Pre- and Post-DFT Combining Space Diversity Receiver for Wideband Multi-Carrier Systems

Pre-and Post-DFT Combining Space Diversity Receiver for Wideband Multi-Carrier Systems

Muhammad Imadur Rahman,Suvra Sekhar Das,Frank H.P.Fitzek,Ramjee Prasad

Center for TeleInFrastruktur(CTiF),Aalborg University,Denmark

e-mail:{imr,ssd,ff,prasad}@kom.aau.dk;ph:+4596358688

Abstract—The basic receive space diversity combining methods,i.e. Maximum Ratio Combining(MRC),Equal Gain Combining(EGC)and Selection Combining(SC),are studied in this work for Pre-and Post-DFT combining wideband multi-carrier receivers.A common system model for both these schemes are devised.The outage and mean capacity and Frame Error Rate(FER)simulations show that Pre-DFT combining techniques performs quite comparatively in indoor scenario,while Post-DFT combining performs far better in outdoor https://www.360docs.net/doc/e76019776.html,pared to Pre-DFT combining schemes in indoor scenario,Post-DFT MRC and SC obtain an increment of1.4and0.9bps/Hz respectively in terms of 10%outage capacity.At10?3of FER,Post-DFT MRC obtains a gain 2dB and4dB in required Signal to Noise Ratio(SNR)for indoor and outdoor scenario respectively.In contrary to performance degradation, Pre-DFT schemes require much less computational complexities and they are simpler to implement compared to Post-DFT schemes.

I.I NTRODUCTION

One well-known way of improving the system performance in hostile wireless channel scenario is exploiting the spatial diversity using multiple receive antennas.Traditionally receive diversity is used in the Base Stations(BSs)to obtain full diversity at the BS receiver.In receive diversity systems,multiple copies of originally single transmitted signal is received via separate antennas.Given that the antennas are suf?ciently separated in space such that the receive signals do not experience any signi?cant spatial correlation, we can easily say that full diversity advantages can be achieved,i.e. the diversity order will be equal to the number of receive branches. This can also be achieved by using transmit diversity schemes,where multiple copies of transmitted signals are received at one single receive antenna simultaneously and full diversity advantages are achieved.The only difference between transmit and receive diversity is that,when Channel State Information(CSI)is not available at the transmitter,then array gain is not possible to obtain,whereas array gain is possible to obtain in the receive diversity system,as we assume that perfect CSI is always available at the receiver[1]. While using the space diversity receiver,three kinds of combiner can be used,namely Maximal Ratio Combining(MRC),Equal Gain Combining(EGC)and Selection Combining(SC).MRC being the most complicated with the best performance,while SC is the least complicated with the worst performance among the three schemes.In Orthogonal Frequency Division Multiplexing(OFDM)systems,these three techniques can be applied either before the Discrete Fourier Transform(DFT)demodulation(i.e.in time-domain on received samples),or after the DFT demodulation(i.e.in frequency-domain on each sub-carrier symbols).We denote the schemes as Pre-DFT and Post-DFT combining respectively.In Post-DFT combining,all the signal processing is performed on sub-carrier basis,thus,best case performance is achieved.While in Pre-DFT combining,all the signal processing needed is performed in time domain,so the duplication of the DFT operation for each receiving antenna branch is not a requirement any more,thus the receiver has lower computational cost compared to Post-DFT techniques[2].

When we implement the above mentioned combining technique after the DFT operation,we can name the receive diversity techniques as Post-DFT MRC,Post-DFT EGC and Post-DFT SC respectively. Their three counterparts in Pre-DFT operation can be called as Pre-DFT Maximal Average Ratio Combining(MARC),Pre-DFT Equal Average Gain Combining(EAGC)and Pre-DFT Antenna Selection Combining(ASC)respectively.

In this work,we have established a system model for Pre-and Post-DFT receive diversity schemes in OFDM systems.Then we have obtained theoretical capacity curves to determine outage and mean capacity of the combining schemes under indoor and outdoor channel conditions.We have obtained FER performance of the Pre-and Post-DFT combining schemes under certain OFDM system parameters. Finally,we have done a preliminary analysis of required system complexity for Pre-DFT MARC and Post-DFT MRC schemes in terms of number of multiplications required.

The rest of this paper is organized as follows.We have presented the system model for Pre-and Post-DFT combining schemes in Sec-tion II.The analysis and simulation results are placed in Section III. Some initial conclusions and the topics to be included in the?nal paper are discussed in Section IV.

II.S YSTEM M ODEL

We adopt a system model that is originally established in[3].The transmitted signal vector for the s th OFDM symbol,s s,is given by:

s s=T cp F H d s(1) where d s is the data vector,F H is Inverse Discrete Fourier Transform (IDFT)matrix,and T cp is the Cyclic Pre?x(CP)addition matrix.The elements of d s is taken from a complex constellation diagram.Since only one antenna is used at the transmitter while multiple antennas are deployed at the receiver,the channel matrix becomes:

G s=

?

G s1,1G s2,1...G s Q,1

?T

(2) with G s q,1is the Channel Impulse Response(CIR)convolution matrix between the transmit antenna and the q th receive antenna.Q denotes number of receive antennas in the system and q denotes receive antenna index.In this case,the dimensions of the CIR convolution matrix will be G s∈C[(N+N g)Q×(N+N g)].N and N g denotes number of OFDM sub-carriers and the CP length in samples respectively.

The received signal vector r s∈C[(N+N g)Q×1]may be expressed as:

r s=G s s s+v(3) where v is the time-domain noise vector at the receiver.At the receiver,the time-domain received signal is down-converted and the CP is removed:

y s=T+cp r s(4)

The

time-domain OFDM symbol vector after CP removal,y s ,can be decomposed into Q vectors,each representing a diversity branch,or one received OFDM symbol at a particular receiving antenna:

y s =?y s 1y s 2...y s Q ?T

(5)At this point,there are two methods for combining the diversity

branches to obtain the desired diversity gain,namely Post-DFT combining and Pre-DFT combining .The former performs OFDM demodulation before combining the diversity branches on sub-carrier basis,while the latter ?rst combines diversity branches over time-domain samples and then the OFDM demodulation is carried out.Both of these diversity combining techniques will be explained in the following sections.A.Post-DFT Combining

Outputed stream

Fig.1.Post-DFT combining technique

Figure 1illustrates the diagram of an OFDM receiver using Post-DFT combining method.The received signal at each antenna branch is DFT demodulated separately before combining.Signal at the output of the DFT demodulator can be expressed as:

z s =Fy s =?z s 1z s 2...z s Q ?T

(6)where F is the DFT-process matrix,analogous to F H given in (1),

and z s q is output sub-carrier vector at the q th

antenna branch:

z s q =?z s q,1z s q,2...z s

q,N ?T

;q =1,2,...,Q (7)The sub-carriers at the output of the OFDM demodulators are

combined separately so as to maximize the instantaneous SNR on sub-carrier basis.For each sub-carrier,the signals from multiple antennas are linearly combined,i.e.:

z s k =αs 1,k z s 1,k +αs 2,k z s 2,k +···+αs Q,k z s Q,k

(8)

where z s k is the k th combined sub-carrier,z s

q,k is de?ned as in (7)

and αs

q,k is the weight factor associated with the k th sub-carrier and the q th antenna (k =1,2,...,N and q =1,2,...,Q ).The complex weight factor can be written as,αs q,k =βs q,k e ?jθs

q,k ,where βs

q,k and θs

q,k are amplitude and phase of the weight factor respectively.The linear combination operation can be expressed in matrix form:

z s =αs z

s

(9)

with αs

∈C N ×NQ

is the weight factor matrix for Q receive antenna

at the s th

OFDM symbol.In case of Post-DFT combining schemes,this matrix is given by:

αs =?αs 1αs 2...αs Q

?

(10)where αs q is the diagonal weight matrix associated with the q th

antenna branch:

αs

q =26664αs q,10 0

0αs q,2...0............00...αs

q,N

37775

(11)For the k th sub-carrier,the choice of weight factors,

{αs 1,k ,αs 2,k ,...,αs Q,k },depends on the combination technique used at the diversity combiner module,namely MRC,EGC and SC.These techniques will be discussed in the following sections.

1)Post-DFT MRC:In MRC,the output is a weighted sum of all

diversity branches,i.e.αs

q,k =0;?q &?k .This multiplication is performed such that each of the signal branches are co-phased (i.e.

all branches have zero phase).The gain factors αs

q,k are selected

so that P Q q =1|βs q,k |2

=1,which implies that the noise power after

diversity combining is not modi?ed.βs

q,k is assigned based on the instantaneous power or the SNR from each received signal branch.If the power is small in any particular branch,then it will be assigned a small gain factor,and vice versa.

2)Post-DFT EGC:The EGC technique is similar to MRC because the diversity branches are co-phased,and all of them contribute to the output of the combiner.However,it is different from MRC that the diversity branches are not weighted corresponding to their instantaneous SNRs.This reduces the complexity of obtaining the SNR for each diversity branch.Equal values are selected for all gain

factors,i.e.e βs q,k =1

√Q

;?q .For example,in case of dual antenna systems,βs 1,k =βs

2,k =0.5;?k .

3)Post-DFT SC:The principle of SC method is to deliver to the detector the diversity branch which yields the highest SNR.In other words,the SC algorithm needs to monitor the SNR levels of all branches,and select the i th branch with the highest SNR value.

The weight factor αs

i,k associated with that branch will be 1,and all

other weight factors,αs

q,k (q =0,1,...,Q ;q =i ),are zeros [4].In practice,it is dif?cult to measure the SNR levels of diversity branches.Therefore,the instantaneous signal plus noise value is usually used instead of the SNR,with the assumption that the noise power (N 0)is constant for all branches.The selected diversity branch can be used for a duration of several OFDM symbols,provided that the duration is less than the channel coherence time [4].B.Pre-DFT Combining

Outputed

stream

Fig.2.An OFDM receiver employing Pre-DFT combining technique

The pre-DFT combining technique combines the time-domain

OFDM samples from multiple receive antennas and delivers the combined samples to the DFT demodulator [5].The process is shown in Figure 2.Clearly,the computational complexity is dramati-cally reduced for pre-DFT combining technique:only one OFDM demodulator and one diversity combiner are required.After the

pre-DFT diversity combining module,the receiver architecture is identical to that of the Orthogonal Frequency Division Multiple Access(OFDMA)receiver discussed in[6].

The output of the pre-DFT combining module is y s∈C N×1,and it can be expressed as:

y s=αs y s(12) where y s is the time-domain OFDM symbol vector after CP removal in(4)andαs∈C N×QN is the weight factor matrix used for linear combination operation on Q antennas.

αs=?

αs0αs1...αs Q?1

?

(13)

whereαs q is the diagonal weight matrix associated with the q antenna

branch:αs q=2

66

64

αs q0 0

0αs q 0

..

.

..

.

..

.

..

.

00...αs q

3

77

75q=0,1,...,Q?1

(14)

It is important to note that the same weight factorαs q is applied for all time-domain samples of an OFDM symbol on the q th antenna branch.This makes the scheme different from the above mentioned post-DFT combining technique.Analogously to post-DFT combining, there are three options for selecting the weight factors for pre-DFT combining scheme,namely Antenna Selection Combining(ASC), Equal Average Gain Combining(EAGC),Maximal Average Ratio Combining(MARC)and Orthogonal Combining(OC).

1)Pre-DFT MARC:The Pre-DFT Maximal Average Ratio Combining(MARC)selects the optimal set of weight factors for Q receive antennas,i.e.{αs0,αs1,...,αs Q},so as to maximize the SNR of the signal after combining.In[5],[7],it is proven that the optimal weight factors are the?rst column of left singular matrix of the CIR

matrix,corresponding to the highest Eigenvalue of CIR matrix,λ2H

1.

In this case,the receive SNR is equal to E s

N0λ2H

1

,

The Pre-DFT MARC requires the estimation of CIR on all receive antenna.The computational complexity in this case is comparable to post-DFT scheme,as each antenna branch requires a DFT processor for frequency-domain channel estimation.In order to reduce the complexity,the correlation matrix of received signals is used instead of CIR matrix[5].

The performance of Pre-DFT MARC scheme depends on type of channel models it is operating on.If channel response is?at over the OFDM spectrum,the Pre-DFT MARC performed equivalently to post-DFT MRC,but with less complexity.However,if the channel is frequency-selective,the performance of Pre-DFT MARC is degraded and less diversity gain is achieved compared to post-DFT MRC[5].

2)Pre-DFT EAGC:All Q weight factors for Q receive antennas, {αs0,αs1,...,αs Q},are set to one in the Pre-DFT Equal Average Gain Combining(EAGC)scheme.As a result,the output signal is the sum of all time-domain signals at different receive antenna.

In order to achieve diversity gain in the Pre-DFT EAGC scheme, the branches must be co-phased before summing.This operation often requires separate channel estimator and equalizer for each receive antenna branch,which increases computational complexity at the receiver,especially when the channel is frequency-selective.However, if the channel coherence time is much longer than the OFDM symbol duration,such requirement could be negligible.

3)Pre-DFT ASC:In the Pre-DFT Antenna Selection Combining (ASC)scheme,only the diversity branch with the highest average SNR is selected and delivered to the OFDM demodulator.In other words,the weight factor for the q th diversity branch,whose average SNR is the highest,is1and all other weight factors are zeros.

In[7],it is shown that the performance of the Pre-DFT ASC scheme is always worse than that of the Pre-DFT MARC scheme, and the receive SNR is equal to max

q

E s

H q 2F.The H q 2F is the squared Frobenius norm of the channel matrix associated with the q th antenna,and,in this case,can be interpreted as the total power gain of the Single Input Single Output(SISO)channel between the transmit antenna and the q th receive antenna[1].

III.A NALYSIS,S IMULATIONS AND D ISCUSSIONS

A.Simulation Parameters

We have used two simulation scenarios as explained in Table I. For all our analysis and simulations,we have con?ned ourselves to the case of dual antenna receiver diversity(i.e.Q=2).We assume that perfect time and frequency synchronization is established.We also assume that perfect channel estimation values for each sub-carrier for both the spatial channels are available at the receiver.We use exponential channel model to generate corresponding CIR and Channel Transfer Function(CTF)of the channel.In our exponential model,channel impulse response in exponentially distributed with decay between the?rst and last impulse as-40dB.

TABLE I

OFDM S IMULATION P ARAMETERS FOR I NDOOR AND O UTDOOR

S CENARIO

Parameters Indoor Outdoor

OFDM sub-carriers,N64512

Data sub-carriers,N data48388

Pilot sub-carriers,N pilot460

Null sub-carriers,N null1264

CP length,N CP16100

OFDM Symbol Duration,T s4μs30.6μs

Useful data period,T data 3.2μs25.6μs

CP period,T CP0.8μs5μs

Data Symbol mapping QPSK

Pilot Symbol mapping BPSK

Channel coding scheme1

2

-rate Convolutional coding System bandwidth20MHz

Carrier frequency 5.4GHz

B.System Capacity Analysis

The theoretical outage capacity of Post-DFT MRC,Post-DFT SC, Pre-DFT MARC and Pre-DFT EAGC are evaluated in this section via a semi-analytical monte-carlo simulation approach.This is done for both indoor and outdoor environment.First,the channel is simulated using the model mentioned above.Then the instantaneous channel capacity is obtained using the simulated CTF based on the following equation:

C=

1

N

N?1

X

k=0

log2(1+ρh k h?k)(15)

whereρis the transmit SNR and h k is the equivalent CTF of k th sub-carrier.Equivalent CTF means the CTF at the particular sub-carrier after the diversity combining.The above instantaneous capacity is derived for each channel realization and then Cumulative Distribution Function(CDF)of instantaneous channel capacity is

Capacity in bps/Hz

P

r

o

b

a

b

i

l

i

t

y

t

h

a

t

q

u

a

n

t

i

t

y

<

a

b

s

c

i

s

s

a

Fig.3.CDF of the corresponding capacity of1×2SIMO System at10dB

SNR in indoor scenario’

Capacity in bps/Hz

P

r

o

b

a

b

i

l

i

t

y

t

h

a

t

q

u

a

n

t

i

t

y

<

a

b

s

c

i

s

s

a

Fig.4.CDF of the corresponding capacity of1×2SIMO System at10dB

SNR in outdoor scenario

plotted in Figure3and4for indoor and outdoor scenario respectively.

For a large number of random channels,the outage and mean capacity

can be determined from these two?gures.In our case,we have

simulated10,000random channels and obtained the CDFs.

As we see from the?gures,Post-DFT schemes always give

better outage capacity values and better mean capacity than Pre-DFT

schemes.Post-DFT MRC and Post-DFT SC depict a10%outage

capacity capacity of2.7and2.2bps/Hz respectively,where as both

Pre-DFT schemes show1.3bps/Hz only.One possible reason for

this is,Post-DFT schemes exploit the available diversity in much

ef?cient way than the Pre-DFT schemes.The mean capacity of Post-

DFT MRC and Pre-DFT MARC are4.1575and3.7702bps/Hz re-

spectively.Higher mean capacity in Post-DFT MRC means a steeper

CDF,which obviously comes from more diversity exploitation.

The difference in outage situation is more evident in outdoor

scenario.Now the mean capacity values are 4.0605bps/Hz and

3.3051bps/Hz for Post-DFT MRC and Pre-DFT MARC respectively.

The differences in10%outage capacity follows the same trend as it

is experienced in indoor scenario.As the combining in done in time-

domain in Pre-DFT schemes,the average gain determination does not

’capture’the effect of all the multipaths.As we have done sub-carrier

by sub-carrier combining for Post-DFT schemes,the diversity order

is fully achieved in those schemes.This is the reason that Post-DFT

schemes always perform better in terms of outage and mean capacity.

10

10

10

10

10

SNR, dB

F

E

P

Fig.5.FER of different Pre-and Post-DFT Combining receiver diversity

C.FER Performance in Indoor and Outdoor Scenario

Figures5shows the Frame Error Rate(FER)performance of the

combining schemes in indoor and outdoor scenarios respectively.In

indoor scenario,a frame consists of64?16?2?0.5=1024source bits,

while in outdoor scenario,it is512?16?2?0.5=8192source bits

in one frame.In indoor scenario,Post-DFT MRC,Pre-DFT MARC

and Post-DFT SC performs almost similar.At10?3of FER,Pre-

DFT MRC obtains a gain2dB in required SNR in this scenario.

While in outdoor scenario,both Post-DFT schemes performs better

than Pre-DFT MARC scheme.This is because of the averaging effect

of Pre-DFT MARC https://www.360docs.net/doc/e76019776.html,pared to indoor scenario,here Pre-

DFT MRC performs even better with respect to10?3of FER,the

gain now becomes4dB.It is interesting to note that even Post-

DFT SC performs better than Pre-DFT MARC in outdoor frequency

selective scenario by nearly1dB at10?3of FER.

While indoor scenario depicts much less severe frequency-

selectivity,the Pre-DFT MARC scheme can still perform on par

with Post-DFT schemes.But,in outdoor scenario,the frequency

selectivity is very severe(5μs of maximum delay spread in our

case),and thus,the average gain at the time-domain samples in Pre-

DFT MARC scheme does not provide enough SNR enhancements,in

other words,the time-domain averaging does not exploit full spatial

diversity available in the system.

D.Implementation Complexity

We compared the complexity of the schemes in terms of number of

multiplications required.Considering that we have a channel which is

time-invariant for considerable amount of time,so that N pkt number

of OFDM symbols can be put in one OFDM packet,then the number

of multiplications required for one OFDM symbol are

X pre=

Q(Q?1)N o

2N pkt

+N pkt

…

N

2

log2N+QN

?

(16)

X post=N pkt

…

QN

2

log2N+QN

?

(17)

where N

2

log2N multiplications are required for FFT module per

OFDM symbol[8].N o is the number of time-domain data samples

that need to be acquired to obtain the correlation matrix.The

computational complexity associated with Eigen analysis for gain

factors is not taken into account,as it only required only once for

complete OFDM packet[5].

Figure

6and 7show the relative processing cost between Pre-DFT MARC and Post-DFT MRC in comparison to N and N pkt respectively for different values of Q .The relative processing cost is

de?ned as X

post X pre

from (17)and (16).In both ?gures,we can see that the processing cost is drastically reduced in Pre-DFT MARC scheme when N or N pkt or Q increases.For N =256,Post-DFT MRC with Q =2and 8require 1.7and 3.3times more processing capabilities respectively compared to Pre-DFT MARC scheme.The same trend is seen when the complexities are also compared in terms of number of symbols per OFDM packet.

Number of Subcarriers, N

R e l a t i v e P r o c e s s i n g C o s t

Fig.6.Relative Processing cost for Pre-DFT MARC and Post-DFT MRC in comparison to number of OFDM sub-carriers;N pkt =150and N o =50.

OFDM symbols/packet, N pkt

R e l a t i v e P r o c e s s i n g C o s t

Fig.7.Relative Processing cost for Pre-DFT MARC and Post-DFT MRC in comparison to number of OFDM symbols/packet;N =64and N o =50.

IV.C ONCLUSION

In this paper,we have brie?y discussed the merits and demerits of Pre-and Post-DFT combining diversity receiver for wideband OFDM systems.We have found that Post-DFT schemes are always superior to Pre-DFT schemes.In indoor scenario,this is not so evident,but this is very evident in outdoor scenario.The advantage that Pre-DFT schemes offer is simpler implementation and lesser computational requirements.This becomes very evident when the number of sub-carriers is very large and when the number of received branches are also high.Thus,the choice between these two types of receive diversity combining schemes are a trade-off problem between performance and system complexity.

In our opinion,the advantages of Pre-DFT schemes in terms of system complexity compared to performance loss with regard to Post-DFT schemes is acceptable in indoor scenario.Where as,in outdoor scenario,the system performance degradation in Pre-DFT schemes

are very high compared to their counterparts.To end,the decision between Pre-and Post-DFT combining reception should be taken based particular system environment,system parameters and expected system performance.

In the current work,the combining schemes are studied when no spatial correlation due to antenna spacing,presence of Line of Sight (LOS)component etc are considered in the channel model.These can be studied as a next step.The effect of channel correlations can be studied for outage and mean capacity.Some innovative techniques to extract more spatial diversity in Pre-DFT techniques can be investigated to reduce the performance gap between Pre-and Post-DFT combining techniques.

A CKNOWLEDGEMENT

The authors would like to thank Daniel V .P.Figueiredo,Huan C.Nguyen and Nicola Marchetti of WINGlab,CTIF,Aalborg University,Denmark for the useful discussions and cooperations in performing this work.

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