Abstract - The combination of orthogonal frequency division multiplexing (OFDM) signal processing and multiple-input multiple-output (MIMO) is regarded as a promising solution for enhancing the data rates of next-generation wireless communication systems operating in frequency- selective fading environments. This alteration of parameters is based on the active monitoring of several factors in the external and internal radio environment, such as radio frequency spectrum, user behavior and network state. It may be tempting to suggest that the solution of this problem lies in simply increasing the transmission power level of each subserviced transmitter. However, increasing the transmission power level of any one transmitter has the undesirable effect of also increasing the level of interference to which the receivers of all the other transmitters are subjected. In this paper we have explained and simulated several fading channels such as Nakagami fading, Ricean fading and Rayleigh fading channel. Also we have compared path gain level in the Rayleigh and Nakagami fading channels. The results show that the consumed power in Nakagami sub-channels is less than consumed power in Rayleigh sub-channels. Keywords: MIMO, OFDM, Nakagami, Rayleigh, Fading channels

I. INTRODUCTION A wireless network consists of a group of nodes that communicate with each other over a wireless medium of propagation. When the channel between two nodes source: Signal Radio Control (SRC) and destination: Destination host (DST) of such a network is in a deep shadow-fading state, increasing the transmit power by the SRC can be too power consuming and result in interference for other co-channel receivers. In this situation a Collaborating node (CN) may be used to relay the packets between the SRC and the DST [1]. We consider a protocol in which the collaborating node is selected from a number of nearby nodes to the SRC, such that it has the host propagation channel to the DST. Recently several authors have considered the problem of imperfect channel state information and its effect on bit and power allocation we show that when an extra node becomes involved in transmitting the power, the total average power consumed in the collaborative network is lower than that of a standard wireless network in the Rayleigh channel. We consider two models for the channels in our analysis; i) Rayleigh fading channels, ii) Nakagami fading channels ,shadowing channels. Let & PR NP be the power of the SRC in the direct channel (single-hop) of a standard wireless network, respectively, of the CN and the SRC in the collaborative network, for the two systems to yield the same received power at the DST. Also MIMO technology has attracted attention in Wireless communications [2, 3] since it offers significant increase in data throughput and link range without additional bandwidth or transmit power. The advantage of MIMO systems has raised prospects for achieving large increases in system throughput. The theoretical performance gains are very large, but considerable

work remains to investigate the performance of MIMO systems in practical scenarios and to find receiver structures which offer an acceptable trade-off between complexity and performance [4]. The main challenge in developing reliable high data rate mobile communications systems is to overcome the detrimental effects of frequency-selective fading in mobile communications channels. These previously proposed OFDM transmitter diversity systems all require a cyclic prefix to be added to the transmitted symbols to avoid ISI and inter channel interference (ICI) in the OFDM symbols, and the number of cyclic prefix symbols has to be equal to or greater than the order of the wireless channels [4]. So combining of these two schemes offers immunity to ISI and high capacity. STBC-OFDM assumes that channel gain is constant during the n OFDM symbols in TXn transmit antennas system. It has weakness under time selective channels; which can be regarded as an advantage under frequency selective channels. The remainder of this paper is organized as follows; channel model is presented in section two, Fading Channel Rayleigh and Nakagami are illustrated in section three, followed by impulse response multi-path fading channel details in section four .The section five of our paper describes the path loss of the channels such as Rayleigh and Nakagami, the section six is dedicated to the simulation results and finally in section seven our contributions are concluded.

II. CHANNEL MODEL In this section we consider a MIMO channel. A point-to-point communication system of TN transmitter and RN receiver antennas is shown in Figure 1.

Figure 1.point-to-point communication system of TN transmitter and RN receiver antennas

This system can be represented by following discrete time

Comparison of the Rayleigh and Nakagami Fading Channels MIMO Multicarrier System

Vahid Fotohabady, Fatin Said, Centre for Telecommunications Research, King's College, Strand London, WC2R 2LS, UK,

[vahid.fotohabadi, fatin.said,]@kcl.ac.uk

2011 Wireless Advanced

978-1-4577-0109-2/11/$26.00 ©2011 IEEE 295

model

11 11 1 1 (1)

h hy x nNt

y h h x nNr Nr NrNt Nt Mr

= +…

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

Or simply as y H x n= + , here x represents the Nt dimensional

transmitted symbol, n is the N R dimensional noise vector, and H

is the t rN N matrix of channel gains. 11h is the gain from transmit

antenna 1 to receive antenna1. We assume a channel bandwidth of B and complex Gaussian noise with zero mean and covariance

matrix 2 In Nrσ , where typically 20N Bnσ = . For simplicity, given

a transmit power constraint P we assume an equivalent model

with a noise power of unity and transmit power 2/P nσ ρ= ,

where ρ can be interpreted as the average SNR per receiver antenna under unity channel gain. This power constraint implies that the input symbols satisfy [5];

*[ ]

1

N tE x xi ii

ρ=∑=

. (2)

In the system design, we considered both discrete Rayleigh and Nakagami fast fading MIMO channel models, where TN

and RN are the number of transmit and receive antennas respectively. In the transmitter, a data stream is demultiplexed into TN independent sub streams. Each sub stream is encoded into transmit symbols using a modulation scheme (e.g. BPSK, QPSK, M-QAM, etc.) at symbol rate 1/T symbol/sec with synchronized symbol timing [4, 5]. The Rayleigh flat fading model is described here. The baseband

RN dimensional received signal vector

. ]( ) [ ( ), ( ), ... ( ) 1 2Tr K r K r K rN KR= (3)

at sampling instant k may be expressed as

( ) . ( ) ( ) r K H x k n K= + (4) where, the formula is too crunched

11 1 ( ) ( )1 1 ,. ( ) ( ) , ( ) .(5)

( ) ( )1

h hN r K n KTH r K X K nK

r K n Kh h NR NRN NN rr T

= = = =

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

( ) [ ( ), ( ), ..., ( ) (6)1 2TX K X K X K XN KT=

The transmit symbol vector with equally distributed transmitted power across the transmitted signal. Here, the superscript T is transposition. H denotes the T RN N channel matrix, whose

elements hm n at the t hm row and nth column is the channel

gain from the thm transmit antenna to the thn receive antenna, and they are assumed to be independent and identically distributed (I.I.D)circularly symmetric complex Gaussian random variables with zero-mean and unit-variance, having uniformly distributed phase and Rayleigh or Nakagami distributed magnitude. A circularly symmetric complex Gaussian random variable is a

random variable a jb c+ ≈ ’2(0, )σ ), in which a and b

are I.I.D real Gaussian distributed as 2(0, / 2)σ . A commonly used channel model in MIMO wireless communication systems is a block fading (also known as quasi-static) channel model where the channel matrix elements, which are I.I.D complex Gaussian (Rayleigh or Nakagami fading) random variables, are constant over a block and change independently from block to block. We drop the index k for the channel gain. The elements of the additive noise vector

( ) [ ( ), ( ),..., ( )] (7)1 2Tn K n K n K nN KT=

are assumed to be also white I.I.D complex Gaussian random variables with zero-mean and unit-variance. From this normalization of noise power and channel loss, the averaged transmitted power which is equal to the average SNR at each receiver antenna is to be no greater than TN [4, 5]. Now we consider the Nakagami fading model for the 2 transmitter and 2 receiver cases which we use in our simulations. The generalization to more than 2 transmitter or receiver antennas is straightforward. The Ricean channel model is given by [6]

11 12

21 2211 12 21 22

1 11 1 1 1

(8)j j

j jF Vh he eK KH Hh he eK K K K

Hθ θ

θ θ= + = ++ + + +

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

where,

11 12 (9 )21 22

j je eH F j je e

θ θ

θ θ=⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

is a fixed matrix (i.e. fixed for each block of a block fading channel) consisting of phase elements corresponding to line-of-sight (LOS) component and is the Nakagami or Rayleigh flat-fading matrix as described before, for instance the matrix element

ijh denotes the Nakagami flat fading channel coefficient from

transmitter i to transmitter j. K is the K-factor of the Nakagami distribution which is proportional to the strength of the LOS component. The Nakagami model was suggested before as a first order approximation to the Nakagami channel. We have decided to use this model in our simulations because under certain reasonable assumptions it gives an accurate approximation to the Nakagami channel. When using this model the following assumptions are valid: (i) the distance between the antennas is sufficiently large; (ii) the subscriber unit is mobile and possibly changing orientations.

With respect to the base station, under these assumptions it is reasonable to state that the phases of the LOS component arriving at the different receiver antennas are random and uncorrelated.

In this work the multipath delay and fading are generated based on the Jake’s fading model. Using this model we composed the

296

channel impulse response of complex samples using uniformly distributed Rice and Nakagami fading channels. The channel impulse response can be described as:

2 2(0, ) (0, ), (10)

2 2k kh N jNk

σ σ= +

where, 2 2 exp(- / )02 1 exp( / )0

kT TrmsSk

T TrmsS

σ σ

σ

=

= − − (11)

III. FADING CHANNEL RAYLEIGH AND NAKAGAMI:

FADING CHANNEL RAYLEIGH, Rayleigh channel has probability density function with the

Rayleigh distribution in formula (12) 2

2

2

2

( ) (0 )

0 (r 0)

r

rr

e rPσ

σ

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

⎧⎪ ≤ ≤ ∞= ⎨⎪ <⎩

(12)

In (12) 2σ is average power when the received signal before the detection, σ is root mean square (rms) of the value of the received signal before the detection and r is amplitude cover of the received signal. [4] The Rayleigh distribution probability density function is shown in Figure 2,

r Figure 2.Rayleigh distribution probability density function FADING CHANNEL NAKAGAMI

In this section the M fading Nakagami channel is described. Phase of the unknown channel is considered. Fading Nakagami model is best fit model for multi-path channels in urban area [4].

(13)

In M fading Nakagami channel when m is equal to 1, the system becomes equivalent to Rayleigh model. The parameter Ω can also be regarded as the average power fading.

X

Figure 3. Probability density function-Nakagami

Figure (3) shows the distribution probability density function Nakagami for m = 1.5. Formula (13) and finally Figure 4 shows the Comparison Probability Density Function- for Rayleigh and Nakagami channel.

Figure 4.Nakagami probability density function (-) k=1/2,

(--) k=1(Rayleigh), (-.-) k=2, (o) k=3

IV- IMPULSE RESPONSE MULTI-PATH FADING CHANNEL The impulse response of multi path channel with complex low pass filter is [7]

22 1

( / )

22

2 2

2( )( )

1[ ],[ ] 2

me mm r

mm rf r e

m

E R mE R

−− Ω=

Γ ΩΩΩ = = ≥

− Ω

297

Where ( ) 2 ( )k c kt f tθ π τ≅ shows domain and path delay time

in thk channel, ( ) ( )k kt and tα τ are path phase and time

delay in the thk channel. In the communication systems where cf are very large, a little delay causes a great phase rotation. In each path, the signal with the local instruments scattered around the reflective approach and makes it appear to the recipient. Since this scatters are random process, so amplitude and phase of each path can be supposed as a randomized variable and therefore the impulse response channel ( , )h t τ is also considered as a random process. Fading random process of channel characteristics is important to determine channel treated [8]. A conventional method in estimating the channel randomly supposing scatters number is high therefore could be using the central limit theorem. This process for some complex Gaussian is obtained channels impulse response. As mentioned if there is a direct path between sender and receiver there is no channel amplitude impulse response with Rayleigh distribution, and the range of distribution channels with Rice will be in phase with both uniformly distributed in the interval (0, 2 π) is. Assuming arrest Wide Sense Stationary (WSS) of channel process, the correlation function as its second equation (15) is defined

*1 2 1 2

1( , , ) [ ( , ) ( , )] (15)2h t E h t h t tφ τ τ τ τΔ = +Δ

Let assume the distribution for 0tΔ = .The Wide Sense Stationary un-correlated Scattering (WSSUS) is a model for multi-channel path .Here lose of the power delay spectrum or the intensity profile is called multi-path channel as shown in (16)

(3.9)

2( ) ( )h k k

kφ τ α δ τ τ= −∑

(16)

This function, in terms of average power output channel delay determines τ. Find the limit of τ Opposed to its value is zero range delay and says a few channel path average values, maximum and rms it is important. Channel parameters using profile are identified include [9]:

0p : Normalized power received to the ratio of power that can be sent, K: Rican coefficient of line of said rmsτ : The amount of delay range multi path, α: profit power path, that using the following relationship is reached:

22 2

0

m = 1 , 2mi i

rms mi Pτ ατ τ τ τ= − =∑

2,max2

0 2 0 ,max

k= (17)k

kk k

PP

αα

α=

−∑

Normally the first signal with zero delay at the receiving point is considered as a maximum range. Delay profiler for a

conventional radio channels are used to shape can form of exponential decrease equation (18) described

( ) k

rmsh k Ce

ττϕ τ

−= (18)

Let assume the channel sample period ( )sτ is in discrete time.

The above equation can be rewritten as shown in equation (19)

( ) k

rm sh k k

sC e k

ττ τϕ τ τ τ

−= = (19)

where the delays kτ uniformly distributed on the channel sample period. Delay range can cause symbol interference such that, digital symbols become adjacent to the delay . The different paths have been interference overlaid and are over lapping. Number of symbols found in a single system single carrier interference is the following [10]

max, Carrier (20)ISI Single

s

NT

τ⎡ ⎤= ⎢ ⎥⎣ ⎦

where sT is the symbol period. Since is The value of sT is very small in a very high rate, the effect of ISI increase. Thus the complexities of implementation mechanisms are compensation, to achieve optimal performance as a significant increase. If the symbol period delay channel posts bigger than maximum of range ( maxτ ), the Channel ISI can create less. In the multicarrier posts method, each period symbol with the number of carriers is increased therefore the ISI is reduced. Thus the number of interfering symbols In the multicarrier systems is as follows

m ax, C arrier (21)ISI Single

C s

NN Tτ⎡ ⎤

= ⎢ ⎥⎣ ⎦

A little remaining ISI can be removed by using guard interval (GI)[11].

V- CELLULAR LAYOUT PATH LOSS OF THE CHANNELS SUCH AS RAYLEIGH AND NAKAGAMI

A wireless communications network simulation scenario with 7 cells has been considered. This scenario is shown in Figure 5. In this arrangement, a central cell and six hexagonal cells around it are assumed. In the central of any cells there is a base station and also in any cells there is a mobile node as well. The distance between mobile node and base station in each cell are different. The radius of each cell is assumed to be 500 meters and also distance between any two adjacent base stations is set to 2 km. The frequency channels in adjacent cells are assumed to be different and non neighbors cells are assumed as co channel. Channels fading are set as Rayleigh and Nakagami. Gain of each mobile node can be calculated as following: 40.0097 /i ig d= .where the id is distance between mobile node and its base station. We also indicate all base stations in the system by a unique index

(14) ( )( , ) ( ) ( ( ))kj tk k

kh t t e tθτ α δ τ τ−= −∑

298

{0,..., 1}b B∈ − and all users in the system by a unique

index {0,..., 1}K K∈ − . The cellular layout spans the interval

[0 B] in the real line, where one base station is located at the spatial coordinate b+1/2 and user k is located at a point belonging to a regular uniform grid of points such that the number of users per cell is K B for all cells and users are symmetrically located in each cell. The channel-path gain for a transmitter-receiver pair located at , [0, ]U V B∈ is given by

0( , ) (22)1 ( , ) / )B

Gg u v

d u v αδ=

+

where we define the modulo distance

( , ) m in z (23)B E zd u v u v B= − +Z

This has the advantage of eliminating the boundary effects.

The parameters 0G , and in (22) denote the SNR at the

cell centre, the path-loss exponent and the loss distance. The function (22) can approximate path loss models used where

gains are 1 / dαα but are clipped to a maximum value for d below some minimum distance [12]. All systems operate under strict per base station power constraints, where

[ ] 0pb f ≥ is the maximum power that station b can use on

subchannel F. The powers are relative to the SNR given by (22), assuming a normalized noise power spectral density

equal to 1, where 1

0

1 [ ] 1F

fpb f

F−

==∑ .

Figure 5.Wireless communications network simulation scenario. VI- SIMULATION RESULTS

Figure 6 illustrates the simulation results for m=1 or Rayleigh fading channel. We assume that the fading type is non-frequency selective. Background noise for all users is

0

152 *10N −= considered. According to the previous formula, if

the distance between users with their base station in each cell is 500meters, the power path gain will be a minimum at

around9

(0.00016 * 10 )−

. Meanwhile, if this distance is 100

meters, the power path gain will be s around 9(0.097*10 )− . In

Figure 6 and 7 it can be seen that the mobile node in cell 2 has the path gain level of Profile can route one path gain better than the rest of users and user index than other users, the farthest distance of the station per cell itself and thus losses and more profile can be a less gain path power. Therefore by game theory we can adjusting and optimize the resource allocation in the system.

Figure 6. Path Gain of each subchannel in Channel fading

Rayleigh Based on the model explained in section III and IV, Figure 7 shows simulation results of fading Nakagami channels with different m values. If m=1, Nakagami fading channel becomes Rayleigh fading channel as stated previously. For any other value of m, the performance of Nakagami sub-channels is better than Rayleigh sub-channels. Also according to the simulations performed for M =2, consumed power in Nakagami sub-channels is less than consumed power in Rayleigh sub-channels. This means that the overall power consumption is reduced. The other hand in the equal condition, user can send value data rate by fading channel Nakagami equal the fading channel Rayleigh but the overall power consumption is reduced.

299

Figure 7. Path Gain of each subchannel in Channel fading Nakagami

M=2 VII- CONCLUSION

For spatially uncorrelated MIMO systems, transmitter diversity is one of the most effective techniques, in particular, for OFDM based systems. We explained and simulated several fading channels such as Nakagami fading, Ricean fading and Rayleigh fading channel. We also compared path gain level in the Rayleigh and Nakagami fading channels. The simulation is performed for both M=1 and M=2, and the results showed promising overall power consumption reduction. It can be concluded that the consumed power in Nakagami sub-channels is less than consumed power in Rayleigh sub-channels. Fading Nakagami model is the best fit model for multi-path channels in urban area. Therefore the employment of these transmission plus using the OFDM systems, “additional diversity gain” the measured delay and fading caused by multipath and Nakagami fading channels, improve system performance by decreasing the consumed power.

ACKNOWLEDGEMENT

The authors would like to thank Dr. R.Delmaghani for giving constructive feedbacks and would also like to thank Dr H. Bovarshad for his suggestions.

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