performance limits of owss: a spectrally efficient wlan system

p

nt

aimedulses,e powera widelization0–40%S andof

r. Then,

eand ishannel

o trulytimation

andulation

channel,of

Digital Signal Processing 15 (2005) 347–366

www.elsevier.com/locate/ds

Performance limits of OWSS: A spectrally efficieWLAN system

V.K. Jaina,∗, D. Divakarana, B.A. Myersb

a University of South Florida, Tampa, FL 33620, USAb Conexant Corporation, Palm Bay, FL 32905, USA

Available online 20 January 2005

Abstract

Orthogonal wavelet spread spectrum (OWSS) is a new wireless LAN signaling schemeat bit rates of 100 Mbps or higher. Starting with power complementary full-tree wavelet pit spreads them using a suitable set of PN codes. The resulting system pulses preserve thcomplementary property and satisfy the generalized Nyquist criterion, while possessing bothtime and a wide frequency support. Benefits from these properties include successful equain spite of deep local fades caused by multipath channels, and high spectral efficiency—3higher than OFDM for comparable net bit rates. This paper presents an overview of OWSexplores thefundamental limits to its performance. In the overview, it discusses the propertiesthe OWSS pulses, the transmit signal spectrum, and the structure of the transmitter/receivethe investigation of fundamental limits to the performance is carried outby formulating the systemas a multi-rate signal processing system, using hierarchical matrices, and thereupon minimizing thtotal mean-square error (TMSE). The TMSE governs the BER performance of the system,defined as the sum of the MSE of the unequalized residual error and the MSE due to the cnoise amplified by the forward equalizer. We formulate the problem at the chip level so as tdiscern the fundamental limits to the performance of the equalizer. This approach enables esof the optimum equalizer for mitigating the effect of the multipath channel, prior to correlationdetection blocks embedded in the FE–DFE loop, and thereby the system performance. Simresults demonstrate its effectiveness. For a 108 Mbps system with a 50 ns delayspreada low BER of 10−5 and spectral efficiency up to 5.2 bits/s/Hz can be achieved at an Eb/No19 dB.

* Corresponding author. Fax: +1 813 974 5250.E-mail address:[email protected] (V.K. Jain).

1051-2004/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2004.12.009

348 V.K. Jain et al. / Digital Signal Processing 15 (2005) 347–366

es ofsedcyencean berposelimits

es, thei-levelmini-alizer. Theof thee FE.ory. Ino

era-ak ofsymbol

de-pled

e thestudyelse-

irelesscation

wide

tion 2eceiverpproachtion.stemmarks

2005 Elsevier Inc. All rights reserved.

Keywords:Wireless LANs (WLANs); Orthogonal wavelet spread spectrum (OWSS); Total mean-squareerror (TMSE); Multi-level matrices

1. Introduction

A recently developed WLAN signaling system called OWSS [1–4] targets bit rat100 Mbps or higher. Unlike the pulses used in OFDM, CDMA, and TDMA, it is baon a new family of pulses which have botha wide time support and a wide frequensupport, and is 30–40% more bandwidth efficient than OFDM [3]. As a consequof the wide frequency support, effective equalization in a multipath environment cachieved using an FE–DFE structure together with the LMS algorithm [4]. The puof this paper is to present an overview of OWSS and to explore the fundamentalto its performance. In the overview, it discusses the properties of the OWSS pulstransmit signal spectrum, and the structure of the transmitter/receiver. Then a multmatrix formulation is employed to model the signal processing system. The totalmum mean-square error (TMSE) for the forward equalizer–decision feedback equ(FE–DFE) structure is derived in a closed form, and thereupon minimized rigorouslyTMSE governs the BER performance of the system, and is the sum of the MSEunequalized residual error and the MSE due to the channel noise amplified by thSimulation results on a 108 Mbps system demonstrate the effectiveness of the theparticular, for a 50 ns delayspread channel, a low BER of 10−5 can be achieved at an Eb/Nof 19 dB.

Although a symbol-level formulation is often performed in the equalization litture, we formulate the problem at the chip level for the following reason. The pethe channel response can occur at instants that are not integer multiples of theinterval (actually, super-symbol interval), and therefore the best delay for optimumtection is not necessarily an integer multiple of the symbol interval. A down-samsymbol-interval formulation could, therefore, easily miss the real optimum. Sincgoal of this paper is to explore the fundamental limits to the performance, wea chip-level formulation. The matter of reduced complexity will be addressedwhere.

It is important to remark that while the wavelet based pulses have been used for wcommunications by other authors [5,6], they have attempted low bit-rate communiusing small constellation symbols. Further, spreading of the pulses for achievingspectrum pulses has not been attempted by these authors.

The paper is organized as follows. For an overview of the OWSS system, Secintroduces the OWSS pulses, discusses the OWSS receiver, performs the ideal ranalysis, and describes the spectrum of the signal. Section 3 discusses a new ato the receiver analysis with multipath channels through a multi-level matrix formulaIt results in the optimum equalizer which is then used to obtain the limits to the syperformance. Simulation results are provided in Section 4, and some concluding reare given in Section 5.

V.K. Jain et al. / Digital Signal Processing 15 (2005) 347–366 349

gonalain,DM

atinter-the-itted

theren

2. OWSS system

2.1. OWSS pulses

As discussed in Refs. [1–4], OWSS signaling pulses are derived from the orthowavelet division multiplexing (OWDM) pulses by spreading them in the wavelet domusing a suitable family of PN codes. Therefore we begin with a discussion of OWpulses.

2.1.1. OWDM pulses from full-tree wavelet filtersConsider that the pulsesϕm(t), m = 0,1, . . . ,M − 1, form an orthonormal set and th

each pulse is orthogonal to itself shifted by non-zero integer multiples of a certainval T . Each basis pulseϕm(t) can then serve to create a ‘virtual’ channel over whichsymbolam is carried. The vector of symbolsA = [a0, a1, . . . , aM−1]T is called a supersymbol, and the intervalT as the supersymbol interval. Then the base band transmsignal becomes

s(t) =∞∑

n=−∞

M−1∑m=0

an,mϕm(t − nT ) =∞∑

n=−∞AT

nϕ(t − nT ). (1)

A synthesis tree for generating the OWDM pulses is shown in Fig. 1. In this exampleare three stages, therefore the tree hasM = 23 = 8 input nodes. More generally, for as-stage tree there would beM = 2s input nodes. The impulse response from theith input

Fig. 1. OWDM synthesizer.


ple

d

or

yquist

as the

ach

notrcomingem inectionignal

g in

Table 1OWDM pulses derived from a 2-stage tree with 4-tap Daubechies filter

Tap weights

Tap 0 1 2 3 4 5 6 7 8 9

Pulse 0 0.2333 −0.4040 −0.2958 0.7623 −0.0792 −0.2958 0.1127 −0.0792 0.0290 0.0167Pulse 1 0.0625 −0.0183 −0.0792 0.2042 −0.4542 0.6708 −0.4208 0.2958 −0.1083 −0.0625Pulse 2 0.0625 −0.1083 −0.5123 −0.0458 0.7288 0.3538 −0.1708 −0.1373 −0.1083 −0.0625Pulse 3 0.0167 −0.0290 −0.1373 −0.0123 0.0792 0.2958 0.6373 0.5123 0.4040 0.2333

Note. For convenience, we will use the variablet to denote both the continuous time variable and the sam(chip) index. Also,M andT will be used interchangeably to denote the block length.

node to the output node is denoted byϕi(t), i = 0,1, . . . ,M − 1. These pulses are calleOWDM pulses.1

The prototype lowpass filterG0(z) can be a Daubechies filter [7], a Jain filter [8,9],some other [7]. As is well known the highpass filterG1(z) is related toG0(z), thereforethe entire tree is specified by the prototype filter. The family of OWDM pulses{ϕi(t)}can be shown to be doubly orthonormal, i.e., these pulses satisfy the generalized Ncriterion:⟨

ϕi(t), ϕk(t − nT )⟩ = δi−kδn for i, k = 0,1, . . . ,M − 1 and alln. (2)

Also, it can be shown that these pulses are power complementary, i.e.,

1

M

M−1∑m=0

∣∣Φm(f )∣∣2 = 1. (3)

In this paper, we use the OWDM pulses generated by through a Daubechies filterprototypeG0(z). An example family, generated from a 4-tap Daubechies filterG0(z) =[0.3415,0.5915,0.1585,−0.0915] and a 2-stage tree, is given in Table 1. Note that epulse is normalized to unit energy.

2.1.2. OWSS pulsesWhile the OWDM pulses do satisfy the generalized Nyquist criterion, they are

broadband and are just as susceptible to deep fades, as are the OFDM pulses. Ovethis deficiency, the OWSS pulses are derived from OWDM pulses by spreading ththe wavelet domain through a suitable family of PN codes. The discussion in this sclosely follows [1]. Refs. [2–4] also provide good resource material. The transmit sfor theith virtual channel is

s(i)(t) =∑n

a(i)n

M−1∑m=0

c(i)m ϕm(t − nT ) =

∑n

a(i)n ψ(i)(t − nT ), (4)

whereψ(i)(t) = ∑M−1m=0 c

(i)m ϕm(t −nT ) = c(i)T φ(t), i = 0, . . . ,M −1, are the new OWSS

pulses. Herec(i), i = 0, . . . ,M − 1, are the code vectors which perform the spreadinwavelet domain. The superscript connotes theith pulse, which is assigned to theith user.

1 In the literature such pulses are also called as ‘wavelet packets’ [5–7].


quist

rit

amard

in

e

struc-cision

Table 2OWSS pulses derived from a 2-stage tree with 4-tap Daubechies filter and Hadamard code

Tap weights for a 2-stage tree

Tap 0 1 2 3 4 5 6 7 8 9

Pulse 0 0.1845 −0.3248 −0.5123 0.4542 0.1373 0.5123 0.0792 0.2958 0.1083 0.0625Pulse 1 0.1083 −0.1875 −0.2958 0.2623 0.5123 −0.4542 −0.1373 −0.5123 −0.1875 −0.1083Pulse 2 0.1083 −0.1875 0.1373 0.5123 −0.6708 −0.1373 −0.3873 −0.0792 −0.1875 −0.1083Pulse 3 0.0625 −0.1083 0.0792 0.2958 −0.1373 −0.5123 0.6708 0.1373 0.3248 0.1875

If Hadamard codes are used, it can readily be shown that the OWSS pulses,ψ(i)(t), i =0, . . . ,M − 1, are also doubly-orthogonal. That is, they satisfy the generalized Nycriterion⟨

ψ(i)(t),ψk (k)(t − nT )⟩ = δi−kδn for i, k = 0,1, . . . ,M − 1 and alln. (5)

Thus, these pulses provide a means for creatingM virtual channels. Further, they inhethe power complementary property from the OWDM pulses, i.e.,

1

M

M−1∑m=0

∣∣Ψ (m)(f )∣∣2 = 1. (6)

Most importantly, they are broadband. That is, each pulse in the family ofM pulses isbroadband [1]. In (6) we have assumed that the spreading is done through a Hadcode. ForM = 4 andM = 8, the corresponding code matrices are

H4 = 1

2

1 1 1 11 −1 1 −11 1 −1 −11 −1 −1 1

,

H8 = 1√8

1 1 1 1 1 1 1 11 −1 1 −1 1 −1 1 −11 1 −1 −1 1 1 −1 −11 −1 −1 1 1 −1 −1 11 1 1 1 −1 −1 −1 −11 −1 1 −1 −1 1 −1 11 1 −1 −1 −1 −1 1 11 −1 −1 1 −1 1 1 −1

. (7)

For the OWDM pulses of Table 1 and codeH4 of Eq. (7), the OWSS pulses are listedTable 2. It is important to remark that these pulses need not be generated on-line;they canbe computed off-line and conveniently stored in a ROM. A more detailed discussion of thbeneficial properties of OWSS pulses is provided in Ref. [1].

2.2. OWSS receiver

The baseband portion of an OWSS receiver is shown in Fig. 2 [4]. The equalizerture consists of two adaptive FIR components: a forward equalizer (FE), and a de


updatea deci-

gener-rya lowere FE, anE areand thede. Of

ase be-as to

sed to

using

Fig. 2. OWSS receiver.

feedback equalizer (DFE). LMS algorithm is used as the adaptation mechanism tothe FE and DFE [4]. The receiver also uses an OWSS correlator, an upsampler andsion device (slicer).

The output of the equalizer is correlated with a user specific OWSS pulse, thusating a statistic for detection. Note also that the correlator generates its output eveM thsample, or chip. Therefore, the decision device and error computations operate atrate compared to the equalizer. Since the DFE operates at the same speed as thupsampler is needed as shown in the figure. Initially, the coefficients of FE and DFobtained through a training phase (using a previously stored sequence of symbols,LMS algorithm for update). Subsequently, the receiver goes into a maintenance mocourse, the equalizer coefficients are updated in this mode as well. The training phgins with an arbitrary set of equalizer weights. These weights tend to converge sominimize the MSE. The final weights obtained at the end of the training are then uinitialize the maintenance phase.

From (4), the composite downlink signal for allU users is given by

s(t) =U∑

i=1

∑l

a(i)l ψ(i)(t − lT ). (8)

We assume that for the uplink case multiple access is performed at the MAC layerCSMA/CA [10].


s. For1,12].

iming

n-

cular,by

S sys-is alsoFDM,sharp

2.3. Ideal receiver analysis

The received signal equals the sum of the signals received from all transmitterthis basic discussion we will ignore channel attenuation and the multipath effects [1Thus, at theith receiver the received signal is

r(t) =U∑

i=1

∑n

a(i)n ψ(i)(t − nT ) + n(t), (9)

wheren(t) denotes additive white Gaussian noise. At the receiver, symbol and block textraction is performed and the received signal is correlated withψ(t − nT ) to detectthe nth supersymbol at time instantnT (actuallynT + τ whereτ is the optimal timingphase [13]). For thekth pulse, the correlator yields

v(k)n = ⟨

r(t),ψ(k)(t − nT − τk)⟩ = a(k)

n

⟨ψ(k)(t − nT − τk),ψ

(k)(t − nT − τk)⟩

+∑l �=n

a(k)l

⟨ψ(k)(t − lT − τk),ψ

(k)(t − nT − τk)⟩

+∑i �=k

∑l

a(i)l

⟨ψ(i)(t − lT − τi),ψ

(k)(t − nT − τk)⟩ + noise

∼= a(k)n + (

IS(k)n + ICk

n

) + N(k)n . (10)

IS(k)n is zero due to the impulsive autocorrelation2 of the OWSS pulses [1]. For the dow

link case as well as for the MAC shared high speed channel case, for which allτi are equal,the inter-channel interference term IC(k)

n is also zero. Thus

v(k)n

∼= a(k)n + N(k)

n . (11)

Therefore, the probability of error is the same as in the simple AWGN case. In partithe probability of bit error for the 64 QAM symbol constellation is approximately given

pbit ≈ 7

12Q

(√2Eb

7No

). (12)

2.4. OWSS spectrum is very compact

If all virtual channels are active, the theoretical baseband spectrum of the OWStems turns out to be perfectly flat [3]. Further, the simulated passband spectrumfound to relatively flat and 30–40% more compact than that of the familiar 802.11a Odue largely to the avoidance of the prefix, elimination of channel coding, and arolloff. These observations are elaborated below.

2 More accurately, the block-wise autocorrelation (i.e., the autocorrelation calculated at a spacing ofM sam-ples) is impulsive.


entaryortho-

torebandernalo-infinite

2.4.1. Theoretical baseband spectrumLet us first consider only a single supersymbol and assume that allM virtual channels

are active. The corresponding signal is

s0(t) =M−1∑i=0

Aiψ(i)(t) ⇒ S0(f ) =

M−1∑i=0

Aic(i)T Φ(f ). (13)

Therefore

E∣∣S0(f )

∣∣2 = ζ 2M−1∑i=0

ΦH (f )c(i)c(i)T Φ(f ) = ζ 2ΦH (f )[MIM ]Φ(f )

= ζ 2M

M−1∑i=0

∣∣Φ(i)(f )∣∣2 = Mζ 2, (14)

whereζ 2 �= E|Ai |2. As an example, for 64 QAM,ζ 2 = 42 [12,13]. Note that in (14) wehave used the properties of the OWDM pulses, namely that they are power complem(see top trace of Fig. 3), and the fact that the Hadmard codes are orthogonal; indeednormal if the code vectors are normalized to unit energy. In the latter case, the facM

on the right hand side of (14) may be replaced by unity. Clearly, from (14) the basspectrum is flat, regardless of the value ofM . A simulated spectrum is shown in the lowtrace of Fig. 3. Although the derivation is not included here, it can be shown in an agous manner that the theoretical baseband spectrum of the complete signal, with ansymbol stream,

s(t) =∑n

M−1∑i=0

Aniψ(i)(t − nT ) =

∑n

ATnψ(t − nT ) (15)

Fig. 3. Baseband OWSS spectrum.


SS are

heitted

gely to) and a

, wedevel-mple,

is also flat. Here, the vector of symbolsAn = [An0,An1, . . . ,An(M−1)]T is called thenthsupersymbol.

2.4.2. Passband spectrumThe passband spectral plots and a suggested emission mask for 108 Mbps OW

not shown due to a lack of space. However, it is found that the signal bandwidth at−10 dBris 18.6 MHz. A direct comparison of 60 Mbps OWSS3 passband spectrum with that of t54 Mbps OFDM is carried out in [3]. For comparable net bit rates, the OWSS transmspectrum turns out to be 30–40% more compact that for the OFDM systems, due larthe avoidance of overhead (prefix, channel coding, and the guard-band zero carrierssharp roll-off of the spectrum—while still achieving low BERs at practical SNRs.

3. Multi-level matrix formulation of OWSS receiver

The OWSS transmitter-receiver system is illustrated in Fig. 4. To minimize its BERneed to minimize the TMSE, which is the subject of discussion here. The theoreticalopment is somewhat complex, therefore to facilitate its understanding a simple exatogether with all the key steps, is provided in Appendix A.

3.1. Optimum FE–DFE receiver

The definitions of the symbol vectora (of lengthD), OWSS transmit filterψ (L taps,assumed to be an integer multiple ofM by zero padding if necessary), multipath channelc(Nc taps), the FE tap weightsw (Nw taps) and the DFE tap weightsb (Nb taps) are givenbelow

a = [a0, a1, a2, . . . , aD−1]T , ψ = [ψ0,ψ1,ψ2, . . . ,ψL−1]T ,

Fig. 4. OWSS transmitter-receiver system.

3 Through adaptive loading described in Section 4, the net bit rate turns out to be about 54 Mbps.


l

.

S

c = [c0, c1, c2, . . . , cNc−1]T , w = [w0,w1,w2, . . . ,wNw−1]T ,

b = [b0, b1, b2, . . . , bNb−1]T . (16)

The symbol stream upsampled by a factorM (of lengthMD = M × D) is given by

a up = [a0, [0 0. . .0]M−1, a1, [0 0. . .0]M−1, a2, [0 . . .0]M−1, . . . ,

aD−1, [0 . . .0]M−1]T

. (17)

The symbol matrixA1 input to the FE, the OWSS filter matrixH1, multipath channematrixC and the additive white gaussian noise (AWGN) matrixN are given by

A1 =

0 · · · 0a up . . . 0

__ a up. . .

...

__ a0

MD×MD

, H1 =

0 . . . 0ψ · · · 0

__ ψ.. .

...

_ _ ψ0

MD×MD

,

C =

0 · · · 0c · · · 0

_ _ c. . .

...

__ c0

MD×Nw

, N =

n0 0 · · · 0n1 n0 · · · 0...

.... . .

...

nMD−1 nMD−2 · · · n0

MD×Nw

(18)

The output of the FE, including the contribution of noise, is

p = (A1H1C + N)w = A1G1w + Nw, G1 = H1C. (19)

The symbol matrix input to the DFE (with delayL due to the correlator) and the OWScorrelator-downsampler matrixH2 are given by

A2 =

0 0 · · · 00L×1 . . . 0

a up 0L×1 · · · ...

_ _ a up

.... . . 0L×1

_ _ · · · a0

MD×Nb

,

H2 =

ψT | 01×M · · · 01×M

01×M ψT | · · · 01×M

01×M 01×M ψT | 01×M

.. .

01×M 01×M 01×M ψT

1×M

D×MD

. (20)

The output of the DFE is given by

f = A2b. (21)

The input to the slicer after the sliding correlation and downsampling operation is

v = H2q = H2(p − f ) = H2(A1G1w + Nw − A2b ). (22)


ws:

The total error signale (assuming perfect detectiona = a) is given by

e = v − a. (23)

Neglecting the effect of noise temporarily, the error signal can be expressed as follo

e = H2[A1G1w − A2b ] − a = V1w − V2b − a,

V1 = H2A1G1 and V2 = H2A2. (24)

Correspondingly, the TMSE is given by

E = E(‖e‖2) = E(eH e ). (25)

Now, to find the optimum FE–DFE to minimize the TMSE, we differentiateJ with respectto w andb and set the two terms to zero.

∇wE = E[2V H

1 (V1w − V2b − a )]

= 2E(V H

1 V1)w − 2E

(V H

1 V2)b − 2E

(V H

1 a) = 0,

∇bE = E[2V H

2 (V1w − V2b − a )]

= 2E(V H

2 V1)w − 2E

(V H

2 V2)b − 2E

(V H

2 a) = 0. (26)

In matrix-vector form, the above equations can be written as[E(V H

1 V1) −E(V H1 V2)

E(V H1 V2) E(V H

2 V2)

] [w

b

]=

[E(V H

1 a )

E(V H2 a )

]⇒ (27a)[

Q2 −Q3QH

3 Q4

] [w

b

]=

[QH

1

0

]. (27b)

Hence the optimumFE andDFE can be estimated (for the noiseless case) as follows:[w

b

]=

[Q2 −Q3

QH3 Q4

]−1 [QH

1

0

]. (28)

Here,

Q1 = E(aH H2A1G1

),

Q2 = E(GH

1 AH1 RA1G1

) = GH1 E

(AH

1 RA1)G1 = GH

1 F1G1,

Q3 = E(GH

1 AH1 RA2

) = GH1 E

(AH

1 RA2) = GH

1 F2,

Q4 = E(AH

2 RA2) = E

(AH

1 RA2) = F3. (29)

A symbol correlator matrix forM symbols is shown below

Pall =

ψT 0 0 · · · 0

0 ψT 0 · · · 0

0 0 ψT · · · 0

......

......

...

0 0 0 · · · ψT

. (30)

M×(M−1)L+M


low

d

We then extract blocksp of sizeM × M . Now let m = L/M . (Note thatL is an integermultiple of M .) So we will getpi blocks of sizeM × M , i = 1,2, . . . , I , whereI = m +(M − 1). Also let Ip = I/m. We then define additional blocks of zeros,pi = ZM×M, i =I + 1, . . . , I + (Ip − 1) × M , so as to complete the polyphase sequences outlined be

1 2 3 · · · I

1+ M 2+ M 3+ M · · · I + M

1+ 2M 2+ 2M 3+ 2M · · · I + 2M...

......

......

1+ (Ip − 1)M 2+ (Ip − 1)M 3+ (Ip − 1)M I + (Ip − 1)M

. (31)

Now using the above polyphase sequences, the multi-level matrixR can be expresse(neglecting edge effects) as

R = HH2 H2

=

pT1

pT2 pT

1

pT3 pT

2 pT1

......

.... . .

pTI

......

. . .

0 pTI

.... . .

...... pT

I

. . .

0 0 0 p1

×

p1 p2 p3 · · · pI 0 0

p1 p2 . . . . . . pI 0 0

p1 . . . . . . . . . pI 0

. . ....

. . ....

. . ....

. . ....

p1


=

K1 J1,1 · · · · · · · · · · · · J1,I−1

J T1,1 K2 J2,1 · · · · · · · · · J2,I−2

... J T2,1

. . .. . .

. . .. . .

......

. . . KM JM,1 · · · JM,I−M

......

. . . JM,1 K1

......

. . .. . .

J T1,I−1 J T

2,I−2 JM,I−M

.. .

KM

MD×MD

.

(32)

Note the repeating blocks whose components are given below

Ki =Ip∑

k=0

pTi+kMpi+kM, Ji,j =

Ip∑k=0

pTi+kMpi+j+kM,

i = 1,2, . . . ,M, j = 1,2, . . . , I − i. (33)

The matrixQ1 can be expressed statistically in terms of the symbol energyζ

Q1 = E(aH H2A1)G1

= ζ 2[Dh0,Dh1, . . . ,DhM−1, (D − 1)hM, . . . , (D − 1)h2,M−1, . . . ,

(D − I )h(I+1),M−1]G1. (34)

The matrixQ2 is given by

Q2 = GH1 E

(AH

1 RA1)G1 = GH

1 F1G1. (35)

The diagonal and off-diagonal terms of the matrixF1 = E(AH1 RA1) are given by

F1(i, i) = ζ 2

[(D

M−

⌈i

M

⌉+ 1

) M∑n=1

Kn −∑

n=mod(i/M)−1

Kn

], (36a)

F1(i, i + j) = ζ 2

[(D

M−

⌈i

M

⌉+ 1

) M∑n=1

Jn,j −∑j

JM−(mod(i/M)−1),j

−∑

n=mod(i/M)−1

Jn,j

], (36b)

F1(i + j, i) = ζ 2

[(D

M−

⌈i

M

⌉+ 1

) M∑n=1

J Tn,j −

∑j

J TM−(mod(i/M)−1),j

−∑

J Tn,j

]
n=mod(i/M)−1


ck

nize

used

mean-

or twodelayedpeak

orous

giveni = 1,2, . . . ,D, j = 1,2, . . . , I − 1 and i + j � D,

elseF1(i, i + j) = F1(i + j, i) = ZM×M, when giveni = 1,2, . . . ,D

thenj > I − 1, i + j � D, (36c)

whereZ is a zero matrix. The matrixQ3 is defined as

Q3 = GH1 E

(AH

1 RA2) = GH

1 F2. (37)

Note that the matrixA1 is of sizeMD × MD (i.e., D × D blocks), and the matrixA2 isof size MD × Nb (D × nb blocks), therefore the final matrix is of sizeMD × Nb (i.e.,D × nb blocks), wherenb �= Nb/M . Also note that the firstl �= L/M block rows ofA2 are

zero rows. ThenF2 = E(AH1 RA2) can be obtained fromF1 by removing the firstl block

columns fromF1, extracting the nextnb block columns and removing the remaining blocolumns.

The matrixQ4 can also be determined similarly:

Q4 = E(AH

2 RA2) = F3. (38)

The final matrixF3 is of sizeNb ×Nb (i.e.,nb ×nb blocks). Also note that the firstl blockrows ofA2 are zero rows. ThenF3 is a submatrix ofF2 and can be obtained fromF2 byremoving the firstl block rows fromF2, extracting the nextnb block rows and removingthe remaining block rows. Now, to consider the effect of AWGN, we will have to recoganother matrixQ5 in our estimation equation (28). Thus,[

w

b

]=

[Q2 + Q5 −Q3

QH3 Q4

]−1 [QH

1

0

]. (39)

The matrixQ5 is similar toQ1 and uses the spectral density of the additive noise,σ 2.

Q5 = E(NH HH

2 H2N) = [RSN ]MD×MD,

SN = diag[(

MDσ 2, (MD − 1)σ 2, . . . σ 2)]MD×MD. (40)

4. Simulation results on performance limits

4.1. Experiment 1 (100 ns delayspread channel)

Consider a 108 Mbps, 64 QAM (18 Msps), four user (M = 4, L = 12) OWSS systemover a 100 ns rms delayspread multipath channel. A Naftali channel model [14] iswhich results in a 9-tap FIR structure. The receiver’s FE–DFE are optimized forNw = 4andNb = 8. We also use an adaptive loading strategy [15,16] based upon the totalsquared error. That is, if the TMSE is below a threshold thr64, 64 QAM transmission isused; if it is at or above thr64 but below another threshold thr16, 16 QAM transmissionis used; otherwise QPSK transmission is used. Table 3 shows the bit error rate fscenarios, each averaged over more than 100 sample channels. In the first casedecision is used which is simulated by accepting only those channels for which theoccurs at the zeroth bin; this is a simplified approach which can be made more rig


.

hishere thedaptive

m,

s thece in

.As be-d error.a-

Table 3Simulation results with adaptive loading, for a 100 ns delayspread channel

Eb/No 64 QAM thr64 16 QAM thr16 QPSK Aggregate BER Bits/s/Hz(dB) (%) TMSE (%) TMSE (%) bit-rate (Mbps)

Delayeddecision

19 59 0.15 12 0.2 29 82.8 7× 10−6 4.222 82 0.15 14 0.2 4 100.1 6× 10−6 5.025 90 0.1 7 0.15 3 103.3 3× 10−7 5.2

Undelayeddecision

19 28 0.15 13 0.2 59 60.8 3× 10−3 3.122 51 0.15 10 0.2 39 76.3 6× 10−4 3.825 58 0.1 17 0.15 25 83.9 2× 10−5 4.3

Note. In the calculation of the spectral efficiency a bandwidth expansion ratio of 1.1 is assumed.

Fig. 5. BER for 108 Mbps 64 QAM with 2:1 selection diversity (over a 100 ns delayspread channel)

either by incorporating the delay into the algorithm itself, or by LMS algorithm [4]. Tdelay is bounded by the channel memory, i.e., eight samples. The second case is wdecision is not delayed. Also shown are the aggregate bit rates as a result of the astrategy. The results are given for the following SNRs: Eb/No= 19, 22, and 25 dB witha 2:1selection diversity[12] at the receiver. Due to a very sharp roll-off of the spectrua bandwidth expansion ratio of just 1.1, and a corresponding bandwidth of 18× 1.1 =19.8 MHz are assumed in the calculation of the spectral efficiency. Figure 5 showBER curve for fixed 64 QAM constellation. Note that this represents the performanthe limit.

4.2. Experiment 2 (50 ns delayspread channel)

Consider again a 108 Mbps, 64 QAM (18 Msps), four user (M = 4) OWSS systemHowever, now we consider a 50 ns rms delayspread multipath channel (5 taps).fore, we use an adaptive loading strategy [15,16] based upon the total mean-squareThe equalizer lengths areNw = 7, Nb = 4. Table 4 shows the bit error rate for the situ


annelsan beLMS

:1 isof 10pectralurse,

ulsesenefi-

ffectived, in aeoreti-i-levelperi-p9 dB.t on a

r stillrenti-

-tap

marder-

Table 4Simulation results with adaptive loading and 2-MRC, for a 50 ns delayspread channel

Eb/No 64 QAM thr64 16 QAM thr16 QPSK Aggregate BER Bits/s/Hz(dB) (%) TMSE (%) TMSE (%) bit-rate (Mbps)

Delayeddecision

19 91.5 0.2 4.5 0.25 5 103.3 10−5 5.2

Note. In the calculation of the spectral efficiency a bandwidth expansion ratio of 1.1 is assumed.

tion where delayed decision is used which is simulated by accepting only those chfor which the peak occurs at the zeroth bin. This is a simplified approach which cmade more rigorous either by incorporating the delay into the algorithm itself, or byalgorithm.

It is also useful to remark that maximal ratio combining (MRC) on a diversity of 2used at the receiver. Using this scheme in a 50 ns delay spread channel, a low BER−5

is achieved at an Eb/No of 19 dB, for an aggregate bit rate of 103.3 Mbps and a sefficiency of 5.2 bits/s/Hz. The results are averaged over 200 sample channels. Of cothis represents performance in the limit.

5. Conclusions

OWSS is a WLAN signaling scheme based on a new family of pulses. These phave both a wide time support and a wide frequency support. The latter property is bcial inasmuch as it leads to inherent robustness to frequency-selective fading and eequalization. The scheme is targeted to deliver data rates of 100 Mbps and beyonbandwidth efficient manner. After presenting an overview, the paper presented a thcal basis for understanding the limits to the performance of OWSS systems. Multmatrix formulation was used in determining the optimum receiver for OWSS. Exments on 108 Mbps (64 QAM) indicate that a BER of 10−5 and spectral efficiencies uto 5.2 bits/s/Hz can be achieved for 50 ns delay spread channels at an SNR of 1For practical implementation, future work should attempt an analogous developmensymbol interval basis. Further, it would be useful to explore MIMO OWSS systems fohigher spectral efficiencies. Also, it would be useful to make a comparison and diffeation with the OFDM based IEEE 802.16 proposed standard.

Appendix A

For a single stage (M = 2) synthesizer, the OWDM pulses generated using a 4Daubechais filter areϕ0 = [−0.3415,0.5915,−0.1585,−0.0915] and ϕ1 = [−0.0915,0.1585,0.5915,0.3415]. These pulses are spread in the wavelet domain by using Hadacodes(1/

√2) × [1,1] and(1/

√2) × [1,−1]. The single stage OWSS pulses thus gen

ated (L = 4), are

ψ(0) = [−0.433,0.75,0.433,0.25],


tap

annel

t

Fig. 6. Frequency response of a 3-tap channel[1,−0.8,1].

ψ(1) = [−0.25,0.433,−0.75,−0.433].For this example, we use the first pulseψ(0). The channel is chosen to be a real 3-channel (Nc = 3), given byc = [1,−0.8,1]. Possessing a notch deeper than−40 dB, thefrequency response of the channel is portrayed in Fig. 6. Clearly, this is a difficult chto equalize.

For simplicity of presentation, we will use only four BPSK symbols (D = 4, ζ 2 = 1)to estimate the equalizer with the specificationNw = 2 andNb = 2. We will also neglecthe effect of noise. The OWSS filter matrixH1 and multipath channel matrixC, definedin (18), are given by

H1 =

−0.433 0 0 0 0 0 0 00.75 −0.433 0 0 0 0 0 00.433 0.75 −0.433 0 0 0 0 00.25 0.433 0.75 −0.433 0 0 0 0

0 0.25 0.433 0.75 −0.433 0 0 00 0 0.25 0.433 0.75 −0.433 0 00 0 0 0.25 0.433 0.75 −0.433 00 0 0 0 0.25 0.433 0.75 −0.433

,

C =

1 0 0 0 0 0 0 0−0.8 1 0 0 0 0 0 0

1 −0.8 1 0 0 0 0 00 1 −0.8 1 0 0 0 00 0 1 −0.8 1 0 0 00 0 0 1 −0.8 1 0 00 0 0 0 1 −0.8 1 00 0 0 0 0 1 0.8 1

.

The correlator matrix, defined in (20), is given by


g (33):

H2 =

−0.4330 0.7500 0.4330 0.2500 0 0 0 00 0 −0.4330 0.7500 0.4330 0.2500 0 00 0 0 0 −0.4330 0.7500 0.4330 0.25000 0 0 0 0 0 −0.4330 0.7500

.

We then extract blocks of sizeM ×M as suggested in Eq. (31),I = L/M +(M−1) = 3,Ip = �I/m� = 2, I + Ip = 5,

p1 =[−0.4330 0.7500

0 0

], p2 =

[0.4330 0.2500

−0.4330 0.7500

],

p3 =[

0 00.4330 0.2500

], p4 =

[0 00 0

], p5 =

[0 00 0

].

The matrixR = HH2 H2 is given by

R =

0.1875 −0.3248 −0.1875 −0.1083 0 0 0 0−0.3248 0.5625 0.3248 0.1875 0 0 0 0−0.1875 0.3248 0.3750 −0.2165 −0.1875 −0.1083 0 0−0.1083 0.1875 −0.2165 0.6250 0.3248 0.1875 0 0

0 0 −0.1875 0.3248 0.3750 −0.2165 −0.1875 −0.10830 0 −0.1083 0.1875 −0.2165 0.6250 0.3248 0.18750 0 0 0 −0.1875 0.3248 0.3750 −0.21650 0 0 0 −0.1083 0.1875 −0.2165 0.6250

.

Neglecting the edge effects, the repetitive component blocks, can be calculated usin

K1 =[

0.3750 −0.2165−0.2165 0.6250

], K2 =

[0.3750 −0.2165

−0.2165 0.6250

],

J1,1 =[−0.1875 −0.1083

0.3248 0.1875

], J1,2 =

[0 00 0

],

J2,1 =[−0.1875 −0.1083

0.3248 0.1875

].

Using (36), we can now generate the matrixF1

F1 =

1.5000 −0.8660 −0.5625 −0.3248 0 0 0 0−0.8660 2.5000 0.9743 0.5625 0 0 0 0−0.5625 0.9743 1.1250 −0.6495 −0.3750 −0.2165 0 0−0.3248 0.5625 −0.6495 1.8750 0.6495 0.3750 0 0

0 0 −0.3750 0.6495 0.7500 −0.4330 −0.1875 −0.10830 0 −0.2165 0.3750 −0.4330 1.2500 0.3248 0.18750 0 0 0 −0.1875 0.3248 0.3750 −0.21650 0 0 0 −0.1083 0.1875 −0.2165 0.6250

.


te

)

. Noteawaytion

00

,WSS

f ICC,

-

press.IEEE

F2 and F3 are submatrices ofF1. F2 can be obtained fromF1 by removing the firstl �= L/M = 2 block columns (4 columns) fromF1, extracting the nextnb block columns(2 columns) and removing the remaining block columns. The final matrixF3 is of sizeNb × Nb (i.e., 2× 2). F3 is a submatrix ofF2 and can be obtained by removing the firsl

block rows (4 rows) fromF2, extracting the nextnb block rows (2 rows) and removing thremaining block rows.

F2 =

0 00 0

−0.3750 −0.21650.6495 0.37500.7500 −0.4330

−0.4330 1.2500−0.1875 0.3248−0.1083 0.1875

, F3 =[

0.7500 −0.4330−0.4330 1.2500

].

We can now calculate the matricesQ1, Q2, Q3, andQ4, as defined in Eqs. (15) and (16

Q1 = [ 3.75 0.3248] , Q2 =[

5.7987 −2.1343−2.1343 1.7119

],

Q3 =[

0.7160 0.5866−0.4584 −0.3729

], Q4 =

[0.7500 −0.4330

−0.4330 1.2500

].

Using the above matrices and Eq. (39), the FE and DFE are estimated to be

w = [1.1302,0.8508] and b = [0.8983,0.5877].From (23), the unequalized error (assuming perfect detectiona = a) is given bye = v − a.Using the above equalizer, the MSE of the unequalized error is found to be 0.010174that the decision boundary for correct detection of a BPSK symbol is a unit distancefrom either BPSK symbol,−1 or 1. An MSE of 0.010174 represents a standard deviaof approximately 0.1, around the transmitted symbols 1 and−1.

For illustration purposes, we used anNw = 2-tap FE. For a longer 5-tap FE and 1symbols for estimation, the optimum equalizer turns out to bew = [1.1394,0.57591,−0.31036,0.037525,0.12544] andb = [0.80517,0.3338], with a corresponding MSE=6.7× 10−5. The feedback equalizer was still maintained at a lengthNb = 2. This successeven for a difficult channel, may be attributed to the wide frequency support of the Opulses. As a reminder, in this simple example, the effect of AWGN was ignored.

References

[1] V.K. Jain, OWSS multiple-access system for 100 Mbps wireless LANs, in: Proc. IEEE Proceedings o2001, pp. 1471–1475.

[2] V.K. Jain, B. Myers, OWSS: A new signaling system for 100–150 Mb/s WLANs, IEEE Wireless Commun. 10 (4) (2003) 16–24.

[3] D. Divakaran, V.K. Jain, B. Myers, Spectral characteristics of OWSS signal, IEEE Commun. Lett., in[4] J. Dholakia, V.K. Jain, B. Myers, Adaptive equalization for 100 Mbps OWSS wireless LANs, in: Proc.

GlOBECOM, 2001, pp. 162–166.


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IEEE

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[7] M. Vettereli, J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, New York, 1995.[8] V.K. Jain, R.E. Crochiere, Quadrature-mirror filter design in time domain, IEEE Trans. Acoust. S

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2000.[12] T.S. Rappaport, Wireless Communications, Prentice Hall, New York, 1996.[13] E.A. Lee, D.G. Messerschmitt, Digital Communication, Kluwer Academic, Dordrecht/Norwell, MA, 1[14] K. Halford, M. Webster, Multipath measurements in wireless LANs, in: Application Note AN9895.1, In

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performance limits of owss: a spectrally efficient wlan system

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