IMP - M - Design and Analysis of Post-Coded OFDM Systems

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008 4907

Design and Analysis of Post-Coded OFDM SystemsS. F. A. Shah and A. H. Tewfik Fellow, IEEE

AbstractThis paper discusses the design and analysis of

post coded OFDM (PC-OFDM) systems. Coded or precodedOFDM systems are generally employed to overcome the symbolrecovery problem in uncoded OFDM systems. We show that PC-OFDM systems are a special case of precoded OFDM systemsthat offer advantageous complexity-performance trade-offs. Inparticular, PC-OFDM systems introduce frequency diversity bymanipulating the OFDM symbols in the time domain so thatthe computational complexity of the system can be significantlyreduced. We discuss the design principles of PC-OFDM trans-mitter that uses upsampling operation and the spreading codesto introduce frequency diversity. We obtain the spreading codeconstruction criterion for minimum error performance and giveexamples of spreading codes for PC-OFDM systems. We alsodescribe the design of low-complexity receiver for PC-OFDMsystems. In particular, our proposed partial spreading schemeresults in a low complexity decoupled detector. The probability oferror analysis of the receiver leads us to postulate different designcriteria. We investigate different choices for detection algorithmssuitable for PC-OFDM receiver and compare their performancethrough simulations over Rayleigh and IEEE UWB channels.

Index TermsOrthogonal frequency division multiplexing(OFDM), precoding, post-coding, spreading codes, frequencydiversity, pulsed OFDM, coding gain, diversity gain.

I. INTRODUCTION

O

RTHOGONAL frequency-division multiplexing

(OFDM) has been proven to be a viable technique

to overcome multipath fading in wireless channels. It hasbeen adopted in many wireless standards, such as digital

audio/video broadcasting, the HIPERLAN/2 standard, the

IEEE 802.11a and g standards for wireless local area networks(WLAN) and is going to be used in various future broadband

wireless communication systems [1]. While OFDM systemsconvert a multipath fading channel into a series of equivalent

flat fading channels, they lack the inherent diversity available

in multipath channels. Theoretically, an uncoded OFDM

system needs a simple receiver due to ISI free channel but

their performance deteriorates severely in the presence of

channel frequency nulls at subcarrier frequencies [2].

To recover symbols at frequency nulls, different codedOFDM systems have been reported that employ some form

of error correction coding [3] or precoding [2], [4]. Error-

correcting codes that have been used with OFDM include

convolutional codes [5], trellis coded modulation [6], turbo

Manuscript received April 22, 2007; revised January 3, 2008 and August23, 2008; accepted September 22, 2008. The associate editor coordinatingthe review of this paper and approving it for publication was D. Huang. Thiswork is partially supported by NSF grant CCR-0313224.

S. Faisal A. Shah is with Azimuth Systems, Inc., Acton, MA 01720, USA(e-mail: [email protected]).

A. H. Tewfik is with the Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455, USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/T-WC.2008.070421

codes [7] and many others. The bit interleaved coded mod-

ulation (BICM) based on convolutional codes used in IEEE802.11 standard for WLAN [5] does not provide sufficient

coding advantage to overcome the deep fades problem. Inaddition, some of these coded OFDM schemes are often com-

putationally intensive and introduce large decoding delays [2]

and hence are practically infeasible.

The second class of coded OFDM systems that has be-

come popular in the literature in recent years is precoded

OFDM systems [2], [8], [4], [9]. In general, precoded OFDMsystems linearly mix the information symbols across the

subcarriers and create a diversity effect by distributing the

effect of channel fades across all the information symbols.

This type of linear combination of information symbols isalso known as spreading transform or spreading codes inthe literature1 [8], [9]. In [8], various choices of spreading

transforms are evaluated and a design of spreading codes based

on rotated Fourier matrix is found to be optimal. Minimum

bit error rate (BER) precoder design based on zero-forcing

equalization for time-invariant channels is presented in [4].

In [2], precoders are designed to achieve optimal performancein Rayleigh fading channels. Beyond Galois field design, the

authors of [2] designed precoders drawn from the real field

as well as complex field. These complex field precoders incur

significant complexity in transmitter and receiver design. To

reduce complexity, a short block spreading is considered in [9]where spreading codes are designed by numerically optimizing

a nonlinear error performance function.

While most of the research related to precoded OFDM con-centrates on the design of precoders to optimize performance,

very little has been done to reduce system complexity. Some

relevant work on low complexity coded OFDM systems is

reported in [10] and [11] in the context of ultra-wideband

(UWB) OFDM systems. In [10], a UWB-OFDM system is

proposed that utilizes short pulses based on Costas sequences

to spread the information symbols across different subcarriers

in the analog domain. A digital equivalent of the pulsed

OFDM proposed in [11] can be seen as repetitive coding thatdoes not have any coding advantage.

Our aim in this paper is to extend the idea of pulsed

OFDM [11] and design extremely low complexity coded

OFDM systems that can achieve near optimal performance.

We will refer to the proposed system as post-coded OFDM

(PC-OFDM) system. The rationale to use the term post-

coding will be explained in Section II. We presented the

initial ideas of PC-OFDM in [12], [13]. In short, PC-OFDMsystems introduce frequency diversity by spreading the in-

formation symbols across all the subcarriers in an efficient

1in contrast to its usual meaning, the word spreading does not refer to

signal bandwidth expansion here

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4908 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008

manner so that the overall computation cost of the system is

significantly reduced. The computation savings in PC-OFDM

come from two sources: 1) smaller size IFFT and FFT areused as compared to frequency domain precoding, and 2) the

special structure of encoding matrices is exploited resulting

in O(N) operations instead of O(N2) operations. To reducethe complexity of PC-OFDM receiver, we consider partial

spreading where the information symbols are spread across

distinct groups of subcarriers. This results in a low complexitydecoupled detector. Our main contributions in this paper are:

1) establishing a one-to-one relation between time domain

postcoding and frequency domain precoding, 2) showing

how time domain postcoding can lower the complexity, 3)

designing the transmitter and receiver to introduce maximumpossible diversity with minimum complexity, 4) analyzing

the probability of error function of the proposed system toobtain a metric that relates performance to the structure of the

spreading code and 5) designing spreading codes that achieve

good performance. In summary, the paper primarily focuses on

reducing the complexity of PC-OFDM transmitter and receiver

without any performance loss.The paper is organized as follows. In Section II, we discuss

the system model and point out the choices of precoding in fre-

quency domain and time domain and their consequences. We

explain the basic architecture of the transmitter in Section III

including the upsampling operation and multiplication with

spreading code. We also establish a relationship between post-coded and precoded OFDM systems, discuss the implications

of low complexity post-coded OFDM systems and introduce

a partial spreading technique. In Section IV, we discuss the

simplified design of receiver using multirate filtering concepts.

Different detector structures for joint detection of OFDM sym-

bols are discussed in Section V. We examine the probability oferror for PC-OFDM systems in Section VI and use it to design

spreading codes for optimal performance in Section VII. InSection VIII, we present a low-complexity detector based on

partial spreading and compare the complexity of PC-OFDM

systems with precoded OFDM systems. Simulation results are

presented and discussed in Section IX.

I I . SYSTEM DETAILS AND PROBLEM FORMULATION

Consider an uncoded OFDM system that is implementedusing an inverse fast Fourier transform (IFFT) at the transmit-

ter and a fast Fourier transform (FFT) at the receiver. Let FN

be the N N FFT matrix with (n, k)th entry given by[FN]n,k = (1/

N) exp{j2(n 1)(k 1)/N} (1)

for n = 1, , N and k = 1, , N. It is well known thatthe use of cyclic prefix (CP) in OFDM systems converts a

multipath fading channel into a set of parallel flat-frequency

channels such that the N1 vector of received OFDM symbolu can be expressed as:

u = HDb+ . (2)

Here, HD := diag[FNh] with h obtained from the concate-nation ofLh channel taps,

{h(l)

}Lh

l=1

, and N

Lh zeros. Here,b is the N 1 vector of modulated information symbols and represents an N1 vector of additive white Gaussian noise.

A

f

F H F

D e t e c t o r /

D e c o d e r

ib

NLNL NLNLNLNL

+

ib

NNL

(a) Frequency domain precoded OFDM (FP-OFDM) system

A

t

F H F

D e t e c t o r /

D e c o d e r

ib

NNL NLNLNLNL

+

ib

NN

(b) Time domain post-coded OFDM (PC-OFDM) system

Fig. 1. Precoded vs. post-coded OFDM systems.

Existing techniques encode the data before the IFFT op-

eration and can be termed as frequency domain precodedOFDM or FP-OFDM in short. A typical FP-OFDM system

is shown in Fig. 1(a). In contrast, we will show in this paper

that the system complexity can be signifi

cantly reduced ifprecoding is applied on OFDM symbols after performing the

IFFT operation as shown in Fig 1(b). Since we are precoding

the time domain OFDM symbols, we will refer to this scheme

as Time Domain Post-coded OFDM (PC-OFDM). The term

post-coded emphasizes the fact that we encode the symbols

after performing the IFFT operation.

For FP-OFDM, the vector of transmitted symbols is given

by

y :=1

K/NFHKAfb (3)

where Af is the frequency domain precoding matrix and

1/K/N is used for normalization. The superscript H in (3)represents the complex conjugate transpose (Hermitian trans-

pose). In contrast, the vector of transmitted symbols for PC-

OFDM is given by

y := AtFHNb. (4)

The design of low-complexity and optimal performance PC-

OFDM systems is tantamount to specifying the structure of

At. In this paper, we discuss in detail the design of Atand subsequently use its structure to design a low-complexity

PC-OFDM receiver. We consider complex field coding forboth FP-OFDM and PC-OFDM, i.e., Af (or At) CKN

with K N, instead of Galois field as it provides moredegrees of freedom [2]. In its simplest form, the design ofPC-OFDM requires K to be an integer multiple of N. In theremainder of this paper, we assume that K = N L where Lis an integer. This should not be considered as a limitation of

PC-OFDM systems because this requirement can be waived

with additional complexity. It is important to note that any

postcoding scheme can be made equivalent to a precoding

scheme by selecting

At =1LFHNLAfFN, (5)

However, the converse is not true since the precoding matrixcorresponding to a post-coded scheme is necessarily circulant

as explained in the next section.



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SHAH and TEWFIK: DESIGN AND ANALYSIS OF POST-CODED OFDM SYSTEMS 4909

III. PC-OFDM TRANSMITTER DESIGN

To overcome the symbol recovery problem in OFDM sys-

tems at frequency nulls in the channel, we propose PC-OFDM

systems with frequency diversity in the following manner:

1. Explicit Frequency Diversity: This can be achieved by

simple repetitive coding that corresponds to a low cost

upsampling operation in the time domain, as done in [11].

2. Implicit Frequency Diversity: In general, repetitive cod-ing alone does not enhance the system performance signifi-

cantly and we need to spread data symbols across differentsubcarriers that results in implicit diversity.

The spreading operation is similar to multi-carrier code di-vision multiple access (MC-CDMA) except that instead of

multiple users we have multiple streams of data from asingle user. We achieve implicit diversity through the use of

spreading codes in the complex field.

Mathematically, the two forms of diversity can be embedded

in the frequency domain precoding matrix Af such that

Af =IN...IN

NLN

Bf , (6)

where the concatenated identity matrices IN account for

repetitive coding and Bf represents the spreading matrix both

in frequency domain. As PC-OFDM performs postcoding intime domain, we substitute Af from (6) into (5) to get

At =1LFHNL

IN

...IN

NLN

BfFN. (7)

Defining a time domain N N spreading matrix as:Bt := F

HNBfFN, (8)

we can rewrite (7) as:

At =1LFHNL

FN...FN

NLN

Bt. (9)

The last equation follows from the fact that the IFFT of anN N matrix that is repeated L times is simply the N-point IFFT of the matrix followed by upsampling by L. Thus,manipulating the FFT matrices on the right side of (9) results

in a N L N degenerate identity matrix of the form:

INL :=1LFHNL

FN...FN

NLN

=e1 e1+L e1+(N1)L

,

(10)

where ei is the standard N L1 column vector with 1 at ithrow and 0 otherwise. For instance, with N = 2 and L = 2the degenerate identity matrix is I4 =

1 00 00 10 0

. It is obvious

that INL can be obtained by upsampling the identity matrix

IN by L, i.e.,INL = ( L) IN, (11)

and we can write (9) in the form

At = ( L)Bt, (12)where (

L) represents upsampling by L. This shows that

PC-OFDM provides explicit frequency diversity using a low-complexity approach by simply upsampling the post-coded

time domain OFDM symbols. Using (6) and (12), we can

write two mathematically equivalent forms of the transmitted

PC-OFDM symbols as

y =1LFHNL

IN...IN

NLN

Bfb = ( L)BtFHNb. (13)

In the following subsection, we outline the guidelines for the

design of the spreading matrix Bt

and its frequency domain

equivalent Bf.

A. Structure of Spreading Codes for PC-OFDM

Consider a PC-OFDM system that employs time domain

postcoding with Bt as the time domain spreading matrix.

From (5), the equivalent spreading matrix in frequency domain

will be

Bf = FNBtFHN. (14)

While designing spreading codes, we limit ourselves to the

case where the spreading matrix Bf leads to [8]:

C1. Bandwidth efficiencyC2. Constant Euclidean distance: To keep the Euclidean

distance among symbols unchanged after spreading.

C3. Low computational complexity: In general, the com-

plexity of spreading operation is O(N2) but it canbe reduced if efficient structures are chosen for the

spreading matrix.

To achieve bandwidth efficiency in PC-OFDM systems, we

constrain Bf to be square shape. To meet C2 and C3, we

propose our design of the spreading matrix in the following

proposition.

Proposition 1: (a) To reduce the complexity of spread-

ing operation in PC-OFDM systems to O(N), we pro-pose Bf to be circulant of the form:Bf = circ [c] (15)

with c = {c(k)}Nk=1.(b) Define a sequence d = {d(n)}Nn=1 such that

d := FHNc. (16)

For constant Euclidean distance, we select d(n) =ej(n) for n = 1, , N.

Proof: To prove 1 (a), we use diagonalization property of

the Fourier matrix and observe from (14) that the circulant

structure ofBf renders Bt as

Bt = diagFHNc

. (17)



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b S/P

d(n) = ej(n)

N-ptIFFT

CPInsertion

P/S D/A

Oscillator

LX X

Fig. 2. PC-OFDM transmitter block diagram.

Since PC-OFDM systems employ time domain postcoding, thediagonal structure ofBt reduces the complexity of spreadingoperation to O(N).

For 1(b) or constant Euclidean distance, the spreadingoperation must be a unitary transform that requires

BfHBf = IN (18)

This results in BHt Bt = IN according to (14). Since Bt isdiagonal with d(n) as the nth diagonal element, the magnitudeof d(n) must be unity or, in general, d(n) = ej(n) for n =1, , N.

Remark 1: It seems that the circulant structure ofBf re-

stricts the degrees of freedom in the selection of the spreadingmatrix but as we will discuss later careful selection ofd can

achieve the same performance as the precoders reported in the

literature, i.e., as a matrix Bf without the circulant restriction.

Remark 2: In the sequel, we will refer to the sequences

c = {c(k)}Nk=1 and d = {d(n)}Nn=1 as the spreading codesinterchangeably. The two sequences form a Fourier transform

pair according to (16). Indeed, it is the phase angle (n) thatdetermines the spreading code.

Remark 3: The peak-to-average power ratio (PAPR) is an

important parameter in the design and implementation of

OFDM systems. The PAPR depends on the magnitude of time

domain samples of an OFDM symbol [1, pg. 13]. Since thespreading codes for PC-OFDM have unit magnitude (|d(n)| =1), they do not alter the magnitude of time domain samples.Thus, the PAPR of PC-OFDM systems remain unchanged after

spreading. This important feature of PC-OFDM systems is a

direct consequence of the diagonal structure ofBt (or circulant

structure ofBf) that was not available with earlier precodedOFDM systems [2].

Figure 2 shows a block diagram of PC-OFDM transmitter

incorporating the explicit diversity in the form of upsampling

by a factor of L and implicit diversity according to thespreading codes d(n) specified by Proposition 1-[b]. It is

obvious that a particular choice of the phase pattern (n) ofthe spreading codes d(n) = ej(n) will affect the spectrum ofd or simply the frequency domain spreading.

B. Partial Spreading

The spreading matrix Bf in (15) is generally a dense matrix

and is capable of spreading the information across all subcarri-

ers. The dense structure ofBf increases the frequency diversity

and provides robustness against spectral nulls. However, this

spreading increases the receiver complexity exponentially with

the increase in the number of OFDM subcarriers. To circum-

vent this problem, we propose PC-OFDM systems with partial

spreading. Assume that the number of subcarriers N can befactored as N = M Q. We will discuss the optimal value of

bH

(

z

)

B

t

N x N

F

L

(a) PC-OFDM transmitter and the channel model

bB

t

N x N

F

H

0

(

z

L

)

H

1

(

z

L

)

H

L - 1

(

z

L

)

.

.

.

z

- 1

z

- L + 1

.

.

.

.

.

.

.

.

.

.

.

.

L

(b) PC-OFDM transmitter with the polyphase channel model

L

H

0

(

z

)

L

H

1

(

z

)

L

H

L - 1

(

z

)

.

.

.

z

- 1

z

- L + 1

.

.

.

.

.

.

.

.

.

.

.

.

bB

t

F

N x N

(c) PC-OFDM transmitter with equivalent channel model

Fig. 3. Simplified model of PC-OFDM transmitter and the channel.

M in Section VIII. For partial spreading, we consider periodicspreading codes of the form

d(ps)(n) = d(nM) for n = 1, , N, (19)where the superscript (ps) indicates partial spreading. The

frequency domain spreading codes can be written as

c(ps)(k) =

c(m) for k = mQ and m = 1, , M0 otherwise

, (20)

where c(m) = 1MM

n=1 d(ps)(n)ej2nm/M. Thus, in caseof partial spreading, the frequency domain spreading matrixBf

(ps) in (15) contains only P non-zero entries in each row.This results in group spreading such that the information is

spread across Q distinct groups ofM subcarriers. For instance,M = 4 and Q = 2 results in the following partial spreadingmatrix

Bf(ps) =

c(0) 0 c(1) 0 c(2) 0 c(3) 00 c(0) 0 c(1) 0 c(2) 0 c(3)

c(3) 0 c(0) 0 c(1) 0 c(2) 00 c(3) 0 c(0) 0 c(1) 0 c(2)

c(2) 0 c(3) 0 c(0) 0 c(1) 00 c(2) 0 c(3) 0 c(0) 0 c(1)

c(1) 0 c(2) 0 c(3) 0 c(0) 00 c(1) 0 c(2) 0 c(3) 0 c(0)

(21)

In Section VIII we will show how partial spreading helps in

reducing the complexity of a detector for PC-OFDM systems.

IV. PC-OFDM RECEIVER STRUCTURE

In this section, we describe the structure of PC-OFDM

receiver and the operations performed at various stages in the

receiver. The first stage in the digital front end of the receiver

separates multiple copies of the received signal generated dueto the upsampling operation at the transmitter. The next stage

combines these diversity branches using an optimal diversity



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~

LH0(z)

LH1(z)

LHL-1(z)

.

.

.

z-1

z-L+1

.

.

.

.

.

....

.

.

.

L

z

z

L

L

.

.

.

.

.

.

PC-OFDM

Demodulator

/Detector

b

bB

tF

NxN

Fig. 4. Equivalent model of PC-OFDM system with polyphase decomposition of channel.

bH

0(z)

H1(z)

HL-1

(z)

.

.

.

.

.

.

.

.

.

Diversity

Combiner

and

Detector

b

FBf

F

F

F

NxN

NxN

Fig. 5. Simplified model of PC-OFDM system.

combining scheme. The third stage implements the detector

as discussed in Section V.To produce a low-complexity PC-OFDM receiver, we con-

sider the transmitted PC-OFDM symbols in the form y =( L)BtFHNb. The upsampling operation at the transmittermanifests itself as frequency diversity at the receiver. To

understand this, we first apply multirate signal processing

concepts to obtain a simplified model for transmitter. If

H(z) denotes the z-transform of channel transfer function inFig. 3(a) then by definition H(z) :=

Lh1l=0 h(l)z

l. To makeuse of the upsampling operation at the transmitter, we use

a polyphase representation of the channel transfer function

given by H(z) =

L1p=0 z

pHp(zL), where we decompose

the channel into L phases and Hp(z) := Lh1l=0 h(lL +p)zlrepresents the pth phase of H(z). Figure 3(b) depicts the PC-OFDM transmitter with the polyphase model of the channel

that can also be redrawn by interchanging upsampling and

filtering (transmission through the channel) operations as

shown in Fig 3(c). The upsampling operation keeps different

phases of the channel separated and the received symbols

appear as if they were transmitted through different phases ofthe channel. Thus, a PC-OFDM transmitter sees an L-branchchannel and provides L copies of the same transmitted symbolat the receiver.

The polyphase decomposition of channel leads us to design

a dual system with downsampling and delay operations at thereceiver as shown in Fig 4. With the help of this structurewe can separate L phases of the received signal and get Lcopies of the transmitted symbols, each having gone through

a different phase of the channel. This results in a simplified

model of PC-OFDM system with L branch channel as shownin Fig. 5. Note that this decomposition also shows that PC-

OFDM effectively implements a frequency domain coding

scheme with very low complexity.

After removing the cyclic prefix at the receiver, the received

symbols at the pth phase or branch of the channel can beexpressed as

up =

HpBtFHNb

+ p,

(22)

where Hp represents the N N circulant matrix of the p-th

phase of {h(l)}Lh1l=0 . For the sake of mathematical conve-nience, substitute Bt with its equivalent precoding matrix in

frequency domain as given by (14) to obtain

up = HpFHNBfb+ p, (23)

The N-point FFT operation at the receiver will render thecirculant channel matrix Hp as diagonal, i.e.,

HpD := FNHpFHN = diag[FNhp], (24)

where hp is the pth phase of the channel {h(l)}Lh1l=0 that iszero-padded to make it N

1. Thus, the demodulated OFDM

symbols at the pth diversity branch of the receiver are givenby:

up = HpDBfb + p, for p = 1, 2, , L (25)Concatenate the received symbols from all diversity branches

to obtain an N L 1 vector u of the formu = HBfb+ , (26)

where H :=

H1D...HLD

is N L N channel matrix and =

1

.

..L

is N L 1 vector of additive white Gaussian noise.It is important to note that if we use the full size (N L-point)

IFFT at the receiver, the channel matrix will appear differently

in the frequency domain but represents the same channel

energy or characteristics and hence the same performance.

V. DETECTION ALGORITHMS FOR PC-OFDM SYSTEMS

In PC-OFDM systems, the task of the detection algorithm

is two-fold: 1) combine different diversity branches (diversity

combining) at the receiver, and 2) unfold the spreading op-

eration (equalization). Recall that the diversity branches in a

PC-OFDM system result from the upsampling operation at thetransmitter. Among different diversity combining techniques,

we consider the maximal ratio combining (MRC) at the PC-

OFDM receiver.The optimal detector for b in (26) is the one that minimizes

the average probability of error. This is achieved by maximum

likelihood (ML) detection that detects the transmitted symbols

based on the following minimization:

b = argminbB

||uHBfb||2, (27)where ||.|| represents the l2 norm and B is the finite set ofsignal constellation. It can be shown that the use of MRC at

the receiver simplifi

es the ML detection criterion in (27) tob = argmin

bB||HHuHHHBfb||2. (28)



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Maximum likelihood detection, though optimum, is a costly

operation and is practically not feasible for large N. In thefollowing subsections we explore the use of three suboptimaldetectors that can be implemented with reduced complexity.

A. Zero Forcing (ZF) Detector

A simple suboptimal detector is the zero forcing (ZF) detec-

tor. Contrary to (28), the ZF detector solves an unconstrainedleast-squares problem of the form:

bZF = argminb

||HHuHHHBfb||2, (29)

and obtains an estimate ofb in the form:

bZF = BfH

L

p=1

HHpDHpD

1 Lp=1

HHpDup

, (30)

where HpD is defined in (24). The data symbols are subse-

quently detected from the estimate bZF using hard decision

according to the modulation scheme used.

B. Successive Interference Cancellation (SIC)

We found through simulations that the performance of ZF

is quite poor. A possible low complexity solution is to apply

the idea of successive interference cancellation (SIC) that was

first proposed for space-time codes in [14]. In successive

interference cancellation, we detect a symbol that corresponds

to the maximum channel gain using ZF detector of (30). As-

suming we made the correct decision, the effect of the detected

symbol is subtracted from the vector of received symbols and

the process is iterated such that we form a better estimate ofeach of the symbols at the end of the iteration. We refer to this

detector as ZF-SIC. Writing (26) in the form u = Gb +

where we define G :=

H1D...HLD

Bf := [g1 gN] with gi as

the ith column ofG. Assuming that G is ordered according tochannel gain, we can summarize ZF-SIC algorithm as shown

in Algorithm 1.

Algorithm 1 ZF-SIC Detector

1: initialization; G0 = G, r0 = u.

2: for i = 1 to N do3: Using Gi1, obtain ZF estimate bZF from (30).4: Use hard decision detector to obtain bi5: Compute ri = ri1 gibi.6: Update: Gi = [gi+1 gN]7: end for

C. Quasi Maximum Likelihood (Q-ML) Detector

The non-linear optimization in (28) is commonly referred

to as an integer least-squares problem that is known tobe unsolvable in polynomial time. An approximate solution

to the optimization in (28) can be found by transforming

the problem to convex optimization. The objective function

F(b) := ||HHuHHHBfb||2 in (28) can be expressed asF(b) =bHBfHHHHHHHBfb

2uHHHHHBfb+ uHHHHu(31)

To simplify notations, we define J := HHHBf that leads usto write

F(b) = tr[JHJbbH] 2uHHJb+ uHHHHu (32)where tr[.] represents the trace operator. For constellationswith |bi|2 = 1, the integer least-squares problem of (28) canbe equivalently written as

b = arg min

tr[JHJX] 2uHHJbsubject to X = bbH, b RNXii = 1, i = 1, , N. (33)

The constraint X = bbH translates into rank-1 criterion forX and makes (33) a nonconvex optimization problem [15].

The semi-definite relaxation in [15] replaces X = bbH witha convex relaxation X

bbH and converts (33) into a semi-

definite programming (SDP) problem of the form

b = arg min

tr[JHJX] 2uHHJbsubject to X bbH, b RNXii = 1, i = 1, , N. (34)

Kisialiou and Luo [16] presented an efficient implementa-

tion of SDP problem in (34) to obtain the quasi maximumlikelihood (Q-ML) solution of (28). The complexity of the

Q-ML detector is O(N3.5). In our simulations, we used theMATLAB scripts for Q-ML provided by the authors of [16].

V I . PROBABILITY OF ERROR ANALYSIS

The probability of error analysis of PC-OFDM systemsis identical to that of space-time coded systems that has

been studied extensively. We adopt the average pairwise

error probability (PEP) technique that has been derived in

similar contexts, e.g., in [2] and [17]. By definition, the PEP

is the probability of erroneously detecting b when b wastransmitted. It has been shown in recent research that the

criteria commonly used to design codes for additive white

Gaussian noise (AWGN) channels have to be adjusted when

dealing with a fading channel (see [18] and references therein).

As we shall see soon, the performance of a code over fading

channels does not depend on the Euclidean distance between

the codewords but it is closely related to the spectrum and theautocorrelation of the spreading codes. In this paper, our maingoal is to design the codes for fading channels. Nevertheless,

it is important to see the system performance over AWGN

channels. Therefore, we consider the probability of error for

AWGN and Rayleigh fading channels separately.

A. AWGN Channels

It is well known that for AWGN channels the Euclidean

distance of the codewords determines the probability of er-ror [19]. Considering ML detection, the PEP of PC-OFDM

systems for AWGN channels can be expressed as

Pr(b b) = Q ||Bf(b b)||

2No

, (35)



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where No/2 is the power spectral density of additive whiteGaussian noise and Q(.) is the Gaussian tail function definedas Q(x) := 1/(

2)

x e

t2/2dt. If we define d := ||Bf(bb)|| as the Euclidean distance between the codewords thensimplifying the square of the norm, we obtain

d2 = (b b)HBfHBf(b b). (36)Thus, the Euclidean distance between the coded symbols, can

be different from the Euclidean distance between the uncodedsymbols. However, the PC-OFDM coding matrix Bf forms

a unitary transform pair (cf. (18)) and hence the Euclidean

distance remains unchanged. Thus, PC-OFDM do not perform

poorly in AWGN channels.

B. Uncorrelated Rayleigh Fading Channels

In order to find the PEP for a Rayleigh fading channel with

Lh taps (see [2] for details), we define a matrix

Ae := (DeV)HDeV (37)

where V is N Lh truncated FFT matrix with [V](k,l) =ej2kl/N and De = diag[Bf(b b)]. Now, for Rayleighfading channels with uncorrelated paths, the PEP is given by

Pr(b b)

1

4No

LGd Gdl=1

ll

L, (38)

where l = E[|h(l)|2] is the variance of the fading chan-nel paths and 1, , Lh are the eigenvalues of Ae. Theparameter Gd is termed as the diversity gain and will bediscussed in the next section. The factor L in the exponent isthe manifestation of the Lth order explicit diversity introducedin PC-OFDM systems through upsampling.

VII. SELECTION OF SPREADING CODES

In this section, we outline the criteria for the design ofspreading codes for Rayleigh fading channels. Our design

criteria is based on minimizing the PEP given by (38). ForPC-OFDM systems, the PEP depends on the following two

factor, the diversity gain Gd and the coding gain Gc that aredefined as

Gd := minDe

rank[(DeV)HDeV], (39)

and

Gc := minDe det[(DeV)HDeV]. (40)

Roughly speaking, the diversity gain represents the slope of

the PEP curve especially at high SNR. It is related to the rank

of Ae [17]. The coding gain controls the shift in the PEP

curve and depends on the product of eigenvalues {l}Lhl=1 ofAe or in otherwords the determinant of Ae [17]. To design

spreading codes with minimum probability of error, we seek

to maximize the minimum of Gd and Gc using the rank andthe determinant criterion, respectively. For BPSK modulation,

we summarize the code design criteria in the following twotheorems.

Theorem 1: (Maximizing Gd using the rank criterion) ThePC-OFDM system achieves the maximum available diversity

gain if the number of non-zero entries in c Lh. In other

words, the spectrum of d should have at least Lh non-zeroentries to maximize Gd.Proof:

Consider the definition of Gd as given in (39) and notethat rank[GHG] = rank[G] for any matrix G. For BPSKmodulation, the minimum of rank occurs when b b = eiwhere ei is the standard N1 column vector with 1 at the ithentry and zero otherwise. Without loss of generality, consider

i = 1 then from (15) we have De = diag[c] and

Gd = rank[(diag[c])V]. (41)

Since V is a full column rank matrix, we can write Gd as

Gd = min{rank[diag[c]], Lh}. (42)Thus Gd achieves the maximum value Lh if number of non-zero entries in c Lh.

Theorem 2: (Maximizing Gc using the determinant crite-rion) Consider a PC-OFDM system with spreading codes of

the form d(n) = ej(n) that satisfies Theorem 1. Define theperiodic autocorrelation of the code as

() :=1

N

Nn=1

d(n)d(n + N) for = 0, , N1 (43)

where .N represents the modulo N operation and repre-sents the complex conjugate.

(a) For BPSK modulation, the matrix Ae in (37) can be

expressed as

Ae =

1 (1) (Lh 1)(1) 1 (Lh 2)

......

. . ....

(Lh 1) (Lh 2) 1

(44)

(b) The PC-OFDM system with maximum coding gain

requires Ae = ILh .

Proof: See Appendix I for proofs.

Remark 4: In essence, Theorem 2(b) requires the first

Lh 1 lags of the periodic autocorrelation of the spreadingcodes to be zero. Since the spreading codes d(n) = ej(n)

for PC-OFDM systems depend on their phase pattern (n),Theorem 2(b) emphasizes the importance of the selection of

the phase pattern.

Remark 5: Another criterion that is commonly used in

the design of space-time codes to maximize Gc is thetrace criterion. Using this criterion, Gc is defined as Gc =minDe tr[(DeV)

HDeV]. For BPSK, it reduces to Gc =tr[Ae] = N for the spreading codes of the form d(n) =ej(n). In other words, the trace ofAe does not depend on thechoice of spreading codes. Thus, the trace criterion does not

help us determine the spreading codes with maximum coding

gain for PC-OFDM systems.

A. Examples of spreading codes

We now present some examples of spreading codes fol-

lowing the design criteria of Theorems 1 and 2. Note thatTheorem 2 only holds for sequences that satisfy Theorem 1.

So, our starting point in the design of spreading codes is to



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find sequences with sufficient number of non-zero entries in

the spectrum for most practical purposes. In this paper, we

consider three such sequences:

1. Maximally flat spreading codes (or Chus Code): The

first sequence we select to maximize the coding gain is

the one that has flat spectrum. A flat spectrum ensures

that statement b in Theorem 2 holds. To design codes

with flat spectrum, we make use of the stationary-phaseconcept (a popular concept in the field of non-linear

frequency modulation [20]) that states that the magnitude

spectrum of the signals of the form d(n) = ej(n)

is proportional to the second derivative of (n) withrespect to n. Thus, the phase pattern (n) proportionalto n2 will result in flat magnitude spectrum. Later, wefound that these codes are similar to Chus code [20]

that also contains an n2 term. For this paper, we used(n) = ejn

2/N for n = 1, , N and refer to thesecodes as maximally flat spreading codes.

2. Costas Sequence: Costas sequences [20] refer to a par-ticular permutation of N consecutive numbers. These se-

quences are another candidate for spreading codes as theypossess good autocorrelation properties. We use Costas

sequence to select the phase pattern of two different

spreading codes. For the first one, we use the spreading

codes of the form d(n) = ejnC where nC refers to theCostas permutation of integers from 1 to N. For example,for N = 8, nC = {2, 6, 3, 8, 7, 5, 1, 4}. This choice resultsin polyphase spreading codes. The second set of spreading

codes we consider are of the form d(n) = ejnC . Thesebinary (biphase) spreading codes simplify the encodingprocess further by limiting d(n) to be +1 or 1.

3. Systematic search for optimal binary sequence: Moti-

vated from the performance of binary spreading codesusing Costas phase pattern, we use a systematic method to

search for binary spreading codes with maximum coding

gain. We limit our search to balanced binary sequences

with equal number of +1s and -1s. For given N, welist balanced sequences and select the one that results in

Ae = ILh for sufficiently large Lh.

The simulation results of the performance of PC-OFDMsystems with these sequences are given in Section IX.

VIII. LOW COMPLEXITY DETECTOR AND COMPLEXITYCOMPARISON

The detection algorithms discussed in Section V have com-

plexity that increases exponentially with the increase in the

number of OFDM sub-carriers N. For instance, the complexityof the ZF suboptimal detector is O(N3). To address the highcomplexity of detectors for PC-OFDM systems, we present

a low complexity detector in this section. We also present a

detailed complexity comparison of PC-OFDM and precoded

OFDM systems.

To reduce the complexity of detectors, we consider a

suboptimal spreading using the partial spreading technique of

Section III-B with N = M Q. In this case, the data symbols arespread across Q distinct groups of M subcarriers. FollowingTheorem 1, if we choose M to be equal (or larger) than thechannel length Lh we can achieve the maximum diversity gain

available in the channel. An optimal choice is to select the

smallest of all M with M Lh and N/M an integer. Wewill now show that partial spreading is capable of reducing thecomplexity of any of the detectors discussed in Section V. This

reduction in complexity comes with a little loss in performance

as we will show in Section IX shortly. The key to low-

complexity detector for partial spreading is the decoupling

algorithm we explain below.

A. Decoupling algorithm for partial spreading

Consider a PC-OFDM system with partial spreading and

N = M Q. Assume that the N 1 vector of data symbolsb in (27) can be divided into Q groups, namely b1, , bQ.Each group {bi}Qi=1 contains M data symbols in a permutedorder and concatenation of all the groups results in

b := [bT1 bTQ]T = PT b, (45)where P is an N N permutation matrix and T representsthe transpose operation. Similarly, we use u1, , uQ todenote the Q groups ofu each representing an M 1 vectorof received symbols in a permuted order such that

u := [uT1 uTQ]T = PT u. (46)We define

H := PTHP (47)

to represent the permuted diagonal entries of the channel ma-

trix and split it into Q groups such that diag[H1 HQ] =H. If we denote the block diagonal spreading code matrix

with Bf D(ps) then it can be written as

Bf D(ps) =

B

(ps)f

B(ps)f. . .

B(ps)f

, (48)

where B(ps)f

is an MM circulant matrix. Now, the detectionrule can be simplified as mentioned in the following proposi-

tion.

Proposition 2: The ML detection of (27) can be decoupled

into Q simpler ML problems of the form

bi = arg minbiB

||ui HiB(ps)f bi||2 for i = 1, , Q, (49)

where bi represents the ML estimate of bi.Proof: First note that in case of partial spreading, the fre-

quency domain circulant spreading matrix can be transformed

into a block diagonal matrix by pre and post multiplicationwith permutation matrices. Thus, the block diagonal spreading

code matrix Bf D(ps) can be expressed as

Bf D(ps) := PT Bf

(ps)P. (50)

Since permutation matrices are orthogonal, we can also write

Bf(ps) = PBf D

(ps)PT . (51)

For better exposition of the proposed decoupling algorithm andwithout loss of generality, we focus on a PC-OFDM system

with L = 1. The same algorithm can be applied to PC-OFDM



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systems with L > 1. Substituting Bf(ps) from (51) into the ML

problem of (27), we obtain

b = arg minbB

||uHPBf D(ps)PTb||2. (52)

Using (45), (46), (47) and the orthogonality of P , we can

write (52) as

b = arg minbB ||

u

HBf D

(ps)b

||2. (53)

Now, the block diagonal structure ofBf D(ps) can help decouple

the original N-dimensional ML problem of (27) into Qsimpler ML problems each of dimension M as given by (49).

Remark 6: The proposed decoupling algorithm can reduce

the complexity of a detector from O(N) to O(QM) whereM is of the order of channel length Lh N. Hence, partialspreading with decoupled detector reduces the complexity

considerably with marginal degradation in the performance.

The steps of the decoupling algorithm can be summarized

as:

1. For a givenBf(ps), use elementary row-column operationsto transform Bf

(ps) into a block diagonal form Bf D(ps) as

given by (50). Also determine P from (50).

2. Compute b, u and H using (45), (46) and (47), respec-tively.

3. Solve a lower complexity ML problem of dimension Mgiven in (49) to estimate bi for i = 1, , Q.

To illustrate the decoupling algorithm, we continue with the

example of Section III-B. The spreading code matrix given

in (21) can be transformed into block diagonal by selecting

P = [e1 e3 e5 e7 e2 e4 e6 e8]. This leads to Bf D(ps) =

B(ps)fB(ps)

f

where B(ps)f = circ[c(1) c(2) c(3) c(4)].B. Complexity and Power Comparison with Precoded OFDM

Systems

To highlight the low complexity of proposed PC-OFDM

systems, we present a detailed complexity and power com-

parison between PC-OFDM and precoded OFDM systems.

The proposed PC-OFDM system is capable of lowering the

implementation cost of coded OFDM system. For instance,

a PC-OFDM transmitter with N source symbols requiresan N-point IFFT module with computational complexity of

O(Nlog N) per N data symbols. In contrast, a redundantprecoded OFDM transmitter [2] with N L N (where L Rand L 1) encoding has a computational complexity ofO(N L log N L). Similarly, the polyphase decomposition ofchannel in PC-OFDM will allow us to use N-point FFTsin all the L branches. This results in total complexity ofO(N L log N) for PC-OFDM receiver while a redundant pre-coded OFDM receiver has a computational complexity of

O(N L log N L) .In addition to the savings in IFFT/FFT modules, the unique

encoding scheme of PC-OFDM is a low cost operation and

requires only O(N) complex multiplications as comparedto

O(N2L) complex multiplications/additions in precoded

OFDM. While the complexity of a detector for PC-OFDM sys-

tem with full spreading is similar to that of precoded OFDM

TABLE ICOMPARISON OF COMPUTATION COST OF DIFFERENT OPERATIONS IN

PRECODED AND POST-CODED OFDM SYSTEMS

PC-OFDM with

Pre-coded PC-OFDM partial spreading

OFDM (full spreading) (N = M Q)

IFFT O(N L logN L) O(N logN) O(N logN)

FFT O(N L logN L) O(NL logN) O(NL logN)

Encoding O(N2L) O(N) O(N)

Detection O(N3.5) O(N3.5) O(QM3.5)

(Q-ML)

TABLE IICOMPARISON OF REQUIRED CLOCK RATE FOR DIFFERENT MODULES

(1/T = CLOCK RATE IN HZ)

Transmitter

IFFT Digital-to-Analog Converter FFT

Pre-coded OFDM L/T L/T L/T

PC-OFDM 1/T L/T 1/T

systems, the use of partial spreading can reduce the complexity

of PC-OFDM systems. Table I compares the computation

cost of FFT/IFFT modules and encoding/decoding operations

for precoded and post-coded OFDM systems. For PC-OFDM

systems with partial spreading (N = QM), the complexityof the Q-ML detector can be reduced from O(N3.5) toO(QM3.5) with M N. It is important to note that thecomplexity of partial spreading detector is much lower than the

complexity of the linear detector. For instance, the complexity

of the ZF detector is O(N3) that is much higher than thatof the partial spreading detector as shown in Table I. The

reduced complexity of PC-OFDM systems make them suitablefor wireless personal area networks.

It is also important to mention that the IFFT/FFT opera-tions in PC-OFDM are performed at the information symbol

data rate. However, in precoded OFDM these operations areperformed after encoding and at a higher sampling rate. Since

power consumption of these DSP modules is proportional to

clock frequency, PC-OFDM saves power by computing the

IFFT/FFT operations at the lower rate. The comparison of

required clock rate for different modules in precoded OFDM

and PC-OFDM systems is shown in Table II.

I X . SIMULATION RESULTS

We perform simulations to compare the bit error rate(BER) of different spreading codes and detection algorithmsdiscussed in the paper. For all simulations, we define SNR as

signal to noise ratio per bit and computed it as Eb/No whereEb is the bit energy and No/2 is the noise variance. We useBPSK modulated symbols and transformed them to OFDM

symbols. All simulation results in this paper correspond to

L = 2 that results in a code rate of 1/2. For Figs. 6, 7and 8, we use an uncorrelated Rayleigh fading channel with

Lh = 5. Thus, signals on each subcarrier undergo indepen-dent Rayleigh fading and additive Gaussian noise. For these

channels we use a cyclic prefix (CP) that is 5 symbols long.

In Fig. 6, we compare the performance of PC-OFDM systemwith different spreading codes mentioned in Section VII-A.

We use N = 16 (FFT size), a Costas permutation pattern given



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0 2 4 6 8 10 12 14

105

104

103

102

101

SNR in dB

Probab

ilityofbiterror

Maximally flat sequence

Costas: ejn_C

Costas: ej n_C

Balanced binary sequence

Fig. 6. BER of PC-OFDM systems with different spreading codes.

0 2 4 6 8 10 12

105

104

103

102

101

100

SNR in dB

Probabilityofbiterror

N=16; L=2

PCOFDM (ZFSIC)

PCOFDM (QML)

Fig. 7. BER of PC-OFDM systems with different detection algorithms.

by nC = {1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6} anda balanced binary spreading code with maximum coding gain

(Ae = I5) is d = [1 + 1 1 1 + 1 1 1 1 1 +1 + 1 + 1 1 + 1 + 1 + 1]. It is obvious from Fig. 6 that allof these spreading codes perform equally good. However, thebalanced binary spreading code with maximum coding gain

requires minimum computations.

We next evaluate the performance of PC-OFDM systems

with different detection algorithms. We consider a linear

detector in the form of ZF-SIC and the Quasi-Maximum

Likelihood detector as explained in Section V. The BER

results of PC-OFDM system with these two detectors are

shown in Fig. 7. It is clear from the figure that ZF-SIC is a

low complexity alternative to Q-ML at a slightly higher error

rate. In Fig. 8, we assess the effect of partial spreading on the

performance of PC-OFDM systems. Partial spreading provides

a trade-off between low complexity linear detectors (e.g. ZF)

and suboptimal spreading. While both ZF detector and partialspreading are capable of reducing the detector complexity, we

have shown in Section VIII-B that partial spreading can reduce

0 2 4 6 8 10 12 14 16 1810

6

105

104

103

102

101

100

101

SNR in dB

Probab

ilityofbiterror

Full spreading (N=M=32) with ZFSIC

Partial spreading (M=8, N=32) with QML

Full spreading (N=M=32) with QML

Fig. 8. BER of PC-OFDM systems with partial spreading.

0 5 10 15

104

103

102

101

100

SNR in dB

Probabilityofbiterror

Pulse OFDM (QML)

CFC

Precoding (Q

ML)PCOFDM (QML)BICM OFDM

Rotated transform precoding

Fig. 9. BER of different coded OFDM systems over UWB channel (CM1).

the complexity significantly. Here, we use simulation results to

evaluate the loss in performance when using partial spreading

or the ZF detector. For partial spreading, we assume M = 8and Q = 4 for a PC-OFDM with N = 32 subcarriers. Fig. 8compares the BER results of PC-OFDM systems with partial

and full spreading using Q-ML detector. The results in Fig. 8

show that the loss in performance due to partial spreading is

marginal. However, the ZF detector with full spreading suffers

severe performance degradation due to suboptimal detection.

This justifies the use of partial spreading as compared to a

linear detector in low complexity PC-OFDM receivers.

In Fig. 9, we compare the BER performance of different

coded OFDM systems over UWB Channels [21] for N = 128and L = 2 using Q-ML. The first system we consider isbit interleaved coded modulation (BICM) OFDM system.

OFDM with BICM is widely used in wireless local area

networks [5]. For BICM, we used rate 1/2 convolution codes

with bit interleaving as recommended in [5] and modulate theencoded and interleaved bits using BPSK. The BER results

for BICM OFDM over UWB channel are shown in Fig. 9.



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Fig. 9 clearly shows that BICM alone performs poorly as

compared to precoded or post-coded OFDM systems. To

compare postcoded and precoded OFDM systems, we considerthe complex field precoders (CFC-precoders) proposed in [2].

For completeness, we examine the performance of precoders

reported in [8] that are based on a rotated transform. We also

computed the BER performance of pulsed-OFDM [11]. The

results are shown in Fig. 9. The slope of the curve shows

that pulsed-OFDM could not achieve the full diversity orderavailable in the system. The comparison between precoded

and PC-OFDM systems shows that the low complexity design

of PC-OFDM systems does not result in any performance loss.

X. CONCLUSIONS

We discussed the design principles for PC-OFDM transmit-

ter and receiver that offers low-complexity equivalent of tradi-

tional precoded OFDM systems. PC-OFDM systems achieved

low-complexity objective by manipulating the OFDM sym-

bols in the time domain. The PC-OFDM receiver separates

and combines different diversity branches and performs joint

detection of data symbols. The proposed partial spreadingscheme for low complexity receivers showed marginal loss in

performance. The probability of error analysis of PC-OFDM

systems enlightened different design criterion for PC-OFDMsystems. We performed simulations to assess different choices

of the spreading codes and the detection algorithms for PC-OFDM systems.

APPENDIX A

PROOF OF THEOREM 2

(a) Recall from (40), that Gc = minDe det[(DeV)HDeV].

In case of BPSK modulation, the minimum of the above

determinant occurs when b b = ei. Without lossof generality, consider i = 1 then from (15) we haveDe = diag[c] and

Gc = det[VH(diag[|c|2])V], (54)

where |c| = [ |c(1)| |c(2)| |c(N)| ]T and |c(k)|represents the magnitude of complex number c(k). Withthis, the Ae matrix for BPSK that corresponds to the

minimum Gc over all De can be expressed as

Ae := VH(diag[|c|2])V. (55)

Since c represents the DFT of d,

|c

|2 represents the

energy spectral density [22] of d. Define a diagonalmatrix containing the energy spectral density sd of d

asSd = diag[|c|2] := diag[sd], (56)

to obtain

Ae = VHSdV. (57)

To simplify (57), let us consider a matrix of the form

P := FHNSdFN. Due to pre and post multiplication withDFT matrices, P is a circulant matrix of the form P =circ[FHNsd]. But F

HNsd represents the inverse DFT of

the energy spectral density ofd. Thus, FHNsd is simply

the autocorrelation ofd and we represent it as [22]

:= FHNsd, (58)

with = [(0) (N 1)]T and (.) as definedin (43). With d(n) = ej(n), we have (0) = 1 and() = (N ) for = 1, , N 1. The circulantmatrix P can be written as

P = circ

1 (1) (N/2 1) (N/2)(N/2 1) (1) (59)

From (57), Ae is a submatrix of P and from (59) it

is obvious that Ae is a Hermitian Toeplitz matrix withentries given by (44).

(b) From (54) and (55), the coding gain can be written as

Gc = det[Ae]. (60)

If 1, , Lh are the eigenvalues of Ae then fromthe properties of the correlation matrix l 0 forl = 1, , Lh. Invoking the relationship between thearithmetic mean (AM) and the geometric mean (GM)

of non-negative numbers, we have

AM of

{l

}Lhl=1

GM of

{l

}Lhl=1

1

Lh

Lhl=1

l Lhl=1

l

1/Lh(61)

where the equality holds if l = l. Note thatLhl=1 l = tr[Ae] = Lh and

Lhl=1 l = det[Ae].

Therefore, the inequality in (61) reduces to

(det[Ae])1/Lh 1. (62)

Taking the log of both sides, we obtain an upper boundon the determinant of the correlation matrix, i.e.,

det[Ae]

1. (63)

The determinant ofAe achieves the maximum value of 1

when l = 1 l. Since Ae is hermitian, its eigen vectorsare orthonormal. Thus, with l = 1 l, Ae = ILh .

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for ultrawideband personal area networks, IEEE Wireless Commun.,vol. 10, pp. 1421, Dec. 2003.

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S. Faisal A. Shah received the B.S. degree fromNED University of Engineering and Technology,Karachi, Pakistan, in 1998 and the M.S. degree fromKing Fahd University of Petroleum and Minerals,Dhahran, Saudi Arabia, in 2001, both in electricalengineering. He received the Ph.D. degree in elec-trical engineering from the University of Minnesota,Minneapolis, MN in 2008. From 2001 to 2004,he was a lecturer in the Department of ElectricalEngineering at University of Sharjah, Sharjah, UAE.From 2004 to 2006, he was a graduate research as-

sistant in the Department of Electrical Engineering, University of Minnesota.He has worked as a DSP intern at KeyEye Communications, Sacramento, CA,from 2006 to 2007. Since February 2008, he has been with Azimuth Systems,Acton, MA where he is a senior DSP engineer working on the design of aMIMO channel emulator for 4G systems. His research spans the fields ofsignal processing and wireless communications, with particular emphasis onOFDMA systems, distributed estimation in wireless sensor networks, adaptivechannel estimation and design of low-complexity DSP algorithms.

Ahmed H Tewfik received his B.Sc. degree fromCairo University, Cairo Egypt, in 1982 and hisM.Sc., E.E. and Sc.D. degrees from the Mas-sachusetts Institute of Technology, Cambridge, MA,in 1984, 1985 and 1987 respectively. Dr. Tewfikhas worked at Alphatech, Inc., Burlington, MA in1987. He is the E. F. Johnson professor of ElectronicCommunications with the department of ElectricalEngineering at the University of Minnesota. Heserved as a consultant to several companies, includ-

ing MTS Systems, Inc., Eden Prairie, MN, Emerson-Rosemount, Inc., Eden Prairie, MN, CyberNova, Milipitas, CA, Macrovision,Santa Clara, CA, Visionaire Technology, Fremont, CA, Ipsos, New York,InterDigital Communications, King of Prussia, PA, Keyeye Communications,Sacramento, CA. Transoma Medical, Arden Hills, MN and St. Jude Medical,Minnetonka, MN. He worked with Texas Instruments and Computing DevicesInternational. From August 1997 to August 2001, he was the President andCEO of Cognicity, Inc., an entertainment marketing software tools publisherthat he co-founded, on partial leave of absence from the University ofMinnesota. His current research interests are in genomics and proteomics,audio signal separation, wearable health sensors, brain computing interfaceand programmable wireless networks.

Prof. Tewfik is a Fellow of the IEEE. He was a Distinguished Lecturer ofthe IEEE Signal Processing Society in 1997 - 1999. He received the IEEEthird Millennium award in 2000. He was elected to the board of governors ofthe IEEE Signal Processing Society in 2005. He was invited to be a principallecturer at the 1995 IEEE EMBS summer school. He was awarded the E.F. Johnson professorship of Electronic Communications in 1993, a Taylorfaculty development award from the Taylor foundation in 1992 and an NSFresearch initiation award in 1990. Prof. Tewfik delivered plenary lectures atseveral IEEE and non-IEEE meetings and taught tutorials on bioinformatics,ultrawideband communications, watermarking and wavelets at major IEEEconferences. He was selected to be the first Editor-in-Chief of the IEEE SignalProcessing Letters from 1993 to 1999. He is a past associate editor of the IEEETrans. on Signal Proc., was guest editor of special issues of that journal, the

IEEE Trans. on Multimedia and the IEEE Journal of Selected Topics in SignalProcessing. He is currently an Associate Editor of the EURASIP Journalon Bioinformatics and Systems Biology. He also served as the president ofthe Minnesota chapters of the IEEE signal processing and communicationssocieties from 2002 to 2005.

IMP - M - Design and Analysis of Post-Coded OFDM Systems

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