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这是一个关于移动通信工程ppt，主要介绍Random Variables Related to Mobile-Radio Signals、Phase-Correlation Characteristics、Characteristics of Random FM、Click-Noise Characteristics、Simulation Models。欢迎点击下载哦。

CHAPTER7 Received-Signal Phase Characteristics

A signal s0(t) received at the mobile unit can be expressed:

s0(t) = m(t)s(t) (7.1)

The long-term-fading factor m(x) is extracted from s0(t) and the resultant

can be expressed:

s(t) = r0ejψ(t) (7.2)

where r0(t) and ψ(t) are the envelope and phase terms, respectively. The

characteristics of r0(t) have been discussed in Chap. 6. The phase and

time derivative of ψ(t), ψ・ (t) = dψ(t)/dt, is the random FM that is

described in this chapter.

Assume that aj(t) is the jth wave arrival, then s(t) represents the sum

of all wave arrivals, as has been shown in Eq. (6.10):

s(t) = ∑a(t)=X1 + jY1 (7.3)

where X1 and Y1 are as defined in Eqs. (6.11) and (6.12), respectively.

Hence ψ1(t) can be defined:

The terms X1 and Y1 of the signal s(t) are two independent Gaussian

variables with zero mean and a variance of σ2. This means that:

E[X1Y1] = 0 (7.5)

E[X12] = E[Y12] = σ2 (7.6)

But when two signals s1 = s(t) and s2 = s(t + τ), expressed

s1(t) = X1 + jY1 = r1ejψ1 (7.7)

and s2(t) = X2 + jY2 = r2ejψ2 (7.8)

are correlated, then E[X1X2] and E[X1Y2] are not necessarily zero. Consequently,

it is first necessary to find the covariance matrix for these

random variables, and then the characteristics of ψ(t) and ψ (t) can beintroduced.

CONTENTS

7.1 Random Variables Related to Mobile-Radio Signals

7.2 Phase-Correlation Characteristics

7.3 Characteristics of Random FM

7.4 Click-Noise Characteristics

7.5 Simulation Models

7.1.1Finding the covariance of random variables

Since the ergodic process is always applied to random variables in the mobile-radio environment, then Eq. (2.80) can also be used here:

From Eq. (7.9), the following expressions are obtained:

Rc(τ) = E[X1X2] = E[Y1Y2] (7.10)

Rs(τ) = E[X1Y2] = −E[Y1X2] (7.11)

Equations (7.10) and (7.11) have been used in Chap. 6 for calculating

the covariances of random variables. Also E[s1s*2] can be expressed as in

Eq. (2.99):

(7.12)

where S( f) df is the average power that lies in the frequency range f,

f + df. S( f) is often given in order to specify the spectrum of a given

noise. Since the signals s1 and s2 shown in Eqs. (7.7) and (7.8) contain

four random variables—X1, Y1, X2, and Y2—then the following expressions

of covariances can be derived from Eq. (7.12):

(7.13)

(7.14)

Following Rice’s notations

R′c(τ) = E[X1X2] = E[Y1Y2] = −E[X1X2] = −E[Y1Y2] (7.15)

R′cs(τ) = E[X1Y2] = E[Y1X2] = −E[X1Y2] = −E[X1Y2] (7.16)

R″c (τ) = −E[X1X2] = −E[Y1Y2] (7.17)

R″cs(τ) = E[Y1X2] = −E[X1Y2] (7.18)

When τ = 0, the moments can be found from these correlation functions:

(7.19)

Thus,

The other functions we can see in the p184

7.1.2Power spectra of a signal s(t )

E[s1s*2] can also be obtained from S( f) in Eq. (7.12). Sometimes it is

much easier to calculate E[s1s*2] by using S( f) than by time-averaging

from Eq. (7.9) as described in Chap. 6. Therefore, the expression of S( f)

must be determined.

Assume that all waves are traveling in the horizontal plane and that the antenna is pointing in a horizontal direction with an azimuthal antenna beam angle of γ. The power contributed to the the received signal by waves arriving in the horizontal plane within the angle dφ is the power arriving in that angular interval that would normally be received by an isotropic antenna of the same polarization:

Ss(φ) dφ = Ap(φ)G2(φ − γ) dφ (7.23)

where A is a constant that will be defined later on, p(φ) is the angular

distribution of wave arrival, and G(φ − γ) is the antenna pattern. The

power spectral density Ss( f) is [2]

Ss( f) = Ss(φ) | dφ /df| (7.24)

The covariance E[s1s*2 ] can be obtained by Eq. (7.12), and the average

power is

(7.25)

7.1.3Covariance matrix

(7.26)

7.2 Phase-Correlation Characteristics

There are several characteristics that can be used to describe ψ(t).

The terms X and Y in Eq. (7.4) are the real and imaginary components

of the mobile-radio signal s(t). These two components are Gaussian relationships, as shown in Eqs. (7.5) and (7.6). Then, the distribution

of the random process ψ(t) can be found from Eq. (2.40):

0 ≤ ψ ≤ 2π others 0 (7.46)

Hence, ψ is uniformly distributed, as shown in Fig. 7.2.

When there are two phase angles, ψ1 and ψ2, of the signals s1 and s2,

then the distribution of ψ1 and ψ2, assuming the two signals are correlated,

is:

(7.47)

where p(r1, r2, ψ1, ψ2) is similar to, but not the same as, the form shown

in Eq. (6.88).

Assuming that

E[X12] = E[Y12] = E[X22] = E[Y22] = σ2 (7.48)

Phase-correlation characteristics of ψ(t)

If there is correlation between s1 and s2 :

0 ≤ ψ1, ψ2 ≤ 2π

others 0 (7.49)

Where

(7.50)

If there is no correlation between s1 and s2, then Rc(τ) and Rcs(τ) are

zero. Thus, Q = 0, from Eq. (7.53), and |Λ|1/2 = σ4, from Eq. (7.45). When

these two values are inserted into Eq. (7.52), p(ψ1, ψ2) becomes:

p(ψ1, ψ2)= 1/4π2=p(ψ1)p(ψ2) (7.51)

which is what is normally expected.

Phase correlation between frequency and time separation

In expressing the phase correlation between frequency separations Δω

and time separations τ, the phase correlation can be expressed:

(7.55)

where p(ψ1, ψ2) is as shown in Eq. (7.52) and the covariance matrix elements are obtained from Eq. (7.44), but with Q expressed differently

from Eq. (7.53), as follows:

(7.56)

Where

φ = tan−1 (−Δω Δ) (7.57)

(7.58)

and Rcs(Δω, τ) = 0

can be obtained from Eqs. (6.123) through (6.125). The integral of Eq.

(7.55) cannot be carried out exactly, but an approximation can be

shown [4]:

(7.59)

where ρc = ρc(Δω, τ) is the normalized correlation ρc = Rc(Δω, τ)/σ2, which

equals sqrt(ρr) in Eq. (6.126), expressed as:

(7.60)

And

(7.61)

The function Ω(ρc) is plotted in Fig. 7.3. Equation (7.59) can be solved

by using the information contained in Fig. 7.3.

If it is assumed that p(ψ) = 1/2π, then the mean average values of ψ1

and ψ2 are:

(7.62)

and therefore the correlation coefficient can be expressed

(7.63)

The function of Eq. (7.63) is plotted in Fig. 7.4.The coherence bandwidth Bc = (Δω)c/2π at values of τ = 0 and ρψ = 0.5 can be found from the data given in Fig. 7.4, where:

ρψ(Bc, 0) = 0.5 (7.64)

And Bc =1 /4πΔ (7.65)

The coherence bandwidth, according to the phase correlation, is onehalf

of the envelope correlation value previously shown in Eq. (6.128).

7.3 Characteristics of Random FM

Probability distribution of random FM

The term p(r, r・, ψ,ψ・ ) has been defined in Eq. (6.61), and therefore the

probability density function (pdf ) of a random FM signal ψ・ , p(ψ・ ), can

be derived from p(r, r・, ψ,ψ・ ) in combination with the following integrals:

(7.66)

From Eq. (6.15) and Eq. (6.55), σ2 and ν2 for an E field signal are expressed: σ2 = E[X2] = N (7.67)

And

Then (7.68)

The correlation coefficient p(ψ) of Eq. (7.68) is plotted in Fig. 7.5(a).

The distribution of random FM can be expressed:

(7.69)

By substituting the results of Eq. (7.67) for σ and ν of an E field signal

into Eq. (7.69), the following is obtained:

(7.70)

The response curve for the correlation coefficient for P(ψ・ ≤ Ψ・ ) is plotted in Fig. 7.5(b).

Power spectrum of random FM

The power spectrum of random FM can be obtained from the autocorrelation function described by Rice [1] and expressed:

(7.71)

Assume that the angular wave arrival is uniformly distributed, that is,

that p(φi) = 1/2π. Then the expression of Rc(τ) in Eq. (7.71) can be found

from Eq. (6.108) for the case of the Ez field. The first and second derivatives, R′c(τ) and R″c (τ), can be obtained as follows:

(7.72)

(7.73)

(7.74)

The power spectrum can be expressed as the Fourier transform of Rψ・ (τ)

as follows:

(7.75)

Equation (7.75) can be integrated either approximately or numerically.

Figure 7.6 shows the curve obtained by integrating the autocorrelation

function over three regions in the time domain [2].

(7.78)

and therefore

(7.79)

The corresponding baseband output noise due to random FM in an

audio band (W1, W2) is:

(7.80)

Hence, the average power of the random FM is a function of vehicular

speed V and the audio bandwidth (W1, W2).

In Fig. 7.6, the power spectrum of the random FM drops very drastically

as soon as reaching the frequency

Therefore, we can use this criterion to calculate the impact of the random

FM versus the mobile speed. The criterion of considering the

impact of random FM is

(7.81)

Level-crossing rate (LCR) of random FM

In order to obtain the lcr of random FM, it is first necessary to find

(7.82)

The lcr of the random FM can then be found [1]:

(7.83)

where b0, b2, and b4 are as shown in Eqs. (7.20) through (7.22).

For the case where the noise is constant, with a value of η across the

frequency band from f − β/2 to f + β/2, then the terms b0, b2, and b4 can

be found by applying Eq. (7.19), as follows:

7.4 Click-Noise Characteristics

As the mobile unit moves through a mobile-radio field, the received signal

fluctuates in both amplitude r and phase ψ. In a conventional FM

discriminator, the fluctuations in phase ψ due to multipath fading generate

noise in the signal. Such noise is called “random FM noise.” The

random FM noise is essentially concentrated at frequencies less than 2

to 3 times the maximum Doppler frequency. Outside this range the

power spectrum falls off at a rate of 1/f. Davis [5] has found that random

FM noise due to fading can be considered a secondary effect in the

reception of mobile-radio signals.

Another kind of noise, called “click noise,” may be found at the output

of the FM discriminator.

Click noise:

7.5 Simulation Models

Many radio-propagation models have been proposed that can be used to predict the amplitude and phase of radio signals propagated within a mobile-radio environment [8.20]. These models can be classified into two general categories. The first category deals only with multipathfading phenomena, whereas the second category deals with both multipath- and selective-fading phenomena. The following paragraphs describe some of the features associated with these two general categories of simulation models.

Rayleigh multipath fading simulator:

(A) Hardware simulator. Based on analyses of the statistical nature of a

mobile fading signal and its effects on envelope and phase, a fading simulator can be configured either from hardware or a combination of hardware and software. Figure 7.12 shows a simple hardware configuration for a Rayleigh multipath fading simulator that consists of two independent Gaussian noise generators (GNG), two variable low-pass filters (VLPF), and two balanced mixers (BM). The cutoff frequency of the low-pass filter is selected on the basis of the frequency fb and the assumed average speed of the mobile vehicle V. The output of the simulator represents the envelope and phase of a Rayleigh-fading signal.

(B) Software simulator. In a software-configured simulator [10], the

model described in Sec. 6.2 is based upon, and Eqs. (6.10) through

(6.13) are used to simulate, a Rayleigh-fading signal. In both Eg. (6.11)

and Eq. (6.12), there are N Gaussian random variables for Ri and Si.

Each of them has zero mean and variance one. Since a uniform angular

distribution is assumed for N incoming waves, we let φi = 2π i/N. To

simplify the simulator model, the direction α of the moving vehicle can be set to α = 0° without loss generosity, then the parameter ξi of the ith

incoming wave is

(7.107)

We have found that summing six or more incoming waves will perform

a Rayleigh fading signal. We use nine evenly angular distributed

waves, each of them is separated by 40° from the next wave arrival.

(7.108)

Let the parameter t in Eq. (7.107) be

t = k ⋅ Δt (7.109)

and Δt is the sample interval that can be set as small as 10 sampling

points in every wavelength.

V ⋅ Δt=λ/10 (7.110)

Now Eq. (6.11) and Eq. (6.12) become

(7.111)

(7.112)

The kth sample point of a Rayleigh envelope is:

(7.113)

Equation (7.113) is the simulated Rayleigh fading signal plotted in

Fig. 7.13 as the increment number k started from one. For our demonstration,a frequency of 850 MHz, the wavelength of which is 0.353 m,

is given, and the sampling interval is λ/10 = 0.0353 cm. Each second

will have m samples depending on the velocity of the mobile unit. The

following table lists the mobile speeds that correspond to the number

of samples per second.

For any different mobile speed, the Rayleigh fading curve shown in Fig.7.13 remains unchanged. Only the number of samples on the x-axis is rescaled according to the different mobile speed conditions indicated above.The application of the software fading simulator (fader) [19] is described as follows. Since the random FM generally does not have an impact on the received signal, only the Rayleigh fading is considered.The modulated signal at the baseband is first digitized at its data rate Rb and then sent through the software fading simulator. For those data bits N1 above the threshold level of the simulator (fader), no errors

occur. Those data bits N2 below the threshold will cause the errors depending on the states M of the modulation (M = 2L). The error probability for each bit is.

M/M-1

Assume that the total number of bits is N; if so, the BER (bit error rate)

can be obtained by

(7.114)

The speed of the vehicle should not have an impact on the BER (see

Sec. 16.14).We may verify this statement by using this simulator. The

simulator also can evaluate the WER (word error rate) or FER (frame

error rate) by grouping a sequence of bits called a frame or word. If FEC (forward error correction) can correct 2 errors in a frame, then the

frame has no error unless 3 or more errors occur. The simple signal

process can generate the FER through this software fader very easily.

Other similar software configuration simulations are shown by

Smith [20] and Arredondo [21]. The main advantage in a softwareconfigured

simulator is the ability to quickly change the operational

parameters by merely making changes in the software program.

Multipath- and selective-fading simulator

A simulator of multipath and selective fading can also be configured

either from hardware or a combination of hardware and software. The

hardware version uses an arrangement of components similar to those

shown in Fig. 7.12; the components are duplicated several times, and a

delay line is added to each assembly [22]. The number of assemblies

and delay lines is determined by the kind of environment that is being

studied. The several delayed output signals from the Rayleigh-fading

simulators are summed together to form a multipath- and selectivefading

signal.

The software version [13, 17] is based on a model developed by Turin

[17], in which the mobile-radio channel is represented as a linear filter

with a complex-valued impulse response expressed:

(7.101)

where δ(t − tk) is the delta function at time tk, and ψk and ak are the

phase and amplitude, respectively, of the kth wave arrival. The terms

ak, ψk, and tk are all random variables. During transmission of a radio

signal s(t), the channel response convolves s(t) with h(t). Figure 7.14

shows the mathematical simulation generated by the software for this

model [14]. The simulator enables evaluation of mobile-radio system

performance without the need for actual road testing, since it simulates

the propagation medium for mobile-radio transmissions.

Thank you

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