From Wireless Communication

Radio Propagation Models



Land-mobile communication is burdened with particular propagation complications compared to the channel characteristics in radio systems with fixed and carefully positioned antennas. The antenna height at a mobile terminal is usually very small, typically less than a few meters. Hence, the antenna is expected to have very little 'clearance', so obstacles and reflecting surfaces in the vicinity of the antenna have a substantial influence on the characteristics of the propagation path. Moreover, the propagation characteristics change from place to place and, if the mobile unit moves, from time to time. Thus, the transmission path between the transmitter and the receiver can vary from simple direct line of sight to one that is severly obstructed by buildings, foliage and the terrain.

In generic system studies, the mobile radio channel is usually evaluated from 'statistical' propagation models: no specific terrain data is considered, and channel parameters are modelled as stochastic variables. The mean signal strength for an arbitrary transmitter-receiver (T-R) separation is useful in estimating the radio coverage of a given transmitter whereas measures of signal variability are key determinants in system design issues such as antenna diversity and signal coding.

Three mutually independent, multiplicative propagation phenomena can usually be distinguished: multipath fading, shadowing and 'large-scale' path loss.

The large-scale effects determine a power level averaged over an area of tens or hundreds of metres and therefore called the 'area-mean' power. Shadowing introduces additional fluctuations, so the received local-mean power varies around the area-mean. The term 'local-mean' is used to denote the signal level averaged over a few tens of wave lengths, typically 40 wavelengths. This ensures that the rapid fluctuations of the instantaneous received power due to multipath effects are largely removed.

Path Loss: Models of "large-scale effects"


The most appropriate path loss model depends on the location of the receiving antenna. For the example above at:

Path-loss law

Figure 1: Average path loss versus distance in UHF bands as measured
in Northern Germany. (a, green): forestrial terrain; (b, orange): open area;
(grey): average of (a) and (b); (black): Egli's model

Free Space Propagation

For propagation distances d much larger than the square of the antenna size divided by the wavelength, the far-field of the generated electromagnetic wave dominates all other components (in the far-field region the electric and magnetic fields vary inversely with distance). In free space, the power radiated by an isotropic antenna is spread uniformly and without loss over the surface of a sphere surrounding the antenna. An isotropic antenna is a hypothetical entity!! Even the simplest antenna has some directivity. For example, a linear dipole has uniform power flow in any plane perpendicular to the axis of the dipole (omnidirectionality) and the maximum power flow is in the equatorial plane (see Appendix 1: Antenna Fundamentals).

The surface area of a sphere of radius d is 4d2, so that the power flow per unit area w(power flux in watts/meter2) at distance d from a transmitter antenna with input accepted power  pT and antenna gain  GT is


Transmitting antenna gain is defined as the ratio of the intensity (or power flux) radiated in some particular direction to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically. When the direction is not stated, the power gain is usually taken in the direction of maximum power flow. The product  GT p  is called the effective radiated power (ERP) of the transmitter. The available power  pR at the terminals of a receiving antenna with gain GR is

where A is the effective area or aperture of the antenna and    (see Appendix 1: Antenna Fundamentals). The wavelength = c / fc with c the velocity of light and fcthe carrier frequency.

While cellular telephone operator mostly calculate in received powers, in the planning of the coverage area of broadcast transmitters, the CCIR recommends the use of the electric field strength E at the location of the receiver. The conversion is .


Show that for a reference transmitter with ERP of 1 kwatt in free space,

As the propagation distance increases, the radiated energy is spread over the surface of a sphere of radius d, so the power received decreases proportional to d2. Expressed in dB, the received power is


Show that the path loss L between two isotropic antennas (GR = 1, GT = 1) can be expressed as

L (dB)= - 32.44 - 20 log ( fc/ 1MHz) - 20 log (d / 1km)

which leads engineers to speak of a "20 log d" law.

Propagation over a Plane Earth

If we consider the effect of the earth surface, the expressions for the received signal become more complicated than in case of free space propagation. The main effect is that signals reflected off the earth surface may (partially) cancel the line of sight wave.


For an isotropic or omnidirectional antenna above a plane earth, the received electric field strength is

with Rg the reflection coefficient and E0 the field strength for propagation in free space. This expression can be interpreted as the complex sum of a direct line-of-sight wave, a ground-reflected wave and a surface wave. The phasor sum of the first and second term is known as the space wave.

For a horizontally-polarized wave incident on the surface of a perfectly smooth earth,

where r is the relative dielectric constant of the earth, is the angle of incidence (between the radio ray and the earth surface) and x = /(2 fc0) with the conductivity of the ground and 0 the dielectric constant of vacuum.

For vertical polarization


Show that the reflection coefficient tends to -1 for angles close to 0. Verify that for vertical polarization, abs( Rc) > 0.9 for <10 degrees. For horizontal polarization, abs( Rc) > 0.5 for < 5 degrees and abs( Rc) > 0.9 for < 1 degree.

UHF Mobile Communication

The relative amplitude F(.) of the surface wave is very small for most cases of mobile UHF communication (F(.) << 1). Its contribution is relevant only a few wavelengths above the ground. The phase difference between the direct and the ground-reflected wave can be found from the two-ray approximation considering only a Line-of-Sight and a Ground Reflection. Denoting the transmit and receive antenna heights as hT and hR, respectively, the phase difference can be expressed as

For large d, one finds, using    the expression


For large d, (d >> 5hT hR ), the reflection coefficient tends to  -1, so the received signal power becomes

For propagation distances substantially beyond the turnover point -- i.e.  -- this expression tends to the fourth power distance law:


Discuss the effect of path loss on the performance of a cellular radio network. Is it good to have signals attenuate rapidly with increasing distance?

Egli's Model (1957)

Experiments confirm that in macro-cellular links over smooth, plane terrain, the received signal power (expressed in dB) decreases with "40 log d". Also a "6 dB/octave" height gain is experienced: doubling the height increases the received power by a factor 4.

In contrast to the theoretical plane earth loss, Egli measured a significant increase of the path loss with the carrier frequency fc. He proposed the semi-empirical model

i.e., he introduced a frequency dependent empirical correction for ranges 1< d < 50 km, and carrier frequencies 30 MHz < fc < 1 GHz.

For communication at short range, this formula looses its accuracy because the reflection coefficient is not necessarily close to -1. For  ,  free space propagation is more appropriate, but a number of significant reflections must be taken into account. In streets with high buildings, guided propagation may occur.

Diffraction loss

If the direct line-of-sight is obstructed by a single object ( of height hm), such as a mountain or building, the attenuation caused by diffraction over such an object can be estimated by treating the obstruction as a diffracting knife-edge.

Figure: Path profile model for (single) knife edge diffraction

This is the simplest of diffraction models, and the diffraction loss in this case can be readily estimated using the classical Fresnel solution for the field in the shadow behind a half-plane. Thus, the field strength in the shadowed region is given by

where E0 is the free space field strength in the absence of the knife-edge and F(v) is the complex Fresnel integral which is a tabulated function of the diffraction parameter

where dT and dR are the terminal distances from the knife edge. The diffraction loss, additional to free space loss and expressed in dB, can be closely approximated by

The attenuation over rounded obstacles is usually higher than Adiff  in the above formula.

Approximate techniques to compute the diffraction loss over multiple knife edges have been proposed by

Total Path loss

The previously presented methods for ground reflection loss and diffraction losses suggest a "Mondriaan" interpretation of the path profile: Obstacles occur as straight vertical lines while horizontal planes cause reflections. That is the propagation path is seen as a collection of horizontal and vertical elements. Accurate computation of the path loss over non-line-of-sight paths with ground reflections is a complicated task and does not allow such simplifications.

Many measurements of propagation losses for paths with combined diffraction and ground reflection losses indicate that knife edge type of obstacles significantly reduce ground wave losses. Blomquist suggested three methods that may be used to find the total loss -- viz.

where Afs is the free space loss, AR is the ground reflection loss and Adiff is the multiple knife-edge diffraction loss in dB values.

Statistical Path Loss Law: 

Most generic system studies address networks in which all mobile units have the same gain, height and transmitter power. For ease of notation, received signal powers and propagation distances can be normalized. In macro-cellular networks
(1 km  < d < 50 km), the area-mean received power can be written as

with r the normalized distance and  the path loss exponent. Theoretical  values are, respectively 2 and 4 for free space and plane, smooth, perfectly conducting terrain. Typical values for irregular terrain are between 3.0 and 3.4 and in forestal terrain propagation can be appropriately described as in free space plus some diffraction losses, but without significant groundwave losses (). If the propagation model has to cover a wide range of distances,  may vary as different propagation mechanisms dominate at different ranges. In micro-cellular nets,  typically changes from approximately 2 to approximately 4 at some turnover distance dg. Experimental values of d are between 90 and 300 m for hT between 5 and 20 m and hR approximately 2m where hand hare, respectively, the heights of the transmitting and receiving and antennas. These values are in reasonable agreement with the theoretical expression  where  is the wavelength of the transmitted wave.

Many models have been proposed and are used in system design:

Multipath in Micro-Cells

The micro-cellular propagation channel typically is Rician : it contains a dominant direct component, with an amplitude determined by path loss, a set of early reflected waves adding (possibly distructively) with the dominant wave, and intersymbol interference caused by the excessively delayed waves, adding incoherently with the dominant wave.


Experiments reported by Egli in 1957 showed that, for paths longer than a few hundred meters, the received (local-mean) power fluctuates with a 'log-normal' distribution about the area- mean power. "Log-normal" means that the local-mean power expressed in logarithmic values -- i.e.,

-- has a normal -- i.e., Gaussian distribution. The probability density function (pdf) of the local-mean power is thus of the form

where s is the logarithmic standard deviation of the shadowing, expressed in natural units.

Depth of Shadowing

For average terrain, Egli reported a logarithmic standard deviation of about 8.3 dB and 12 dB for VHF and UHF frequencies, respectively. Such large fluctuations are caused not only by local shadow attenuation by obstacles in the vicinity of the antenna, but also by large-scale effects leading to a coarse estimate of the area-mean power.

This log-normal fluctuation was called large-area shadowing by Marsan, Hess and Gilbert; over semi-circular routes in Chicago, with fixed distance to the base station, it was found to range from 6.5 dB to 10.5 dB, with a median of 9.3 dB. Large-area shadowing thus reflects shadow fluctuations if the vehicle moves over many kilometres.

In contrast to this, in most papers on mobile propagation, only small-area shadowing is considered: log-normal fluctuations of the local-mean power over a distance of tens or hundreds of metres are measured. Marsan et al. reported a median of 3.7 dB for small area shadowing. Preller and Koch measured local-mean powers at 10 m intervals and studied shadowing over 500 m intervals. The maximum standard deviation experienced was about 7 dB, but 50% of all experiments showed shadowing of less than 4 dB.

Implications for Cell Planning

If one extends the distinction between large-area and small-area shadowing, the definition of shadowing covers any statistical fluctuation of the received local-mean power about a certain area-mean power, with the latter determined by (predictable) large-scale mechanisms. Multipath propagation is separated from shadow fluctuations by considering the local-mean powers. That is, the standard deviation of the shadowing will depend on the geographical resolution of the estimate of the area-mean power. A propagation model which ignores specific terrain data produces about 12 dB of shadowing. On the other hand, prediction methods using topographical data bases with unlimited resolution can, at least in theory, achieve a standard deviation of 0 dB. Thus, the standard deviation is a measure of the impreciseness of the terrain description. If, for generic system studies, the (large-scale) path loss is taken of simple form depending only on distance but not on details of the path profile, the standard deviation will necessarily be large. On the other hand, for the planning of a practical network in a certain (known) environment, the accuracy of the large-scale propagation model may be refined. This may allow a spectrally more efficient planning if the cellular layout is optimised for the propagation environment.

With shadowing, the interference power accumulates more rapidly than proportionally with the number of signals!! The accumulation of multiple signals with shadowing is a relevant issue in the planning of cellular networks.

Combined model by Mawira (Netherlands' PTT Research)

Mawaira modelled large-area and small-area shadowing as two independent superimposed Markovian processes:

Multiple log-normal signals

In cellular networks, interference does not come from only one source but from many co-channel transmitters. In a hexagonal reuse pattern the number of interferers typically is six.

At least two different methods are used to estimate the probability distribution of the joint interference power accumulated from several log-normal signals. Such methods are relevant to estimate the joint effect of multiple interfering signals with shadowing. Fenton and Schwartz and Yeh both proposed to approximate the pdf of the joint interference power by a log-normal pdf, yet neither could determine it exactly.


The method by Fenton assesses the logarithmic mean and variance of the joint interference signal directly as a function of the logarithmic means and variances of the individual interference signals. This method is most accurate for small standard deviations of the shadowing, say, for less than 4 dB.

Schwartz and Yeh

The technique proposed by Schwartz and Yeh is more accurate in the range of 4 to 12 dB shadowing, which corresponds to the case of land-mobile radio in the VHF and UHF bands. Their method first assesses the logarithmic mean and variance of a joint signal produced by cumulation of two signals. Recurrence is then used in the event of more than two interfering signals. A disadvantage of the latter method is that numerical computations become time consuming.

Table Mean mt and standard deviation st (both in dB) of the joint power of n signals with uncorrelated shadowing, each with mean 0 dB and with identical standard deviation. Networks, with 0, 6, 8.3 and 12 dB of shadowing of individual signals.

         0 dB               6 dB              8.3 dB             12 dB
n     mt     st         mt       st        mt        st        mt       st
1    0.00 0.00       0.00     6.00       0.00     8.30       0.00     12.00
2    3.00 0.00       4.58     4.58       5.61     6.49       7.45      9.58
3    4.50 0.00       6.90     3.93       8.45     5.62      11.20      8.40
4    6.00 0.00       8.43     3.54      10.29     5.08      13.62      7.66
5    7.00 0.00       9.57     3.26      11.64     4.70      15.37      7.13
6    7.50 0.00      10.48     3.04      12.69     4.41      16.74      6.74

Besides these methods, by Fenton and Schwartz and Yeh, a number of alternative (and often more simplified) techniques are used. For instance in VHF radio broadcasting, signals fluctuate with location and with time according to log-normal distributions. Techniques to compute the coverage of broadcast transmitters are in CCIR recommendations.

Outage probabilities for systems with multiple Rayleigh fading and shadowed signals can however be computed easily without explicitly estimating the joint effect of multiple shadowed signals.

Multipath Reception

The mobile or indoor radio channel is characterized by 'multipath reception': The signal offered to the receiver contains not only a direct line-of- sight radio wave, but also a large number of reflected radio waves.

These reflected waves interfere with the direct wave, which causes significant degradation of the performance of the network. A wireless network has to be designed in such way that the adverse effect of these reflections is minimized.

Although channel fading is experienced as an unpredictable, stochastic phenomenon, powerful models have been developed that can accurately predict system performance.

Most conventional modulation techniques are sensitive to intersymbol interference unless the channel symbol rate is small compared to the delay spread of the channel. Nonetheless, a signal received at a frequency and location where reflected waves cancel each other, is heavily attenuated and may thus suffer large bit error rates.

Models for multipath reception

Narrowband Rayleigh, or Rician models mostly address the channel behaviour at one frequency only. Dispersion is modelled by the delay spread.

The effect of multipath reception

Rician fading

The model behind Rician fading is similar to that for Rayleigh fading, except that in Rician fading a strong dominant component is present. This dominant component can for instance be the line-of-sight wave. Refined Rician models also consider

Besides the dominant component, the mobile antenna receives a large number of reflected and scattered waves.

PDF of signal amplitude

The derivation is similar to the derivation for Rayleigh fading. In order to obtain the probability density of the signal amplitude we observe the random processes I(t) and Q(t) at one particular instant t0. If the number of scattered waves is sufficiently large, and are i.i.d., the central limit theorem says that I(t0) and Q(t0) are Gaussian, but, due to the deterministic dominant term, no longer zero mean. Transformation of variables shows that the amplitude and the phase have the joint pdf

Here, is the local-mean scattered power and C2/2 is the power of the dominant component. The pdf of the amplitude is found from the integral


where I0(..) is the modified Bessel function of the first kind and zero order, defined as


Show that the total local-mean power is

Rician factor

The Rician K-factor is defined as the ratio of signal power in dominant component over the (local-mean) scattered power. Thus

Expressed in terms of the local-mean power and the Rician K-factor, the pdf of the signal amplitude becomes


Show that for a large local-mean signal-to-noise ratio , the probability that the instantaneous power p drops below a noise threshold tends to

Rician Channels

Examples of Rician fading are found in

Rayleigh fading

Rayleigh fading is caused by multipath reception. The mobile antenna receives a large number, say N, reflected and scattered waves. Because of wave cancellation effects, the instantaneous received power seen by a moving antennna becomes a random variable, dependent on the location of the antenna.

In case of an unmodulated carrier, the transmitted signal has the form


Next we'll discuss the basic mechanisms of mobile reception.

Effect of Motion

Let the n-th reflected wave with amplitude cn and phase arrive from an angle relative to the direction of the motion of the antenna.

The Doppler shift of this wave is


where v is the speed of the antenna.

Phasor representation

Figure: Phasor diagram of a set of scattered waves (in blue), resulting
a Rayleigh-fading envelope (in black)

The received unmodulated signal r(t) can be expressed as

An inphase-quadrature representation of the form

can be found with in-phase component

and quadrature phase component


This propagation model allows us, for instance,

If the set of reflected waves are dominated by one strong component, Rician fading is a more appropriate model.

Doppler spectrum

Doppler shifts

We consider a Rayleigh fading signal. Let the n-th reflected wave with amplitude cn and phase arrive from an angle relative to the direction of the motion of the antenna.

The Doppler shift of this wave is

where v is the speed of the antenna.

Such motion of the antenna leads to phase shifts of individual reflected waves, so it affects the amplitude of the resulting signal. It is often assumed that the angle is uniformly distributed within [0, 2 ]. This allows us to compute a probability density function of the frequency of incoming waves. Assuming that the number of waves is very large, we can obtain the Doppler spectrum of the received signal.

Rate of Rayleigh Fading

Consider the following sample of a set of reflected waves:

Figure: Phasor diagram of a set of scattered waves (in blue), resulting
a Rayleigh-fading envelope (in black)

Let the n-th reflected wave with amplitude cn and phase arrive from an angle relative to the direction of the motion of the antenna.

If the mobile antenna moves a small distance d, the n-th incident wave, arriving from the angle with respect to the instantaneous direction of motion, experiences a phase shift of

 (2 d/) cos(  )

Thus all waves experience their own phase rotation. The resulting vector may significantly change in amplitude if individual components undergo different phase shifts.

Figure: Phasor diagram of a set of scattered waves after antenna displacement (in blue)
and before motion (in light blue), resulting a Rayleigh-fading envelope (in black)

In mobile radio channels with high terminal speeds, such changes occur rapidly. Rayleigh fading then causes the signal amplitude and phase to fluctuate rapidly.

If d is in the order of half a wave length (/2) or more, the phases of all incident waves become mutually uncorrelated, thus also the amplitude of the total received signal becomes uncorrelated with the amplitude at the point of departure.

Doppler Shifts

Each reflected wave experiences its own Doppler shift. If an unmodulated carrier is being transmitted, a spectrum of different components is received.


The normalised covariance L(d) of the electric field strength for an antenna displacement d is of the form

with J0(.) the zero-order Bessel function of the first kind.
The signal remains almost entirely correlated for a small displacement, say d< /8, but becomes rapidly independent for larger displacements, say for d> /2.

Figure: Auto-covariance L(d) of the electric field strength in a Rayleigh-fading channel
versus the normalised antenna displacement d/ in horizontal direction.

The antenna displacement can also be expressed in the terminal velocity v and the time difference T between the two samples (d = v T). So with fm the maximum Doppler shift (fm = v fc / c).

Analog Transmission over Fading Channels

This page covers analogue AM, SSB, PM and FM modulation.

Amplitude modulation

Various methods exist to transmit a baseband message m(t) using an RF carrier signal
c(t) = Ac cos ( c t + ). In linear modulation, such as Amplitude Modulation (AM) and Single Side band (SSB) the amplitude Ac is made a linear function of the message m(t).

Figure: Phasor Diagram for AM with tonal modulation.

AM has the advantage that the detector circuit can be very simple. This allows inexpensive production of mediumwave broadcast receivers. The transmit power amplifier, however, needs to be highly linear and therefor expensive and power consuming.

For mobile reception of AM audio signals above 100 MHz, the spectrum of channel fluctuations due to fading and in the message overlap. Hence the Automatic Gain Control in the receiver IF stages can not distinguish the message and channel fading. AGC will thus distort the message m(t).

AM is only rarely used for mobile communication, although it is still used for radio broadcasting.

Single Side Band

In the frequency power spectrum of AM signals we recognize an upper side band and a lower side sideband, with frequency components above and below the carrier at fc. In Single Side Band transmission, the carrier and one of these side bands are removed.

An SSB message can be recovered by multiplying the received signal by cos( ct + ). If the local oscillator has a phase offset ( - ), the detected signal is a linear combination of the orginal message m(t) and a 90 degree phase-shifted version (its Hilbert transform). The human ear is not very sensitive to such phase distortion; therefor the detected signal sounds almost identical to m(t), despite any phase offset. However, such phase shifts make SSB unsuitable for digital transmission.

The effect of a frequency error in the local oscillator is more dramatic to analog speech signals. Its effect can best be understood from the frequency domain description of the SSB signal. A frequency shift of all baseband tones occurs. In this case, the harmonic relation between audio tones is lost and the signal sounds very artificial.

SSB is relatively sensitive to interference, which requires large frequency reuse spacings and severely limits the spatial spectrum efficiency of cellular SSB networks.

AGC to reduce the effect of amplitude fades substantially affects the message signal. Furthermore, SSB requires very sharp filters, which are mostly sensitive to physical damage, temperature and humidity changes. This makes SSB not very attractive for mobile communication.


In phase modulation, the transmit signal has the constant-amplitude form

where the constant is called the phase deviation.


Show that for Narrowband Phase Modulation (NBPM) with constant << 1, Phase modulation can be approximated by the linear expression s(t) = Ac cos (c t) + m(t) cos (c t). Compare NBPM with AM


For frequency deviation , the transmit signal is of the form

That is, FM can be implemented by a PM modulator, and an integrator for the message signal.

For a message bandwidth W, the transmit bandwidth BT can be approximated by the Carson bandwidth

BT= 2 ( + W)

In the event of 2W < <10 W, a better approximation is BT = 2 ( +2 W)


Find BT for FM broadcasting with =75 kHz and W = 15 kHz. Cellular telephone nets with speech bandwidth W = 3000 Hz typically transmit over BT = 12.5 or 25 kHz. Find the frequency deviation . Compare the use of 12.5 and 25 kHz bandwidth in terms of implementation difficulty, audio quality, vulnerability to interference, reuse factors needed and spectrum efficiency.

After frequency-nonselective multipath propagation, a received FM signal

Reception above the FM-Threshold

If the signal-to-noise ratio is sufficiently large, the received signal is dominated by the wanted signal. The effect of noise can be approximated as a liniear additive distrurbance. The minimum signal-to-noise ratio for which this assumption is reasonable is called the FM capture threshold.


In non-linear modulation, such as phase modulation (PM) or frequency modulation (FM), the post-detection signal-to-noise ratio can be greatly enhanced as compared to baseband transmission or compared to linear modulation. This enhancement occurs as long as the received pre-detection signal-to-noise ratio is above the threshold. Below the threshold the signal-to-noise ratio deteriorates rapidly. This is often perceived if the signal-to-noise ratio (C/N) increases slowly: a sudden transition from poor to good reception occurs. The signal appears to "capture" the receiver at certain C/N. A typical threshold value is 10 (10 dB) C/N at RF. The audio SNR where capture occurs depends on the frequency deviation.

Effects of Rayleigh fading on FM reception

In a rapidly fading channel, the events of crossing the FM capture threshold may occur too frequently to be distinguished as individual drop outs. The performance degradation is perceived as an average degradation of the channel. The capture effect and the FM threshold vanish in such situations.

Effect of Amplitude variations

Fluctuations of the signal-to-noise ratio cause fluctuations of received noise power and fluctuations of the amplitude of the detected wanted signal. Some analyses assume that the difference between the detected signal and the expected signal is perceived as a noise type of disturbance. It is called the signal-suppression 'noise', even though disturbances that are highly correlated with the signal are mostly perceived as 'distortion' rather than as noise.

Effect of Random FM

For large local-mean signal-to-noise ratios Random FM is the only remaining factor. For voice communication with audio passband 300 - 3000 Hz, the noise contribution due to random FM leads to a SNR of


with S the audio power at the detector output. This does not depend on additive predection noise. Wideband transmission (large frequency deviation) is thus significantly less sensitive to random FM than narrowband FM.

Threshold Crossing Rate

The average number of times per second that a fading signal crosses a certain threshold is called the threshold crossing rate. Lets enlarge the following (orange) signal path, at the (yellow) instant when it crosses the (purple) threshold.

Figure: Threshold crossing. Threshold R is crossed with derivative dr/dt.

The above crossing of the threshold R with width dr lasts for dt seconds. The derivative of the signal amplitude, with respect to time, is dr / dt.

If the signal always crosses the threshold with the same derivative, then:

Average number of crossings per second * dt = Probability that the amplitude is in the interval [R, R + dr].

The probability that the signal amplitude is within the window [R, R + dr] is known from the probability density of the signal amplitude, which can for instance be Rayleigh, Rician or Nakagami. Moreover, the joint pdf of signal amplitude and its derivative can be found. For a Rayleigh-fading signal.

The expected number of crossings per second is found by integrating over all possible derivatives.

Threshold Crossing Rate in Rayleigh-fading channel versus fade duration for n = 1, 2, and 6 Rayleigh-fading interfering signals and for a constant noise floor.
Normalized to the Doppler Spread.

The TCR curve has a maximum if the local-mean-power is about as large as the threshold noise or interference power. If the signal is on average much stronger than the threshold, the number of threshold crossings (i.e., deep fades) is relatively small. Also, if the signal is much weaker than the threshold, the number of crossings is small because signal "up-fades" are unlikely.

Fade Duration

The mobile Rayleigh or Rician radio channel is characterized by rapidly changing channel characteristics. If a certain minimum (threshold) signal level is needed for acceptable communication performance, the received signal will experience periods of

It is of critical importance to the performance of mobile data networks that the used packet duration is selected taking into account the expected duration of fades and non-fade intervals.

Fade and non-fade duration for a sample of a fading signal.

Average Fade Duration

We use:

Outage Probability = Average number of fades per second * Average fade duration

where the average number of fades per second is called the threshold crossing rate.

Expressions for Average (Non-) Fade Duration

In a Rayleigh fading channel with fade margin M, the average nonfade duration (ANFD) is

where fD is the Doppler spread. M is the ratio of the local-mean signal power and the minimum (threshold) power needed for reliable communication.

Average non-fade duration in Rayleigh-fading channel versus fade margin for n = 1, 2, 3, 4, 5 and 6 Rayleigh-fading interfering signals. Normalized by dividing by the Doppler Spread.

The curve for n = 6 closely resembles the curve the ANFD in an interference-free but noise-limited channel.


Calculation of the distribution of non-fade periods is tedious, but has been elaborated by Rice. Because of the shape of the Doppler spectrum, fade durations that coincide with a motion of about half a wavelength are relatively frequent.

The average fade duration (AFD) is


Average fade duration in Rayleigh-fading channel versus fade margin for n = 1, 2, 3, 4, 5 and 6 Rayleigh-fading interfering signals. Normalized by dividing by the Doppler Spread.

Experiments revealed that at large fade margins, the fade durations are approximately exponentially distributed around their mean value.

Delay Spread

Because of multipath reflections, the channel impulse response of a wireless channel looks likes a series of pulses.

Figure: Example of impulse response and frequency transfer function of a multipath channel.

We can define the local-mean average received power with excess delay within the interval (T, T + dt). This gives the "delay profile" of the channel.

The delay profile determines to what extent the channel fading at two different frequencies f1 and f2 are correlated.

Some definitions

For a digital signal with high bit rate, this dispersion is experienced as frequency selective fading and intersymbol interference (ISI). No serious ISI is likely to occur if the symbol duration is longer than, say, ten times the rms delay spread.

Typical Values

In macro-cellular mobile radio, delay spreads are mostly in the range from Trms is about 100 nsec to 10 microsec. A typical delay spread of 0.25 microsec corresponds to a coherence bandwidth of about 640 kHz. Measurements made in the U.S., indicated that delay spreads are usually less than 0.2 microsec in open areas, about 0.5 microsec in suburban areas, and about 3 micros in urban areas. Measurements in The Netherlands showed that delay spreads are relatively large in European-style suburban areas, but rarely exceed 2 microsec. However, large distant buildings such as apartment flats occasionally cause reflections with excess delays in the order of 25 microsec.

In indoor and micro-cellular channels, the delay spread is usually smaller, and rarely exceed a few hundred nanoseconds. Seidel and Rappaport reported delay spreads in four European cities of less than 8 microsec in macro-cellular channels, less than 2 microsec in micro-cellular channels, and between 50 and 300 ns in pico-cellular channels.

Resolvable Paths

A wideband signal with symbol duration Tc (or a direct sequence (DS)-CDMA signal with chip time Tc), can "resolve" the time dispersion of the channel with an accuracy of about Tc. For DS-CDMA, the number of resolvable paths is

N = round (Tdelay / Tchip ) + 

where round(x) is the largest integer value smaller than x and Tdelay is total length of the delay profile. A DS-CDMA Rake receiver can exploit N- fold path diversity.

Coherence Bandwidth

One can define 'narrowband' transmission defined in the time domain, considering interarrival times of multipath reflections and the time scale of variations in the signal caused by modulation. A signal sees a narrowband channel if the bit duration is sufficiently larger than the interarrival time of reflected waves. In such case, the intersymbol interference is small.

Transformed into constraints in the frequency domain, this criterion is found to be satisfied if the transmission bandwidth does not substantially exceed the 'coherence' bandwidth Bc of the channel. This is the bandwidth over which the channel transfer function remains virtually constant.

For a Rayleigh-fading channel with an exponential delay profile, one finds

Bc = 1/(2 Trms)

where Trms is the rms delay spread.

Scatter Function

The scatter function combines information about Doppler shifts and path delays. Each path can be described by its

Thus we can plot the received energy in a two dimensional plane, with Doppler shift on one horizontal axis and delay on the other horizontal axis.

Reciprocity - differences between uplink and downlink

Two-way communication requires facilities for 'inbound', i.e., mobile-to-fixed, as well as 'outbound', i.e., fixed-to-mobile communication. In circuit-switched mobile communication, such as cellular telephony, the inbound and outbound channel are also called the 'uplink' and 'downlink', respectively. The propagation aspects described on other pages are valid for inbound and outbound channels. This is understood from the reciprocity theorem:

If, in a radio communication link, the role of the receive and transmit antenna are functionally interchanged, the instantaneous transfer characteristics of the radio channel remain unchanged.

In mobile multi-user networks with fading channels, the reciprocity theorem does not imply that the inbound channel behaves identically as the outbound channel. Particular differences occur for a number of link aspects:

Indoor Wireless RF Channel

There are several causes of signal corruption in an indoor wireless channel. The primary causes are signal attenuation due to distance, penetration losses through walls and floors and multipath propagation.

Effect of distance

Signal attenuation over distance is observed when the mean received signal power is attenuated as a function of the distance from the transmitter. The most common form of this is often called free space loss and is due to the signal power being spread out over the surface area of an increasing sphere as the receiver moves farther from the transmitter.

In addition to free space loss effects, the signal experiences decay due to ground wave loss although this typically only comes into play for very large distances (on the order of kilometers). For indoor propagation this mechanism is less relevant, but effects of wave guidance through corridors can occur.


Multipath results from the fact that the propagation channel consists of several obstacles and reflectors. Thus, the received signal arrives as an unpredictable set of reflections and/or direct waves each with its own degree of attenuation and delay. The delay spread is a parameter commonly used to quantify multipath effects. Multipath leads to variations in the received signal strength over frequency and antenna location.

Rate of fading

Time variation of the channel occur if the communicating device (antenna) and components of its environment are in motion. Closely related to Doppler shifting, time variation in conjunction with multipath transmission leads to variation of the instantaneous received signal strength about the mean power level as the receiver moves over distances on the order of less than a single carrier wavelength. Time variation of the channel becomes uncorrelated every half carrier wavelength over distance.

Fortunately, the degree of time variation within an indoor system is much less than that of an outdoor mobile system. One manifestation of time variation is as spreading in the frequency domain (Doppler spreading). Given the conditions of typical indoor wireless systems, frequency spreading should be virtually nonexistent. Doppler spreads of 0.1 - 6.1 Hz (with RMS of 0.3 Hz) have been reported.

Some researchers have considered the effects of moving people. In particular it was found by Ganesh and Pahlavan [9] that a line of sight delay spread of 40 ns can have a standard deviation of 9.2 - 12.8 ns. Likewise an obstructed delay spread can have a std. dev. of 3.7 - 5.7 ns.

For wireless LANs this could mean that an antenna place in a local multipath null, remains in fade for a very long time. Measures such as diversity are needed to guarantee reliable communication irrespective of the position of the antenna. Wideband transmission, e.g. direct sequence CDMA, could provide frequency diversity.

Indoor RF propagation at 2.4 GHz in Cory Hall at UC Berkeley.

For the infopad project at the University of California, Berkeley propagation measurements have been conducted in Cory Hall, the home of the EECS department.

The measured rms delay spreads ranged from 16 - 52 ns. Delay spread increased with transmitter/receiver distance as well as room dimensions. The delay spread in the hallway was relatively constant when compared to the other rooms. The path loss drop off rates were 1.72 in the hallway, 1.99 in room 307 and 2.18 in room 550. Significant signal transmission occurs through the walls. For the U.C. Berkeley Cory Hall building, this suggests that walls can not be used as natural cell boundaries.

Indoor Measurement Environment

We made measurements in three rooms and one hallway of Cory Hall as follows:

In all of the tests, the channel was kept stationary while the transfer function H'(f) was measured in the sense that no non-essential movement occurred during a single sweep of frequencies except that required to operate the test equipment. To average out fading, the receive and transmit antennas were slightly varied over position for each given separation distance between the two antennas. Thus, for each separation distance a set of measurements were taken where we define a measurement as being the response from a single sweep over frequency of the network analyzer. During the course of a set of measurements, the amount of position variation between each individual measurement was approximately one half of the carrier wavelength. This ensured that the various measurements of the set would be uncorrelated.

Delay Spread

Essentially we treat the instantaneous impulse response as a pdf of various values of excess delay where excess delay is defined as all possible values of delay in the instantaneous impulse response after subtracting off the delay of the fastest arriving signal component. We then numerically calculate the mean excess delay and the rms delay spread of this pdf. In determining delay spread, we calculate a power level as the area under the instantaneous impulse response curve.

The first point to note is that the rms value generally increases with distance. Heuristically this can be explained by the idea that with greater distance, the long distanced reflections are relatively stronger (compared to the line-of-sight) to their contribution to an increased delay spread is more significant.

Note further that the smallest delay spreads are found in the hallway. The width of the hallway was less than 8 feet as compared to over 12 feet for Room 400 well above 20 feet for the other two rooms. These small dimensions in conjunction with the relative absence of obstacles resulted in a delay spread that was essentially independent of distance. Heuristically we can say that increasing the distance did not result in a greater number of multipath reflections. Consistent with the above is the fact that delay spread is directly proportional to the general dimensions of the rooms within which measurements were taken. Thus, Room 550 produced the greatest delay spreads and likewise Room 550 was the largest room.

FIGURE: RMS Delay Spread vs. Antenna Separation Distance

The Figures below exhibit instantaneous impulse responses from measurements in Room 550. These two figures serve as examples of different respective arrival times for the first and the strongest received signal component arrival times. Measurements have been taken at a 35 foot received-transmit separation distance.

FIGURE: Impulse Response

FIGURE: Impulse Response

Path Loss

We typically expect received power to be a function of distance as follows

p = dn

where d represents distance. In the case of ideal free space loss, n = 2. We call n the path loss rate.

As the three graphs show, signal strength drops off faster as dimensions of the room increase. In the case of the hallway, the drop off rate was significantly smaller than 2.0 which is the value for free space loss. The explanation for this is that the hallway acts as a waveguide given its small width. Note that the received power levels in these Figures are in dB (not in dBm) and are with respect to +10 dBm. To obtain absolute power values in the figure, 10 dB must be added to each component. Note further that n is based on log-log data although the graphs shown are log-linear.

Rician K Factor

Each K-factor is the result from an average of eight measurements taken from the same transmitter/receiver separation distance. Note that the K-factors are independent of distance and the three results shown are very close to one another. In all cases the presence of a line of sight path was the same. Given that the above measurements all had a LOS path available, one would expect that the ratio of the strongest received signal component to the reflected components would be the same. Our measurements bear this out.

Discussion for Wideband CDMA System

Delay Spread and Resolvable Paths

One of the advantages of direct sequencing spread spectrum is the ability to distinguish between differing signal path arrivals. These resolvable paths can then be used to mitigate corruption caused by the channel. Delay spread is directly related to the number of resolvable paths. The rms delay spread in Cory Hall ranged from 16 to 52 ns. Given the proposed InfoPad downlink CDMA chip rate of several tens of Megachips per second, we arrive at the following:

For CDMA systems with smaller chip rates the channel would behave as a narrowband, i.e., frequency nonselective channel.

Path Loss, Wall Penetration and Cell Layout

An important issue for indoor cellular reuse systems is the possibility of interference from users in adjacent cells. In designing cells it would be convenient if natural barriers such as walls and ceilings/floors could be used as cell boundaries. With these thoughts in mind, we took measurements through walls of Cory Hall.

The instantaneous impulse response was taken on the 5th floor of Cory Hall. The transmitter was located in room 550 and the receiver was located in the corridor near the freight elevator. The received powers shown are with respect to +10 dBm so that absolute values can be obtained by adding 10 to magnitudes shown. Clearly the received signal strength through the wall is significant and shows that some of the walls in Cory Hall can not serve as cell boundaries. Note that certain walls in Cory consist of different materials than those on the 5th floor. Most notably room 473 was once an anechoic chamber and is essentially a metal cage. With these exceptions, we reiterate that most walls in Cory can not serve as cell boundaries.

Appendix 1: Antenna Fundamentals

An antenna radiation pattern or antenna pattern is defined as "a mathematical function or graphical representation of the radiation properties of the antenna as a function of space coordinates. In most cases, the radiation pattern is determined in the far-field region ( the region where electric and magnetic fields vary inversely with distance) and is represented as a function of the directional coordinates." The radiation property of most concern is the two- or three-dimensional spatial distribution of radiated energy as a function of an observer's position along a path or surface of constant distance from the antenna. A trace of the received power at a constant distance is called a power pattern. A graph of the spatial variation of the electric or magnetic field along a contant distance path, is called a field pattern.

An isotropic antenna is defined as "a hypothetical lossless antenna having equal radiation in all directions." Clearly, an isotropic antenna is a ficticious entity, since even the simplest antenna has some degree of directivity. Although hypothetical and not physically realizable, an isotropic radiator is taken as a reference for expressing the directional properties of actual antennas. A directional antenna is one "having the property of radiating or receiving electromagetic waves more effectively in some directions than in others." The term is usually applied to an antenna whose maximum directivity is significantly greater than that of a linear dipole antenna. The power pattern of a half-wave linear dipole is shown below.

The linear dipole is an example of an omnidirectional antenna -- i.e. an antenna having a radiation pattern which is nondirectional in a plane. As the figure above indicates, a linear dipole has uniform power flow in any plane perpendicular to the axis of the dipole and the maximum power flow is in the equatorial plane. One important class of directional antennas are form from linear arrays of linear dipoles as illustrated below. The most famous and ubiquitous member of this class is the so called Uda-Yagi antenna shown on the right.


As example of array patterns, the highly directional power pattern of an end-fire linear array of 8 half-wave linear dipoles is shown below.

Like the Uda-Yagi antenna, the end-fire array picture here has is its maximum maximum power flow in a direction parallel to the line along which the dipoles are deployed. The power flow in the opposite or back direction is negligible!

Antenna Terms:

Some of this material was taken from Antenna Theory: Analysis and Design (2nd edition), Constantine A. Balanis, John Wiley and Sons, Inc. (1997)

Freely adapted from a portion of Jean-Paul M. G. Linmartz's
Wireless Communication, The Interactive Multimedia CD-ROM,
Baltzer Science Publishers, P.O.Box 37208, 1030 AE Amsterdam,
ISSN 1383 4231, Vol. 1 (1996), No.1