inter               environments. This work begins with the results of
Symbol Interference (ISI) due to the dispersive
measurements made with a sounder that uses the          E{H (t1 ; f1 ) * H (t2 ; f 2 )} =
chirp technique for sounding.                           ∞ ∞

                                                        ∫ ∫ E{h(t ;τ ) * h(t ;τ )}e
                                                                                                     − 2πf1τ 1 + j 2πf 2τ 2
                                                                    1       1           2    2               e                dτ 1dτ 2
                                                        − ∞− ∞
Multipath fading channels are usually classified
into flat fading and frequency selective fading                     (2)
according to their coherence bandwidth relative to
the one of the transmitted signal. Coherence            By considering the channel to have uncorrelated
bandwidth is defined as the range of frequencies        scattering (US) and to be wide sense stationary
over which two frequency components remain in a         (WSS), the subscript for τ is eliminated and f1 and
strong amplitude correlation. Physically, it defines    f2 can be replaced by f + ∆f and t1 and t2 replaced
the range of frequencies over which the channel         by t + ∆t, then:
can be considered “flat”. The analytic issue of
coherence bandwidth was first studied by Jakes [1]      R H (∆t ; ∆f ) =                 ∫R      h   (∆t ;τ )e − j 2π∆fτ dτ
where by assuming homogeneous scattering, his                                           −∞
work revealed that the coherence bandwidth of a                  (3)
wireless channel is inversely proportional to its       In (3) RH and Rh represents the correlation of
root-mean-square (rms) delay spread. The same           random variations in the channels transfer function
issue was subsequently studied by various authors       and its impulse response respectively. If there are
[4], [8], [9], [10]. Since many practical channel       US, then ∆t is 0 then:
environments can significantly deviate from the
homogeneous assumption, various measurements
were conducted to determine multipath delay
                                                        Rh (0;τ ) = E h (0;τ )      {                   2
                                                                                                            }= E {h(τ ) }            2

profiles and coherence bandwidths [19], [20], [21],                 (4)
[22], aiming to obtain a more general formula for
coherence bandwidth. In this work the variations of     substituting into (3) gives:
this formula are reviewed and compared with

                                                                                ∫ E {h(τ )                  }e
actual results and a comparison is provided.                                                           2         − j 2π∆fτ
                                                        R H ( ∆f ) =                                                           dτ
The rest of this document is structured as follow; in                           −∞
part II the theoretical foundations of the channel                  (5)
impulse response frequency selective fading and
coherence bandwidth are reviewed. Also in this
part, the characteristics of the three environments     where
                                                                  E h (τ )
                                                                            is the average Power Delay
sounded are described. In part III, the frequency       Profile PDP of the channel. So, under the above
selective fading evaluation and analysis are            conditions, RH is the Fourier transform of the
presented. Plots of the dependency of fading deep       average PDP.
and frequency separation of two specific points in
the response are studied. At part IV, data about the    2.2 Coherence bandwidth.
relationship between delay spread and coherence         The multipath effect of the channel, the arrival of
bandwidth are provided. At the end in part V,           different signals in different time delays causes the
conclusions and future work are mentioned.              statistical properties of two signals of different
                                                        frequencies to become independent if the frequency
II. MATHEMATICAL BACKGROUND                             separation is large enough. The maximum
                                                        frequency separation for which the signals are still
2.1 The wideband channel model.                         strongly correlated is called coherence bandwidth
The radio propagation channel is normally               (Bc). Besides to contribute to the understanding of
represented in terms of a time-varying linear filter,   the channel, the coherence bandwidth is useful in
with complex low-pass impulse response, h(t, τ). Its    evaluating the performance and limitations of
time-varying low-pass transfer function is [4] [6]      different modulations and diversity models.
[8] [10]:
               ∞                                        The coherence bandwidth of a fading channel is
               ∫ h(t;τ )e τ dτ
                         − j 2πf
H (t , f ) =                                            probed by sending two sinusoids, separated in
               −∞                                       frequency by ∆f = f1- f2 Hz, through the channel.
          (1)                                           The coherence bandwidth is defined as ∆f, over
Where τ represents delay, using (1) the frequency       which the cross correlation coefficient between r1
correlation function for the channel can be written     and r2 is greater than a preset threshold, say, η0=
as:                                                     0.9. Namely:
                Cov( r1, r 2)                                                                                 ⎛2 λ ⎞ π
C r1,r 2 =                      = η0                                                               (1 + λ ) E ⎜      ⎟
               var(r1) var(r 2)                                                                               ⎜1 + λ ⎟ − 2
                                                                                     ρ ( s, τ ) =             ⎝      ⎠
         (6)                                                                                                     π
Then, using (2)                                                                                                  2
                                                                                                J (ω τ )
                                 ∞   ∞                                               = λ2 = 0 2m 2
R( s, τ ) = r1r 2 = ∫                                                                           1+ s σ
                                     ∫ r1r 2 p(r1, r 2)dr dr
                                                                             1   2
                                                                                     It is possible to see in this expression that the
Where p(r1,r2) is
                                                                                     correlation decreases with frequency separation.
                                                                                     This formula has been substituted by several
                   2π 2π
                                                                                     practical expressions some of them are the
p( r1, r 2) =      ∫ ∫ p(r1, r 2, θ , θ
                   0        0
                                                    1        2   )dθ1dθ 2            following [4], [8], [9], [10].

             (8)                                                                                       1                     1
                                                                                     BC =0.9 =             (13) BC =0.5 =           (14)
                                                                                                 50σ rms             5σ rms
      r1r 2       ⎡   r + r ⎤ ⎛ r1r 2 λ ⎞ 2              2
=              exp⎢−              I ⎜
                              2 ⎥ 0⎜
                                            2 ⎟
                                                                                                   1                 1
    µ (1 − λ )
     2      2
                  ⎣ 2 µ (1 − λ ) ⎦ ⎝ µ 1 − λ ⎠                                       BC =0.9   =         (15) BC =          (16)
                                                                                                 8σ mean           2πσ rms

Where I0(x) is the modified Bessel function of zero                                  In general
order. Then, substituting (8) in (7) and integrating                                             k
                                                                                     BC =                  (17)
               π   1 1                                                                         σ rms
R( s,τ ) = b0 F ( − ,− ;1; λ2 )
          2        2 2
         (9)                                                                         It will be shown, comparing with practical
                                                                                     measurements that none of these expressions are
this may also be expressed as                                                        accurate and it is difficult to obtain a
                                                                                     comprehensive expression for all environments.
               ⎛ λ2 ⎞
R( s, τ ) = b0 ⎜1 + ⎟                                                        (10)    2.3 Sounder systems          characteristics     and
           2 ⎜ ⎝   4⎟
                    ⎠                                                                environment description.

                                                                                     The sounder system used to make the
                            R ( s, τ ) − r1 r 2                                      measurements of this work was developed at
ρ ( s, τ ) =
                   [r  1
                                − r1
                                             ][ r
                                                             − r2
                                                                         ]           UMIST in Manchester UK and is described in [2]
                                                                                     and [3]. This sounder uses the FMCW or chirp
                                                                                     technique. The generated chirp consists of a
                                                                                     linearly frequency modulated signal with a
                         ⎛2 λ ⎞                                                      bandwidth of 300 MHz and a carrier frequency of
R( s,τ ) = b0 (1 + λ ) E ⎜     ⎟
                         ⎜1+ λ ⎟                                             (11)    2.35 GHz. The chirp repetition frequency is 100
                         ⎝     ⎠                                                     Hertz, which allows having 50-Hertz Doppler
                                                                                     range measurements. The receiver has the same
                                                                                     architecture than the transmitter. But in the
Where E(x) is the complete elliptic integral of the                                  receiver, the generated chirp is not transmitted but
second kind. The expansion of the hyper geometric                                    mixed with the incoming signal from the antenna,
function gives a good approximation to (9). After                                    which are the multi-path components of the
several reductions and considerations, the                                           transmitted chirp. This mixing allows having the
correlation coefficient becomes                                                      multi-path components at low frequencies, these
                                                                                     low frequencies can be sampled, digitized and
                                                                                     stored in a computer to perform the required

                                                                                     The three environments where the measurements
                                                                                     took place were the following: 1) Indoors, in
different floors around a building in an eight stories   MHz were calculated. Figure 1 shows a typical
building. Each floor form a rectangle with four          channel transfer function. In figure 2 are shown the
long corridors; two 66 m long and two 86 m long,         fading characteristics for a 20 MHz sub channel, in
the width of the corridors is 3 m, the total covered     this figure there are 15 different lines, each one
area was 912 square meters and the ceiling height        corresponds to a 20 MHz sub channel in a 300
is 5 meters. The transmitter was static and the          MHz bandwidth. To get this figure, the following
receiver was moved around the corridors and in           has been done; first, the fading information of the
different floors, measurements were taken at             complete 300 MHz bandwidth was divided in 15
specific distances in each corridor. 2) From a           segments of 20 MHz each. Then, each point on that
building to different building. These two buildings      segment, that represents a 60 kHz sample, was
are eight stories high, they have about the same         measured and the result was compared to the next
high and they are separated by about 200 meters.         sample, then the next sample was compared and so
The transmitter was located in the top of one of the     on up to the complete 20 MHz bandwidth was
buildings and the receiver was moved around              compared. After that, a second 20 MHz sub
specific locations inside the second building in all     channel was applied the same procedure and so on
eight different floors. 3) An urban environment          up to the end of the 300 MHz bandwidth. To form
around the city center, a commercial area with           figure 3, the same procedure was followed, but in
several high rising buildings. Each location in each     this case the sub channel bandwidth was of 5 MHz,
environment was sampled during one second and            then this figure has 60 different lines which come
100 impulse responses were stored for this specific      from the 300 MHz total bandwidth.
location. When the measurements were taken,
every location was sampled with 100 impulse              Figure 2 shows that the maximum fading deep
responses, an impulse response (IR) was taken on         within the 20 MHz sub channel is lower than 14 dB
that specific location every 10 milliseconds.            in all the bandwidth. On the other hand, in the 5
                                                         MHz bandwidth the maximum fading deep was of
III. FREQUENCY SELECTIVITY                               18 dB. It is possible to see in figure 2, that most of
PARAMETERS                                               the lines follows a pattern, which means that the
                                                         fading in all sub channels is about the same. In
To carry out the calculation of frequency selective      figure 2, most of the lines stay below 5 dB and only
fading, each processed IR was split in samples of        a few lines go higher than 6 dB; this could mean
60 kHz, which was the minimum sampled                    that deep fading in this bandwidth is rare.
frequency, each 300 MHz bandwidth was sampled
5000 times every 10 milliseconds, this means that a
sample was taken every 2 microseconds or every
60 kHz. Each IR was averaged over the complete
second, meaning that every frequency was
averaged 100 times for each IR. The level of the
frequency response of each 60 kHz segment was
calculated and recorded

In each environment the channel transfer function
for about 50 different locations were calculated and
recorded. Each transfer function has 5000 sampled
frequencies, i.e. 5000 segments for each location.
Every sample represents the signal level on that
segment, relative to the maximum over the                Figure 1. Typical channel transfer function.
complete 300 MHz bandwidth. The outdoors
environment has a bandwidth of 120 MHz, the              Figure 3 shows the calculations of fading
indoors and indoor to outdoor environments has           characteristics for the second environment with 15
300 MHz bandwidth.                                       different 20 MHz sub channels, this figure shows
                                                         that there are more dispersion of the lines, which
The fading characteristics in each of the                means there are more deep fading in the responses
environments were calculated. For each of the            of the IR. Since this environment is the
locations the fading characteristics that were           measurement of the propagation for the penetration
calculated were; average delay spread, RMS delay         of the signal in different floors in a building, higher
spread, coherence bandwidth, channel transfer            delay spread, (time dispersion) than the former
function and frequency correlation. Using the            environment was expected and therefore more
channel transfer function for each IR, specific          fading.
fading characteristics for sub channels of 5 and 20
Another way to look at the statistics of the fading is   correlation coefficient, the coherence bandwidth
to calculate the CDF of this parameter. To make          (Bc) is lower than 10 MHz most of the locations.
the calculations of these CDF’s figures, the mean        This is corroborated in figure 6, this figure shows
of all locations in the involved environment were        the average Bc for all locations in the indoors
used. Figure 4 shows the CDF of the building-to-         environment. Figure 7, shows the RMS delay
building environment for both, the 20 and 5 MHz          spread for all locations for the same environment.
bandwidths, this figure shows that the fading deep       Quick calculations comparing figure 11 results and
for a 20 MHz sub channel is below 7 dB for 90%           expression (13) show that, few calculated values of
of the time. On the other hand, for the 5 MHz sub        the versions of expression (13) match with the
channel, the fades are below 5 dB for 90% of the         measured values of figure 7.

Figure 2 Indoor to outdoor fading Characteristics
for a 20 MHz sub channel                                 Figure 4. Fading CDF for indoor to outdoor
                                                         for a 5 MHz sub channel

Figure 3. Outdoor to indoor fading Characteristics
For a 5 MHz sub channel
                                                         Figure 5. Coherence bandwidth for indoors

Figure 5 shows the frequency correlation of all
locations in the indoors environment. To make this
figure the following was done; first the PDP of all
locations was calculated. Then a Fourier transform
was performed on the PDP, which gave us the
frequency correlation for all locations. Then the
frequency correlation for each location was plotted
in figure 5. On this figure, the thick and dashed line
is the line for the maximum coherence bandwidth,
when the transmitter and receiver are connected
directly. In figure 5, one can see that at 0.9
Figure 6. Average of coherence bandwidth for

Figure 8 shows the frequency correlation for the
outdoor to indoor environment, this figure shows
that in this environment the Bc at frequency
correlation of 0.9 is higher than the indoor
environment, although the delay spread is not
different is both environments. Figure 9, shows the
average Bc for the outdoors to indoors
environment, one can see in this figure, that the
coherence bandwidth is higher than the indoor
environment, which was an expected result, but the
difference is higher than expected. In indoors the
coherence bandwidth is not bigger than 20 MHz in        Figure 8. Coherence bandwidth for outdoor to
average. In the other hand, in the outdoor to indoor,   indoor
the average is about 100 MHz, here is relation of 5
to 1. The difference in RMS delay spread is 100 nS
versus 200 nS, there is a relation of 2 to 1.

                                                        Figure 9. Average of coherence bandwidth for
                                                        indoor to outdoor

Figure 7. RMS delay spread for indoors                    Table 1, shows the comparisons of Bc for the
                                                        three environments with the different versions of
Figure 11 shows the frequency correlation for the       expressions 13 -16 and measured results.     This
outdoors environment. Figure 12, shows the Bc at        table shows that the values of the expressions are
frequency correlation of 0.9. In this case the Bc can   always lower than the measured results, which
not be compared to the Bc for the other two             induce to conclude that the expressions were
environments, since in this environment a lower         underestimated, at least in these environments.
bandwidth is evaluated, 120 MHz instead of 300          Moreover, it is possible to conclude that these
MHz.      Despite this difference and observing         expressions were deduced with not enough
figures 11 and 12, Bc is not significantly lower        measured results. Also, table 1 show that the
even when we have higher distances and higher           relationship between delay spread and coherence
delay spread. In outdoors the Bc is not bigger than 2   bandwidth, not necessarily is a single constant.
MHz in average. In the other hand, the RMS delay
spread is 1.5 µS in average.
                                                         Figure 11. Coherence bandwidth for outdoors

Figure 10. RMS delay spread for outdoors to
                                                         Figure 12. Average of coherence bandwidth for

In this work the results of analysis of frequency
                                                         Table 1. Coherence bandwidth calculations
selective fading on two indoor and one outdoor
environment have been presented. The three               Value from     Indoors   Outdoors      Outdoors
environments analyzed demonstrate that the fading                                 to indoors
is within specific limits, these results could help to   (13)           400 kHz 200 kHz         30 kHz
the designers of adaptive receivers to estimate the      (14)           4 MHz     2 MHz         300 kHz
channel more accurately. The division of the             (15)           3.3 MHz 2.5 MHz         250 kHz
channel impulse bandwidth in segments of 20 and          (16)           3.2 MHz 1.6 MHz         212 kHz
5 MHz bandwidths, allow the calculation of fading        Measured0.9    5.3 MHz 12 MHz          300 kHz
in the bandwidth of interest for OFDM and                Measured0.5    19 MHz 72 MHz           5.6 MHz
WCDMA transmission. Plots of the frequency
selective fading will help for this assessment. The
analysis of coherence bandwidth show that the
expressions accepted in the literature for its
calculation are not accurate and the accepted direct
relationship between delay spread and coherence
bandwidth is not simple. Also, additional work is
require on try to determine how much the
combined effect of Doppler spread, time variability
and frequency offsets affects the transmission on
multi-carrier signals as the ones on OFDM y

                                                         Figure 13. RMS delay spread for outdoors


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