jakes model

1Model of the WirelessRadio Channel 1

IN THIS CHAPTER THE MOBILE WIRELESS RADIO CHANNEL

AND ONE IMPLEMENTATION OF THE CHANNEL IS DISCUSSED.

THE CHANNEL IS MODELED AS A FREQUENCY SELECTIVE

WSSUS CHANNEL DESCRIBED BY A DELAY-DOPPLER POWERSPECTRUM. THE CONCLUSION OF THIS CHAPTER IS A DIS-

CRETE TIME/FREQUENCY COVARIANCE FUNCTION FOR USEIN THE RECEIVER STRUCTURE.

In the design and implementation of a receiver for wireless communications, it isimportant to have an understanding of the wireless radio channel, since it influencesthe received signal significantly. The derived channel model in this study is alsoused for verification. However, since a model of the wireless radio channel is asimplification, it does not necessarily follow that a receiver derived with the mindon one specific channel model works properly, when it is tested using a real wirelessradio channel.

The channel is modeled with the parameters from the LTE downlink SISO OFDMsystem based on the signal model derived in Chap. ??. The carrier frequen- LTE CONSIDERATIONScies for downlink have been allocated in ten bands in the area 0.8 GHz 2.7 GHz[Dahlman et al.(2007), page 14]. A typical downlink is 4.5 MHz (5 MHz spectrumallocated), 15 kHz subcarrier spacing and based on the block sizes for the LTE down-link we chose 120 subcarriers and 56 OFDM symbols [Dahlman et al.(2007), page59]. Realistically, the environment plays an important role in the behavior of thechannel. The channel has different characteristics depending on if it is indoor, out-door, urban, hilly ect.. A generic channel model based on a specific delay-Dopplerpower spectrum is derived. The model chosen is power delay profile of the COST2071 typical urban model power delay profile and Jakes Doppler spectrum (also

1The COST 207 model is an average from extensive collected data from different locations in Europe.The model considers four different scenarios, each one characterized by a specific profile: typical urban

2 CHAPTER 1. MODEL OF THE WIRELESS RADIO CHANNEL

Mobile-station Base-station

Fig 1.1: NON-STATIC OUTDOOR URBAN ENVIRONMENT WITH MULTI-PATH FADING.

known as Clarkes spectrum). The advantage of this model is its simplicity and itsgenerality. The channel model is first described in continuos time. From this modela discrete time/frequency model is developed to describe the channel in each sub-carrier. The channels between base-stations are assumed uncorrelated, which is arealistic assumption since they are not normally in physical proximity.

1.1 An Uncorrelated Scattered Model of the Multi-path Channel

Because the radio waves propagate along different paths from the transmitter to thereceiver, the channel is modeled as a set of multi-paths with different characteris-tics. Different effects occur in the channel, which are (i) path loss, exponentiallydecaying with the distance the wave travelled, (ii) short-term fading, due to smallfluctuations in the channel caused by phase changes in each signal contribution,and (iii) shadow fading, which is due to physical changes in the channel. Only theshort-term fading is considered in this model. The statistical properties of short-term fading is approximated to Rayleigh fading. For fading with one strong com-

(TU), bad urban (BU), rural area (RU) and hilly terrain (HT).

1.1. AN UNCORRELATED SCATTERED MODEL OF THE MULTI-PATH CHANNEL 3

ponent (line of sight), Rician fading gives a better approximation2. As we onlyconsider the small scale fading, we assume that the stochastic process behind the thetime/frequency description of the channel is wide sense stationary (WSS), and that WSSUS CHANNELthe auto-correlation for the delay/Doppler function is zero in all instances, wherei , and i , , which means uncorrelated scattering (SU). The channel modelshowing these characteristics is thus described as the WSSUS channel.We describethe channel as a tap delay line model, given by a time-variant channel impulse re- TAP DELAY LINE MODELsponse [Tse and Viswanath(2005), page 41] and [Molisch(2005), page 121], whichusing a base-band representation is described as

h(t,) =Ntaps

i=1

i(t)( i(t)) =Ntaps

i=1

i(t)exp(j2pi fci(t))( i(t)),(1.1)

where i(t) is the ith tap and () is the Kronecker delta function, i(t) is the time-variant ith delay, fc is the carrier frequency, i(t) is the time-variant complex am-plitude of the ith tap and Ntaps is the number of taps in the model. The amplitude ofeach tap i(t) is modelled as a zero-mean complex Gaussian process, thus |h(t,)|is Rayleigh distributed.

From [Bello(1963)] we use the notation to express the description of the channelin the different domains and the connection between them, see Fig. 1.2. The time-variant transfer function is the Fourier transform of the impulse response (1.1) fromthe delay domain to the frequency domain f

H(t, f ) =Z

d h(t,)exp( j2pi f ).(1.2)

A statistical description of the channel is obtained by the correlation function intime and frequency, given by the auto-correlation function of H(t, f ) over time t andfrequency f [Molisch(2005)]

H(t, f )H(t + t, f + f ) f , t = RH(t, f ),(1.3)

where x denotes the expected value with respect to x and t and f is the shiftin time and frequency respectively. Notice, for the WSS channel the first momentis constant. The time correlation function and the frequency correlation function of

2Rician fading is a generalization of Rayleigh fading, where one strong component is present.


h(t,) Rh(t,)

H(t, f ) RH(t, f ) s(,) SH(,)

P(, f ) RP(, f )

WSS US

Fig 1.2: RELATIONSHIP BETWEEN CHANNEL FUNCTIONS IN THE WSSUS CHANNELMODEL. THE FIGURE SHOWS FOUR SYSTEM DESCRIPTIONS, AND THEIR SECOND OR-DER DESCRIPTIONS. THE SYSTEM DESCRIPTIONS ARE CONNECTED AS FOURIER PAIRS,SIMILARLY THIS APPLIES FOR THE SECOND ORDER DESCRIPTIONS. THE FOURIER PAIRSARE MARKED WITH , WHERE REPRESENT THE FOURIER TRANSFORMED. WE ONLYCONSIDER THE PARTS OF THE MODEL MARKED IN BLACK.

the channel are obtained from RH(t, f ) [Molisch(2005)] according to

RH(t,0) = RH(t)(1.4)RH(0, f ) = RH( f ).(1.5)

Averring across many channels given many different locations it is assumed that

RH(t, f ) = RH(t)RH( f ).(1.6)

The assumption is not true for one instantaneous channel, since the time and fre-quency correlation functions here cannot be assumed independent. Because of theUS assumption the second moment of the scattering function is

s(1,1)s(2,2), = SH(1,1)(1)( 1).(1.7)

The power spectrum is also obtained from the correlation function in (1.3), seeFig. 1.2, by the Fourier transform in the time direction and the inverse Fourier trans-form in the frequency direction of the correlation function [Durgin(2003)]

SH(,) =Z

d f exp( j2pi f )

Z

dt RH(t, f )exp( j2pit).(1.8)

1.2. SELECTED CHANNEL MODEL 5

Transmitter

Ring of scatters

Receiver

Fig 1.3: MODEL OF THE CHANNEL WITH NO LINE OF SIGHT BETWEEN TRANSMITTERAND RECEIVER. THE SCATTERS ARE EQUALLY DISTRIBUTED IN ALL DIRECTIONS ANDTHE RECEIVER IS MOVING. THE SIGNAL FROM ALL DIRECTIONS HAS ON AVERAGE THESAME POWER AND DELAY. THIS MODEL OF THE DOPPLER SPECTRUM IS GIVEN BY(1.12).

It follows from the factorization in (1.6) that SH(,) also factorize: SH(,) =SH()SH().

1.2 Selected Channel Model

In this section a channel model is chosen and described on the form SH()SH() andRH[n]RH[l], where in the latter expression RH[n] and RH[l] are discrete-timeand frequency correlation functions across subcarriers l and time interval betweenOFDM symbols n, respectively. For one subcarrier l in one OFDM symbol n, we as-sume a frequency flat and time constant channel. We will later see from the behaviorof the correlation function RH(t)RH( f ), that this assumption is reasonable. Theconsidered model for the Doppler domain with one transmitter and one receiver is DOPPLER SPECTRUMdepicted in Fig. 1.3. We assume that all scatters on average are distributed uniformlyon a circle around the receiver, with the same average power from all directions. In-voking the central limit theorem, we can assume that the power from one directionis complex Gaussian distributed. If we assume that the receiver moves with a certainspeed, the number of components coming from each direction will take the tub formof Jakes spectrum [Jakes(1974)]. For the non-spacial channel model, this is thusthe Doppler spectrum SH(), which specifies the average power from different fre-quency shifts over infinite many different implementations of the channel described


by the spectrum. Jakes spectrum is computed from the definition of the Dopplershift. The Doppler shift is caused by the velocity of the receiver

=fcc0

cos() Hz,(1.9)

where c0 = 3 108 ms is the speed of the impacting radio wave, and fc is the carrierfrequency3 and is the azimuth of arrival of the radio wave. The Doppler shiftsfrom and is the same, we thus limit the probability density function (pdf) top() = U (0,pi). Thus for each frequency in the Doppler dispersion is given

f = fc(

1 ||c0

cos())

,(1.10)

where = ||cos(). Transforming this uniform distribution into the Doppler do-main we get

p() =dd

U (0,pi).(1.11)If we consider the Doppler spectrum as the pdf in (1.11) and by realizing that themaximum Doppler shift max is caused by = 0 or = pi, we thus by simple differ-entiation and insertion find the average energy from each Doppler shift

SH() =

1pimax

1( max )

2 max max,

0 elsewhere,(1.12)

The spectrum can be interpreted such that infinite many contributions, which arevery small, come from 0o and 180o, which creates two singularities in max andmax4.

The power delay profile (pdp) SH() is generated by a set of delayed paths. ForPOWER DELAY PROFILEthe tap delay line model, each cluster has an exponential profile. The pdp is typicallydifferent depending on the environment. To simplify the model, SH() is approxi-mated by an exponential decaying function [Molisch(2005), page 121], the physicalexplanation of this phenomenon are still not solved, however this has been verifiedby extensive measurements campaign [Molisch(2005)]. For the indoor scenario,models explaining have been made [Steinbock(2008)], where it was shown that thisbehavior can be explained by reverberation effects in the environment. A simpledescription of the pdp profile is given as

3For LTE fc is in the range of 1 GHz.4This is obvious a purely theoretic phenomenon.


Doppler Frequency [Hz]

Del

ay[

s]

Pow

er[d

B]Power [dB]

4030 50

10

10

0

1

3

2

4

6

760

0

5

maxmax

Fig 1.4: THE DELAY-DOPPLER POWER SPECTRUM SH(,), WHICH CORRESPONDS TO ANAVERAGE OVER MANY CHANNEL REALIZATIONS. FOR ONE LOCALIZED CHANNEL THEDELAY-DOPPLER SPECTRUM CANNOT FACTORIZE ACCORDING SH(,) = SH()SH() BE-CAUSE THE DELAY AND THE DOPPLER FREQUENCY IS NOT INDEPENDENT. IN THE PLOTTHE DELAY-SPREAD IS = 1 s.

SH() =

1

exp(

)0

0 0 > ,(1.13)

where is the delay spread. The delay spread is defined as the square root of thesecond moment of the pdp. The delay-Doppler power spectrum SH(,) is then DELAY-DOPPLER POWER

SPECTRUMdefined from the disjoint specters (1.12) and (1.13) as

SH(,) =

exp( )

pimax

1( max )

2 0 and max max

0 else.(1.14)

The power delay profile (pdp) is depicted in Fig. 1.4. From the power spectrumSH(,), we find the autocorrelation function RH(t, f ). The problem is considereddisjoint, so first the autocorrelation function RH(t) is derived from the Dopplerspectrum (1.12) by the inverse Fourier transform [Tse and Viswanath(2005)]

RH(t) =Z

d SH()exp( j2pit).(1.15)

By solving the integral, we get the second moment of the time transfer function


Time [s]R H

(t)|[d

B]

5

10

15

20

25

30

0

0 0.050.040.030.020.010.05 0.04 0.03 0.02 0.01

Fig 1.5: THE AUTOCORRELATION FUNCTION RH(t) FOR A MAXIMUM DOPPLER FRE-QUENCY OF max = 92.6 Hz. WE DEFINE THE COHERENCE TIME AS THE TIME WHERETHE BESSEL FUNCTION HAS DECREASED FROM 1 TO 0.05 [TSE AND VISWANATH(2005),PAGE 39], IN THIS CASE Tc = 3.97 ms. THE COHERENCE BANDWIDTH IS PROPORTIONALTO THE MAXIMUM DOPPLER SHIFT.

RH(t) = J0(2pitmax),(1.16)

where J0() is the Bessel function of the zeroth order, given by

J0(2pitmax) =1pi

Z pi0

d exp( j2pitmax cos) .(1.17)

The auto-correlation function is depicted in Fig. 1.5. The discrete function RH[n]is obtained letting Nsn = t, where Ns is the time between two OFDM symbols,CORRELATION FUNCTION

OVER OFDM SYMBOLS thus

RH[n] = J0(2piNsnmax).(1.18)

By the Fourier transform of the power delay profile SH(), we find the time correla-tion function RH( f ) of the channel as

RH( f ) = 1

Z

0d exp

(

)exp( j2pi f )

RH( f ) = 11 + j2pi f ,(1.19)

which has the characteristic of a low-pass filter, see Fig. 1.6. The width of the low-pass filter is proportional to the delay spread . The auto-correlation function forCORRELATION FUNCTION

OVER SUBCARRIERS the subcarriers l is thus a discrete function given by

RH[l] =1

1 + j2piLspl ,(1.20)

where Lsp is the subcarrier spacing. The time/frequency correlation function is

RH[n,l] =1

pi(1 + j2piLspl

) Z pi0

d exp( j2piNsnmax cos) .(1.21)


Frequency [MHz]

|[R H

(f)]

|[d

B]

13 24

0

05 1 32 4 5

5

10

15

Fig 1.6: THE AUTOCORRELATION FUNCTION RH( f ) WITH A DELAY SPREAD = 6 s.THE COHERENCE BANDWIDTH IS Bc = 0.2 MHz.

Thus in the selected channel model we end up with a description, where the channelis characterized by two parameters; the delay spread and the maximum Dopplerfrequency max, notice that this is not equal to the Doppler spread. For the COST207 typical urban channel model (which has the considered exponential profile) thedelay spread is approx. 1 s (30 dB on 7 s, [Molisch(2005)]). The approxi-mate coherence bandwidth is thus Bc 15 = 0.2 MHz [Durgin(2003)], see alsoFig. 1.6. For the general case a linear relationship exist between the delay spreadand the coherence bandwidth, from [Fleury(1996)] the uncertainty relation betweenthe coherence bandwidth and the delay spread was stated.

Bc &1

2pi.(1.22)

Over the coherence bandwidth the channel is assumed constant5. Thus across a smallnumber of neighboring subcarriers with a spacing of 15 kHz, the channel frequencyresponse can still be assumed flat. However over the whole bandwidth of 1.8 MHzthe channel is frequency selective.

5The coherence bandwidth is considered here as a 3 dB limit.

11

References

[Bello(1963)] Bello, P. (1963). Characterization of Randomly Time-Variant LinearChannels. IEEE Transactions on Communications, 11(4), 360393.

[Dahlman et al.(2007)] Dahlman, E., Parkvall, S., Skold, J., and Beming, P. (2007).3G Evolution: HSPA and LTE for Mobile Broadband. Elsevier.

[Durgin(2003)] Durgin, G. D. (2003). Space-Time Wireless Channels. PrenticeHall, first edition.

[Fleury(1996)] Fleury, B. (1996). An uncertainty relation for WSS processes and itsapplication to WSSUS systems. IEEE Transactions on Communications, 44(12),16321634.

[Jakes(1974)] Jakes, W., editor (1974). Microwave Mobile Communications. WileyInterscience.

[Molisch(2005)] Molisch, A. (2005). Wireless Communications. Wiley, secondedition.

[Steinbock(2008)] Steinbock, G. (2008). Modeling of the Specular and DiffuseMultipath Components of the Indoor Radio Channel. Master thesis, AalborgUniversity.

[Tse and Viswanath(2005)] Tse, D. and Viswanath, P. (2005). Fundamentals ofWireless Communication. Cambridge university press.

Model of the Wireless Radio Channel[-0.4cm]An Uncorrelated Scattered Model of the Multi-path ChannelSelected Channel Model

References

jakes model

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