Homework #5

1. Write a function generating the Jakes time-varying flat Rayleigh fading channel coef-

ficients assuming bit rate Rb = 12.2Kbps, vehicle velocity v = 120Km/h, and carrier

frequency f0 = 2.115GHz. When sampling the process, start at time t = 500Tb.

(a). Plot the gain (dB) of channel coefficients over 2000 bit interval.

(b). Evaluate E{|c(t)|2}.(c). Numerically evaluate E{c(t)c∗(t + ∆t)} and plot it as a function of ∆t.

(d). Plot J0(2πfd∆t) as a function of ∆t.

Note:

• You should write a Matlab function (we need to use it later). Pass v, fc, Tb to

the function. The function should return a vector, which contains the channel

coefficients corresponding to each bit interval. We have to assume that channel

is constant over one bit interval.

• Assume the maximum Doppler shift

• J0(·) is the zeroth-order Bessel function of the first kind

• To generate J0(2πfd∆t) in Matlab, you may use function “besselj(0, 2πfd∆t)”.

• After you get the function working properly, change velocity v and observe the

channel variations for each case.

1

ECE669 52

Wireless channels (continued)ECE669 53

Wireless channels (continued)

Fading channel simulators

• Flat (frequency nonselective) Rayleigh channels:

� Fading channel process c(t) is modeled as anormalized, zero-mean complex-valuedwide-sense stationary Gaussian process.

ECE669 54

Wireless channels (continued)

� Spaced-time correlation function Φ(∆t)

Φ(∆t) = E{c(t)c∗(t + ∆t)},

∗ E{·}: statistical expectation.∗ ∗: complex conjugate.∗ Φ(0) = 1: due to the normalization of the channel.

ECE669 55

Wireless channels (continued)

• Continue time-varying function c(t) is usuallyapproximated by piece-wise constants.

• How channel changes from one interval to the otheris determined by the spaced-time correlationfunction.

ECE669 56

Wireless channels (continued)

• Jakes’ model:

Φ(∆t) = J0(2πfd∆t)

� fd: the maximum Doppler shift of the channel.� J0(·): the zeroth order Bessel function of the first

kind.

ECE669 57

Wireless channels (continued)

• The channel fading process whose spaced-timecorrelation is modeled by Jakes’ model can begenerated by

c(t) =C0√

2S0 + 1[Xc(t) + jXs(t)]

Xc(t) = 2S0∑

n=1

[cos(φn)cos(ωnt)] +√

2cos(φN)cos(ωmt)

Xs(t) = 2S0∑

n=1

[sin(φn)cos(ωnt)] +√

2sin(φN)cos(ωmt)

ECE669 58

Wireless channels (continued)

� S0: the total number of sinusoids (Jakes suggeststhat S0 = 8 sinusoids will give a pretty goodapproximation)

� Other parameters:

ωm = 2πfd,

ωn = ωmcos(2πn/S).

ECE669 59

Wireless channels (continued)

• Relationship between S and S0:

� S = 2(2S0 + 1) (we use S0 = 16 sinusoids for oursimulations).

� φN = 0, φn = πn/(S0 + 1).� C0 = 1 for normalization.

• The corresponding statistics of Xc(t), Xs(t) and c(t):

E{X2c (t)} = S0,

E{X2s (t)} = S0 + 1,

E{|c(t)|2} = C0 = 1.

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In Section II.D, we also use the same analysis result to study several

important special cases to provide insights and explore connections with well known

existing results. The following special cases are considered: quasi-static Rayleigh

fading channels, fast Rician fading channels and fully correlated fast Rayleigh fading

channels. In particular, the role of channel estimation errors, antenna correlation

and user signal correlation on detector performance are quanti�ed. In addition,

we quantify the e�ect of the number of training symbols on the performance of

our MAP receiver with a Kalman �lter. Finally, we compare our MAP receiver

with two other adaptive receivers, the adaptive channel predictor and the adaptive

MMSE combiner.

Finally, in Section II.E, we brie y summarize the results.

II.A Signal Model

In this section, we �rst discuss in detail how the fading process is mod-

eled and generated in simulation. Then we establish the digital signal model used

throughout this chapter. We end this section with a short discussion on the inter-

ference and a summary of the notations used.

II.A.1 Fading Processes

Let x(t) be the complex fading process for the desired user. Experiments

indicate that the complex fading coe�cient x(t) is a random quantity that changes

slowly over time [64]. So the mathematical nature of x(t) is a narrowband random

process which has correlation over time. In the case of Rayleigh fading, x(t) is

a complex Gaussian narrowband process, which can be modeled as the output of

a low pass �lter excited by temporally white complex Gaussian noise. The low

pass �lter is often referred as the shaping �lter, because it determines the power

spectrum shape and the temporal correlation function of the fading process. In

the most widely used Jakes' model, x(t) is assumed to have the following temporal

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correlation function,

Efx(t)xH(t� �)g = J0(2�Fd�); (II.1)

where J0(?) is the zeroth Bessel function of the �rst kind and Fd is the physical

Doppler frequency 2. The corresponding power spectrum isX(f) =1q

1� ( f

Fd)2; jf j �

Fd. In current systems, typical Fd ranges from 5Hz to 300Hz, depending on the

speci�c situation. For example, for a carrier frequency fc of 2GHz and a mobile

speed v of 30 m/sec (68 mile/hour), Fd =vfc

c=

30� 2� 109

3� 108= 200Hz. If Fd is

bigger than 100Hz, it is often referred to as \fast fading". A common method to

generate Jakes' x(t) is to sum up several sinusoids [66, 67], as �rst suggested by

Jakes [64]. However, this method in fact generates a deterministic process [68] and

the temporal correlation property of such a process is brought into question in [69].

In the following, we will discuss how to generate the fading process for simulation

with a digital receiver.

In the digital receiver we still use parameters in the above example. Now

suppose that the symbol duration is limited to Ts. If the baud rate Rbaud(=1

Ts)

is 40k/per second, then the fading rate normalized to data rate is fd =Fd

Rbaud

=

200

40� 103=

1

200. Roughly speaking, the channel does not change much over 200

symbols. Since the fading is so slow at the symbol level, we can often neglect the

change of the fading process over one symbol duration Ts and assume that the

fading process remains constant over a symbol, i.e., x(t) = x(nTs)�= xn, for nTs �

t < (n + 1)Ts. According to Eq.(II.1), the correlation function is EfxnxHn�mg =

J0(2�fdm), where fd =Fd

Rbaud

= FdTs is the normalized Doppler frequency. The

power spectrum of xn is x(f) =1q

1� ( ffd)2; jf j � fd.

To generate xn, a straightforward way is to pass white complex gaussian

2In fact, Doppler frequency alone does not account for all the dynamics of the channel. For example,

�xed wireless links also slowly change over time due to the movement of nearby re ectors such as tree

leaves and pedestrians [65].

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noise un through the following �lter,

h(f) =qx(f) =

1

4

s1� (

f

fd)2; jf j � fd; (II.2)

which is the square root of the power spectrum of xn. This �lter is highly nonlinear

and approximation has to be sought. Since the �lter in Eq.(II.2) has in�nite impulse

response, it is natural to try to �nd an IIR �lter to approximate this �lter. In [70],

a third order AR �lter is shown to �t the spectrum above accurately. More recently,

32 order IIR �lters have been developed in the wireless industry to generate the

fading process [71]. Throughout this thesis, we will use a general AR �lter to

approximate an arbitrary shaping �lter, i.e., our shaping �lter is

h(f) =el(�j2�f)

1�l�1Xi=0

�ie(l�i)(�j2�f)

; (II.3)

where (l � 1) is the order of the �lter and �i; i = 0; : : : ; l � 1 are the coe�cients of

the �lter. In the temporal domain, the �lter is

xn+l =l�1Xi=0

�ixn+i + un; (II.4)

where un is a complex white Gaussian noise process. One advantage of this approach

is that �eld data can be �t into this parametric model. In fact, the correlation in

Eq.(II.1) is derived from mathematical models rather than measured from real

data. It is well possible that di�erent correlation functions may arise when the

re ecting environment deviates from the assumed mathematical model. In order

to get optimal tracking performance, the receiver requires knowledge of temporal

variation characteristics of the channel such as Eq.(II.2) or Eq.(II.3). Therefore,

modeling of the channel is of much interest and an active research area (see [72] and

references therein). To retain the exibility of the analysis and to accommodate

various channel models, we will assume the order and coe�cients of the �lter to be

known. In simulations in this thesis, the temporal correlation function is speci�ed

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using Jakes' model as shown by Eq.(II.1). Then the coe�cients of the shaping �lter

are calculated using the Yule-Walker equation [27, 73].

Stacking Eq.(II.4) for m antennas in vector form, we have the following

model to characterize the dynamics of the vector channel response,

Xn+l =l�1Xi=0

FiXn+i +Un;

where Fi; i = 0; : : : ; l�1 arem�m diagonal matrices whose diagonal entries are the

coe�cients of m shaping �lters of order (l� 1). Un is a temporally white Gaussian

noise vector, whose elements are input to m shaping �lters. Q1 = EfUnUHn g can

be non-diagonal, corresponding to correlated fading channels. We point out that in

its most general form, the spatial-temporal correlation function is a function of both

temporal interval � and distance d, i.e., Efx(t; s)xH(t��; s�dg = R(�; d). However,

a common engineering practice is to assume that this correlation function can be

decoupled as the product of the temporal correlation and the spatial correlation [74,

75]. That is, R(�; d) = R(�)R(d). Though not necessarily true, this assumption can

reduce the complexity of the problem in many cases. The model above also implies

this assumption. Finally, we note that all the correlation functions involved are the

correlation of the complex fading coe�cients, not that of the fading amplitude. For

transformation between these two quantities, see [76].

Note that this model can accommodate even more general scenarios than

mentioned above. For example, by making Fi time dependent, i.e., Fi(n), we can

have a time-varying model. Another possible extension is that diagonal elements

in Fi are not necessarily identical, so the e�ect of possible di�erent fading modes

can be incorporated.

II.A.2 Vector Signal Model

Suppose that we use �(t) as waveform for a BPSK signal of interest, where

�(t) has support over [0,T]. Furthermore,R T0 �2(t)dt = 1.

We assume Rayleigh at fading channels. So the received signal at a single

9

Rayleigh Faded Carrier Envelope

Raylei gh Faded Ca rrie r Enve lope (dB)fd = 5Hz

-30

-25

-20

-15

-10

-5

0

5

10

0.0 1.0 2.0 3.0 4.0 5.0Time (s)

Leve

l abo

ve m

ean

(dB)

Rayleigh Faded Carrier Phasefd = 5Hz

0.00.51.01.52.02.53.03.54.04.55.05.56.06.5

0.00 1.00 2.00 3.00 4.00 5.00

Time ( s)

Phas

e An

gle

(rad

)

The envelope of a received multi-path signal will typically have a Rayleigh distribution

Typical time varying amplitude and phase of a Rayleigh faded channel are shown in the two graphs

Doppler Frequency (Fd) refers to Doppler shifts of different components of the receive signal, if either the transmitter, receiver or reflecting objects are in motion

10

Rayleigh Fading Simulator (Jakes' Model)

Weighted sum of 9 sinusoidsComplex output form to model amplitude and phase changeImplemented as ROM look-up and scaling factor multiplierFrequency adjusted using Phase AccumulatorDoppler frequency (Fd) selected at 5Hz (worst case) for indoor channel modeling