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Project Assignment for Mobile & Satellite

Communication Systems

Autumn 2010/11

Eleni Chatzikyriakou

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SIMULATIONS OF SMALL-SCALE FADING CHANNELS

In the following, we are going to simulate various situations of fading channels. Specifically, the

multipath behaviour of signals is going to be analyzed. Multipath components appear on the propagation of a

signal whose route is obstructed, and are caused by reflections, scattering and other mechanisms. When

different multipath components interfere with each-other, small scale fading appears.

For most of our real-world scenarios, the 3G standard is going to be used with QPSK modulation, which

contains 2 bits per symbol. UMTS specifies the bands 1900-1980 MHz, 2010-2025 MHz and 2110-

2170 MHz for 3G terrestrial communications. The simplified frequency of 2010 MHz has been

chosen for the simulation of GSM channels with an equivalent symbol time, Ts of 49.7 ns. Symbol

period in QPSK is, as expected, twice the bit period. Delay profiles for the simulations are chosen

from COST259 standard. Rayleigh fading channel block of simulink is used for simulating the

fading channel.

How fading works

Generally, the multipath components can be thought of as samples in the impulse response of the delay

channel. The time elapsed between adjacent samples (paths), in relation to the symbol period of the baseband

signal, characterizes a fading channel as either flat or frequency selective. Specifically, when all the delayed

components arrive within the interval of the symbol period, no change is observed at the receiver, so the

channel is considered flat. If, on the other hand, the delay between two adjacent samples is more than the

symbol period, the channel will undergo attenuations outside the latter interval that will be perceived at the

receiver as changes in frequency. This would be a frequency selective fading channel. These definitions are

important for us because, as will be seen later in this report, there is currently no observed delay profile fast

enough to stay “inside” the symbol time of a GSM channel, thus fading is always frequency selective. The

statistical measure, rms delay spread, which is the standard deviation of the samples in the impulse response,

aids this process of characterization of a channel.

Figure 1. Attenuation in the frequency response of a channel [1]

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Another measure that contributes to the characterization of a channel is coherence bandwidth, Bc.

Calculation of the coherence bandwidth depends on how strong we decide a correlation between two discrete

frequencies should be. Error! Reference source not found. shows signal attenuation in the frequency

domain [1]. Coherence bandwidth is the interval Δf. An approximate relationship between coherence

bandwidth and rms spread is Βc = 1/5στ, for a frequency correlation function above 0.5, or Βc = 1/50στ if

above 0.9.

Any change of the position of the transmitter or receiver as well as the objects around them could also

affect a fading channel. This stems from the introduction of Doppler shift. Doppler shift “pushes” the signal

in different frequencies which eventually leads to faster signal strength fluctuations. Table 1 contains a list

of Doppler shift values calculated for 2010 MHz channel frequency by the following equation,

𝑓𝑑 =𝑣𝑓

𝑐𝑐𝑜𝑠𝜃

Where v is the speed of the origin, f is the signal frequency and c is the speed of light. Depending on the

angle of the direction of the origin in relation to the receiver, Doppler shift spreads from a maximum value

fs+fd (0° angle – approaching directly) to a negative value fs-fd (180° degrees angle – opposite direction).

Doppler spread, Bd is the interval between maximum and minimum Doppler shifts and is equal to 2*fd.

Speed Doppler Shift

Pedestrian 1 m/s 6.7 Hz

Vehicle 33.3m/s (120 km/h) 223Hz

Train 55.5m/s (200km/h) 371 Hz

Table 1. Doppler shift values for 2010 MHz frequency used in the simulation scenarios

The well-documented time-varying nature of the multipath channel [2] introduces the concept of

coherence time, Tc, which is the interval over which the channel is considered invariant. Since the Doppler

spread is responsible for the variance of the impulse response, an approximate relationship between

coherence time and Doppler spread can be defined as Tc = 1/Bd. This is also a statistical measure and

depends on the desirable time correlation function value. If chosen above 0.5, then Tc = 9/(16*π*Bd). If

coherence time is greater than the rate of change of the channel (symbol period), then slow fading will occur,

otherwise, fast fading.

Scenario 1: Pedestrian moving in build-up environment

This scenario is based on the COST259 Typical Urban delay profile. In build-up environments, the

non-lossy surfaces of the buildings can cause reflections on paths that arrive at late intervals, yet

they are not attenuated enough to be ignored. Figure 2 contains the gain and delay vectors used in

the Rayleigh fading channel block, as well as the simulink model. Unless otherwise stated, this

model will be used in all simulations. The input values for the Random Integer Generator and

Rayleigh fading blocks are shown in Figure 3.

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Delay (μs) Attenuation (dB)

0 -5.7

0.217 -7.6

0.512 -10.1

1.230 -16.3

1.818 -21.5

2.048 -23.5

Figure 2. COST259 Typical Urban delay profile (left) simulink fading channel model (right)

Figure 3. Input values for model in figure 7

Next, the mean excess delay and rms delay spread are calculated from the input parameters with

the equations shown in Table 2. The same table contains all the values that will help us in the

characterization of the channel.

Figure 4 contains the impulse and frequency responses. Some immediate conclusions are made. It

is obvious from the frequency spectrum the channel is frequency selective. Also, the impulse

response contains so many attenuated points between successive samples that they seem like a

continuous line. This big ‘distance” between the samples would effectively allow a RAKE receiver

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to distinguish between them. With very slow alterations in the time and frequency domain

spectrums, it can safely be assumed that it is a slow fading channel. Using the results from Table 2

we can now prove our speculations mathematically.

Mean excess delay 𝜏̅ =𝛴𝑘𝑃(𝜏𝜅)𝜏𝜅

𝛴𝑘𝑃(𝜏𝑘)

Rms delay spread 𝜎𝜏 = √𝜏2̅̅ ̅ − (𝜏̅)2

Where 𝜏2̅̅ ̅ =∑ 𝑃(𝜏𝑘)𝜏𝑘

2𝑘

∑ 𝑃(𝜏𝑘)𝑘

Pedestrian in Urban environment

Rms delay spread στ 0.227 μs

Doppler spread Bd 13.4 Hz

Coherence time Tc 13.3 ms

Coherence Bandwidth Bc 881 KHz

Channel Bandwidth Bs 2010 MHz

Symbol period Ts 49.7 ns

Bs >Bc, Ts < στ (frequency selective)

Ts << Tc, Bs >> Bd (slow fading)

Table 2. Calculating delay spread values for urban scenario

Figure 4. Impulse and frequency responses of the fading channel

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Scenario 2: Vehicle moving on a motorway in a hilly environment

The lossy surfaces of a hilly environment introduce paths of very late times. This is in contrast to the

urban scenario, where the signals meet more obstructions and eventually ‘die out’ before reaching the

receiver. These late paths of the hilly environment become apparent in the delay profile as though they come

from different transmitters (the first group completely attenuates before a second arrives).

Fading is again frequency selective since rms delay spread is even greater than the previous scenario, and

of course, greater than the symbol time. It is worth noting that for both the coherence time, as well as the

coherence bandwidth, a correlation function above 0.5 was chosen. Under these conditions, the channel has

slow fading variations. The same would be true even if correlation function was decided to be above 0.9.

Vehicle in hilly environment

Delay (μs) Attenuation (dB) Rms delay spread στ 2.8 μs

0 -3.6 Doppler spread Bd 446 Hz

0.356 -8.9 Coherence time Tc 401 μs

0.916 -17.3 Coherence Bandwidth Bc 71 KHz

15 -17.6 Channel Bandwidth Bs 2010 MHz

17 -24.1 Symbol period Ts 49.7 ns

Bs > Bc, Ts < στ (frequency selective)

Ts < Tc, Bs > Bd (slow fading)

Table 3. Hilly environment channel characterization

Figure 5. Impulse (top) and frequency responses (bottom) of scenario 2

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Scenario 3: Train passing through rural environment

In a rural environment, most of the multipath components arrive from nearly LOS directions because

there are fewer buildings and other tall obstructions. This results in a smaller rms delay spread value. The

speed of a train is even greater than that of a vehicle, so Doppler spread is also greater. We can clearly see

from the graphs below that even in the best case of outdoor rms delay spread, fading continues to be

frequency selective. As the signal traverses even a few meters of outdoor distance, reflections and scattering

are bound to happen. This is one of the problems microcells aim to solve [3].

If we wanted to achieve a flat fading channel with the same frequency, we could switch to an indoor

scenario to reduce the delay spread to some nanoseconds. In that case, Doppler spread would also have to be

reduced to reflect realistic values. So fading would continue to be slow, but this new delay spread would

render the channel flat. That, however, would not be a realistic scenario, since it’s not quite often that you see

a base station inside a building.

Fast fading with this particular frequency is even more unrealistic to simulate. In order to achieve a

coherence time less than the symbol period, Doppler shift would have to be 1.8 MHz, which corresponds to

332.617 km/h. That is faster than the fastest recorded speed of a spacecraft. This speed would have to be

even greater if we wanted Doppler spread to be greater than the channel bandwidth (2010 MHz).

Train in rural environment

Delay (μs) Attenuation (dB) Rms delay spread στ 0.072 μs

0 -5.2 Doppler spread Bd 742 Hz

0.042 -6.4 Coherence time Tc 241 μs

0.129 -9.3 Coherence Bandwidth Bc 2.7 MHz

0.245 -13.1 Channel Bandwidth Bs 2010 MHz

0.410 -18.5 Symbol period Ts 49.7 ns

Bs > Bc, Ts < στ (frequency selective)

Ts << Tc, Bs >> Bd (slow fading)

Table 4. Channel characteristics for scenario 3

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Figure 6. Impulse and frequency responses of scenario 3

Scenario 3b. Train in rural environment - flat fading

The best way to turn this scenario into a flat fading channel is to reduce the operating frequency

(900 KHz), and of course abandon the GSM prototype. For reasons of simplicity, QPSK remains as

the modulation type, although down to these frequencies, an analog modulation scheme would be

more appropriate. Table 5 lists the new parameters of the channel.

Symbol period has clearly increased to a value much greater than the delay spread, so the channel

has become flat. This is evident in the impulse response in Figure 6 which indicates the proximity

of the multipath components in relation to the symbol period. This is further confirmed by the

frequency response of the channel (Figure 8) which is clearly flat.

Decreasing the frequency has also decreased Doppler spread. We immediately conclude from this

that it is virtually impossible to simulate a fast fading channel. In a slow channel, signal strength

remains increased for a very long interval, aiding communication between transmitter and receiver.

On the other hand, prolonged deep fades could seriously affect communication, a problem that can

be dealt with in frequency selective fading, based on the fact that multipath components do not

attenuate simultaneously. In Figure 7, the simultaneous attenuation of the multipath components is

shown. At the top we can see each component experiencing attenuation higher than 10dB, while on

the right the phase trajectory is close to zero.

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Train in rural environment (flat fading)

Delay (μs) Attenuation (dB) Rms delay spread στ 72 ns

0 -5.2 Doppler spread Bd 0.33 Hz

0.042 -6.4 Coherence time Tc 542 ms

0.129 -9.3 Coherence Bandwidth Bc 2.7 MHz

0.245 -13.1 Channel Bandwidth Bs 900 KHz

0.410 -18.5 Symbol period Ts 1.1 μs

Bs < Bc, Ts > στ (flat fading)

Ts < Tc, Bs > Bd (slow fading)

Table 5. Channel characteristics of scenario 3b

Figure 7. Multipath components (top), impulse response (bottom left) and phase trajectory (right) of scenario 3b

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Figure 8. Frequency response of scenario 3b.

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CHANNEL PERFORMANCE SIMULATIONS

A simulink model that adds AWGN and error rate detection will be devised next. We are going to

simulate a full communication channel (transmitter to receiver) and calculate its BER. Fading

models of scenarios 3 and 3b are going to be used with frequencies of 2010 MHz and 200 KHz.

Modulation is, again, QPSK. As a 4-array technique, QPSK includes 2 bits per symbol. In order to

get the BER we have to convert the symbol to binary form. The Integer to Bit Converter block is

going to be used before the modulation and the results will be fed to the Error Calculation block

along with the output of the demodulation.

The existence of a fading channel adds complexity to the demodulation process. This is because a

fading channel introduces complex gain, which is amplitude as well as phase distortion. As we are

using a phase modulation technique, amplitude distortion does not affect the signal. Phase

distortion, however, can be quite a destructive factor causing the signal to be interpreted completely

wrong at the receiver giving very high error rates. To combat this situation in a real world scenario,

RAKE receivers or equalizers would be used. For the purposes of this simulation, the Complex

phase difference and Complex phase shift blocks of simulink are going to be used. Shifting the

signal backwards we fix the phase rotation introduced by the Rayleigh channel without affecting the

amplitude. The model is depicted in Figure 9.

Figure 9. Simulink model of BER calculation

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We can now use MATLAB’s bertool to produce BER graphs for different frequencies and fading

channels. The results are shown in Table 6.

Graph Symbol Operating frequency Channel characteristics

2010 MHz Frequency Selective, slow

200 KHz Flat, slow

Table 6. Simulation operating frequencies

As can be seen from Figure 10, the two frequencies produce almost the same output for

Eb/N0<10dB. The higher BER of the second simulation may be the result of a deep fade.

In a flat fading channel, errors are mainly caused by fading and interference from external

resources. In contrast, in a frequency selective fading channel, errors are also the result of inter-

symbol interference. If the paths in the latter case do not add destructively, the signal is good and

errors decrease. If they also add constructively, they increase the power of the signal. Since we

made ideal signal equalization, completely recovering the phase of the initial signal after fading,

multipath did not affect our simulations significantly. Errors were, thus, the result of fading with

combination of the AWGN, which had more impact on the flat channel than on the frequency

selective.

All three curves start decreasing more slowly after 28 dB. This means that in higher Eb/N0 values,

no matter how much we increase the signal power, there will be no further improvement over the

error rate, or the improvement will be random and minimal.

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Figure 10. BER graphs for scenarios 3 and 3b

Reducing the error rate usually means increasing signal strength. This however should be done

within reasonable limits. Antenna power levels should be kept within safety standards. For mobile

communications, they should also be kept low for portability issues. Batteries are preferred to be as

small as possible.

For frequency selective channels, which is the case in most of today’s models, equalization is an

important factor that greatly affects channel performance. Providing better equalization techniques

seems to be an effective mitigation method for the lower levels.

The third way to reduce error rates is by changing the modulation method used. BPSK, and also

QPSK, provide the lowest probability of error among the phase modulation techniques, but spread

spectrum provides immunity to interference further decreasing BER values.

References

[1] M.A. Ingram Notes on Coherence Bandwidth. Georgia Tech College of Engineering [Online]

http://www2.ece.gatech.edu/research/labs/sarl/tutorials/ECE4606/22-CoherenceBandwidth.pdf [Dec

2010].

[2] T.S. Rapapport. Wireless Communications: Principles and Practice. Prentice Hall, 2002.

[3] J. Volakis. Antenna Engineering Handbook. McGraw-Hill Professional, 2007.