Wireless Signal Propagation
In the previous chapter, we learned how the bits are transmitted to the wireless channel. In this chapter we are going to discuss how these bits travel or propagate through the wireless channel before reaching to the receiver.
3.1 Wireless Radio Channel
Wireless radio channel is different than a wired channel. The radio channel is “open” in the sense that it does not have anything to protect or guide the signal as it travels from source to destination. As a result, the signal is subject to many issues, which we must be aware of to understand how the receiver will receive the signal. These issues will be discussed in this chapter.
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3.2 Antenna
Transmitter converts electrical energy to electromagnetic waves and the receiver converts those electromagnetic waves back to electrical energy. It is important to note that the same antenna is used for both transmission and reception. Therefore a device can use the same antenna for both transmitting bits and receiving bits.
Depending on how the antennas radiate or receive power, there can be three types of antennas as illustrated in Figure 3.1. An antenna is called omni-directional if the power from it radiates in (or it receives power from) all directions. A directional antenna, on the other hand, can focus most of its power in the desired direction. Finally, an isotropic antenna refers to a theoretical antenna that radiates or receives uniformly in each direction in space, without reflections and losses. Note that, due to reflections and losses in practical environments, the omni-directional antenna does not radiate or receive in all directions uniformly.
An isotropic transmitting antenna cannot produce much power at the receiver because the power is dissipated to all directions and wasted. Given that the receivers are likely to be contained in some space, for example in a horizontal plane rather than in a sphere, antennas are designed to control the power in a way so that the receivers receive more power compared to a theoretical isotropic antenna. Antenna gain refers to the ratio of the power at a particular point to the power with isotropic antenna, which gives a measure of power for the antenna. Antenna gain is expressed in dBi, which means “decibel relative to isotropic”. For example, if an antenna is advertised as 3 dBi, it means that it will produce twice as much power than an isotropic antenna. Note that an isotropic antenna will have a gain of 0 dBi.
Figure 3.1 Different types of antennas
Example 3.1
How much stronger a 17 dBi antenna receives (transmits) the signal compared to the isotropic antenna?
Power of isotropic antenna = Piso
Power of 17 dBi antenna = P
17 = 10log10(P/Piso)
Thus P/Piso = 101.7 = 50.12, i.e., the 17 dBi antenna will receive (transmit) the signal 50.12 times stronger than the isotropic antenna albeit using the same transmit power.
Antennas are designed to transmit or receive a specific frequency band. For example, antennas used in wireless routers operating with 2.5/5 GHz are too small to receive TV signals operating with 700MHz. Fundamentally, end-to-end antenna length must be half the wavelength so electrons on the antenna can travel back and forth in one cycle. Small consumer devices, such as mobile phones, may hide their antennas within the device, but the fundamental relationship between antenna size and frequency exists, i.e., we would need larger antennas for lower frequencies, and vice versa.
3.3 Reflection, Diffraction, Scattering
When the transmitting antenna transmits a signal, the signal can reach the receiver antenna directly in a straight path if there is a line-of-sight (LOS) between the transmitter and the receiver. However, the signal also bounces from many other objects around us and the bounced signals reach the receiver by traveling different paths.
Figure 3.2 shows that there are three types of bouncing that can happen for the signal transmitted by a car antenna on the street. Reflection happens when the signal hits a large solid object such as a wall. Diffraction happens when the signal bounces off a sharp edge, such as a corner of a block. Finally, a signal may hit very small objects, such as a thin light post or even dust particles in the air, which causes scattering. Note that reflection and diffraction are more directional, but scattering is more omni- directional. There are complex mathematical formulas to capture the effect of reflection, diffraction, and scattering on the received signal at the receiving antenna, but those are outside the scope of this book. Our main objective here is to be aware of
the fact that the transmitted signal can reach the receiver in many different ways, which may cause certain issues when designing the communication protocols.
Figure 3.2 Reflection, Diffraction, and Scattering
Now let us try to understand the effect of these signal bouncing phenomena on wireless communication. Reflection happens when the surface is large relative to the wavelength of the signal. When the reflected signal reaches the receiver, it may have a phase shift. Depending on the phase shift, the reflected signal may actually cancel out the original signal (destructive), or strengthen it (constructive).
Similarly, diffraction happens when the edge of an impenetrable body is large relative to signal wavelength, but the phase shift is calculated differently than a reflection.
Finally, scattering happens when the size of the object is in the order of the wavelength. This means a light post can cause scattering for low frequency signals (large wavelength), but would cause reflection for very high frequency signals, such as 60 GHz. However, for 60 GHz, very tiny objects, such as snowflakes, hailstones, can cause scattering.
An interesting outcome of reflection, diffraction, and scattering is that the receiver can still receive the signal even if there is no LOS between the transmitter and the receiver. This is a great advantage for wireless communications. For example, it is not possible to have a LOS to the wifi access point or router located in the garage or in a central location from every room in the house. We however can still receive signals from the AP. It is because of this bouncing property of wireless signals. On the other hand, when we have LOS, we do not have to depend on signal bouncing, but the reflection, diffraction and scattering then actually cause some form of interference with the LOS signal.
3.4 Channel Model
Now that we have some appreciation of the signal propagation through the radio channel and how it can get affected by different physical phenomena, we need to find a way to predict or estimate the signal that may be received at a given location given certain transmissions. This is called channel modeling.
Figure 3.3 shows that there is a transmitter mounted on a tower to transmit signals to subscriber devices, which may be located at anywhere around the tower. Power profile of the received signal at the subscriber station can be obtained by convolving the power profile of the transmitted signal with the impulse response of the channel. Note that convolution in time is multiplication in frequency.
Mathematically, after propagating through the channel H, transmitted signal x
becomes y, i.e.,
Where H(f) is channel response, and n(f) is the noise. Note that x, y, H, and n are all
functions of the signal frequency f.
Figure 3.3 Channel model
3.5 Path Loss
When a signal travels through space, it loses power. This is called path loss or signal attenuation. As a result of path loss, the power of a signal at the receiver (received power) is usually only a fraction of the original or input power used at the transmitter to generate the signal.
Path loss depends on the length of the path travelled by the signal. The larger the distance between the transmitter and receiver, the higher the path loss, and vice versa. Clearly, the path loss must be estimated and factored in properly when a wireless link is designed. Different path loss models are used for estimating path loss for different scenarios. Two popular path loss models are the Frii’s model designed for free space with no reflections, and the 2-ray model that takes reflections from the ground into consideration. Frii’s model is used as a guide for ideal scenarios whereas 2-ray is a more practical model used widely in wireless communications. We will examine 2- ray later in the chapter after discussing multipath reflections.
𝑦(𝑓) = 𝐻(𝑓). 𝑥(𝑓) + 𝑛(𝑓) (3.1)
In free space without any absorbing or reflecting objects, the path loss depends on the distance as well as on the frequency (or wavelength) according to the following Frii’s law:
𝑃 =𝑃 𝐺 𝐺 % * = 𝑃 𝐺 𝐺 % * (3.2)
! ” ” ! 4𝜋𝑑 ” ” ! 4𝜋𝑓𝑑
Where PR and PT are the received and transmitted powers (in Watts), respectively, while GT and GR are transmitter and receiver antenna gains in linear scale, respectively. We see that, for a given frequency, path loss increases as inverse square of distance, which is sometimes referred to as the d-2 law (path loss exponent = 2). It is also observed that path loss increases as inverse square of the frequency, which means that the signal power attenuates more rapidly for higher frequency signals, and vice versa.
Figure 3.4 Power spreading in space and received power calculation for Frii’s Law
Equation (3.2) shows path loss in linear scale. For the convenience of calculating the link budget, however, path loss is actually measured in dB. By converting Equation (3.2) in dB, we obtain:
Receive Antenna
Isotropic Point Power Source
𝑃”# =𝑃”# +𝐺”# +𝐺”# +10𝑙𝑜𝑔 6 !$$! %&4𝜋𝑑
Where 𝑃”# and 𝑃”# refer to receive and transmit powers, respectively, in dBm, while
𝐺$ and 𝐺! are the antenna gains in dBi. Thus, the path loss is obtained as:
𝑃𝑎𝑡h𝐿𝑜𝑠𝑠 =𝑃”# − 𝑃”# =−𝐺”# − 𝐺”# −10𝑙𝑜𝑔 6 7 (3.4) $ ! $ ! %&4𝜋𝑑
For isotropic antennas (𝐺”# and 𝐺”# are both 0 dB), path loss is reduced to the $!
following simple formula:
λ’ 4π𝑑 4π𝑓𝑑 Pathloss(𝑑𝐵) = −10𝑙𝑜𝑔%& F4π𝑑J = 20𝑙𝑜𝑔%& F λ J = 20𝑙𝑜𝑔%& F 𝑐 J
= 20𝑙𝑜𝑔%&(𝑑) + 20𝑙𝑜𝑔%&(𝑓) + 20𝑙𝑜𝑔%& F4πJ
= 20𝑙𝑜𝑔%&(𝑑) + 20𝑙𝑜𝑔%&(𝑓) − 147.55 𝑐 (3.5)
where d is in meter, f in Hz, and 𝑐 = 3 × 10( 𝑚/𝑠. Equation (3.5) implies that for free-space propagation, the received power decays with distance (transmitter-receiver separation) or frequency at a rate of 20 dB/decade, i.e., the signal loses 20 dB for every decade (tenfold) increase in distance or frequency.
A simple explanation for Frii’s path loss formula in Equation (3.2) can be given using a sphere around an isotropic point power source at the center radiating a power of PT as shown in Figure 3.4. Basically, the power from the source spreads in space in all directions equally. As such, the power density on the surface of the sphere decreases with increasing sphere radius, d. With 4𝜋d2 being the area of the sphere, we have a power density of PT/4𝜋d2. Therefore, the total power received at an antenna located at the sphere surface becomes equal to the power density times the antenna area. We have learned that antenna size is dependent on the frequency or the wavelength.
Given that the ideal antenna has an area of 𝜆2/4𝜋, the received power at the antenna is equal to PT2 & 3$, which is given by the Frii’s law in Equation (3.2) for isotropic
antennas with unit gains.
Example 3.2
If 50W power is applied to a 900 MHz frequency at a transmitter, find the receive power at a distance of 100 meter from the transmitter (assume free space path loss with unit antenna gains).
Unit antenna gain means: GT=GR=1.
We have d=100m, f=900×106 Hz, PT=50W, c=3×108 m/sec, and 𝜋 = 3.14
𝑃 = 𝑃 F 𝑐 J’ = 3.5𝜇𝑊
Example 3.3
What is the received power in dBm at 10 meters from a 2.4GHz wifi router transmitting with 100mW of power (assume free space path loss with unit antenna gains)?
Unit antenna gain means: GT=GR=0dBm.
We have d=10m, f=2.4×109 Hz, PT=100mW=20dBm, c=3×108 m/sec, and 𝜋 = 3.14
𝑃𝑎𝑡h𝑙𝑜𝑠𝑠 = 20𝑙𝑜𝑔%& F4𝜋𝑓𝑑J 𝑑𝐵 = 60 𝑑𝐵 𝑐
𝑃 =𝑃 − 𝑝𝑎𝑡h𝑙𝑜𝑠𝑠=20−60=−40𝑑𝐵𝑚 !$
3.6 Receiver Sensitivity
If the path loss is too much, the SNR at the receiver could be too low for decoding the data. The noise at the receiver is a function of the channel bandwidth, i.e., larger the bandwidth, the higher the total noise power, and vice versa. The noise is also sensitive to the circuits and hardware of the receiver and the operating temperature, which is called the noise figure of the receiver. Receiver sensitivity refers to the minimum received signal strength (RSS) required for that receiver to be able to decode information. Noise, bandwidth and modulation affect the receiver sensitivity. For example, Bluetooth specifies that, at room temperature, devices must be able to achieve a minimum receiver sensitivity of -70dBm to -82dBm [BTBLOG].
Example 3.4
increase the coverage with low transmit power, a manufacturer produced Bluetooth chipsets with a receiver sensitivity of -80 dBm. What is the
maximum communication range that could be achieved for this chipset for a transmit power of 1 mW? Assume Free Space Path Loss with unit antenna gains.
Bluetooth frequency f = 2.4 GHz, PT= 1 mW, PR=-80 dBm=10-8 mW We have
PR=PT( c )2 4πdf
d=c PT/PR 4π f
= 99.5 meter
3.7 Multipath Propagation
As wireless signals reflect from typical objects and surfaces around us, they can reach the receiver through multiple paths. Figure 3.4 illustrates the multipath phenomenon and explains its effect at the receiver. Here we have a cellular tower transmitting radio signals omnidirectionally. A mobile phone antenna is receiving not just one copy of
the signal (the LoS), but another copy of the same signal that is reflected from a nearby high-rise building (NLoS). We make two observations:
• The LoS signal reaches the receiver first followed by the NLoS copy. This is due to the longer path length of the NLoS signal compared to the LoS path.
• The signal strength for LoS is higher compared to that of NLoS. This is
because the NLoS signal travels further distance and hence attenuates more compared to the LoS.
Figure 3.5 considers only a single NLoS path. In reality, there are many NLoS paths due to many reflecting surfaces. For multipath, there are also phase differences among the received signals copies due to the differences in their travelling time (different paths have different lengths). Such phase differences, however, are not shown in Figure 3.5 as it illustrates the signals only as simple impulses.
Figure 3.5 Effect of multipath
3.8 Inter-Symbol Interference
One problem with multipath is that the receiver continues to receive the signal well after the transmitter has finished transmitting the signal. This increases the time the receiver has to dedicate to decode one symbol or one bit, i.e., the symbol interval has to be longer than the ideal case when no NLoS paths exist. If we do not adjust the symbol interval adequately, then the signals from the previous symbol will enter into the next symbol interval and interfere with the new symbol. As a result, even if there were no other transmitters, the same transmitter would interfere with its own signal at the receiver. This phenomenon is called inter-symbol interference.
The process of inter-symbol interference is illustrated in Figure 3.6 with two short pulses, dark and light at the transmitter, which become much wider at the receiver due
to multipath. We can see that the dark symbol, which was transmitted before the light, is interfering with the light symbol. To reduce this interference at the receiver, the transmitter has to use much wider symbol intervals. As a result of having to widen the bit intervals at the receiver, we have to reduce the data rate or bits per second as the data rate is inverse of the symbol length.
Figure 3.6 Inter-symbol interference
3.9 Delay Spread
Now let us examine the effect of multipath more closely. Recall that when a single pulse is transmitted, multiple pulses arrive at the receiver. As a result, the transmitter cannot transmit two pulses quickly one after the other. Otherwise the late arrivals will collide with the new transmission.
One good thing, however, is that the subsequent arrival of the signal copies are attenuated further and further. So we really do not have to wait too long, but just enough so the next arrivals are below some threshold power.
The time between the first and the last versions of the signal above the power threshold is called the delay spread. The concept of delay spread is illustrated in Figure 3.7. One thing to notice here is that the amplitude of the late arrivals can actually fluctuate, although they on average consistently diminish with time.
Figure 3.7 Multipath Propagation and Delay Spread
3.10 2-ray Propagation Model and d-4 Power Law
Earlier we learned that, in the absence of any multipath (no reflector), Frii’s formula can be used to estimate the received power at the receiver where the power decreases as square of distance, which is the d-2 law. Later, significant measurements were done in real environments, which revealed that the attenuation follows a d-n law where n is called the path loss exponent and varies from 1.5 to 5 [Munoz2009].
It was also found that the antenna heights significantly affect the received power when multipaths are present. Based on this observation, a new propagation law was derived, which is called the 2-ray model or d-4 Power Law, which is illustrated in Figure 3.8. This model, which considers 1 LoS and 1 reflection from the ground, considers a transmitter at height ht and a receiver at height hr separated by distance d. The received power is then described as:
𝑃 =𝑃 𝐺 𝐺 2*#*$3$ (3.6) ! “”!)%
And, the path loss in decibel,
𝑝𝑎𝑡h𝑙𝑜𝑠𝑠 (𝑑𝐵) = 40𝑙𝑜𝑔%&(𝑑) − 20𝑙𝑜𝑔%&(h*h+) (3.7)
From Equation (3.7), we see that with the 2-ray model, the received power decays with distance (transmitter-receiver separation) at a rate of 40 dB/decade. It is interesting to note that the 2-ray model is independent of the frequency. However, the 2-ray pathloss of Equation (3.7) is valid only when the distance is greater than a threshold (cross-over distance), i.e., when 𝑑 ≥ 𝑑,+-./:
𝑑,+-./ = 4 Fh*h+J = 4 Fh*h+𝑓J (3.8) 𝜆𝑐
The 2-ray model shows that the higher the base station antenna, the higher the received power at the mobile device on the ground. This explains why the radio base stations are mounted on high towers, on the roof top, and so on.
Figure 3.8 2-ray propagation model and d-4 power law
Example 3.5
A 2m tall user is holding his smartphone at half of his height while standing 800m from a 10m high base station. The base station is transmitting a 1.8GHz signal using a transmission power of 30dBm. What is the received power (in dBm) at the smartphone? Assume unit gain antennas.
We have ht = 10m, hr=1m, d=500m, f=1.8×109 Hz, PT=30dBm, c=3×108 m/sec 𝑑,+-./ = 4Fh*h+𝑓J = 240𝑚
This means that the 2-ray model can be applied to estimate the pathloss at 800m.
𝑝𝑎𝑡h𝑙𝑜𝑠𝑠 = 20𝑙𝑜𝑔%& 6h*h+7 = 87.96dB
The received power = 30 – 87.96 = -57.96dBm (approx.)
3.11 Fading
One interesting aspect of multipath we discussed earlier is that the multipath signals can be either constructive or destructive. It depends on how the phase changes happen due to reflection. As Figure 3.9 shows, if the phases are aligned, multipath can increase the signal amplitude. On the other hand, the multipath can cancel out the signal if totally out of phase.
Sometimes by moving the receiver only a few centimetres can cause big differences in signal amplitude due to changes in multipath. This is called small scale fading.
3.12 Shadowing
Figure 3.9 Small Scale Fading
If there is an object blocking the LoS, then the power received will be much lower due to the blockage. Figure 3.10 shows how the power suddenly decreased due to the shadowing effect when the received is moved.
Figure 3.10 Shadowing
3.13 Total Path Loss
Now we see that there are many pheno
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