Statistical Models for Multipath Fading Channels

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Table of Contents

  1. 5.7 Statistical Models for Multipath Fading Channels
  2. 5.7.1 Clarke’s Model for Flat Fading
  3. 5.7.2 Simulation of Clarke and Gans Fading Model
  4. 5.7.3 Level Crossing and Fading Statistics
  5. 5.7.4 Two-ray Rayleigh Fading Model
  6. 5.7.5 Saleh and Valenzuela Indoor Statistical Model
  7. 5.7.6 SIRCIM and SMRCIM Indoor and Outdoor Statistical Models
  8. Relevant NI products
  9. Buy the Book

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.

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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 E0 is the real amplitude of local average E-field (assumed constant), Cn is a real random

variable representing the amplitude of individual waves, η is the intrinsic impedance of free

space   and fc, and is the carrier frequency. The random phase of the nth arriving component

θn is given by

The amplitudes of the E-and H-field are normalized such that the ensemble average of the

Cn ’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

Ez, Hx, and Hy 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




Both  Tc(t) and Ts(t) are Gaussian random processes which are denoted as

Tc and , Ts respectively, at any time t. Tc and Ts 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, Ez(t), is given by

Since Tc and  Ts 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 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 fc, then the instantaneous frequency of the received

signal component arriving at an angle  is obtained using Equation (5.57)

where fm 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


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.

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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 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 fm. 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 -fm to fm), 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 (fm).  The value used for N is usually a power of two.

2. Compute the frequency spacing between adjacent spectral lines


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 Tc(t) and Ts(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.

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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, fm, 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 NR, 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

fm 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 fm 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 fm 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.


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 Nr = 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.

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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.

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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.

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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, Xl,

within a 1 m local area, the topography Sm which is either line-of-sight (LOS) or obstructed,

the large-scale T–R separation distance Dn, and the particular measurement location Pn.

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), A2i  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, Np, arriving at a certain location is a function of X, Sm,

and Pn, 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

Ti in a particular environment Sm is denoted as PR (Ti ,Sm ). 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 S1 corresponds to the LOS topography, and S2 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, NP(X,Sm,Pn), 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

NP 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


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].

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8. Relevant NI products

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