Mobile Radio Propagation

Mobile Radio Propagation

By

Tanveer Hasan Assistant Professor

Mobile Radio Propagation

• Power P in dBm:

𝑃 𝑑𝑏𝑚 = 10𝑙𝑜𝑔 𝑃 1 𝑚𝑊

• Difference of two powers in dB: Let P₁ = 10 nW, P₂ = 1 W 𝑃₂

𝑃₁ = 1 𝑊

10 𝑛𝑊 = 1 𝑊

10 × 10⁻⁹ 𝑊 = 10⁸ 10𝑙𝑜𝑔 𝑃₂

𝑃₁ = 10𝑙𝑜𝑔 1 𝑊

10 × 10⁻⁹ 𝑊 = 10𝑙𝑜𝑔 10⁸ = 80 𝑑𝐵

Mobile Radio Propagation

• We can also write

10𝑙𝑜𝑔 𝑃₂

𝑃₁ = 10𝑙𝑜𝑔

𝑃₂ 1𝑚𝑊

𝑃₁ 1𝑚𝑊

= 10𝑙𝑜𝑔 𝑃₂

1𝑚𝑊 − 10𝑙𝑜𝑔 𝑃₁ 1𝑚𝑊

= 𝑃₂ 𝑑𝐵𝑚 − 𝑃₁[𝑑𝐵𝑚]

• Power P in dBW:

Mobile Radio Propagation

• If P = 100 W, then

𝑃 𝑑𝑏𝑊 = 10𝑙𝑜𝑔 𝑃

1 𝑊 = 10𝑙𝑜𝑔 100𝑊

1 𝑊 = 20 𝑑𝐵𝑊

• If P = 100 W, then

𝑃 𝑑𝑏𝑚 = 10𝑙𝑜𝑔 𝑃

1 𝑚𝑊 = 10𝑙𝑜𝑔 100𝑊

10⁻³ 𝑊 = 50 𝑑𝐵𝑚 𝑃 𝑑𝑏𝑚 = 𝑃 𝑑𝑏𝑊 + 30

𝑃 𝑑𝑏𝑊 = 𝑃 𝑑𝑏𝑚 − 30

Mobile Radio Propagation

• Problem: Calculate P[dBm] and P[dBW] for P = 2.5 pW = 2.5 x 10^-12 W.

𝑃 𝑑𝑏𝑚 = 10𝑙𝑜𝑔 𝑃

1 𝑚𝑊 = 10𝑙𝑜𝑔 2.5 × 10⁻¹²𝑊

10⁻³ 𝑊 = −86.0206 𝑑𝐵𝑚 𝑃 𝑑𝑏𝑊 = 10𝑙𝑜𝑔 𝑃

1 𝑊 = 10𝑙𝑜𝑔 2.5 × 10⁻¹²𝑊

1 𝑊 = −116.0206 𝑑𝐵𝑊

Mobile Radio Propagation

• Mobile Radio Channel: Radio channels are extremely random and require careful analysis. The speed of motion also affects how rapidly the signal level fades as mobile moves in space.

• Propagation models: Propagation models predict the average received signal strength at a given distance from the transmitter, as well as the variability of the signal strength in close spatial proximity to a particular location.

• Large-scale Propagation Models: Propagation models that predict the mean signal strength for an arbitrary transmitter-receiver (T-R) separation distance are called large-scale propagation models and are useful in estimating the radio

coverage area of a transmitter.

Mobile Radio Propagation

• Small-scale Propagation Models or Fading models: The propagation models that characterize the rapid fluctuations of the received signal strength over very short travel distances (a few wavelengths) or short time durations (on the order of seconds) are called small-scale or fading models.

• Path Loss: The path loss is defined as the difference (in dB) between the effective transmitted power and the received power. It represents signal attenuation as a

positive quantity measured in dB.

𝑃𝐿 𝑑𝐵 = 10𝑙𝑜𝑔 𝑃_𝑡

𝑃_𝑟 = 𝑃_𝑡[𝑑𝐵𝑚] − 𝑃_𝑟[𝑑𝐵𝑚]

Free Space Propagation Model

1. The free space propagation model is used to predict received signal strength when the transmitter and receiver have a clear line-of-sight path between them.

2. Satellite communication systems and microwave line-of-sight radio links typically undergo free space propagation.

3. The free space power received by a receiver antenna which is separated from a radiating transmitter antenna by a distance d, is given by the Friis free space equation

𝑃_𝑟 𝑑 = 𝑃_𝑡𝐺_𝑡𝐺_𝑟𝜆² (4𝜋)²𝑑²𝐿

Free Space Propagation Model

where 𝑃_𝑡 is the transmitted power, 𝑃_𝑟 𝑑 is the received power at the T-R separation (d), 𝐺_𝑡 is the transmitter antenna gain, 𝐺_𝑟 is the receiver antenna gain, 𝐿 is the system loss factor not related to propagation (≥1), and 𝜆 is the wavelength in meters.

𝜆 = 𝑐

𝑓_𝑐 = 2𝜋𝑐 𝜔_𝑐

where f_c is the carrier frequency in Hertz, 𝜔_𝑐 is the carrier frequency in radians per second, and c is the speed of light given in meters/s.

Free Space Propagation Model

4. This equation can also be written as

𝑃_𝑟 𝑑 = 𝑃_𝑟 𝑑₀ 𝑑₀ 𝑑

2

where 𝑑₀ is some reference distance in the far field (d₀ > d_f, the Fraunhofer distance) of transmitting antenna and 𝑃_𝑟 𝑑₀ is the power received at distance 𝑑₀.

5. The above equation can also be written as 𝑃_𝑟 𝑑 [𝑑𝐵𝑚] = 10𝑙𝑜𝑔 𝑃_𝑟 𝑑₀

+ 20𝑙𝑜𝑔 𝑑₀

𝑑 ≥ 𝑑₀ ≥ 𝑑_𝑓

Free Space Propagation Model

or

𝑃_𝑟 𝑑 𝑑𝐵𝑚 = 𝑃_𝑟 𝑑₀ 𝑑𝐵𝑚 + 20𝑙𝑜𝑔 ^𝑑0

𝑑

Radio Wave Propagation

1. Reflection 2. Diffraction 3. Scattering

Reflection: Reflection occurs when a propagating electromagnetic wave impinges upon an object which has very large dimensions compared to the wavelength of the propagating wave. Reflections occur from the surface of the earth and from

buildings and walls.

Radio Wave Propagation

Diffraction: Diffraction occurs when the radio path between the transmitter and receiver is obstructed by a surface that has sharp edges. The secondary waves resulting from the obstructing surface are present throughout the space and even behind the obstacle, giving rise to a bending of waves around the obstacle, even when a line-of-sight path does not exist between transmitter and receiver.

Scattering: Scattering occurs when the medium through which the wave travels consists of objects with dimensions that are small compared to the wavelength, and the number of obstacles per unit volume is large. Scattered waves are produced by

Ground Reflection (2-ray) Model

1. The 2-ray ground reflection model is a useful propagation model that is based on geometric optics. It considers both the direct path and a ground reflected

propagation path between transmitter and receiver.

Ground Reflection (2-ray) Model

2. This model is reasonably accurate for predicting the large-scale signal strength over distances of several kilometres.

3. In most mobile communication systems, the maximum T-R separation distance is at most only a few tens of kilometers, and the earth may be assumed to be flat.

4. The total received E-field, E_TOT, is then a result of the direct line-of-sight component, E_LOS, and the ground reflected component, E_g.

Ground Reflection (2-ray) Model

5. The path difference between LOS and ground reflected path

∆≈ 2ℎ_𝑡ℎ_𝑟 𝑑

6. At large distance (𝑑 ≫ ℎ_𝑡ℎ_𝑟) the received power can be proved to be 𝑃_𝑟 = 𝑃_𝑡𝐺_𝑡𝐺_𝑟 ℎ_𝑡²ℎ_𝑟²

𝑑⁴

Ground Reflection (2-ray) Model

7. At large distance (𝑑 ≫ ℎ_𝑡ℎ_𝑟) received power falls off with distance raised to the fourth power, or at a rate of 40 dB/decade. This is a much more rapid path loss as compared to free space.

8. At large values of d, the received power and path loss become independent of frequency.

9. The path loss for the 2-ray model (with antenna gains) can be expressed in dB as 𝑃𝐿 𝑑𝐵 = 40 log 𝑑 − (10𝑙𝑜𝑔𝐺_𝑡 + 10𝑙𝑜𝑔𝐺_𝑟 + 20𝑙𝑜𝑔ℎ_𝑡 + 20𝑙𝑜𝑔ℎ_𝑟)

Ground Reflection (2-ray) Model

Example: A mobile is located 5 km away from a base station and uses an antenna with a gain of 2.55 dB to receive cellular radio signals. Suppose transmitted power is 1 watt, find the received power at the mobile using the 2-ray ground reflection model. The height of the transmitting antenna is 50 m and the receiving antenna is 1.5 m above ground. Also calculate the path loss at the location of mobile.

Solution: G_t = 1, G_r = 10^2.55/10 =10^0.255 = 1.8 h_t = 50 m, h_r = 1.5 m, d = 5 km = 5 x 10³ m

P_r = 1 W x 1 x 1.8 x 50² x 1.5² / (5 x 10³)⁴ = 1.62 x 10^-11 W (-77.9 dBm) PL(dB) = 40log(5000) – (10 log (1) + 2.55 +20log(75)) = 107.9 dB

Ground Reflection (2-ray) Model

𝑃𝐿 𝑑𝐵 = 10𝑙𝑜𝑔 𝑃_𝑡

𝑃_𝑟 = 10𝑙𝑜𝑔 1𝑊

1.62 × 10⁻¹¹ = 107.9 dB 𝑃_𝑡 𝑑𝐵𝑚 = 10log ^𝑃𝑡

1𝑚𝑊 = 10 log ^1𝑊

0.001𝑊 = 30𝑑𝐵𝑚

𝑃𝐿 𝑑𝐵 = 𝑃_𝑡 𝑑𝐵𝑚 − 𝑃_𝑟 𝑑𝐵𝑚 = 30 − −77.9 = 107.9 𝑑𝐵

Knife-edge Diffraction Model

1. Estimating the signal attenuation caused by diffraction of radio waves over hills and buildings is essential in predicting the field strength in a given service area.

Knife-edge Diffraction Model

2. When shadowing is caused by a single object such as a hill or mountain, the

attenuation caused by diffraction can be estimated by treating the obstruction as a diffracting knife edge.

3. The electric field strength, E_d, of a knife-edge diffracted wave is given by 𝐸_𝑑

𝐸₀ = 𝐹 𝑣 = (1 + 𝑗)

2 න

𝑣

∞

𝑒𝑥𝑝 −𝑗𝜋𝑡² /2 𝑑𝑡

Where, E is the free space field strength in the absence of both the ground and the

Knife-edge Diffraction Model

4. The Fresnel integral, F(v), is a function of the Fresnel-Kirchoff diffraction parameter v, defined as

𝑣 = ℎ 2(𝑑₁ + 𝑑₂) 𝜆𝑑₁𝑑₂

5. The diffraction gain due to the presence of a knife edge, as compared to the free space E-field, is given by

𝐺_𝑑 𝑑𝐵 = 20𝑙𝑜𝑔 𝐹(𝑣)

Knife-edge Diffraction Model

6. The approximate solution for above equation is

𝐺_𝑑 𝑑𝐵 = 0 𝑣 ≤ −1

𝐺_𝑑 𝑑𝐵 = 20 log 0.5 − 0.62𝑣 − 1 ≤ 𝑣 ≤ 0 𝐺_𝑑 𝑑𝐵 = 20 log 0.5 𝑒𝑥𝑝 −0.95𝑣 0 ≤ 𝑣 ≤ 1

𝐺_𝑑 𝑑𝐵 = 20 log 0.4 − 0.1184 − 0.38 − 0.1𝑣 ² 1 ≤ 𝑣 ≤ 2.4 𝐺_𝑑 𝑑𝐵 = 20log 0.225

𝑣 𝑣 > 2.4

Knife-edge Diffraction Model

Example: Compute the diffraction loss for the folowing three cases λ = 1/3 m, d₁ = 1km, d₂=l km, and (a) h = 25 m, (b) h = 0 m (c) h = -25 m

Solution: (a) 𝑣 = ℎ ^{2(𝑑1+𝑑2)}

𝜆𝑑₁𝑑₂ = 25 2(1000+1000)

1

3∗1000∗1000 = 2.74 𝐺_𝑑 𝑑𝐵 = 20log ^0.225

𝑣 =20log ^0.225

2.74 =-21.7 dB (b) 𝑣 = ℎ ^{2(𝑑1+𝑑2)}

𝜆𝑑₁𝑑₂ = 0 2(1000+1000)

1

3∗1000∗1000 = 0

𝐺_𝑑 𝑑𝐵 = 20 log 0.5 − 0.62𝑣 = 20 𝑙𝑜𝑔 0.5 − 0.62 ∗ 0 = -6 dB

Multiple Knife-edge Diffraction

• In practice the propagation path may consist of more than one obstruction. The series of obstacles can be replaced by a single equivalent obstacle so that the path loss can be obtained using single knife-edge diffraction model.

Log-distance Path Loss Model

1. The average large-scale path loss (in dB) for T-R separation (d) can be derived as

𝑃_𝑟(𝑑) = 𝑃_𝑟(𝑑₀) 𝑑₀ 𝑑

𝑛

𝑃_𝑡/𝑃_𝑟 𝑑 = 𝑃_𝑡/𝑃_𝑟(𝑑₀) 𝑑 𝑑₀

𝑛

or 𝑃𝐿 𝑑 = 𝑃𝐿 𝑑₀ + 10𝑛𝑙𝑜𝑔 ^𝑑

𝑑₀ (1)

ഥ ^𝑑

Log-distance Path Loss Model

path loss exponent (n) indicates the rate at which the pathloss increases with distance, d₀ is the close-in reference distance which is determined from

measurements close to the transmitter.

2. The bars in above equations denote the ensemble average of all possible path loss values for a given value of d.

3. When plotted on a log-log scale, the path loss is a straight line with a slope equal to 10n dB per decade. The value of n depends on the specific propagation environment.

Log-distance Path Loss Model

5. Log-normal Shadowing: The model in equation (1) does not consider the fact that the surrounding clutter may be different at two different locations having the same T-R separation.

6. Measurements have shown that at any value of d, the path loss PL(d) at a particular location is random and distributed log-normally (normal in dB) about the mean distance dependent value. That is

𝑃𝐿 𝑑 = 𝑃𝐿 𝑑 + 𝑋_𝜎 = 𝑃𝐿 𝑑₀ + 10𝑛𝑙𝑜𝑔 ^𝑑

𝑑₀ + 𝑋_𝜎 (2) and

Log-distance Path Loss Model

where 𝑋_𝜎, is a zero-mean Gaussian distributed random variable (in dB) with standard deviation 𝜎 (also in dB).

7. The log-normal distribution describes the random shadowing effects which occur over a large number of measurement locations which have the same T-R separation, but have different levels of clutter on the propagation path. This phenomenon is referred to as log-normal shadowing.

8. The close-in reference distance d₀, the path loss exponent n, and the standard deviation 𝜎, describe the path loss model for an arbitrary location having a

Small-scale Fading

1. Small-scale fading, or simply fading, is used to describe the rapid fluctuation of the amplitude of a radio signal over a short period of time or travel distance, so that large-scale path loss effects may be ignored.

2. Fading is caused by interference between two or more versions of the transmitted signal which arrive at the receiver at slightly different times.

3. These waves, called multipath waves, combine at the receiver antenna to give a resultant signal which can vary widely in amplitude and phase, depending on the distribution of the intensity and relative propagation time of the waves and the bandwidth of the transmitted signal.

Small-Scale Multipath Propagation

Multipath in the radio channel creates small-scale fading effects. The three most important effects are:

• Rapid changes in signal strength over a small travel distance or time interval

• Random frequency modulation due to varying Doppler shifts on different multipath signals

• Time dispersion (echoes) caused by multipath propagation delays.

Doppler Shift

1. Consider a mobile moving at a constant velocity v, along a path segment having length d between points X and Y, while it receives signals from a remote source

Doppler Shift

2. The difference in path lengths traveled by the wave from source S to the mobile at points X and Y is ∆ 𝑙 = 𝑑 𝑐𝑜𝑠𝜃 = 𝑣∆𝑡𝑐𝑜𝑠𝜃. Where ∆𝑡 is the time required for the mobile to travel from X to Y, and 𝜃 is assumed to be the same at points X and Y since the source is assumed to be very far away.

3. The phase change in the received signal due to the difference in path lengths is therefore

∆𝜑 = ^2𝜋∆𝑙

𝜆 = ^{2𝜋𝑣∆𝑡}

𝜆 𝑐𝑜𝑠𝜃 (1)

Doppler Shift

4. The apparent change in frequency, or Doppler shift, is given by 𝑓_𝑑 = ¹

2𝜋 Δ𝜑

Δ𝑡 = ^𝑣

𝜆 𝑐𝑜𝑠𝜃 (2)

Equation (2) relates the Doppler shift to the mobile velocity and the spatial angle between the direction of motion of the mobile and the direction of arrival of the wave.

5. It can be seen from equation (2) that if the mobile is moving toward the direction of arrival of the wave, the Doppler shift is positive (i.e., the apparent received frequency is increased), and if the mobile is moving away from the direction of arrival of the wave, the Doppler shift is negative (i.e. the apparent received

Doppler Shift

Example: Consider a transmitter which radiates a sinusoidal carrier frequency of 1850 MHz. For a vehicle moving 60 mph, compute the received carrier frequency if the mobile is moving (a) directly towards the transmitter, (b) directly away from the transmitter, (c) in a direction which is perpendicular to the direction of arrival of the transmitted signal.

Solution: f_c = 1850 MHz, v = 60 mph = 60 × ¹⁶⁰⁰

3600 = 26.67𝑚/𝑠 𝑐 3 × 10⁸

Doppler Shift

a) 𝜃 = 0, 𝑓_𝑑 = ^26.67

0.162 cos0 = 164.63 Hz

𝑓 = 𝑓_𝑐 + 𝑓_𝑑 = 1850 MHz + 0.00016 MHz = 1850.00016 MHz b) 𝜃 = 180, 𝑓_𝑑 = ^26.67

0.162 cos180 = −164.63 Hz

𝑓 = 𝑓_𝑐 − 𝑓_𝑑 = 1850 MHz − 0.00016 MHz = 1849.99984 MHz c) 𝜃 = 90, 𝑓_𝑑 = ^26.67

0.162 cos90 = 0 Hz

𝑓 = 𝑓_𝑐 + 𝑓_𝑑 = 1850 MHz + 0 = 1850 MHz

Power Delay Profile

Power Delay Profile: power delay profile of a multipath channel is a plot of average received power, P_r(τ) as a function of excess delay, τ. Where excess delay τ is measured with respect to the first received multipath component.

Time Dispersion Parameters

• The mean excess delay, rms delay spread, and excess delay spread (X dB) are multipath channel parameters that can be determined from a power delay profile.

The time dispersive properties of wide band multipath channels are most commonly quantified by their mean excess delay ( ҧ𝜏) and rms delay spread (𝜎_𝜏).

• The mean excess delay is the first moment of the power delay profile and is defined to be

ҧ𝜏 = σ_𝑘 𝑃(𝜏_𝑘)𝜏_𝑘 σ_𝑘 𝑃(𝜏_𝑘)

Time Dispersion Parameters

• The rms delay spread is the square root of the second central moment of the power delay profile and is defined to be

𝜎_𝜏 = 𝜏² − ( ҧ𝜏)² where

𝜏² = σ_𝑘 𝑃(𝜏_𝑘)𝜏_𝑘² σ_𝑘 𝑃(𝜏_𝑘)

Time Dispersion Parameters

The maximum excess delay (X dB) of the power delay profile is defined to be the time delay during which multipath energy falls to X dB below the maximum. In other words, the maximum excess delay is defined as 𝜏_𝑋 − 𝜏₀, where 𝜏₀ is the first arriving signal and 𝜏_𝑋 is the maximum delay at which a multipath component is within X dB of the strongest arriving multipath signal (which does not necessarily arrive at 𝝉_𝟎).

Time Dispersion Parameters

• Coherence Bandwidth: Coherence bandwidth (B_c) is a statistical measure of the range of frequencies over which the channel can be considered "flat" (i.e., a channel which passes all spectral components with approximately equal gain and linear phase). Two sinusoids with frequency separation greater than B_c are affected quite differently by the channel. If the coherence bandwidth is defined as the bandwidth over which the frequency correlation function is above 0.9, then the coherence bandwidth is approximately

1

Time Dispersion Parameters

• If the definition is relaxed so that the frequency correlation function is above 0.5, then the coherence bandwidth is approximately

𝐵_𝑐 ≈ 1 5𝜎_𝜏

Time Dispersion Parameters

Example: Calculate the mean excess delay, rms delay spread, and the maximum excess delay (10 dB) for the multipath profile given in the figure. Estimate the 50% coherence bandwidth of the channel. Would this channel be suitable for AMPS or GSM service without the use of an equalizer?

Time Dispersion Parameters

Solution 𝑃 𝜏₀ = 0.01, 𝑃 𝜏₁ = 0.1, 𝑃 𝜏₂ = 0.1, 𝑃 𝜏₃ = 1 𝜏₀ = 0, 𝜇𝑠, 𝜏₁= 1 𝜇𝑠, 𝜏₂ = 2 𝜇𝑠, 𝜏₃ = 5 𝜇𝑠

ҧ𝜏 = σ_𝑘 𝑃(𝜏_𝑘)𝜏_𝑘

σ_𝑘 𝑃(𝜏_𝑘) = 0.01 × 0 + 0.1 × 1 + 0.1 × 2 + 1 × 5

0.01 + 0.1 + 0.1 + 1 = 4.38 𝜇𝑠 𝜏² = σ_𝑘 𝑃(𝜏_𝑘)𝜏_𝑘²

σ_𝑘 𝑃(𝜏_𝑘) = 0.01 × 0² + 0.1 × 1² + 0.1 × 2² + 1 × 5²

0.01 + 0.1 + 0.1 + 1 = 21.07 𝜇𝑠² 𝜎_𝜏 = 𝜏² − ( ҧ𝜏)²= 21.07 − (4.38)²= 1.37 𝜇𝑠

Time Dispersion Parameters

Maximum excess delay (10 dB) = 6 𝜇𝑠 𝐵_𝑐 ≈ 1

5𝜎_𝜏 = 1

5 × 1.37 = 0.146 MHz = 146 kHz

• Since 𝐵_𝑐 is greater than 30 kHz. AMPS will work without an equalizer. However, GSM requires 200 kHz bandwidth which exceeds 𝐵_𝑐 thus an equalizer would be needed for this channel.

Doppler Spread and Coherence Time

1. Delay spread and coherence bandwidth describe the time dispersive nature of the channel. Doppler spread and coherence time are parameters which describe frequency dispersive nature of the channel. Doppler spread B_D is a measure of the spectral broadening caused by the time rate of change of the mobile radio channel.

2. If the baseband signal bandwidth is much greater than B_D, the effects of Doppler spread are negligible at the receiver. This is a slow fading channel.

3. Coherence time T_C, is used to characterize the time varying nature or the frequency dispersiveness of the channel.

Doppler Spread and Coherence Time

𝑇_𝐶 = 0.423 𝑓_𝑚

where f_m is the maximum Doppler shift given by 𝑓_𝑚 = ^𝑣

𝜆

4. Coherence time is actually a statistical measure of the time duration over which the channel response is essentially invariant.

5. If the reciprocal bandwidth of the baseband signal is greater than the coherence time of the channel, then the channel will change during the transmission of the

Types of Small-Scale Fading

• T_s: symbol duration

• 𝐵_𝑠 = 1/𝑇_𝑠

• 𝜎_𝜏: rms delay spread

• T_c: coherence time

Types of Small-Scale Fading

Level Crossing Rate (LCR)

1. The level crossing rate (LCR) and average fade duration of a Rayleigh fading signal 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.

2. 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

Level Crossing Rate (LCR)

where f_m is the maximum Doppler frequency and 𝜌 = R/R_rms is the value of the specified level R, normalized to the local rms amplitude of the fading envelope.

3. Equation (1) gives the value of N_R, the average number of level crossings per second at specified R.

4. The level crossing rate is a function of the mobile speed as is apparent from the presence of f_m in equation (1).

5. There are few crossings at both high and low levels, with the maximum rate 𝜌 = 1/ 2

Average Fade Duration

1. 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

ҧ𝜏 = 𝑒^𝜌2 − 1 𝜌𝑓_𝑚 2𝜋

2. The average duration of a signal fade helps determine the most likely number of signaling bits that may be lost during a fade.

3. Average fade duration primarily depends upon the speed of the mobile, and decreases as the maximum Doppler frequency f becomes large.

Average Fade Duration

4. 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.

5. This is useful for relating SNR during a fade to the instantaneous BER which results.

Example For a Rayleigh fading signal, compute the positive-going level crossing rate for 𝜌 = 1, when the maximum Doppler frequency (f_m) is 20 Hz. What is the maximum velocity of the mobile for this Doppler frequency if the carrier frequency is 900 MHz?

Solution: The number of zero level crossings is

𝑁_𝑅 = 2𝜋𝑓_𝑚𝜌𝑒^−𝜌2 = 2𝜋 × 20 × 1 × 𝑒⁻¹ =18.44 crossings per second

The maximum velocity of the mobile can be obtained using the Doppler relation 𝑓_𝑚 = 𝑣/𝜆.

Therefore velocity of the mobile at 20 Hz is

Diversity

• Diversity: Diversity is a technique used to compensate for fading channel impairments. Diversity is usually employed to reduce the depth and duration of the fades experienced by a receiver in a flat fading (narrowband) channel. Diversity techniques can be employed at both base station and mobile receivers. The four common types of diversity techniques are mentioned below.

• Spatial Diversity: The most common diversity technique is called spatial diversity, whereby multiple antennas are strategically spaced and connected to a common receiving system. While one antenna sees a signal null, one of the other

Diversity

• Antenna Polarization Diversity: At the base station, space diversity is less practical than at the mobile because the narrow angle of incident fields requires large antenna spacing. Hence at base station orthogonal polarization diversity is more feasible. Although this provides only two diversity branches; it allows the antenna elements to be co-located.

• Frequency Diversity: Frequency diversity transmits information on more than one carrier frequency. The logic behind this technique is that frequencies separated by more than the coherence bandwidth of the channel will not experience the same fades. Frequency diversity is often employed in microwave line-of-sight links

Diversity

• Time Diversity: Time diversity repeatedly transmits information at time spacing that exceed the coherence time of the channel, so that multiple repetitions of the signal will be received with independent fading conditions, thereby providing for diversity. One modern implementation of time diversity involves the use of the RAKE receiver for spread spectrum CDMA, where the multipath channel provides redundancy in the transmitted message.