1. 5.7 Statistical Models for Multipath Fading Channels
Several multipath models have been suggested to explain the observed statistical nature of a
mobile channel. The first model presented by Ossana [Oss64] was based on interference of
waves incident and reflected from the flat sides of randomly located buildings. Although
Ossana’s model [Oss64] predicts flat fading power spectra that were in agreement with measurements
in suburban areas, it assumes the existence of a direct path between the transmitter and
receiver, and is limited to a restricted range of reflection angles. Ossana’s model is therefore
rather inflexible and inappropriate for urban areas where the direct path is almost always
blocked by buildings or other obstacles. Clarke’s model [Cla68] is based on scattering and is
widely used.
2. 5.7.1 Clarke’s Model for Flat Fading
Clarke [Cla68] developed a model where the statistical characteristics of the electromagnetic
fields of the received signal at the mobile are deduced from scattering. The model assumes a
fixed transmitter with a vertically polarized antenna. The field incident on the mobile antenna is
assumed to be comprised of N azimuthal plane waves with arbitrary carrier phases, arbitrary azimuthal
angles of arrival, and each wave having equal average amplitude. It should be noted that
the equal average amplitude assumption is based on the fact that in the absence of a direct lineof-
sight path, the scattered components arriving at a receiver will experience similar attenuation
over small-scale distances.
Figure 5.19 Illustrating plane waves arriving at random angles.
Figure 5.19 shows a diagram of plane waves incident on a mobile traveling at a velocity ,
in the x-direction. The angle of arrival is measured in the x-y plane with respect to the direction
of motion. Every wave that is incident on the mobile undergoes a Doppler shift due to the
motion of the receiver and arrives at the receiver at the same time. That is, no excess delay due to
multipath is assumed for any of the waves (flat fading assumption). For the nth wave arriving at
an angle to the x-axis, the Doppler shift in Hertz is given by
where is the wavelength of the incident wave.
The vertically polarized plane waves arriving at the mobile have E and H field components
given by
where E_{0} is the real amplitude of local average E-field (assumed constant), C_{n} is a real random
variable representing the amplitude of individual waves, η is the intrinsic impedance of free
space and f_{c}, and is the carrier frequency. The random phase of the nth arriving component
θ_{n} is given by
The amplitudes of the E-and H-field are normalized such that the ensemble average of the
C_{n} ’s is given by
Since the Doppler shift is very small when compared to the carrier frequency, the three
field components may be modeled as narrow band random processes. The three components
E_{z}, H_{x}, and H_{y} can be approximated as Gaussian random variables if is sufficiently large.
The phase angles are assumed to have a uniform probability density function (pdf) on the interval
Based on the analysis by Rice [Ric48] the E-field can be expressed in an in-phase and
quadrature form
where
and
Both T_{c}(t) and T_{s}(t) are Gaussian random processes which are denoted as
T_{c} and , T_{s} respectively, at any time t. T_{c} and T_{s} are uncorrelated zero-mean Gaussian random variables
with an equal variance given by
where the overbar denotes the ensemble average.
The envelope of the received E-field, E_{z}(t), is given by
Since T_{c} and T_{s} are Gaussian random variables, it can be shown through a Jacobean
transformation [Pap91] that the random received signal envelope r has a Rayleigh
distribution given by
where
5.7.1.1 Spectral Shape Due to Doppler Spread in Clarke’s Model
Gans [Gan72] developed a spectrum analysis for Clarke’s model. Let
denote the fraction of the total incoming power within of the angle and let A denote the average
received power with respect to an isotropic antenna. As N → ∞, approaches a continuous,
rather than a discrete, distribution. If is the azimuthal gain pattern of the mobile
antenna as a function of the angle of arrival, the total received power can be expressed as
where is the differential variation of received power with angle. If the
scattered signal is a CW signal of frequency f_{c}, then the instantaneous frequency of the received
signal component arriving at an angle is obtained using Equation (5.57)
where f_{m} is the maximum Doppler shift. It should be noted that
is an even function of
If S(f) is the power spectrum of the received signal, the differential variation of received
power with frequency is given by
Equating the differential variation of received power with frequency to the differential
variation in received power with angle, we have
Differentiating Equation (5.70), and rearranging the terms, we have
Using Equation (5.70), can be expressed as a function of f as
This implies that
Substituting Equation (5.73) and (5.75) into both sides of (5.72), the power spectral density
S(f) can be expressed as
where
The spectrum is centered on the carrier frequency and is zero outside the limits of
Each of the arriving waves has its own carrier frequency (due to its direction of arrival)
which is slightly offset from the center frequency. For the case of a vertical
antenna and a uniform distribution over
to the output spectrum is given by (5.76) as
In Equation (5.78), the power spectral density at is infinite, i.e., Doppler components
arriving at exactly 0° and 180° have an infinite power spectral density. This is not a
problem since is continuously distributed and the probability of components arriving at
exactly these angles is zero.
Figure 5.20 shows the power spectral density of the resulting RF signal due to Doppler
fading. Smith [Smi75] demonstrated an easy way to simulate Clarke’s model using a computer
simulation as described Section 5.7.2.
After envelope detection of the Doppler-shifted signal, the resulting baseband spectrum
has a maximum frequency of It can be shown [Jak74] that the electric field produces a
baseband power spectral density given by
where K[•] is the complete elliptical integral of the first kind. Equation (5.79) is not intuitive
and is a result of the temporal correlation of the received signal when passed through a nonlinear
envelope detector. Figure 5.21 illustrates the baseband spectrum of the received signal after
envelope detection.
The spectral shape of the Doppler spread determines the time domain fading waveform and
dictates the temporal correlation and fade slope behaviors. Rayleigh fading simulators must use a
fading spectrum such as Equation (5.78) in order to produce realistic fading waveforms that have
proper time correlation.
Figure 5.20 Doppler power spectrum for an unmodulated CW carrier [from [Gan72] © IEEE].
Figure 5.21 Baseband power spectral density of a CW Doppler signal after envelope detection.
3. 5.7.2 Simulation of Clarke and Gans Fading Model
It is often useful to simulate multipath fading channels in hardware or software. A popular simulation
method uses the concept of in-phase and quadrature modulation paths to produce a simulated
signal representing Equation (5.63) with spectral and temporal characteristics very close to
measured data.
As shown in Figure 5.22(b), two independent Gaussian low pass noise sources are used to
produce in-phase and quadrature fading branches. Each Gaussian source may be formed by summing
two independent Gaussian random variables which are orthogonal (i.e., g=a+jb, where a and b are real
Gaussian random variables and g is complex Gaussian). By using the spectral filter defined by
Equation (5.78) to shape the random signals in the frequency domain, accurate time domain
waveforms of Doppler fading can be produced by using an inverse fast Fourier transform (IFFT)
at the last stage of the simulator.
Smith [Smi75] demonstrated a simple computer program that implements Figure 5.22(b).
His method uses a complex Gaussian random number generator (noise source) to produce a
baseband line spectrum with complex weights in the positive frequency band. The maximum
Figure 5.22 Simulator using quadrature amplitude modulation with (a) RF Doppler filter and
(b) baseband Doppler filter.
frequency component of the line spectrum is f_{m}. Using the property of real signals, the negative
frequency components are constructed by simply conjugating the complex Gaussian values
obtained for the positive frequencies. Note that the IFFT of each complex Gaussian signal
should be a purely real Gaussian random process in the time domain which is used in each of the
quadrature arms shown in Figure 5.24. The random valued line spectrum is then multiplied with
a discrete frequency representation of having the same number of points as the noise
source. To handle the case where Equation (5.78) approaches infinity at the passband edge,
Smith truncated the value of by computing the slope of the function at the sample frequency
just prior to the passband edge and increasing the slope to the passband edge. Simulations
using the architecture in Figure 5.22 are usually implemented in the frequency domain
using complex Gaussian line spectra to take advantage of easy implementation of Equation
(5.78). This, in turn, implies that the low pass Gaussian noise components are actually a series of
frequency components (line spectrum from -f_{m} to f_{m}), which are equally spaced and each have
a complex Gaussian weight. Smith’s simulation methodology is shown in Figure 5.24.
Figure 5.23 Simulator using quadrature amplitude modulation with (a) RF Doppler filter and
(b) baseband Doppler filter.
Figure 5.24 Frequency domain implementation of a Rayleigh fading simulator at baseband
To implement the simulator shown in Figure 5.24, the following steps are used:
1. Specify the number of frequency domain points (N) used to represent
and the maximum Doppler frequency shift (f_{m}). The value used for N is usually a power of two.
2. Compute the frequency spacing between adjacent spectral lines
as
This defines the time duration of a fading waveform,
3. Generate complex Gaussian random variables for each of the N ⁄ 2 positive frequency
components of the noise source.
4. Construct the negative frequency components of the noise source by conjugating positive
frequency values and assigning these at negative frequency values.
5. Multiply the in-phase and quadrature noise sources by the fading spectrum .
6. Perform an IFFT on the resulting frequency domain signals from the in-phase and quadrature
arms to get two N-length time series, and add the squares of each signal point in time
to create an N-point time series like under the radical of Equation (5.67). Note that each
quadrature arm should be a real signal after the IFFT to model Equation (5.63).
7. Take the square root of the sum obtained in Step 6 to obtain an N-point time series of a
simulated Rayleigh fading signal with the proper Doppler spread and time correlation.
Several Rayleigh fading simulators may be used in conjunction with variable gains and
time delays to produce frequency selective fading effects. This is shown in Figure 5.25.
By making a single frequency component dominant in amplitude within and at
f = 0, the fading is changed from Rayleigh to Ricean. For a multipath fading simulator with
many resolvable components, this mtheod can be used to alter the probability distributions of the
individual multipath components in the simulator of Figure 5.25. One must take care to properly
Figure 5.25 A signal may be applied to a Rayleigh fading simulator to determine performance in
a wide range of channel conditions. Both flat and frequency selective fading conditions may be
simulated, depending on gain and time delay settings.
implement the IFFT such that each arm of Figure 5.24 produces a real time domain signal as
given by T_{c}(t) and T_{s}(t) in Equations (5.64) and (5.65).
To determine the impact of flat fading on an applied signal s(t), one merely needs to multiply
the applied signal by r(t) the output of the fading simulator. To determine the impact of
more than one multipath component, a convolution must be performed as shown in Figure 5.25.
4. 5.7.3 Level Crossing and Fading Statistics
Rice computed joint statistics for a mathematical problem which is similar to Clarke’s fading model
[Cla68], and thereby provided simple expressions for computing the average number of level crossing
and the duration of fades. The level crossing rate (LCR) and average fade duration of a Rayleigh
fading signal are two important statistics which are useful for designing error control codes
and diversity schemes to be used in mobile communication systems, since it becomes possible to
relate the time rate of change of the received signal to the signal level and velocity of the mobile.
The level crossing rate (LCR) is defined as the expected rate at which the Rayleigh fading
envelope, normalized to the local rms signal level, crosses a specified level in a positive-going
direction. The number of level crossings per second is given by
where is the time derivative of r (t) (i.e., the slope), is the joint density function of
r and at r = R, f_{m}, is the maximum Doppler frequency, and is the value of the
specified level R, normalized to the local rms amplitude of the fading envelope [Jak74]. Equation
(5.80) gives the value of N_{R}, the average number of level crossings per second at specified.
The level crossing rate is a function of the mobile speed as is apparent from the presence of
f_{m} in Equation (5.80). There are few crossings at both high and low levels, with the maximum
rate occurring at (i.e., at a level 3 dB below the rms level). The signal envelope
experiences very deep fades only occasionally, but shallow fades are frequent.
The average fade duration is defined as the average period of time for which the received
signal is below a specified level R. For a Rayleigh fading signal, this is given by
where Pr[r ≤ R] is the probability that the received signal r is less than R and is given by
where is the duration of the fade and T is the observation interval of the fading signal. The probability
that the received signal r is less than the threshold R is found from the Rayleigh distribution as
where p(r) is the pdf of a Rayleigh distribution. Thus, using Equations (5.80), (5.81), and
(5.83), the average fade duration as a function of ρ and f_{m} can be expressed as
The average duration of a signal fade helps determine the most likely number of signaling
bits that may be lost during a fade. Average fade duration primarily depends upon the speed of
the mobile, and decreases as the maximum Doppler frequency f_{m} becomes large. If there is a
particular fade margin built into the mobile communication system, it is appropriate to evaluate
the receiver performance by determining the rate at which the input signal falls below a given
level R, and how long it remains below the level, on average. This is useful for relating SNR
during a fade to the instantaneous BER which results.
Example 5.9
Find the average fade duration for a threshold level of ρ = 0.707 when the
Doppler frequency is 20 Hz. For a binary digital modulation with bit duration
of 50 bps, is the Rayleigh fading slow or fast? What is the average number
of bit errors per second for the given data rate. Assume that a bit error
occurs whenever any portion of a bit encounters a fade for which ρ < 0.1.
Solution
The average fade duration can be obtained using Equation (5.84).
For a data rate of 50 bps, the bit period is 20 ms. Since the bit period is
greater than the average fade duration, for the given data rate the signal
undergoes fast Rayleigh fading. Using Equation (5.84), the average fade
duration for ρ = 0.1 is equal to 0.002 s. This is less than the duration of one
bit. Therefore, only one bit on average will be lost during a fade. Using Equation
(5.80), the number of level crossings for ρ = 0.1 is N_{r} = 4.96 crossings
per seconds. Since a bit error is assumed to occur whenever a portion of a
bit encounters a fade, and since average fade duration spans only a fraction
of a bit duration, the total number of bits in error is 5 per second, resulting in
a BER = (5/50) = 0.1.
5. 5.7.4 Two-ray Rayleigh Fading Model
Clarke’s model and the statistics for Rayleigh fading are for flat fading conditions and do not
consider multipath time delay. In modern mobile communication systems with high data rates, it
has become necessary to model the effects of multipath delay spread as well as fading. A commonly
used multipath model is an independent Rayleigh fading two-ray model (which is a specific
implementation of the generic fading simulator shown in Figure 5.25). Figure 5.26 shows a
block diagram of the two-ray independent Rayleigh fading channel model. The impulse
response of the model is represented as
whereand are independent and Rayleigh distributed, and are independent and
uniformly distributed over and is the time delay between the two rays.
By setting equal to zero, the special case of a flat Rayleigh fading channel is obtained as
By varying , it is possible to create a wide range of frequency selective fading effects.
The proper time correlation properties of the Rayleigh random variablesand are guaranteed
by generating two independent waveforms, each produced from the inverse Fourier transform
of the spectrum described in Section 5.7.2.
Figure 5.26 Two-ray Rayleigh fading model.
6. 5.7.5 Saleh and Valenzuela Indoor Statistical Model
Saleh and Valenzuela [Sal87] reported the results of indoor propagation measurements between
two vertically polarized omnidirectional antennas located on the same floor of a medium sized
office building. Measurements were made using 10 ns, 1.5 GHz, radar-like pulses. The method
involved averaging the square law detected pulse response while sweeping the frequency of the
transmitted pulse. Using this method, multipath components within 5 ns were resolvable.
The results obtained by Saleh and Valenzuela show that: (a) the indoor channel is quasistatic
or very slowly time varying, and (b) the statistics of the channel impulse response are independent
of transmitting and receiving antenna polarization, if there is no line-of-sight path
between them. They reported a maximum multipath delay spread of 100 ns to 200 ns within the
rooms of a building, and 300 ns in hallways. The measured rms delay spread within rooms had a
median of 25 ns and a maximum of 50 ns. The large-scale path loss with no line-of-sight path
was found to vary over a 60 dB range and obey a log-distance power law (see Equation (4.68))
with an exponent between three and four.
Saleh and Valenzuela developed a simple multipath model for indoor channels based on
measurement results. The model assumes that the multipath components arrive in clusters. The
amplitudes of the received components are independent Rayleigh random variables with variances
that decay exponentially with cluster delay as well as excess delay within a cluster.
The corresponding phase angles are independent uniform random variables over
The clusters and multipath components within a cluster form Poisson arrival processes with different rates.
The clusters and multipath components within a cluster have exponentially distributed interarrival
times. The formation of the clusters is related to the building structure, while the components
within the cluster are formed by multiple reflections from objects in the vicinity of the
transmitter and the receiver.
7. 5.7.6 SIRCIM and SMRCIM Indoor and Outdoor Statistical Models
Rappaport and Seidel [Rap91a] reported measurements at 1300 MHz in five factory buildings
and carried out subsequent measurements in other types of buildings. The authors developed an
elaborate, empirically derived statistical model to generate measured channels based on the discrete
impulse response channel model of Equation (5.12) and wrote a computer program called
SIRCIM (Simulation of Indoor Radio Channel Impulse-response Models). SIRCIM generates
realistic samples of small-scale indoor channel impulse response measurements [Rap91a]. Subsequent
work by Huang produced SMRCIM (Simulation of Mobile Radio Channel Impulseresponse
Models), a similar program that generates small-scale urban cellular and microcellular
channel impulse responses [Rap93a]. These programs are currently in use at over 100 institutions
throughout the world, and have been updated to include angle of arrival information for
microcell, indoor, and macrocell channels [Nuc99], [Lib99].
By recording power delay profile impulse responses at intervals on a 1 m track at many indoor
measurement locations, the authors were able to characterize local small-scale fading
of individual multipath components, and the small-scale variation in the number and arrival
times of multipath components within a local area. Thus, the resulting statistical models are
functions of multipath time delay bin the small-scale receiver spacing, X_{l},
within a 1 m local area, the topography S_{m} which is either line-of-sight (LOS) or obstructed,
the large-scale T–R separation distance D_{n}, and the particular measurement location P_{n}.
Therefore, each individual baseband power delay profile is expressed in a manner similar
to Equation (5.12), except the random amplitudes and time delays are random variables
which depend on the surrounding environment.
Phases are synthesized using a pseudo-deterministic model which provides realistic
results, so that a complete time varying complex baseband channel impulse response
may be obtained over a local area through simulation.
In Equation (5.87), A^{2}_{i} is the average multipath receiver power within a discrete excess delay
interval of 7.8125 ns.
The measured multipath delays inside open-plan buildings ranged from 40 ns to 800 ns.
Mean multipath delay and rms delay spread values ranged from 30 ns to 300 ns, with median
values of 96 ns in LOS paths and 105 ns in obstructed paths. Delay spreads were found to be
uncorrelated with T–R separation but were affected by factory inventory, building construction
materials, building age, wall locations, and ceiling heights. Measurements in a food processing
factory that manufactures dry-goods and has considerably less metal inventory than other factories
had an rms delay spread that was half of those observed in factories producing metal products.
Newer factories which incorporate steel beams and steel reinforced concrete in the building
structure have stronger multipath signals and less attenuation than older factories which used
wood and brick for perimeter walls. The data suggested that radio propagation in buildings may
be described by a hybrid geometric/statistical model that accounts for both reflections from
walls and ceilings and random scattering from inventory and equipment.
By analyzing the measurements from fifty local areas in many buildings, it was found that
the number of multipath components, N_{p}, arriving at a certain location is a function of X, S_{m},
and P_{n}, and almost always has a Gaussian distribution. The average number of multipath components
ranges between 9 and 36, and is generated based on an empirical fit to measurements.
The probability that a multipath component will arrive at a receiver at a particular excess delay
T_{i} in a particular environment S_{m} is denoted as P_{R} (T_{i} ,S_{m} ). This was found from measurements
by counting the total number of detected multipath components at a particular discrete
excess delay time, and dividing by the total number of possible multipath components for each
excess delay interval. The probabilities for multipath arriving at a particular excess delay values
may be modeled as piecewise functions of excess delay, and are given by
where S_{1} corresponds to the LOS topography, and S_{2} corresponds to obstructed topography.
SIRCIM uses the probability of arrival distributions described by Equation (5.88) or (5.89) along
with the probability distribution of the number of multipath components, N_{P}(X,S_{m},P_{n}), to
simulate power delay profiles over small-scale distances. A recursive algorithm repeatedly compares
Equation (5.88) or (5.89) with a uniformly distributed random variable until the proper
N_{P} is generated for each profile [Hua91], [Rap91a].
Figure 5.27 shows an example of measured power delay profiles at 19 discrete receiver
locations along a 1 m track, and illustrates accompanying narrowband information which SIRCIM
computes based on synthesized phases for each multipath component [Rap91a]. Measurements
reported in the literature provide excellent agreement with impulse responses predicted by
SIRCIM.
Using similar statistical modeling techniques, urban cellular and microcellular multipath
measurement data from [Rap90], [Sei91], [Sei92a] were used to develop SMRCIM. Both large
cell and microcell models were developed. Figure 5.28 shows an example of SMRCIM output
for an outdoor microcell environment [Rap93a].
8. Relevant NI products
Customers interested in this topic were also interested in the following NI products:
- RF and Communication Hardware and Software
- Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
- LabVIEW Graphical Programming Environment
For the complete list of tutorials, return to the NI RF and Communications Fundamentals Main page.
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