Wireless communication systems are designed to recover received signals that are damaged by wireless channel impairments. In classical communication theory, wireless channels were unknown and people think unpredictable noises occur in wireless channels. However, many scientists and engineers investigated wireless channels and developed different types of wireless channel models. They can be categorized as additive white Gaussian noise, jitter, phase shift, path loss, shadowing, multipath fading, interference, and so on. They give the transmitted signals different types of damages. In addition, transmission distance, mobility, geographical feature, weather, antenna position, and signal waveform affect the parameters of wireless channel models. For example, the channel model of short range wireless communications is different from the one of cellular systems. An urban area channel model is different from that of a rural one. A mobile user has more severe frequency offset due to the Doppler effect. Therefore, it is important to understand wireless channels when designing wireless communication systems. In this chapter, we look into wireless channels and their characteristics.

3.1 Additive White Gaussian Noise

In Chapter 2, we have defined noise as Gaussian distribution due to central limit theorem. When the noise power has a flat power spectral density, we call it white noise and its single-sided power spectral density (P_wn(f) (W/Hz)) can be denoted as follows:

(3.1)

where N₀ is constant. This equation means the white noise has an equal power spectral density for all frequency bands. This white noise affects most types of signals independently so that we usually call it Additive White Gaussian Noise (AWGN). If a wireless channel is given, we can calculate capacity of the wireless channel. The channel capacity of single antenna systems for AWGN channel is defined as follows:

Channel capacity, C_awgn, means the amount of information we can send through the given channel. We call this equation Shannon-Hartley theorem [1]. The average received signal power over the channel noise (P_r/N₀W) is simply the signal to noise ratio (SNR) and C_awgn/W is simply the normalized channel capacity. Figure 3.1 illustrates their relationship.

In Figure 3.1, we can observe two areas: theoretically impossible area and theoretically possible area. When we want to transmit a signal with data rate, R, it is not possible to communicate reliably in this area (R > C_awgn). The curve represents the limitation of liable transmission under the given channel. When the average received power is large enough (SNR  1), the capacity reaches the saturation point in the power and regards as a linear function in the bandwidth. Therefore, we call this area bandwidth limited regime. The capacity equation can be expressed approximately as follows:

(3.2)

On the contrary, when the average received power is small enough (SNR  1), the capacity reaches the saturation point in the bandwidth and regards as a linear function in the power. Therefore, we call this area power limited regime. The capacity equation can be expressed approximately as follows:

(3.3)

For infinite bandwidth, . (3.3) can be rewritten as follows:

(3.4)

where E_b is a bit energy. We can obtain the following equation:

(3.5)

We call this value (0.694 or −1.6 in dB) Shannon limit. This value means it is not possible to communicate reliably below this value.

3.2 Large-Scale Path Loss Models

An electromagnetic wave propagates from a transmitter to a receiver over the air. During this propagation, the energy of the electromagnetic wave is reduced. Basically, we can express this process as path loss models. One of simple channel models is Friis transmission model [2]. It describes a simple free space channel without obstructions between a transmitter and a receiver. In this model, the electromagnetic waves spread outward in a sphere, and the received power, P_r, is given as follows:

(3.6)

where P_t is the transmitted power, G_t and G_r are the transmitter antenna gain and the receiver antenna gain, respectively, λ is the wavelength, and R is the distance between a transmitter and a receiver. The Friis transmission model is illustrated in Figure 3.2.

The isotropic antenna of the transmitter radiates the Effective Isotropic Radiated Power (EIRP) uniformly. It is represented by

(3.7)

and the antenna of the receiver collects the transmitted power as much as the antenna size (or the effective area) of the receiver. We call it the antenna aperture, A_e. The received power is related to this antenna aperture and the surface area of the sphere, 4πR². The received power is proportional to A_e/4πR². Therefore, we can present the received power, P_r, as follows:

(3.8)

The relationship between the receiver antenna gain and effective aperture is

(3.9)

Combining (3.8) and (3.9), we can obtain (3.6). This equation means the received power depends on the square of the distance proportionally because the other parameters are basically given. In practical systems, the received power decreases more quickly due to reflection, diffraction, and scattering. These mechanisms are governed by the Maxwell equations. Reflection is the change in the direction of an electromagnetic wave when it encounters an object. The behavior of the reflection is explained by Fresnel equations. Diffraction is the bending of light when it passes the edge of an object. The behavior of diffraction is explained by Huygen’s principle. Scattering is the production of other small electromagnetic waves when it encounters an object with a rough surface. The process of the scattering is quite complex and some types of scattering are Rayleigh scattering, Mie scattering, Raman scattering, Tyndall scattering, and Brillouin scattering.

Figure 3.3 illustrates the two-ray ground reflection model.

When considering the reflection from ground, we can use the two-ray ground reflection model as follows:

(3.10)

where h_t and h_r denote the transmitter antenna height and receiver antenna height, respectively. Unlike Friis transmission equation, the received power is affected by the antenna heights. The fourth power of the distance, d⁴, is included in the equation so that the received power decreases more quickly according to the distance. This equation does not depend on the wavelength.

In practical systems, we use empirical path loss models. The received power of the empirical models is expressed as follows:

(3.11)

where P₀ is the power at the reference distance, d₀, and γ is the path loss exponent which typically has 2–8 according to carrier frequency, environment, and so on. The path loss is defined as the difference between the transmitted power and the received power as follows:

From Definition 3.2, the path loss of (3.11) is

(3.12)

where PL₀ is the path loss at the reference distance d₀ (dB). When a propagating electromagnetic wave encounters some obstruction such as wall, building, and tree, reflection, diffraction, and scattering happen and the electromagnetic wave is distorted. We call this effect shadowing or slow fading and use the modified path loss model as follows:

(3.13)

where χ denotes a Gaussian distributed random variable with standard deviation, σ, and represents shadowing effect.

3.3 Multipath Channels

When an electromagnetic wave propagates in the air, reflection happens and it produces some reflected waves as shown in Figure 3.4. Although the receiver wants to have the original signal only, it receives the reflected waves together with the original signal. The reflected signals travel different paths and they have different amplitudes and phase noises. Thus, many signals with different delays and amplitudes arrive at the receiver.

The power delay profile is widely used to analyze multipath wireless channels. It is described by several parameters such as the mean excess delay, maximum excess delay, root mean square (rms) delay spread, and maximum delay spread. Among them, the maximum delay spread, τ_max, and the rms delay spread, τ_rms, are important multipath channel parameters. The maximum delay spread means the total time interval we can receive the reflected signals with high energy. A high value of τ_max means a highly dispersive channel. However, the maximum delay spread can be defined in several ways. Therefore, the rms delay spread is more widely used and indicates the rms value of the delay profile. We use this parameter to decide coherence bandwidth. One example of the power delay profile is shown in Figure 3.5.

The multipath channel is modeled as a linear Finite Impulse Response (FIR) filter. The impulse response, h(t, τ), of the multipath channel is

(3.14)

where a_i(t) and denote amplitude of a multipath and the Dirac delta function, respectively. N means N different reflected paths. One example is illustrated in Figure 3.6.

We can observe two different reflected paths in the delay profile and represent this using a three-tap FIR filter as shown in Figure 3.7. In this figure, delay, τ₁ and τ₂, can be implemented by shift-registers and amplitude, a₀, a₁, and a₂, can be implemented by multiplications (or amplifiers).

From the power delay profile, we can analyze the multipath channel.

If a wireless channel is not constant over a transmit bandwidth and severely distorted, it would be very difficult to establish a reliable radio link. Thus, we define the coherence bandwidth (B_c) as follows:

It simply means the range of frequencies we use to send a signal without serious channel distortions. The rms delay spread, τ_rms, is inversely proportional to the coherence bandwidth as follows:

(3.15)

Therefore, the larger τ_rms means the wireless channel becomes more frequency selective channel. The more accurate definition of B_c is related to the frequency correlation function. For correlation greater than 0.9 [3],

(3.16)

For correlation greater than 0.5 [3],

(3.17)

When we design a symbol structure of wireless communication systems, the symbol length should be defined according to τ_rms. For example, if the symbol length is greater than 10 τ_rms, we assume the wireless communication system does not need to mitigate Inter Symbol Interference (ISI). If it is less than 10 τ_rms, we must avoid ISI using several techniques such as an equalizer. If it is much less than 10 τ_rms, a reliable communication is not possible.

Typically, the rms delay spread of cellular systems is less than 8 µs. The urban or mountain area has longer than 8 µs due to more reflections. The rms delay spread of indoor small cells is less than 50 ns.

A wireless channel is varying according to time, movement, environment, and so on. If a wireless channel changes every moment, we will not know wireless channel characteristics and cannot design wireless communication systems. Therefore, we assume a wireless channel is constant during certain time. The coherence time means the certain time. It is defined as follows:

Similar to the power delay profile, the Doppler power spectrum provides statistical information about the coherence time. The Doppler power spectrum is affected by a user mobility while the power delay profile is affected by a multipath. A moving wireless communication device produces the Doppler shift (Δf ). The Doppler shift is expressed as follows:

(3.18)

where v, λ, θ, and c denote the speed of a moving wireless communication device, the wavelength of the carrier electromagnetic wave, the angle of arrival with respect to a direction of the moving wireless device, and the speed of light, respectively. When a receiver moves in the opposite direction of the electromagnetic wave (θ = 0°), the Doppler shift is

(3.19)

When a receiver moves in the same direction of the electromagnetic wave (θ = 180°), the Doppler shift is

(3.20)

When a receiver moves in perpendicular to the electromagnetic wave (θ = 90°), the Doppler shift is

(3.21)

Therefore, the maximum Doppler shift (  f_m) means cos θ = 1 and is represented as follows:

(3.22)

and the coherence time (T_c) is inversely proportional to the Doppler spread as follows:

(3.23)

For correlation greater than 0.5 [3],

(3.24)

If the symbol duration is greater than the coherence time, the wireless channel changes during symbol transmission. We call this fast fading. On the other hands, if the symbol duration is less than the coherence time, the wireless channel does not change during symbol transmission. We call this slow fading.

The Rayleigh fading distribution is widely used to model a small-scale fading and describes a statistical time varying model for the propagation of electromagnetic waves. When the time variant channel impulse response in a flat fading channel is given by

(3.25)

where a(t) and φ(t) denote the magnitude and phase rotation, respectively, the probability density function of the Rayleigh distribution is represented by

(3.26)

where σ² denotes the time average power of the received signal. One example of the Rayleigh fading is illustrated in Figure 3.8.

When there is one strong multipath signal such as a line-of-sight signal, we can use the Rice distribution. It is expressed as follows:

(3.27)

where a₀ and I₀ denote the peak magnitude of one strong multipath and the modified Bessel function of the first kind. Every natural phenomenon in time domain has corresponding phenomenon in frequency domain. Sometimes we explain the power delay profile in frequency domain through the Fourier transform. The multipath fading results in significant power losses, frequency and phase offsets, and so on. Therefore, a wireless communication system designer should understand fading environments clearly and select suitable mitigation techniques. The diversity techniques are very important techniques to overcome fading channels. Basically, this technique uses the fact that each wireless channel experiences different fading. For example, some part of a signal passing through channel 1 can be damaged seriously but same part of a signal passing through channel 2 may not be damaged. A receiver can recover the transmitted signal through observing multiple signals passing through different channels. More detail will be dealt in Chapter 5. The type of small-scale fading is illustrated in Figure 3.9 where B_s and T_s denote the signal bandwidth and time interval, respectively.

The capacity, C_{freq sel} (bits/s/Hz), of the frequency selective fading channel [4] is given by

(3.28)

where is the sub-channel gain and can be found by Lagrangian method as follows:

(3.29)

where λ_L is the Lagrangian multiplier. We call this power allocation water-filling. One example of the water-filling algorithm is illustrated in Figure 3.10.

In the fast fading channel, we transmit a signal in time interval LT_c and have approximately same channel gain for lth coherence time period. Therefore, the average capacity, C_fast (bits/s/Hz), is

(3.30)

When L is large, the capacity is

(3.31)

The outage capacity is used for slow-fading channels because the capacity is a random variable and there is delay limitation. The outage capacity, C₀ (bits/s/Hz), is simply defined as the maximum data rate without errors. The outage probability, p_out(C₀), can be represented as follows:

(3.32)

For example, means the frame error rate is 0.1 with ideal code if the wireless communication system frame has 10 bits/s/Hz spectral efficiency.

3.4 Empirical Wireless Channel Models

The channel model we looked into the previous sections is to provide theoretical backgrounds but it does not include measurement data from real wireless environments. Therefore, many scientists and engineers extensively measure wireless channel parameters in real wireless environments and develop empirical wireless channel models such as Hata model [5], Walfisch-Ikegami model [5], Erceg model [6], ITU model [7], and so on. Among them, the ITU-R recommendation M.1225 is widely used as empirical channel models of cellular systems. Especially, Worldwide Interoperability for Microwave Access (WiMAX) by the Institute of Electrical and Electronics Engineers (IEEE) and Long Term Evolution (LTE) by the 3rd Generation Partnership Project (3GPP) use these models. The wireless channel models are classified into three test environments such as indoor office, pedestrian, and vehicular environment. Their path loss model for the indoor office environment [7] is expressed as follows:

(3.33)

where d_indoor and n denote the distance (m) between a transmitter and a receiver and the number of floors, respectively. The path loss model for the pedestrian test environment [5] is

(3.34)

where d_pedestrian and f denote the distance (km) between a base station and a mobile station and carrier frequency, respectively. The path loss model for the vehicular test environment [5] is

(3.35)

where d_vehicular, h_b, and f denotes the distance (km) between a base station and a mobile station, the base station antenna height (m), and carrier frequency, respectively. In addition, they have two different delay spreads: channel A (Low delay spread) and channel B (High delay spread). Their parameters are summarized in Tables 3.1, 3.2, 3.3, and 3.4.

3.5 Problems

• 3.1. Consider a communication system with SNR = 10 dB and Bandwidth = 3 kHz. Calculate the channel capacity for AWGN.
• 3.2. Compare the channel capacity between a single antenna system and a multiple antenna system.
• 3.3. Define the channel capacity using the maximum mutual information.
• 3.4. The required channel capacity for AWGN and bandwidth are 100 kbps and 10 MHz, respectively. Find the required SNR.
• 3.5. Calculate the received power using the Friis transmission equation, when designing the wireless communication link with the following parameters:

  Transmission power	25 W
  Transmitter antenna gain	10
  Receiver antenna gain	15
  Carrier frequency	2 GHz
  Distance between transmitter and receiver	1 km

• 3.6. A base station transmits a signal with the following parameters and the required received power is 1 mW. If a mobile user is located at the cell edge and path law obeys the Friis transmission model, then find the cell radius:

  Transmission power	25 W
  Transmitter antenna gain	1.5
  Receiver antenna gain	1
  Carrier frequency	1 GHz

• 3.7. Consider a cellular system with the following parameters and the measured power at d₀ = 0.1 km follows the Friis transmission model. When the minimum required power is 1 mW and the mean of the received power is log normally distributed with standard deviation 10 dB, find the cell radius which satisfies connection probability 0.8:

  Transmission power	20 W
  Carrier frequency	2 GHz
  Transmitter antenna gain	1.6
  Receiver antenna gain	1
  Transmitter antenna height	20 m
  Receiver antenna height	1.5 m

• 3.8. Consider a unit delay filter with the following:
  

  Find the frequency response of the filter.

• 3.9. Consider a filter with the following:
  

  Find the frequency response of the filter when L = 1, 3, 5, and 10.

• 3.10. The multipath channel is modeled as the following FIR filter:
  

  Find the z-transfer function and the phase response.

• 3.11. Consider the following power delay profile:

  

  (i) Find the coherence bandwidth. (ii) Do we need an equalizer when signal bandwidth is 200 kHz? (iii) Find the coherence time when the carrier frequency is 1.8 GHz and a user mobile at a speed of 100 km/h.

• 3.12. Describe the advantages and disadvantages of empirical channel models.

References

1. [1] C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of the IRE, vol. 37, no. 1, pp. 10–21, 1949.
2. [2] H. T. Friis, “A Note on a Simple Transmission Formula,” Proceedings of the IRE, vol. 34, pp. 254–256, 1946.
3. [3] T. S. Rappaport, Wireless Communications: Principles & Practice, Prentice Hall, Upper Saddle River, NJ, 1996.
4. [4] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, Cambridge, 2005.
5. [5] European Cooperative in the Field of Science and Technical Research EURO-COST 231. “Urban Transmission Loss Models for Mobile Radio in the 900- and 1,800 MHz Bands (Revision 2),” COST 231 TD(90)119 Rev. 2, The Hague, the Netherlands, September 1991.
6. [6] V. Erceg, L. J. Greenstein, S. Y. Tjandra, S. R. Parkoff, A. Gupta, B. Kulic, A. A. Julius, and R. Bianchi, “An Empirically Based Path Loss Model for Wireless Channels in Suburban Environments,” IEEE Journal on Selected Areas in Communications, vol. 17, no. 7, pp. 1205–1211, 1999.
7. [7] ITU-R Recommendation M.1225, “Guidelines for Evaluation of Radio Transmission Technologies for IMT 2000,” February 1997. http://www.itu.int/rec/R-REC-M.1225/en (accessed April 20, 2015).