1
Lecture 11: Narrowband Signal Fast Fading Characteristics in the
Space, Time and Frequency Domains
Below we will continue the description of the main characteristics of multipath links
which based on the statistical presentation of signal envelope and phase fading, by
introducing various PDF or CDF presentations described in Lecture 10. These
functions determine the multipath characteristics in the space and time domains.
Moreover, these statistical functions which describe the signal fading phenomena play
an important role in power-budget design in narrowband wireless networks for
various terrestrial links [1-4]. Therefore, for optimal selection of propagation
parameters of mobile radio communication links in built-up areas, and for effective
prediction of signal fluctuations within them, it is very important to examine the
spectral properties of signal fluctuations and the correlation between such fluctuations
in the space and time domains.
11.1. Power Spectra in Space and Time Domains
Let us now extend the 2D-statistical description of signal fading presented in Topic VI
based on Clarke’s 2D-model [5] and briefly present the 3D statistical model proposed
by Aulin [6] which was described in more detail in [7-9].
11.1.1. Angle-of-Arrival PDF for E- and H-field Components
Aulin generalised Clarke’s model by the inclusion of the angle  i which there was
always zero (see Fig. 10.7). In the 3D-case the angle  i describes spatial orientation
2
of the ith incoming wave (Fig. 11.1) and  i is now its projection on the XY-plane.
As above in Topic X, we assume, following [6-8] , that both  i and  i are uniformly
and independently distributed at the ranges [0,  / 2] and [0, 2] , respectively. The
same distribution is assumed for the ith wave phase  i . In this case instead of (10.12)
(see Topic X) for each electrical field component with number i, we can obtain a more
general expression at the arbitrary point ( x 0 , y 0 , z0 ) according to [6-9], if an
unmodulated carrier was transmitted:

Ei (t )  Ai cos  c t  k  x0 cos  i cos  i  y0 sin  i cos  i  z0 sin  i    i

(11.1)
and the resulting E-field component of the received electromagnetic wave is given by
N
E (t )   E i (t )
(11.2)
i 1
where k  2 /  ,  is the wavelength. For the case when the electric field
component is aligned along the z-axis formula (11.1) reduces to the case presented in
[5, 9]:

E zi (t )  Ai cos  c t  k  x0 cos  i  y0 sin  i    i

(11.3)
The same expressions can be obtained for the magnetic field components of each
multipath wave with number i. Without any loss of generality of the problem let us
present their 2D-expressions, i.e., for the case when all of them lie in the XY-plane
[5]:
H xi (t )  

Ai
cos  c t  k sin  i  x 0 cos  i  y 0 sin  i    i
120

(11.4)
H yi (t ) 

Ai
cos  c t  k cos  i  x 0 cos  i  y 0 sin  i    i
120

The resulting H-field components of the received electromagnetic wave are given in
the same manner as (11.2), that is,
3
N
H x (t )   H xi (t );
i 1
N
H y (t )   H yi (t )
(11.5)
i 1
Formulas (11.2)-(11.5) determines the total electromagnetic field at the receiver and
cover both the cases of verical and horizontal wave polarization. As follows from
(11.2)-(11.5) and was shown in [5, 9], all three electromagnetic field components are
randomly but independently oriented along the corresponding axis in the space
domain and due to random summation of each component with random phase  i and
spatial angle  i they are mutually uncorrelated.
If we now assume, as in Topic X, that the receiver moves with velocity v in
the XY-plane making an angle  to the x-axis, then, after unit time, the coordinates of
the receiver are {v cos  , v sin  , z0 } and the Doppler shift of each ith component of
E- and H-field can be presented as
f Di 
i v
 cos(    i ) cos  i
2 
(11.6)
All three components in this 3D-case can be expressed by the same in-phase and
quadrature components S(t) and C(t) according to (10.14), but with new parameters
 i and  i in them:
i 
2 v
cos(    i ) cos  i

(11.7)
2 z 0
sin  i   i

(11.8)
i 
Expressions (11.7)-(11.8) reduce to the 2D Clarke’s model [5], if all wave
components lie in the XY-plane, that is, the angle  i
is zero and Fig. 11.1 is
transformed to Fig. 10.7. If we now assume that waves arrive from all angles in the
azimuth XY-plane with equal probability, we obtain, according to [6], the PDFs of
angle-of-arrival  and  :
4
PDF( ) 
PDF () 
1
2
cos 
,
2 sin  m
(11.9)
|| | m | 

2
(11.10)
PDF () = 0
elsewhere
Formula (11.10) can be simplified for the “quasi-3D-case” when the majority of
incoming waves travel in the vertical plane in a nearly horizontal direction with  i
having a mean value of 0 0 . This case is more realistic for mobile communication
links whose antenna height is less than the heights of the local obstructions
surrounding it
PDF () 
  

cos   ,
4| m |  2  m 
|| | m | 

2
(11.11)
PDF () = 0
elsewhere
Both cases, the three-dimensional (11.10) and the quasi-two-dimensional (11.11), are
presented in Fig. 11.2a and Fig. 11.2b for  m  100 , 150 , 300 and 450 , respectively.
Here each  m relates to the corresponding maximum Doppler shift f m . As follows
from Fig. 11.2a, in the 3D-case there are sharp discontinuities at    m , a result
which does not seem to be very realistic, except at very small values of  m of the
order of a few degrees. A more realistic PDF according to (11.11) is presented by Fig.
11.2b, where the PDF is limited to  m continuously, which of course is dependent
on the local surroundings.
5
11.1.2. Power Baseband Spectra Distribution
In narrowband communication links the most energy of the received signal is
concentrated near the carrier frequency, f c , and the power spectra is defined as the RF
spectra of such a narrowband signal. At the same time, as was shown in Topic IX, the
real narrowband signal can be converted into the complex baseband signal
presentation which has its frequency content centered around zero frequency (see Fig.
11.3). Below we will describe narrowband fast fading through the power baseband
spectra distribution in the space and time domains. In this purpose we introduce the
autocorrelation function K (t , t  )  E (t )  E (t  ) of signal fading in the time
domain [10]. It can be done by the use, for example, the E-field components
presentation according to [11] in terms of a time delay  , as
E (t )  E (t  )  C(t )  C(t  ) cos  c   S (t )  S (t  ) sin  c  
(11.12)
 c( ) cos  c   s( ) sin  c 
Here operator  describes procedure of averaging, so we can rewrite the mean square
 
value of amplitude Ai , introduced earlier by (11.11), as E Ai2  Ai2  E 0 / N .
According to definition (11.12) the correlation properties of signal fading are fully
described by functions c( ) and s( ) [6]:
c(  ) 
E0
cos  c  ,
2
s( ) 
E0
sin  c 
2
(11.13)
Follow the assumption of a homogeneous distribution of incoming waves over the
angles of arrival and phase (see above) we immediately obtain for the 3D-statistical
model (see Fig. 11.1) proposed by Aulin [6] that s( )  0 , and
c( ) 
E0
2

 I 2f
0

m
 cos PDF ()d
(11.14)
6
where I 0  w is the zero-order Bessel function, PDF() is described by (11.10)(11.11) and f m is the maximum Doppler shift f m  v /  . For the particular case of
Clarke’s 2D model (Fig. 10.7), when PDF() is the delta-function, from (11.14) we
have
c0 (  ) 
E0
I 0 (2 f m )
2
(11.15)
As well-known [10], the power spectrum of the resulting received signal can be
obtained as the Fourier transform of the temporal autocorrelation function (11.12).
Taking the Fourier transform of the component (11.15) for the autocorrelation
function (7.12) we finally, according to Clarke’s 2D model, have the corresponding
complex baseband power spectrum in the frequency domain:
W0 ( f ) 
E0
4 f m
1
 f 
1  
 fm 
W0 ( f )  0,
2
, | f | f m
(11.16)
elsewhere
As follows from the illustration presented in Fig. 11.4, the 2D spectrum (we denote it
by the continuous curve) is strictly band-limited within the range | f |  f m but the
power spectral density becomes infinite at f   f m . From Aulin’s 3D model more
complicated baseband power spectral density distributions in frequency domain were
obtained in [6] by use of (11.10) and (11.14) (which we denote as W2 ( f ) ), and can be
evaluated only numerically according to [6] and analytically by use of (11.11) for the
“quasi-3D-case” denoted as W1 ( f ) [9]:
7
W1 ( f )  0,
| f | f m
W1 ( f ) 
E0
,
4 sin  m f m
W1 ( f ) 
E0
fm
f m cos  m | f |  f m
2
2

1 2 cos  m  1  ( f / f m ) 

sin

,
2
2
1

(
f
/
f
)
m


(11.17)
| f |  f m cos  m
The form of the three-dimensional baseband spectra, W1 ( f ) and
W2 ( f ) , is also
shown in Fig. 11.4 for  m  450 by dotted and dashed curves, respectively.
Comparison with the 2D spectrum presented in Fig. 11.4 by continuous curve shows
that all three spectra are strictly bandlimited to | f |  f m , but in 3D-cases the power
spectrum density is always finite. The spectrum W1 ( f ) from (11.17) is actually
constant for f m cos  m | f |  f m , whereas the spectrum W2 ( f ) according to (11.11)
does not have this non-realistic flatness. In contrast, the 2D-spectrum W0 ( f ) is infinite
at | f |  f m . As was mentioned in [9] the low-frequency content is greatly increased
even when  m is small.
Now let us compare the spectra of the different field components; earlier we
talked only about the E z component. For Clarke’s statistical 2D model [5], the
baseband signal power spectrum of the E z component is given by formula (11.16). A
similar analysis, as above, using (11.4) and (11.5) leads to expressions for the
baseband spectra W0 ( f ) distribution in the frequency domain for the magnetic field
components H x and H y :
8
E0
 f 
W0 ( f ) 
1  
 fm 
4 f m
2
for H x
(11.18)
W0 ( f ) 
E0
4 f m
 f 
 
 fm 
2
 f 
1  
 fm 
2
for H y
The form of these signal power spectra versus the radiated frequency is presented in
Fig. 11.5 with the spectrum of the E z component included for comparison. As is
shown in the illustrations, the E z component, like the H y
component, has a
minimum at the radiated carrier frequency and both are strictly limited to infinity at
the range  f m around the carrier frequency while the H x component has maximum at
the carrier frequency and is strictly limited to zero at the range  f m . We conclude
therefore that the RF signal spectrum of the resulting received signal is strictly
bandlimited to the range  f m around the carrier frequency and within those limits
the power spectral density depends on the PDFs associated with the spatial angles of
arrival  and  . The limits of the Doppler spectrum can be quite high. Frequency
shifts of such a magnitude can cause interference with the message information.
9
11.2. Spatial and Temporal Distribution of Signal Envelope and
Phase
In narrowband links the receivers usually do not detect the components C(t) and S(t)
introduced above. They record the envelope and phase of the complex signal E(t)
according to (11.1)-(11.2), which contain all information about the real signal. The
random envelope x(t) of the complex signal (11.2) can be given as
x(t )  C(t )  S (t )
(11.19)
where the components C(t) and S(t) are determined by (10.14) in Topic X.
Rayleigh signal fading. The PDF and CDF of the Rayleigh fast fading
distribution were defined by (11.6) and (11.7), respectively, where x(t) within them is
now the signal envelope. These functions are fully described in terms of the single
constant  (i.e., the mean power) by introducing the so-called mean value or
expectation
(rms 
(x mean  x
x 2  1414
.
)
 1253
. ) ,
the
rms
as
a
root
of
mean-square
.  ), which were
and the median ( x median  x M  1177
determined by (10.23)-(10.25) and presented in Fig. 10.9 (see Topic X). We must
note that both the PDF and CDF can be also expressed in terms of these parameters
instead of  . In fact, in terms of rms we have
 x 2
2x
PDF ( x )  2 exp  2
x
 x



(11.20)
 X 2 
CDF ( X )  1  exp  2 
 x 
In terms of mean value we have
10
 x 2 
x
exp
 2 
2x 2
 4x 
(11.21)
 X 2 
CDF ( X )  1  exp 
2 
 4x 
(11.22)
PDF ( x ) 
In terms of the median value we have
 x 2 ln 2 
2 x ln 2
PDF ( x) 
exp

2
x 2M
 2x M 
CDF ( X )  1  2

 2
X
xM
(11.23)
(11.24)
The same presentations can be obtained for the received signal phase. In fact,
according to the definition of signal phase ( t ) through the signal components C(t)
and S(t) [7-9] we have:
 S (t ) 
(t )  tan 1 

 C(t ) 
(11.25)
As was assumed above for Rayleigh distributed signal, its phase distribution is
uniform within the interval [0, 2] , that is, PDF()  1 / 2  . In the other words, the
resulting signal at the receiver is random and takes on all values in the range [0, 2]
with equal probability. Hence the mean value of the phase is
2
E       PDF ()d  
(11.26)
0
.  and variance  0.577 .
and in this case rms  1155
Rician signal fading.
microcell
Usually in urban communication links created for
land environments there is an LOS component or dominant specular
reflected component, which determines the coherent part of the resulting received
signal detected at the receiver, together with the multiple scattered and reflected
components which form the incoherent part of the resulting received signal [11, 12].
11
As was shown earlier, the fast signal fading in such a communication link can be
described by a Rician statistics presented by (10.29), (10.31) and (10.32). In this case
there will be a deeper fading with respect to that for Rayleigh fading with the LOS
component equal to zero (Fig. 10.11). The joint PDF of the resulting signal envelope
and phase with the dominant specular component amplitude A can be given as [6, 9]
PDF ( x, ) 
 x 2  A 2  2 Ax cos  
x
exp


2 2
2 2


(11.27)
which after integrating over the phase  reduces to (10.14) which describes the Rician
envelope PDF and limits to that for Reyleigh in the special case of A=0.
In the terms of the Rician K-factor one can obtain from (10.29), according to
(10.30a), the following formula for the Rician envelope PDF:
PDF ( x ) 
 10 K /10 2
 2 x  10 K /10 
2 x  10 K /10
2 
exp

x

A

I

 0

2
A
A2

 A
 


(11.28)
This function is shown in Fig. 11.6 for various values of K-factor: K  0, K  1 and
K  1 . It is clear that for K  0 (11.27) limits to the Rayleigh PDF while for
K  1 it becomes Gaussian with a mean value x  A . The signal phase PDF can
be presented as
 x 2 cos2    
 A 2 
1
 A cos 
 A cos   
PDF () 
exp 2 1 
exp
  1  erf 

2
  2  
2
2

 2
 
 2  
(11.29)

As follows from (11.29), the phase will be uniformly distributed in the range , 

if A  0 , that is PDF()  1 / 2 (the Rayleigh statistics). Conversely, for
A /   1 the phase will tend to that of the dominant specular component which is
defined by the Gaussian statistics.
12
7.3. Effect of Specular Reflection on Power Spectra
Now we will consider briefly the effect of the dominant specular reflected component
(with respect to the effects of multiple scattered and reflected components of the
resulting signal) on the baseband signal power spectra which takes place in the general
case of Rician signal fading. As follows from the geometry of the Doppler problem
presented in Fig. 10.7 (Topic X), a horizontally propagating dominant component
arriving at an angle  0 with respect to the direction of mobile motion causes a
Doppler shift of f m cos 0 . The resultant baseband power spectrum therefore contains
an additional component with the shape of a  -function as presented in Fig. 11.7a.
This leads to two components at normalized frequencies f / f m  1  cos  0 in the
base-band envelope spectrum which are shown in Fig. 11.7b on a logarithmic
frequency scale. As follows from envelope spectrum presentation, its upper limit
ramains at f / f m  2.
11.4. Level Crossing and Fading Statistics
If the reader looks back at Fig. 10.10 or Fig. 10.11, then it is clear that the signal
envelope is subject to fast fading. In mobile communication links when one deals with
dynamic fading (see definitions in Topic X) due to the motion of any
receiver/transmitter the picture of envelope fading also varies. In other words, in a real
situation in mobile communication the fading rate and the envelope amplitude are
functions of time. As was mentioned above, the deep signal fades which are described
by first-order Rayleigh statistics (see Fig. 10.10 or 10.11, case K=0) occur only rarely
13
and at short distances from the base station while shallow fades which are described
by Rician statistics (see Fig. 10.10 or 10.11, cases K>1) are much more frequent
within the microcell communication links. For cellular network designers it is very
important to obtain at the quantitative level a description of the rate at which fades of
any depth occur and of the average duration of a fade below any given depth.
Therefore the level crossing rate (LCR) and the average fade duration (AFD) of a
fading signal are two important statistical parameters which are useful for mobile link
design and, mostly, for designing various coding schemes in digital networks. The
required information there is provided in terms of LCR and AFD. This subject is not
treated in our book which does not deal with digital information, but rather with CW
and impulse information. Nevertheless, we will briefly describe these statistical
parameters based on a simple 2D Clarke’s model [5] and on Rayleigh short-range
statistics with usually constant mean signal level. For the reader who is interested in
this subject, we refer to the specific literature [7-9, 15-17, 19-21]. The manner in
which both required parameters, LCR and AFD, can be defined, is illustrated in Fig.
11.8a. As follows from Fig. 11.8a, the LCR at any specified level X is defined as the
expected rate at which the received signal envelope crosses that level in a positivegoing or negative-going direction. To find this expected rate we need information
about the joint PDF of the specific level X and the slope of the envelope curve x(t),
x& dx / dt , that is about PDF(X, x&). In terms of this joint PDF the LCR is defined as
the expected rate at which the Rayleigh fading envelope, normalized to the local rms
signal level, crosses a specified level X, let us say in a positive-going direction. The
number of level crossings per second, or the level-crossing rate (LCR), N X , can be
defined as [7, 9]:
14


N X   x  PDF ( X , x )dx  2 f m exp   2

(11.30)
0
where, as above, f m  v /  is the maximum Doppler frequency shift and

X
X

2   rms
(11.31)
is the value of specific level X, normalized to the local rms amplitude of fading
envelope (according to Rayleigh statistics rms  2   , see Topic X). Because f m is
a function of mobile speed v, the value N X also depends, according to (11.30), on
this parameter. For Rayleigh deep fades there are few crossings at both high and low
levels (see Fig. 10.11, the case K=0 and Fig. 11.7) with the maximum rate occurring at
  1 / 2 , i.e., at the level 3 dB below the rms level (see Fig. 11.8b). We can also
express N X in terms of the median value x M :
 X  ( X / x M )2
N X  2 ln 2 f m 
 2
 xM 
(11.32)
The average fade duration (AFD),  , is defined as the average period of time for
which the received signal envelope is below a specific level X (see Fig. 11.8a). For a
Rayleigh fading signal with deep fades, this is given by
 
1
CDF ( X )
NX
(11.33)
Here CDF(X) describes the probability of the event that the received signal x(t) does
not exceed a specific level X , that is
CDF ( X )  Pr( x  X ) 
1 n
 i
T i 1
(11.34)
15
where  i is the duration of the fade (see Fig. 11.8a) and T is the observation interval of
the fading signal. According to the Rayleigh PDF and CDF defined by (10.21) and
(10.22) the AFD can be also expressed as a function of  and f m
in terms of the rms value:
 
 
exp  2  1
2  f m
(11.35a)
in terms of the median value:
2
 

 1  X  1


2 ln 2  x M 
( X / x M )2
(11.35b)
The latter AFD presentation does not depend on the Doppler frequency shift and
hence on the mobile antenna movements. To someone interested in digital systems
design we only will point out that knowing the average duration of a signal fade helps
to determine the most likely number of signaling bits that may be lost during the fade.
AFD primarily depends upon the speed of the mobile and decreases as the maximum
of Doppler shift becomes larger according to (11.35a). It is very important to
determine also the rate at which the input signal inside the mobile communication link
falls below a given level X, and how long it remains below this level (see Fig. 11.8c).
This is useful for relating the signal-to-noise ratio (S/N) during a fade to the
instantaneous bit error rate (BER) which results. All these aspects are presented in
more detail in [7-9, 15-17].
16
11.5. Statistics of Signal Phase Variation
Above we presented statistical descriprion of signal envelope fading phenomena using
the 2D Clarke’s [5] and 3D Aulin’s [6] scattering models based on the assumption
that the received signal consists of a large number of randomly-phased components
and on the conclusion that the PDF and CDF of the signal envelope follows a
Rayleigh distribution. Now we will show how to describe the random phase variations
of the received signal in multipath environments based on such a statistical
description. We must note that to describe problems of signal phase random spatial
and temporal variations and the phase difference random variations, one need to use
more general, than Clarke’s, models following the material presented in [6-9], because
Clarke’s model can be used only for “flat” or “quasi-flat” scattering problems
(compare Fig. 10.7 and Fig. 11.1).
11.5.1. Statistical Description of Phase Difference
Variations of phase difference between multiray components of the received signal
which arrive at a given receiving point at different times, or between signals at
spatially-separated locations at the same time, can be described by the use of Aulin’s
3D statistical model. If we consider, firstly, the phase difference between the signals at
a given receiving point as a function of time delay  , then the PDF that describes
random phase difference variations can be expressed as [6, 7]:
PDF ( ) 
Here
2
1
1   2 ( ) 1  y  y(   cos y)
4 2
(1  y 2 ) 3/ 2
(11.36)
17
( ) 
c(  )
,
c(0)
y  ( ) cos 
(11.37)
where c( ) and c(0) are defined by (11.14) for   0 and   0 , respectively.
Assuming, as above, that angles-of-arrival are uniformly distributed over the range
[0, 2] , i.e., PDF()  1 / 2  , we can, from (11.44), determine the phase difference
distribution between the signals at two spatially-separated points using the timedistance relation   v  . The results of deriving the PDF( ) distribution is shown
in Fig. 11.9, according to [6, 7, 9], versus the phase difference for various normalized
separation distances between receiving points,  /   0.1  0.5. It is clear that for the
case of coincident points (   0 ) the PDF( ) limits to zero everywhere except at
  0, 2 , while for the other extreme case of    the phase difference  is
uniformly distributed with PDF( )  1 / 2 .
11.5.2. Random Phase Variations
In mobile communication links, due to mobile antenna motion the phase  of the
received signal randomly varies in the time domain with changes of mobile antenna
location. This effect is equivalent to a random phase modulation, an analog of
frequency modulation because the time derivative of phase  causes frequency
modulation and appears as noise to the receiver [7, 9]. This derivative can be
presented simply by the use of expression (7.23), that is
 
d d  1  S (t ) 

 tan 
dt dt 
C
(
t
)


(11.38)
Follow [6, 9] we can present the PDF of phase variations in the time domain in terms
of the maximum Doppler frequency shift f m as
18

1
1  2
PDF () 
f m 8  2
1
2
   
  
 f m  
3 / 2
(11.39)
The corresponding CDF which defines the probability of the event that the phase
 , can be given as
derivative  does not exceed some specific level 


2
 )  1 1   1   
PDF (
2 2
2
f m 2  2 f m 

1 / 2



(11.40)
Investigations of these two functions have shown that the highest probabilities of
phase difference distribution occur for small values of the phase derivative  .
The spectrum of random phase (frequency) modulation can be found, as
above, from the Fourier transform of the autocorrelation function of  according to
[6] through the well-known quadratures:
2
1  c( )  c( )   c( ) 


 
K (t , t   )   (t ) (t   )  
 ln 1 
2  c( ) 
c( )   c(0) 


(11.41)
where c( ) and c( ) denote first and second order derivatives of function c( ) with
respect to  . In [7] the expression for the random spectrum of phase modulation
which is based on Clarke’s 2D statistical model is presented. We do not present this
complicated formulas because it cannot be used for a realistic case of multipath
mobile links. More interesting to present here is the power spectrum of random phase
(frequency) modulation obtained in [18] by Davis who studied this problem in some
detail. The power spectrum of random phase (frequency) modulation is plotted in Fig.
11.10 on normalised frequency scales. We note that in contrast to the strictly
bandlimited power spectrum of the signal envelope (the Doppler spectrum presented
in Fig. 11.6) there is a finite probability of finding the frequency of the random phase
(frequency) modulation at any value. Nevertheless, the signal power is largely
19
confined to f  2 f m 10 log( f / 2 f m )  0 from where it falls off as 1 / f and is
insignificant beyond f  5 f m . As was mentioned in [9], the majority of signal power
is therefore confined to the audio band and the larger excursions, being associated
with the deep fades (Rayleigh fading) occur only rarely.
20
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