widely-linear-minimum (1)

Widely Linear Minimum Mean Square Error

Estimation for Unique Word-OFDM

Muga Vamshi Krishna,[email protected]

August 7th,2014

Contents

0.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.3 Mathematical Recaptulation . . . . . . . . . . . . . . . . . . . . 1

0.4 Systematic Coded UW-OFDM . . . . . . . . . . . . . . . . . . . 3

0.4.1 Non Systemtic Coded UW-OFDM . . . . . . . . . . . . . 3

0.5 System model and Optimum linear data estimators . . . . . . . 4

0.6 Widely Linear Data Estimation for Systematic Coded UW-

OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.7 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 6

0.8 Review of WLMMSE for Non Systematic UW-OFDM . . . . . . 9

0.8.1 Optimum Code Generator Matrices for Real Data Vec-

tors in Combination with WLMMSE Estimation . . . . . 9

0.8.2 Improved and extended G matrices for the BPSK scheme 10

0.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

0.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1

List of Figures

1 Simulated Setup for the UW-OFDM System . . . . . . . . . . . 7

2 Performance comparison of the WLMMSE ans LMMSE at two

information rates . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 LMMSE v/s WLMMSE comparison for QPSK schemes . . . . . 8

4 Improved G matrices for the BPSK scheme . . . . . . . . . . . . 12

5 Equivalent BPSK and QPSK schemes . . . . . . . . . . . . . . . 14

6 Mixed/Combined Modulation Schemes . . . . . . . . . . . . . . 15

2

Abstract

The present report is a summary of my internship work carried out at the In-

stitute of Signal processing,Johannes Kepler University on the UW-

OFDM project.The report presents a brief introduction to conventional

OFDM schemes,Unique Word OFDM and Estimation for UW-OFDM.The

basic focus has been on studying the Widely Linear Minimum mean

Square Error(WLMMSE) estimator and its implications on the UW-

OFDM scheme.A brief introduction is also given on some novel modulation

schemes that take the advantage of the WLMMSE estimator and relevant

simulations are presented.

0.1 Notation

The following notation are used throughout this report.Bold face letters with

lower case represent vectors (e.g.x,y,...), and bold face letters with upper case

indicate matrices (e.g., X, Y,...).Conjugation, conjugate transposition, and

transposition are denoted by (.)∗, (.)H , and(.)T respectively. Frequency do-

main variables are represented by placing a tilde above the variable (e.g., y,

Y ). The expectation operator is denoted by E.. Xi,j denotes the element in

the ith row and jth column of the matrix X. The inverse of the matrix X is

denoted by X−1.InandOn denote an n*n identity and zero matrix, respec-

tively. The set of complex numbers is denoted by C.

0.2 Introduction

Orthogonal Frequency Division Multiplexing (OFDM) has become an impor-

tant modulation technique in the field of telecommunications. Currently, a

major research is being done on implementing the OFDM technique in cel-

lular mobile communication systems. The most conventional approach to

OFDM is the cyclic prefix OFDM where the last sample of each OFDM sym-

bol is copied onto the guard interval. In [1] a new OFDM signalling scheme

was proposed where the usual cyclic prefixes was built by unique words in-

stead of cyclic prefixes. The most important feauture of this scheme is that

the unique word is part of the DFT interval.

UW-OFDM has proven to be an effective signalling technique because of its

many appealing features: 1) The idea of substituting CPs by UWs in OFDM

has the advantage of choosing the UW sequence such that it can effectively

be used for synchronization and channel parameter estimation purposes, thereby

inherently serving as a pilot sequence 2) The redundancy present in the fre-

quency domain can be exploited to reduce the infuence of noise on the data,

thus leading to a substantial improvement over the CP-OFDM in terms of

data estimation.

The report will initially recaptulate the UW-OFDM approach briefly.This will

be followed by a description of the estimators used for the scheme.Results will

then be presented comparing various estimators followed by the conclusion.

0.3 Mathematical Recaptulation

Let xuεCNu×1 be a predefined sequence which we call unique word. This unique

word shall form the tail of each time domain OFDM symbol vector. For the

investigations in this report the particular shape of the UW is irrelevant, so

we assume xu = 0 if not stated otherwise explicitly. Hence, an OFDM time

domain symbol vector x, as the result of a length-N-IDFT (inverse discrete

1

Fourier transform), consists of two parts and is of the form x=

[xTd

0T

]εCNx1.

As in conventional OFDM, the QAM data symbols (denoted by the vector

dεCNd×1) and the zero subcarriers (usually at the band edges and at DC) are

specified in frequency domain as part of the vector x, but here in addition

the zero-word is specified in time domain as part of the vector x = F−1N x.

Here, FN denotes the length-N-DFT matrix with elements[FN

]k,l

=e−j2piN kl for

k,l=0,1,2,...N-1. The system of equations x = F−1N (x) with the introduced fea-

tures can be fulfilled by introducing redundancy in the frequency domain. We

do that by defining code words

c=Gd (1)

with the help of appropriate complex valued generator matrices GεC(Nd+Nr)×Nd

, where Nr = Nu. The frequency domain symbol x is finally built by inserting

the zero subcarriers which can be modelled by x = Bc, where Bε[0, 1]N×(Nd+Nr)

consists of zero-rows at the positions of the zero subcarriers, and of appropri-

ate unit row vectors at all other positions. To produce the zero-UW in time

domain for every possible data vector d a valid code generator matrix has to

fulfill

F−1N BG =

[∗0

](2)

Note that F−1N B is composed of those columns of F−1

N that correspond to the

non-zero entries of the OFDM frequency domain symbol x. Let WεCNr×(Nd+Nr)

be the matrix built by the Nr lowermost rows of F−1N B. Then the constraint

(2) can also be formulated as

WG=0 (3)

which says that the columns of a valid G have to lie in the nullspace of W.

2

0.4 Systematic Coded UW-OFDM

In the original UW-OFDM concept presented in [1]-[3], the G matrix was

chosen as

G=P

[I

T

](4)

where Pε[0, 1](Nd+Nr)×(Nd+Nr) is a carefully chosen permutation matrix. Let

M=WP=[M1 M2

]where M1εC

Nr×Nd and M2εCNr×Nr , then the constarint

(3) is fulfilled by choosing T = −M−12 M1. We call r=Td the vector of redun-

dant symbols, hence a codeword can be written as C = P[dT rT

]T. Conse-

quently, this approach leads to code symbols c with dedicated data and ded-

icated redundant elements. We therefore refer to this approach as systematic

coded UW-OFDM.

0.4.1 Non Systemtic Coded UW-OFDM

In this concept, code generator matrices G are proposed in such a way that it

distributes the redundancy over all sub carriers. A possible choice of G is

G=A

[I

T

](5)

with a non singular A εR(Nd+Nr)×(Nd+Nr).Defining M again as M = WA =[M1 M2

], the constraint in (3) is fullfiled by choosing

T=−M−122 M21 (6)

In [4] the approach to find optimised generator matrices is completely de-

scribed.

After the above initial briefing of the UW-OFDM scheme, we now shift our

focus to the optimum linear data estimators necessary at the receiver end.

3

Before we move on to presenting the data estimators, the mathematical model

used is firstly described.

0.5 System model and Optimum linear data

estimators

Referring back to [1]-[4] the received symbol yε(Nd+Nr)×1 (already excluding

the zero sub carrier symbols) can be modeled as

y = HGd+ v (7)

where the diagonal matrix HεR(Nd+Nr)×(Nd+Nr) contains the channel frequency

response on its main diagonal and v represents a zero mean Gaussian(frequency

domain) noise vector with the covariance matrix Cvv = σ2dI. Further more we

assume the data vector to be of zero mean and covariance matrix Cdd = σ2dI

In [3],[4] linear data estimators of the form d = Ey have been derived and

studied. The linear minimum mean square error (LMMSE) estimator and the

covariance matrix of its error vector e=d− d are given by

ELMMSE = (GHHHHG+σ2vσ2dI)GHHHand (8)

Cee = σ2v(GHHHHG+

σ2vσ2dI)−1 (9)

0.6 Widely Linear Data Estimation for Sys-

tematic Coded UW-OFDM

The WLMMSE can outperform the LMMSE mainly for improper data con-

stellations. The estimated data model for the WLMMSE is given by

4

d = E1y + E2y∗ (10)

where the matrices E1 and E2 are given by

E1 = (Cdy − CdyC−∗yy C

∗yy)P

−1yy (11)

E2 = (Cdy − CdyC−1yy Cyy)P

−∗yy (12)

where Pyy = Cyy − CyyC−∗yy C

∗yy.

The error covariance matrix reads as

Cee = Cdd − (Cdy − CdyC−∗yy C

∗yy)P

−1yy C

Hdy − (Cdy − CdyC

−1yy Cyy)P

−∗yy C

Hdy (13)

For data vectors consisting of elements with proper constellations like QPSK,

16-QAM or 64-QAM the complementary covariance matrices Cdd Cyy and the

complementary cross covariance matrices Cdy become zero matrices and the

WLMMSE estimator degenrates to the LMMSE estimator i.e E1=E, E2=0.

In the present report our major focus would be on the BPSK scheme that

has real constellations and is improper by nature. For such data vectors and

under the assumption of zero mean and uncorrelated data symbols the co-

variance matrix Cdd=E[ddH ] and the complementary data covariance matrix

Cdd=E[ddT ] are identical i.e Cdd = Cdd = σ2dI. For the complementary cross

covariance matrix we have Cdy = C∗dy, and the WLMMSE estimator simplifies

to E2 = E∗1 or

d=2Re{E1y

}(14)

Also the error covariance matrix (13) simplifies and can be written as

Cee = Cdd−2Re(Cdy − CdyC−∗yy C

∗yy)P

−1yy C

Hdy

Inserting the linear model equation (7) the covariance matrices for data vec-

tors with elements having real constellations can easily shown to be

5

Cyy = σ2dHGGHHH + σ2vI (15)

Cyy = σ2dHGGTHT (16)

Cdy = σ2dGHHH (17)

Cdy = σ2dGTHT (18)

Results are now shown that prove the performance gain that can be achieved

for the WLMMSE estimator when using the BPSK modulation scheme. The

simulated conditions and setup are firstly presented below, followed by the

matlab simulations.

0.7 Simulation Setup

Fig.1 shows the block diagram of the simulated UW-OFDM system.After as-

sembling the OFDM symbol, which is composed of d,r, and a set of zero sub-

carriers, the IFFT (inverse FFT) is performed. Finally, the UW is added in

the time domain. At the receiver, the FFT operation is followed by a sub-

traction of the UW influence. This is followed by a resorting following which

the estimators are applied. The remaining parameters are similar to the setup

mentioned in [1].

We now compare the performance of the WLMMMSE and LMMSE schemes

at the same information or data rate.

In Fig.2 the BER performance is studied in indoor multipath scenarios. For

each BER curve, the average over 10000 random channel realizations, feautur-

ing(on average) an rms delay spread of 100ns and all being normalized such

that the receive power is independent of the actual channel. Perfect channel

knowledge is assumed at the receiver. For BPSK the gain of the WLMMSE

6

Figure 1: Simulated Setup for the UW-OFDM System

Figure 2: Performance comparison of the WLMMSE ans LMMSE at two in-

formation rates

7

Figure 3: LMMSE v/s WLMMSE comparison for QPSK schemes

estimator over the LMMSE estimator is around 1dB with an outer code at

both the data rates, respectively (again measured at a BER of 10−6). Based

on this,the BPSK/WLMMSE can be regarded as a much superior scheme

when compared to the BPSK/LMMSE scheme. Also from [1] it was proved

that the novel UW-OFDM using the LMMSE achieves a gain of about 0.9dB

and 0.6dB(measured at BER of 10−6) at the coding rates of r=34 and r=1

2

over the conventional IEEE802.1a scheme. To conclude, we can say that the

WLMMSE/UW-OFDM schem is able to achieve a gain of about 1.9dB and

1.6dB at r=34 and r=1

2 over the IEEE802.1a scheme.

Also,in Fig.3 the BER performance(same setup as above) clearly depict the

fact that the performance of the QPSK/WLMMSE and QPSK/LMMSE are

almost same, which is what was expected theoretically.

8

0.8 Review of WLMMSE for Non Systematic

UW-OFDM

In [4] the non systematic approach to the UW-OFDM scheme was described

in detail. We breifly recaptulate that approach and its implications to the

WLMMSE. In this approach, the symbol energy is allowed to spread over all

the codeword sysmbols and new cost functions are defined based on the sum

of the error variances at the output of the data estimator.In [4], it was also

shown that these constrained optimisation problems for the BLUE,LMMSE

are solved globally if and only if G fulfills (3) along with

GHG = s2I (19)

where s are all the identical singular values of G. In the following, we only

consider normalized optimum code generator matrices such that s2 = 1 or

GHG = I. This normalization implies that the operation c=Gd becomes

energy-invariant. To conclude, we can say that G matrices satisfying (19)

along with (3) produce the most optimum error covariance matrices for the

BLUE and LMMSE. In the next section we employ a similar strategy to the

WLMMSE and look for optimum code generator matrices.

0.8.1 Optimum Code Generator Matrices for Real Data

Vectors in Combination with WLMMSE Estima-

tion

Our initial aim would be to first define the cost function for the WLMMSE.

We employ the similar strategy used in [4] and define JWLMMSE as

JWLMMSE = tr[Cdd − 2Re[(Cdy − CdyC−∗yy C

∗yy)P

−1yy C

Hdy]] (20)

with

9

Cyy = σ2dGGH +

σ2dtrG

HGCNd

I

Cyy = σ2dGGT

Cdy = σ2dGH

Cdy = σ2dGT

Pyy = Cyy − CyyC−∗yy C

∗yy

Our initial strategy would be to check if the same functions optimising the

JLMMSE, JBLUE i.e G satisfying GHG = I could also optimise the JWLMMSE.

Although these matrices were found to optimise the function in [5], it is highly

interesting to note that solutions exist that donot satisfy GHG = I. Based

on the mathematical relations given in [5] we can conclude that for real data

constellations the WLMMSE increases the choices available for our code gen-

erator matrix G.To conclude, it can be seen that any matrix built by an ar-

bitrary subset of Nd vectors out of the 2Nd vectors g0, jg0, g1, jg1, g2, jg2, ....

forms an optimum code generator matrix for the WLMMSE. Here, it is im-

portant to note that the columns g0, g1, g2, .. are obtained using the relation

GHG = I.

With this above theoretical perspective, we now investigate how improve-

ments can be brought within the BPSK scheme with the combined usage of

the WLMMSE estimator for the UW-OFDM scheme.

0.8.2 Improved and extended G matrices for the BPSK

scheme

Let G =[g0 g1 g2 g3 ... gNd−1

]be an optimum generator matrix fulfill-

ing GHG = I. In Sec.0.2.8 we have shown that any arbitrary subset of Nd

vectors out of the 2Nd vectors g0, jg0, g1, jg1, ..., gNd+1, jgNd + 1 forms an op-

timum code generator matrix for real data constellations. A natural conse-

quence of this is the ability to form extended generator matrices Ge for real

data vectors de with length Nd < Nde ≤ 2Nd by chosing Nde vectors out of

g0, jg0, g1, jg1, ..., gNd+1, jgNd + 1. By the same arguments as used in Sec.0.2.8

10

any of these matrices will fulfill the necessary conditions to be optimal.

With this simple construction of expanded generator matrices the raw data

rate can now be configured extremely flexible. Also our choice of columns of

G for the same original size i.e 36 is also now broadened. This gives us many

new schemes that can be analysed for their performance. It must be noted

that all of the extended schemes are possible only for the BPSK scheme.

It is also quite intutive that instead of extending these G matrices, we can

also downsize them, the advantage here being that G matrices in the down-

sized case can also be produced for both the QPSK and the BPSK cases un-

like the extended ones which was only possible for the BPSK case. Another

possibility could also be to use data vectors constructed from a combination

of the BPSk and the QPSK schemes. These schemes known as mixed modu-

lation schemes are also quite interesting to study as they increase the number

of possibilities. To give a complete view of these possibilities, the following

table is shown. All the possible modulation schemes are shown at a particular

data rate, the correspoding estimator is also shown within this table.Also, five

different scenarios for choosing the columns in G were studied, they are:

G1 =[g0 g1 ... g35

]G2 =

[g0 g1 ... g17

]....downsized to 18

G3 =[g0 g1 .. g17 jg0 jg1 ... jg17

]G4 =

[g0 g1 .. g34 g35 jg0 jg1 ... jg17

]....size increased to 54

G5 =[g0 g1 .. g34 g35 jg0 jg1 ... jg34 jg35

]....size increased to 72

We then simulated all of these schemes and then the results were studied to

determine the best performing scheme at a particular data rate.

The table is presented here:

From fig.4 we can clearly conclude that the G2 matrix is the most optimised

version for the BPSK scheme. It outperforms the BPSK/LMMSE scheme by

about 2dB at both the coding rate of 0.75 and 0.5. A theoretical explana-

tion to its performance lies in the fact that orthogonality between 18 columns

11

Information bits Modulation scheme Estimator G Size Coding Rate G Columns Simulation

18 BPSK LMMSE 36 0.5 G1 Fig.2

18 BPSK WLMMSE 36 0.5 G1 Fig.4


18 QPSK LMMSE/WLMMSE 18 0.5 G2 Fig.5



27 BPSK LMMSE 36 0.75 G1 Fig.2



27 18BPSK+18QPSK WLMMSE 36 0.5 G1 Fig.6



Table 1: Table of simulated schemes

Figure 4: Improved G matrices for the BPSK scheme

12

is maintained even afterpassing through the channel(or mathematically after

multplication with the channel matrix). This is however not the case for the

original matrix where although all the columns are orthogonal,their orthog-

onality is lost after passing through the channel. This explanation can also

be generalised to any G matrix size and it can be concluded that the most

optimised BPSK scheme would be the one where half the columns of G are

built from the elements[g0 g1 g2 ... g35

]and other half from the elements[

jg0 jg1 jg2 ... jg36

].

It can also be cleverly shown that this scheme quite closely relates to the

QPSK through a simple decomposition of the matrix i.e

for the QPSK scheme, we have

cQPSK=Gd implies

cQPSK=[g0 g1 g2 ... g35

]

a1 + b1j

a2 + b2j

a3 + b3j

...

a36 + b36j

where aj, bjare symbols corresponding to the real and imaginary parts of the

QPSK scheme.

Now cQPSK after the matrix multiplication is given by,

cQPSK=g0 × (a1 + b1j) + g1 × (a2 + b2j) + ....+ g35 × (a36 + b36j)

Now we can observe that the same above matrix can also be obtained if we

take a BPSK scheme and then use a G matrix double the one used above.

This cBPSK can be given by

cBPSK=[g0 g1 g2 ... g35 jg0 jg1 jg2 ... jg35

]

a1

a2

a3

...

a36

After multiplication we can clearly see that cQPSK=cBPSK , which means both

13

Figure 5: Equivalent BPSK and QPSK schemes

these schemes are theoretically equivalent.

Also from fig.5, it is quite clear how these BPSK and QPSK schemes are

equivalent.

The WLMMSE estimator also allows us to investigate if improved combined

modulation schemes can be studied at a particular information rate.One such

scheme that was studied is shown in fig.6 where it is compared with other

schemes at the similar data rate.

From all the above simulations, few interesting observations can also be drawn

which can be arguable issues.These are :

1.What factor should decide the ’size of the G matrix?’ and the ’Number of

real constellations to be incorporated in it?’

2.What ’Modulation scheme is to be chosen?’

Earlier simulations have clearly shown us that reducing the G matrix size

is a good option for improving the performance.But this would mean using

modulation schemes higher than the QPSK which are generally not desir-

able.The other alternative to this could be forming some columns with BPSK

and some with a higher modulation scheme. The exact position of these can

14

Figure 6: Mixed/Combined Modulation Schemes

only be studied from simulations and cannot be predicted theoretically.

0.9 Conclusion

1.The WLMMSE estimator clearly out performs the LMMSE schemes for

modulations using real constellations.

2.The WLMMSE estimator extends the choice for the columns of ’G’ that can

now further improve the performance of this estimator.

3.Degenerating the columns of ’G’ for proper constellations seems to out beat

other schemes, although the exact limit to Good modulation scheme v/s G

column size is yet to be studied.

4.Many combined modulation schemes that use real constellations can be in-

spected that could further use the properties of the WLMMSE estimator.

15

0.10 References

[1] M. Huemer, C. Hofbauer, J.B. Huber, “The Potential of Unique Words in

OFDM,” in Proceedings of the 15th International OFDM-Workshop,Hamburg,

Germany, pp. 140-144, September 2010.

[2] A. Onic, M. Huemer, ”Direct vs. Two-Step Approach for Unique Wordv-

Generation in UW-OFDM,” In the Proceedings of the 15th InternationalvOFDM-

Workshop, Hamburg, Germany, pp.145-149, September 2010.

[3] M. Huemer, A. Onic, C. Hofbauer, “Classical and Bayesian Linear Data

Estimators for Unique Word OFDM,” in IEEE Transactions on Signal Pro-

cessing, vol. 59, no. 12, pp. 6073-6085, Dec. 2011.

[4] M. Huemer, C. Hofbauer, J. B. Huber, “Non-Systematic Complex Number

RS Coded OFDM by Unique Word Prefix,” in IEEE Transactions on Signal

Processing, vol. 60, no. 1, pp. 285-299, Jan 2012.

[5] Widely Linear Data Estimation and Bit Grained Rate Adaption in Unique

Word OFDM.(Not published yet)

16

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