Chapter 3



Channel








Mind map:





Mind map for 
Chapter 3








In this chapter, the classification of the channel, the channel models, the noise in the channel and the capacity of channel will be introduced and discussed.






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3.1The classification of channels

According to the transmitted media, channels can be classified into two categories：   wireless channel and wired channel.

3.1.1Wireless channels

The transmission of signals in wireless channel is achieved by the propagation of electromagnetic waves in space. In principles, the electromagnetic wave of any frequency can be produced. For effectively transmitting and receiving, the frequency of the magnetic wave used for communications is usually rather high in practical applications.

According to different communication ranges, frequencies and locations, the electromagnetic wave propagation can be classified into three types：   the line of sight (LOS：  视线) propagation, the ground wave(地波) and the sky wave(天波)(or called ionosphere reflection wave：  电离层反射波),as shown in Table 3.1.1. The sketch maps of these three propagations are shown in Figure 3.1.1.


Table 3.1.1The comparison of three propagation modes



Type
Frequency range
Propagation mode
Propagation range
Ground 
wave
< 2MHz
Along the curved ground surface
Over hundreds to thousands of km
Sky wave
2~30MHz
Many reflections between

ground and flayer
More than 10000km
LOS
>30MHz
Line of sight
D=50h(km)

→Satellite communication




Figure 3.1.1Sketch of three propagations



The propagation of electromagnetic wave in the atmosphere is influenced by the atmosphere. The relationship between the attenuation characteristics of the atmosphere and the frequency are shown in the following Figure 3.1.2.



Figure 3.1.2Attenuation of oxygen(氧气) and vapor(水蒸气)(concentration 7.5g/m3)


3.1.2Wire channel 

There are three kinds of wired channels：   symmetrical cables, coaxial cables and optical fibers.

Symmetrical cables is also called twist wire (双绞线).The telephone channel is built using twist pairs for signal transmission. A twisted pair consists of two solid copper conductors, each of which is encased in a polyvinylchloride (PVC：  聚氯乙烯) sheath. Twisted pairs are usually made up into cables, as in Figure 3.1.3, with each cable consisting of many pairs in close proximity to each other. Twisted pairs are naturally susceptible to electromagnetic interference (EMI：  电磁干扰).



Figure 3.1.3Twist wire


Typical coaxial cable (同轴电缆) has a characteristic impedance of 50 or 75 ohms. The composition of coaxial cable is shown in Figure 3.1.4. Compared to a twistedpair cable, a coaxial cable offers a greater degree of immunity to EMI. The standard bite rate is 10Mb/s, which is higher than twisted pairs. The applications of coaxial cables are as the transmission medium for local area networks (LAN) and in cabletelevision systems.



Figure 3.1.4Coaxial cable



The optical fiber is widely used for the transmission of light signals from one place to another. It can be classified into two categories：   multimode and singlemode, as given in Figure 3.1.5.

Optical fibers have unique characteristic that make them highly attractive as a transmission medium. They offer the following unique characteristics：  

 Enormous potential bandwidth. 

 Low transmission losses.

 Immunity to EMI.

 Small size and weight.

 Ruggedness and flexibility.



Figure 3.1.5The structure of singlemode and multimode optical fiber


There are two minimum loss points at 1.31μm and 1.55μm from the following Figure 3.1.6. Therefore, these two wavelengths are widely used.



Figure 3.1.6The relationship between loss and wavelength






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3.2Channel models

We have introduced two types of channels, how to describe the channels in mathematical tool is the content of this part. There are two basic channel models：   one is for modulation, another one is for coding.

3.2.1Modulation channel model (调制解调模型)

The basic modulation channel is defined as


eo(t)=f［ei(t)］+n(t)(3.2.1)


where ei(t) is the signal voltage at the channel input terminal,eo(t) is the signal voltage at the channel output, and n(t) is the noise voltage. Noise n(t) always exists in the channel. It is usually called the additive noise because “+”. The model is illustrated in Figure 3.2.1.



Figure 3.2.1Modulation channel model



f(·) is the function between ei(t) and eo(t). For simplicity, we usually assume f［ei(t)］=k(t)ei(t).

k(t) is a complicated function and it reflects the characteristics of the channel.


k(t)= time variant    random  parameter  channels(随参信道)
constant      constant  parameter  channels(恒参信道)


3.2.2Coding channel model (编码信道模型)

The input and output signals of the coding channel are digital sequences in Figure 3.2.2.



Figure 3.2.2Coding channel


Error usually happens at the output because of interference. Therefore, the best method to describe this model is the error probability (错误概率). It is also called the transfer probability.


P(0/0)transmitting0andreceiving 0
P(1/1)transmitting1andreceiving 1Correct transfer probability


P(1/0)transmitting0andreceiving 1
P(0/1)transmitting1andreceiving 0Error transfer probability


For binary systems：  


P(0/0)=1-P(1/0)
P(1/1) =1-P(0/1)


The model in Figure 3.2.3 is the simple binary coding memoryless channel model, in which the occurrence of errors in adjacent symbols is independent.



Figure 3.2.3Binary coding channel model


3.3Influence of the channel characteristics on 
transmission (for modulation model)
3.3.1Influence of constant parameter channel on signal transmission 

The main transmission characteristics of the transmission function are usually described by the amplitudefrequency characteristics (幅频特性) and phasefrequency characteristics (相频特性).

In practice, phasefrequency characteristic can also be described by group delay (群延迟).

The amplitude characteristic can be described by insertion loss (插入损耗). Figure 3.3.1 show the ideal amplitudefrequency and phasefrequency characteristics. 



Figure 3.3.1The ideal amplitudefrequency and phasefrequency characteristics



The definition of group delay is 


τ(ω)=dφ(ω)dω



The plots of Figure 3.3.2 clearly illustrate the dispersive nature of the telephone channel.



Figure 3.3.2Characteristic of typical telephone connection


3.3.2Influence of random parameter channel of signal transmission

There are there common characteristics：  

(1) Transmission attenuation of the signal is varying with time；   

(2) Transmission delay of the signal varies with time；  

(3) Signal arrives at the receiver over several paths, i.e. multipath propagation phenomenon  
exists.

Multipath will be discussed for its great influence on the quality of the signal transmission, 
the model of multipath is illustrated in Figure 3.3.3.



Figure 3.3.3Model of a multipath channel


Suppose the transmitting signal is Acosω0t, which is a common signal model in a communication system. When the signal propagates to the receiver over n paths, then the received signal R(t) may be written as：  


R(t)=∑ni=1μi(t)cosω0［t－τi(t)］=∑ni=1μi(t)cos［ω0t+φi(t)］(3.3.1)

where


μi(t)  is the attenuation of ith path
τi(t) is the delay of ith path
φi(t)=－ω0τi(t) are random varying


Equation (3.3.1) can also be written as： 


R(t)=∑ni=1μi(t)cosφi(t)cosω0t－∑ni=1μi(t)sinφi(t)sinω0t(3.3.2)


Let


Xc(t)=∑μicosφi(t)
Xs(t)=∑μisinφi(t)


Then


R(t)=Xc(t)cosω0t－Xs(t)sinω0t=V(t)cos［ω0t+φ(t)］



where


V(t)=X2c(t)+X2s(t)—envelope
φ(t)=arctanXs(t)Xc(t)  —phase


Comparing this eq. with the narrowband random process yields：  


X(t)=Xccosωct－Xssinωct=ax(t)cos［ωct+φx(t)］


So R(t) can be regarded as a narrowband signal with random varying envelope and phase, as shown in Figure 3.3.4.



Figure 3.3.4The narrowband signal


To simplify the problem, here we only discuss two paths of the multipath propagation with the same attenuation and different delays.

Suppose the transmission signal is f(t), the received signals are Aft－τ0 and Af(t-τ0-τ), respectively, where A is a constant.

Their corresponding Fourier transforms are

Input：  


f(t)F(ω)
Af(t－τ0)AF(ω)e－jωτ0
Af(t－τ0－τ)AF(ω)e－jω(τ0+τ)

Output：  


Af(t－τ0)+Af(t－τ0－τ)AF(ω)e－jωτ0(1+e－jωτ)


Therefore, the transfer function of the two paths channel is


H(ω)=Ae－jωτ0(1+e－jωτ)
H(ω)=Ae－jωτ0(1+e－jωτ)
=A1+e－jωτ
=A1+cosωτ－jsinωτ=A(1+cosωτ)2+sin2ωτ=2Acosωτ2

The curve drawn according to the above equation is shown in Figure 3.3.5.



Figure 3.3.5Multipath effect



From the above curve, we consider that fading is related to the frequency, it is called frequency selective fading (频率选择性衰落).

Figure 3.3.6 illustrates the effect of Rayleigh fading on the waveform of the received signal, whose amplitude and phase components vary randomly with time.



Figure 3.3.6Effect of Rayleigh fading on a sinusoidal wave






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3.4Channel capacity (continuous channel)

The information capacity of a continuous channel of bandwidth BHz, with the addictive white Gaussian noise of PSD n02 and limited in B, is given by


Ct=Blog21+Sn0B(b/s)(3.4.1)

where S is the average transmitted power.

There are three key system parameters：   channel bandwidth (B), average transmitted power (S) and the noise power spectral density (n0/2).

(1)  When S increases or n0 decreases, C increases.

(2)  When B→∞, Ct approaches to the following limit：  


limB→∞Ct=limx→0Sn0Bn0Slog21+Sn0B
=limx→0Sn0log2(1+x)1/x=Sn0log2e≈1.44Sn0


The channel capacity approaches 1.44 times of signal power to noise PSD ratio.

Example 3.4.1：   A frame of black and white TV image is composed of 300 thousand pixels, each pixel has 10 levels of brightness and these 10 levels occur at equal probabilities. If the image is transmitted at a rate of 25 frames per second, the image signal to noise ratio is required to reach 30dB, find the required transmission bandwidth.

Answer：   Since each pixel takes 10 possible levels with equal probability, the information content of each pixel Ip is


Ip=log210=3.32(b/pixel)


The information content If of each image frame is 


If=30000×3.32=9.96×105(b/frame)


Since there are 25 frames of image per second, the required information transmission rate is


If×25=9.96×25×105=24.9×106(b/s)


According to the formula


C=Blog2(1+S/N)
24.9×106=Blog2(1+103)=9.96B



The bandwidth is：   

B=24.9×106/9.96=2.5(MHz)



Summary and discussion

Channel plays a very important role in the communication system, which also is the basis of the following chapters. The classification of the channel is given in this chapter. Three main wireless channels are the line of sight (LOS：  视线) propagation, the ground wave(地波) and the skywave(天波). There are also three main kinds of wired channels：   symmetrical cables, coaxial cables and optical fibers. Nowadays, these channels are widely used, and they have affected our daily lives, due to their wide range of applications.

It is important to understand two mathematics models of channel. Modulation channel model is usually regarded as analog channel. The multiplicative noise and additive noise are used to reflect the channel effect. Multiplicative noise k(t) can cause signal distortion，including linear distortion, nonlinear distortion, time delay and attenuation. Depending on the k(t) being a constant or time varying, constant parameter channel and random parameter channel are distinguished. The influence of random parameter channel on signal is multipath effect, which can cause the frequency selective fading of the signal. Additive noise always exists in communication systems. Thermal noise is usually called white noise. We mainly discuss the influence of the white noise, especially the Gaussian white noise in this book.

Coding channel model is usually viewed as digital channel. Both additive noise and multiplicative noise influence the coding channel. Error probability is used to describe this kind of channels, which is also called the transfer probability.

There is a ShannonHartley theorem (香农定理) about continuous channel capacity：  


Ct=Blog21+Sn0B(b/s)


Ct represents maximal information rate which can be transmitted by the channel, and its unit is b/s. The bandwidth and SNR can be traded off. If the bandwidth is increased then the SNR decreases and the capacity remains unchanged. This tradeoff relationship has very steering significance in the design of communication systems. This trade off cannot be naturally achieved. It required that the signal is modulated or encoded to increase its occupied bandwidth, and then it is sent to the channel for transmission corresponding to demodulation for decoding at the receiver.


Homework

3.1Assume a wireless link uses lineofsight propagation for communication, and the heights of the transmitting antenna and the receiving antenna are both 80m.Find the maximum communication distance.

3.2A voicegrade channel of the telephone network has a bandwidth of 3.4kHz.

(1)  Calculate the information capacity of the telephone channel for a signaltonoise ratio of 30dB.

(2) Calculate the minimum SNR required to support information transmission through the telephone channel at the rate of 9600b/s.

Vocabulary and terminologies


attenuation衰减

copper铜

channel信道

electromagnetic电磁的

fiber光纤

interference干扰

oxygen氧气

propagation传播

surface表面

terminal终端

vapor水蒸气

wireless无线

channel capacity信道容量

coaxial cable同轴电缆

constant parameter channel恒参信道

EMI电磁干扰

frequency selective fading频率选择性衰落

ground wave地波

group delay群延时

insert loss插入损耗

multipath effect多径效应

random parameter channel随参信道

sky wave天波

transfer probability转移概率

twist wire双绞线