ofdm-cdm with spatial pre-coding in fading channels

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEuro. Trans. Telecomms. 2008; 19:611–618Published online 3 June 2008 in Wiley InterScience(www.interscience.wiley.com). DOI: 10.1002/ett.1306

OFDM-CDM with spatial pre-coding in fading channelsy

Stefan Kaiser*

DoCoMo Communications Laboratories Europe GmbH, Landsberger Strasse 312, 80687 Munich, Germany

SUMMARY

This paper investigates the potential of orthogonal frequency division multiplexing-code divisionmultiplexing (OFDM-CDM) for its application in future broadband mobile radio systems. The focus is onthe performance analysis of OFDM-CDM with spatial pre-coding and its comparison with OFDM. Codedivision multiplexing (CDM) is an efficient coding scheme without rate loss. The investigations take intoaccount the spatial diversity and pre-coding schemes cyclic delay diversity (CDD), spatial phase coding(SPC), equal gain transmission (EGT) and maximum ratio transmission (MRT). SPC can closely approachthe performance of optimum pre-coding schemes, whereas SPC requires less complexity and overhead withrespect to channel estimation and feedback information. Imperfections due to channel estimation are takeninto account in the analysis. It can be shown that OFDM-CDM with spatial pre-coding applying singlesymbol detection as well as multi-symbol detection outperforms OFDM with spatial pre-coding. Copyright# 2008 John Wiley & Sons, Ltd.

1. INTRODUCTION

Broadband mobile radio standards like 3GPP long-term

evolution (LTE) and WiMAX have orthogonal frequency

division multiplexing (OFDM) as common transmission

technology for robust high rate data transmission [1].

The requirements on the spectral efficiency further

increase when it comes to the specification of IMT-

Advanced systems. The core of an IMT-Advanced air

interface is again OFDM [2]. Besides exploiting advanced

channel coding and multi-antenna schemes, further efforts

are necessary to increase the efficiency of OFDM schemes.

Here, efficiency relates to reduced power consumption

since battery lifetime is a critical issue in broadband

mobile communication systems.

In this paper the performance improvements achievable

with code division multiplexing (CDM) [3] in future

mobile radio systems are analysed. CDM is an efficient

coding scheme without rate loss (the rate is equal to 1).

Field experiments [4] have proven the suitability of apply-

ing spreading [5, 6] in broadband OFDM systems. The

focus of this paper is on a comparison of conventional

OFDM with OFDM-CDM, where the effects of different

spatial diversity and pre-coding schemes are investigated.

Results are presented for the spatial signal processing

schemes cyclic delay diversity (CDD) [7], spatial phase

coding (SPC) [8], equal gain transmission (EGT) and

maximum ratio transmission (MRT) [9, 10]. Effects due

to imperfections of the channel estimation are taken into

account in this paper. The performance results presented

for OFDM are also valid for the multiple access scheme

OFDMA.

One main difference of the spatial pre-coding schemes

and its performance is the amount of channel knowledge

required at the transmitter as well as at the receiver. The

optimum scheme requires full channel knowledge about

the channel from each transmit antenna to the receive

antenna. Suboptimum solutions with only partial channel

knowledge can closely approach the performance of

the optimum scheme. With SPC the phase relation of the

Copyright # 2008 John Wiley & Sons, Ltd. Accepted 5 May 2008

*Correspondence to: Stefan Kaiser, DoCoMo Communications Laboratories Europe GmbH, Landsberger Strasse 312, 80687 Munich, Germany.E-mail: [email protected] previous edition of the paper has been presented in the 6th International Workshop on Multi-Carrier Speed Spectrum (MC-SS 2007)

signals between multiple transmit antennas is modified

such that the probability of constructive superposition of

the signals at the receive antenna is increased. Compared

to EGT or MRT the required overhead and complexity for

channel estimation with SPC is by a factor of 2 smaller in a

two transmit antenna system.

The results show that CDM can improve the perfor-

mance of OFDM schemes with and without spatial diver-

sity exploitation. OFDM-CDM is a promising evolution

towards future OFDM based mobile radio systems like

IMT-Advanced.

The paper is organised as follows. The OFDM-CDM

transmitter and receiver are introduced in Section 2.

The investigated spatial diversity and pre-coding

schemes CDD, SPC, EGT and MRT are detailed in

Section 3. Section 4 briefly describes the applied channel

estimation concept. The performance comparison between

OFDM-CDM and OFDM with and without spatial diver-

sity is presented in Section 5. Finally, Section 6 sum-

marises the results.

2. OFDM-CDM TRANSMISSION SYSTEM

2.1. Transmitter

An OFDM-CDM system with two transmit antennas is

investigated. The transmitter applying SPC is shown in

Figure 1. After channel encoding and symbol mapping the

data symbols are spread with a symbol specific spreading

code of length L. K subsequent spread data symbols are

superimposed (multiplexed) before spatial pre-coding, that is

sl ¼XKk¼1

dðkÞzðkÞl ð1Þ

where d(k) is the kth data symbol and zl(k) is the lth chip of

the kth spreading sequence. CDM without rate loss is

achieved when K is equal to L. The resulting chip sequence

is s¼ (s1, s2, � � � , sL)T. The symbol (�)T denotes the transpo-sition of a vector. In order to reduce the complexity of the

mobile receiver the M&Q modification is applied which is

in detail explained in Reference [1]. The M&Q modifica-

tion achieves a spreading code length L much smaller than

the total number of sub-carriers Nc. This reduces the com-

plexity especially of multi-symbol detectors. The symbols

transmitted on the two transmit antennas after pre-coding

are sl(1) and sl

(2), where (m) is the transmit antenna index

m¼ 1, 2. The pre-coding is given by

sl ¼ slwlcl ¼ sð1Þl ; s

ð2Þl

� �T

ð2Þ

where cl¼ (cl(1), cl

(2))T is the spatial pre-coding vector and

wl the power normalisation coefficient. According to

Equation (1), the lth chip to be transmitted is given by sl.

After pre-coding the symbol sl(m) on the mth antenna is

modulated on sub-carrier n, n¼ 1, � � � , Nc, by applying

OFDM with Nc subcarriers. The OFDM operation also

includes the insertion of a cyclic extension as guard inter-

val. F is the feedback information required for pre-coding.

2.2. Receiver

The OFDM-CDM receiver with SPC is shown in Figure 2.

The received signal after inverse OFDM is given by

rl ¼ cð1Þl H

ð1Þl þ c

ð2Þl H

ð2Þl

� �wlsl þ Nl

¼ Hlsl þ Nl

ð3Þ

where

Hl ¼ cð1Þl H

ð1Þl þ c

ð2Þl H

ð2Þl

� �wl ð4Þ

represents the superimposed pre-coded channel and Nl

the additive noise affecting sl. The channels from the two

transmit antennas to the receive antenna are given by the

complex-valued fading coefficients Hl(1) and Hl

(2), respec-

tively.

Channel estimation is required for pre-coding as well as

for data detection. The feedback information F to the trans-

mitter contains the information for pre-coding. The detection

comprises despreading, data symbol demapping and channel

decoding. The OFDM-CDM receiver has additional com-

plexity in the detector compared to an OFDM receiver.

Figure 1. OFDM-CDM transmitter with spatial phase coding. Figure 2. OFDM-CDM receiver.

612 S. KAISER

Copyright # 2008 John Wiley & Sons, Ltd. Eur. Trans. Telecomms. 2008; 19:611–618DOI:10.1002/ett

For data detection two different schemes are applied in this

paper.

2.2.1. MMSE equalisation

Single symbol detection with minimum mean square error

(MMSE) equalisation is applied as low-complexity detec-

tion scheme. Equalisation according to the MMSE criter-

ion minimises the mean square value of the error between

the transmitted signal and the output of the equaliser. The

equalisation coefficient based on the MMSE criterion

results in

Gl ¼ Hl

Hlj j2þ�2ð5Þ

The variance of the noise affecting sl is given by �2.

2.2.2. Iterative soft interference cancellation

Multi-symbol detection with soft interference cancellation

(soft IC) [11] is applied as powerful detection scheme. The

principle of soft IC is to take reliability information about

the detected interference into account in the interference

cancellation process. Soft values instead of hard values are

subtracted. The channel decoding is included in the iterative

process and reliability information about the interference is

obtained from a soft-in/soft-out channel decoder.

3. SPATIAL DIVERSITYAND PRE-CODING

The performance of CDM in broadband OFDM systems is

analysed for a variety of spatial diversity and pre-coding

schemes. A transmitter with two antennas is considered.

The spatial pre-coding block (see Figure 1) encodes the

data symbol sn according to

sn ¼ snwncn ¼ sð1Þn ; sð2Þn

� �T

ð6Þ

where wn is the power normalisation coefficient and cnis the pre-coding vector. Equation (6) differs from Equa-

tion (2) only in referring to the subcarrier index instead

of the chip index of the spreading code. The vector snrepresents the two symbols to be transmitted in parallel

on the two transmit antennas. The spatial diversity and

pre-coding schemes are introduced in the following.

Transmit antenna selection diversity is not included in

the analysis since SPC shows better performance with real

channel estimation at less complexity.

3.1. Cyclic delay diversity (CDD)

With CDD the same signal is simultaneously transmitted

from two transmit antennas. In order to increase the fre-

quency selectivity of the channel, the signal on the second

antenna is cyclically delayed as proposed in Reference [7].

The cyclic delay is equivalent to a subcarrier dependent

phase shift �n. The pre-coding vector for CDD on subcar-

rier n results in

cn ¼ 1; ej�n� �T ð7Þ

The total transmit power is equally split between both

antennas and has to be normalised by

wn ¼ 1ffiffiffi2

p ð8Þ

At the receiver antenna only the superimposed channel Hn

has to be estimated. No feedback information is required.

3.2. Spatial phase coding (SPC)

The principle of SPC is to achieve a constructive superpo-

sition of the signals from the different transmit antennas at

the receive antenna without the necessity to estimate the

two channels from the two transmit antennas to the receive

antenna. Only one channel has to be estimated, which is

the superimposed channel Hn.

By comparing the absolute value of the superimposed

channel jHnj with a predefined threshold � (see Figure 3),

SPC detects a destructive superposition at the receiver with

high probability and indicates to the transmitter over a feed-

back channel that the phase relation of the transmitted signals

should in this case be changed by p. A straightforward solu-

tion is to flip the phase of the signal at one transmit antenna

by p. Thus, in the subsequent transmission the channels

superimpose constructively at the receive antenna. The

assumption is that the phase relation between both channels

is quasi-constant between subsequent OFDM symbols. This

is a typical assumption in OFDM systems. If jHnj falls belowthe threshold� due to variations of the channel over the time,

the phase will be flipped again.

Figure 3. Principle of spatial phase coding (SPC).

OFDM-CDM WITH SPATIAL PRE-CODING 613


The spatial phase encoder has two possible states. These

are termed State A and State B. Depending on the actual

state, the data symbols sn to be transmitted are pre-coded

in different ways. The SPC vector cn for the two states is

defined as

cn ¼ ð1; 1ÞT State A

ð1; e�jpÞT State B

�ð9Þ

The property of SPC is that the receiver evaluates the

received signal and decides if the pre-coder should remain

in the actual state or should perform a state change. The

receiver does not need to know the actual state of the spa-

tial phase pre-coder. It is sufficient to evaluate a predefined

flipping criterion and based on this to indicate to the trans-

mitter via a feedback channel to flip the phase on one

antenna compared to the previous transmission, that is to

perform a state change, or to remain in the actual state.

The criterion for a state change is defined as follows:

F ¼ no state change Hnj j5�state change Hnj j < �

�ð10Þ

where F is the feedback information from the receiver to

the transmitter indicating if a state change is necessary or

not. Since it is only binary feedback information, 1 bit is

sufficient for the feedback information F. The performance

of SPC depends on the choice of the threshold � [8].

With SPC using two transmit antennas the power nor-

malisation factor wn results in

wn ¼ 1ffiffiffi2

p ð11Þ

3.3. Equal gain transmission (EGT)

EGT is a pre-coding scheme which requires knowledge

about the phase of the channels Hn(1) and Hn

(2) and trans-

mits the signals at the antennas with a phase shift such that

they superimpose constructively at the receiver antenna.

The pre-coding vector is

cn ¼ 1; e jan� �T ð12Þ

where an is the phase difference between Hn(1) and Hn

(2).

The transmit power is equally split between both antennas,

that is

wn ¼ 1ffiffiffi2

p ð13Þ

The phase of the channels Hn(1) and Hn

(2) has to be esti-

mated at the receiver and the relative phase difference

between both channels has to be fed back to the transmit-

ter. EGT is a lower bound for SPC.

3.4. Maximum ratio transmission (MRT)

The optimum pre-coding scheme is MRT. The pre-coding

coefficient of MRT is

cn ¼ Hð1Þ�n ;Hð2Þ�

n

� �T

ð14Þ

Compared to the previous schemes, MRT allows for an

unequal distribution of the transmit power between both

antennas. The power normalisation factor results in

wn ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2m¼1

HðmÞn

�� 2s ð15Þ

The full channel information about Hn(1) and Hn

(2) has to

be estimated at the receiver. Thus, MRT requires the most

complex channel estimation with the highest overhead of

all schemes investigated in this paper. Moreover, MRT

requires full feedback of amplitude and phase of each

channel, which exceeds the amount of feedback informa-

tion required by the previous schemes.

3.5. Reference: single transmit antenna (1Tx)

The single antenna scheme is given by choosing the pre-

coding vector equal to

cn ¼ ð1; 0ÞT ð16Þand the normalisation coefficient equal to

wn ¼ 1 ð17ÞNo feedback information is required.

4. CHANNEL ESTIMATION

Channel estimation in OFDM systems can efficiently be

realised by two-dimensional channel estimation [12].

The principle is based on sampling and filtering the chan-

nel in time and frequency. This can significantly reduce the

overhead due to pilots compared to channel estimation

schemes exploiting only the correlation of the channel in

either time or frequency. The two-dimensional filtering

applied in this paper is based on two cascaded one-

dimensional filters. The first filtering is performed in

frequency direction on OFDM symbols containing pilots.

The second filtering is performed in time direction on all

614 S. KAISER


subcarriers. A rectangular pilot grid is chosen. The pilot

spacing has to fulfil the sampling criterion in time and fre-

quency direction. The maximum Doppler frequency in the

channel is given by fD,max and the maximum delay spread

is �max. The pilot spacing Nf in frequency direction has to

fulfil

Nf4Ts

�max

ð18Þ

and the pilot spacing Nt in time direction should fulfil

Nt41

2fD;maxðTg þ TsÞ ð19Þ

where Ts is the OFDM symbol length and Tg is the guard

interval length. A practically proven value of the sampling

rate is the selection of approximately two times oversam-

pling to achieve a reasonably low complexity with respect

to the filter length and performance.

A Wiener filter is applied in each filter direction. This

filter minimised the mean square error (MSE) between

the actual channel and the estimated channel, that is it

minimises the MSE given by

MSE ¼ E Hn � Hn

�� 2n oð20Þ

where Hn is the estimate of the channel coefficient Hn.

5. SIMULATION RESULTS

5.1. Parameters

An OFDM system with two transmit antennas and one

receive antenna is investigated. The transmission band-

width is 2MHz and the carrier frequency is 2GHz. The

multicarrier modulation is realised by OFDM occupying

Nc¼ 512 subcarriers. An OFDM frame consists of 24

OFDM symbols. The guard interval duration is 5 ms.Results are presented for a rate 2/3 coded transmission

using convolutional codes with memory 6. Soft decision

decoding with log-likelihood ratios optimised for

OFDM-CDM is applied [3]. QPSK is chosen for symbol

mapping. For spreading Walsh Hadamard codes of length

L¼ 8 are used. K¼ 8 data symbols are superimposed, that

is all schemes are fully loaded and have the same spectral

efficiency. For data detection either single symbol detec-

tion with MMSE equalisation or multi-user detection with

soft IC (one iteration) is applied.

In order to compare the performance differences

between different spatial diversity and pre-coding schemes

without side effects, effects due to synchronisation and

feedback errors are omitted by assuming that these compo-

nents are perfect. It is explicitly stated in the text when the

effects of imperfect channel estimation are taken into

account. Otherwise the channel estimation is assumed to

be perfect.

As propagation channel the COST 207 typical urban

(TU) channel model is taken [13]. The maximum velocity

of the mobile user is 30 km/h which corresponds to a Dop-

pler frequency of 55.6 Hz. A classical Doppler spectrum is

assumed [14]. The channels from different transmit anten-

nas to the receive antenna are assumed to be uncorrelated.

The signal-to-noise power ratio (SNR) given in the follow-

ing results refers to Et/N0 where Et is the total energy per

bit at the transmitter and N0 is the one-sided noise power

spectral density.

5.2. Results and discussion

Figure 4 shows the bit error rate (BER) versus the SNR for

OFDM-CDM and OFDM with MRT. OFDM-CDM is

applied with soft IC. These curves serve as lower bounds

for the further investigations shown in this paper since

MRT is the optimum pre-coding scheme. The results

are presented for fully loaded systems. Additionally, the

single symbol bound for OFDM-CDM with MRT is

shown. The single symbol bound represents the perfor-

mance with perfect interference cancellation. It can be

Figure 4. BER versus SNR for OFDM-CDM and OFDM withMRT, COST 207 TU channel, fully loaded, R¼ 2/3, QPSK.



observed that OFDM-CDM outperforms OFDM by sev-

eral dB in SNR especially at low BERs. Moreover, the pre-

sented OFDM-CDM scheme requires only a 0.5 dB higher

SNR than the single symbol bound showing that soft IC

with one iteration can cancel most of the interference.

The BER performance of OFDM-CDM with soft IC

and OFDM, both with and without CDD, is compared in

Figure 5. The systems are fully loaded. OFDM-CDM with

only one transmit antenna (no spatial diversity) performs

the same as OFDM with CDD. Moreover, the results show

that OFDM-CDM can take more advantage of CDD than

OFDM. While OFDM improves its performance by about

1.7 dB at a BER of 10�5, OFDM-CDM improves its per-

formance by about 3 dB at the same BER. The reason is

that OFDM-CDM can better exploit the artificial fading

introduced by CDD.

The effects of spatial pre-coding with SPC on the chan-

nel transfer function for an OFDM system with 512 sub-

carriers and two transmit antennas in the COST 207 TU

channel are shown in Figure 6. A snapshot of the absolute

value of the superimposed channel coefficient jHnj is

plotted over the 512 subcarriers. The channel jHnj withoutSPC (original) is normalised to EfjHnj2g ¼ 1. The dashed

line shows the original channel coefficients while the solid

line shows jHnj after SPC. The threshold � is chosen equal

to 0.9. It can be observed that in cases where the original

channel is in a deep fade, the channel after SPC often is

even enhancing the transmitted signal.

Figure 7 shows the BER versus the SNR of OFDM-

CDM and OFDM both applying SPC. The systems are

fully loaded. The performance of OFDM-CDM is shown

with MMSE single symbol detection and multi-symbol

detection applying soft IC. Additionally, the performance

Figure 5. BER versus SNR for OFDM-CDM and OFDM withCDD, COST 207 TU channel, fully loaded, R¼ 2/3, QPSK.

Figure 6. Channel transfer function for the COST 207 TUchannel with and without SPC, �¼ 0.9.

Figure 7. BER versus SNR for OFDM-CDM and OFDM withSPC, COST 207 TU channel, fully loaded, R¼ 2/3, QPSK,�¼ 0.4.

616 S. KAISER


with perfect interference cancellation is presented, which

corresponds to the case K¼ 1. It can be observed that

OFDM-CDM outperforms OFDM in all scenarios.

The performance of OFDM-CDM with CDD, SPC,

EGT and MRT is show in Figure 8. OFDM-CDM is

applied with soft IC. The system is fully loaded. The

results show that CDD achieves a performance improve-

ment of about 3 dB at a BER of 10�5 compared to the

1Tx scheme. CDD is the least complex spatial diversity

scheme since it requires no feedback channel. The least

complex pre-coding scheme with feedback channel is

SPC. SPC can improve the performance of 1Tx OFDM-

CDM by about 6 dB at a BER of 10�5. SPC requires only

the estimation of the superimposed channel Hn at the

receive antenna while EGTand MRT have to estimate both

channels Hn(1) and Hn

(2). Thus, EGT and MRT require

twice the overhead and complexity for channel estimation

than SPC. Additionally, SPC needs only a 1 bit feedback

information, while EGT and MRT require a soft feedback

information per transmit antenna, which is a significantly

higher overhead for the feedback channel. EGT and MRT

can improve the performance of 1Tx OFDM-CDM by

about 7.5 dB and 8.2 dB, respectively, at a BER of 10�5.

However, it can be expected that the gains with MRT

and EGTwill become smaller when a more realistic quan-

tised feedback will be applied.

For the simulation results shown in the following with

pilot symbol aided channel estimation, the pilot spacing

is six subcarriers in frequency direction and three OFDM

symbols in time direction. The filtering is performed by

two times one-dimensional Wiener filtering where for

the filtering in each dimension five taps are used. For

CDD and SPC only the superimposed channel Hn has to

be estimated and not the individual channels from each

transmit antenna. For MRT the overhead and complexity

for channel estimation is twice that of the other schemes

since both individual channels have to be estimated.

Figure 9 shows the BER versus SNR of OFDM-CDM

with different spatial diversity and pre-coding schemes.

The system is fully loaded. OFDM-CDM is applied with

soft IC. At a BER of 10�6 a performance gain of about

4 dB can be achieved with spatial transmit diversity apply-

ing CDD compared to the 1Tx scheme. Additional 3 dB

can be gained by exploiting a feedback channel for pre-

coding with SPC and MRT. It is interesting to observe that

with pilot symbol aided channel estimation SPC and MRT

perform nearly the same. The reason is that MRT has

higher degradations due to channel estimation since with

MRT two channels have to be estimated while all other

schemes have to estimate only the superimposed channel.

In Table 1, the gain in SNR with OFDM-CDM com-

pared to OFDM is shown for CDD, SPC and MRT at a

BER of 10�6. The results are shown for pilot symbol aided

channel estimation. It can be observed that CDM achieves

the highest gains with CDD since CDD increases the fre-

quency selectivity of the channel which can be efficiently

Figure 8. BER versus SNR for OFDM-CDM with CDD, SPC,EGT and MRT, COST 207 TU channel, fully loaded, R¼ 2/3,QPSK, �¼ 0.4.

Figure 9. BER versus SNR for OFDM-CDM with pilot symbolaided channel estimation, COST 207 TU channel, fully loaded,R¼ 2/3, QPSK, �¼ 0.3.



exploited by spreading. As reference the comparison with

the single symbol bound (K¼ 1) is shown to indicate the

additional potential of CDM if for example more iterations

are applied in the soft interference canceller. The gap

between OFDM-CDM with soft IC (one iteration) and

the single user bound is larger with pilot symbol aided

channel estimation compared to the scenario with perfect

channel estimation.

The gain in SNR with spatial pre-coding compared to a

1Tx scheme is presented in Table 2 for an OFDM-CDM

system at a BER of 10�6. The system is fully loaded

(K¼ 8). Results are shown for a system with perfect chan-

nel estimation (perfect CE) and a system with pilot symbol

aided channel estimation (real CE). The results show that

with real channel estimation SPC achieves the same gains

as MRT.

6. CONCLUSIONS

The combination of OFDM with CDM has been presented

and analysed with different spatial transmit diversity and

pre-coding schemes. Under investigation have been

CDD, SPC, EGT and MRT. The performance of an

OFDM-CDM system has been compared to that of a con-

ventional OFDM system. It could be shown that OFDM-

CDM outperforms OFDM. The achievable gains depend

on the chosen spatial diversity and pre-coding scheme,

respectively. The highest gains could be achieved with

CDD. OFDM-CDM required an up to 3 dB lower SNR

to achieve a BER of 10�5 compared to OFDM. Spatial

diversity and pre-coding schemes can reduce the required

SNR at a BER of 10�5 of OFDM-CDM systems by up to

8 dB compared to a scheme with one transmit antenna.

Effects due to imperfect channel estimation have been

taken into account in the analysis. It could be shown that

low complexity SPC performs the same as optimum MRT

when both apply pilot symbol aided channel estimation

since SPC is more robust to channel estimation imperfec-

tions. The presented performance analysis showed that

OFDM-CDM is a promising technique without rate loss

which can in combination with spatial diversity and pre-

coding further increase the efficiency of future broadband

OFDM systems like LTE and IMT-Advanced.

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Table 1. Gain in SNR with OFDM-CDM compared to OFDM atBER¼ 10�6, R¼ 2/3, pilot symbol aided channel estimation.

CDD (dB) SPC (dB) MRT (dB)

K¼ 8 3.8 1.6 1.6K¼ 1 7.5 4.2 4.4

Table 2. Gain in SNR with spatial pre-coding compared to a 1Txscheme at BER¼ 10�6 for OFDM-CDM, R¼ 2/3, K¼ 8.

CDD (dB) SPC (dB) MRT (dB)

Perfect CE 3.3 6.1 8.8Real CE 3.8 7.0 7.0

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ofdm-cdm with spatial pre-coding in fading channels

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