### !

## ୯!ҥ!Ҭ!೯!ε!Ꮲ!

## !!!!!!!

### !

### ႝߞπำᏢسᅺγ!

### ᅺγፕЎ!

! !### ׇӈϐᙯඤୱౢғݤϷځӧ

### ӭᒡΕӭᒡр҅ҬϩᓎӭπسϐᔈҔ

### !

### Transform Domain Approach for Sequence Design

### and

### Its Applications to MIMO-OFDM Systems

### ࣴ ز ғǺጰໜ Student: Lung-Sheng Tsai

### Ꮴ௲Ǻػቺ റγ Advisor: Dr. Yu Ted Su

!### ׇӈϐᙯඤୱౢғݤϷځӧ

### ӭᒡΕӭᒡр҅ҬϩᓎӭπسϐᔈҔ

### Transform Domain Approach for Sequence Design and

### Its Applications to MIMO-OFDM Systems

### ࣴ ز ғǺጰໜ Student: Lung-Sheng Tsai

### Ꮴ௲Ǻػቺ റγ Advisor: Dr. Yu T. Su

### ୯ҥҬ೯εᏢ!

### ႝߞπำᏢسᅺγ!

### ᅺγፕЎ!

### A Thesis

### Submitted to Institute of Communication Engineering

### College of Electrical Engineering and Computer Science

### National Chiao Tung University

### in Partial Fulfillment of the Requirements

### for the Degree of

### Master of Science

### in

### Communication Engineering

### June 2004

### ׇӈϐᙯඤୱౢғݤϷځӧ

### ӭᒡΕӭᒡр҅ҬϩᓎӭπسϐᔈҔ

!### ࣴزғǺጰໜ!! ! ! ! ! Ꮴ௲Ǻػቺറγ!

### !

### ୯ҥҬ೯εᏢႝߞπำᏢسᅺγ!

### !

### !

### ύЎᄔा!

### !

### ӭᒡΕӭᒡрس)NJNP*མଛ҅Ҭϩᓎӭπ)PGEN*מೌڀԖ࣬ε

### ޑወΚёаၲډ׳ଯޑᒡໆǴӧჴሞسБय़ςԖ࣬ӭޑࣴزԋ

### ݀ǴՠϝԖ٤ᜢᗖ܄ޑᚒࡑլܺǶҁፕЎख़ᗺӧܭ MIMO-OFDM

### س܌ሡޑૻဦीǶ

### ӧҁፕЎύǴךॺගрسϯޑᙯඤୱׇӈౢғݤǹԜБݤёа

### ౢғಔڀԖؼӳԾ࣬ᜢ(autocorrelation)ᆶϕ࣬ᜢ(cross-correlation)܄

### ፦ޑׇӈǶ೭٤ׇӈΨёҗԖज़ૻဦဂᗺ(finite constellation points)

### җϸӛᚆණഡճယᙯඤٰౢғǶ٤ςԖޑׇӈёаҔךॺ܌ගрޑБ

### ԄౢғǴฅԶךॺගрޑׇӈၨᙑԖޣڙၨϿޑज़ڋǶ୷ܭךॺ܌ගр

### ޑཷۺǴ೭٤ࡌҥᆢׇӈޑБݤёаᇸܰӦۯ՜ډӭᆢତӈׇӈ

### )multi-dimensional array sequences*ޑࡌᄬǶ

### ճҔҁЎ܌ϟಏޑཥׇӈǴךॺගрΑёҔܭ MIMO-OFDM سޑ

### ߞဦ่ᄬǹךॺΨፕΑ୷ܭԜ่ᄬჹᔈޑᓎᅆ౽ෳаϷ೯ၰ

### ෳᄽᆉݤǶҗႝတኳᔕޑ่݀ёޕךॺ܌ගрޑբݤޑዴၲډΑന٫

### ޑਏૈ߄Ƕ!

**Transform Domain Approach for Sequence Design**

**and Its Applications to MIMO-OFDM Systems**

Student : Lung-Sheng Tsai Advisor : Yu T. Su

Department of Communication Engineering National Chiao Tung University

**Abstract**

Multiple antenna based Multiple-Input Multiple-Output (MIMO) systems employ-ing Orthogonal Frequency Division Multiplexemploy-ing (OFDM) have the potential of achievemploy-ing the capacity promised by information theoretical prediction. Though much progress to-ward a practical high rate MIMO-OFDM system has been made, many related system design issues remain to be settled. This thesis sets forth to solve the critical issue of the preamble design for MIMO-OFDM systems.

We present a systematic method based on the frequency (transform) domain char-acterization to generate a new family of sequences with the desired autocorrelation and cross-correlation properties. Sequences having the desired properties can then be gen-erated by taking inverse transform of some ﬁnite constellation points (BPSK, QPSK, ... etc.). We also demonstrate that some existing sequences can easily be generated by our approach but our new family of sequences renders less constraints. The proposed approach can easily be extended to synthesize two dimensional arrays or even higher dimensions waveforms that possess the desired multi-dimensional correlation properties. A preamble structure based on our new sequence family is suggested and algorithms for frequency oﬀset and channel estimations in MIMO-OFDM systems are developed. Both theoretical analysis and computer simulation show that these algorithms yield optimal performance.

**Contents**

**English Abstract** **i**

**Contents** **ii**

**List of Figures** **iv**

**1** **Introduction** **1**

**2** **Orthogonal Sequences and Related Properties** **4**

2.1 Welch bound (Sarwate bound) . . . 5

2.2 A new set of orthogonal sequences . . . 7

2.2.1 Notation and deﬁnitions . . . 7

2.2.2 FZC sequences . . . 8

2.2.3 Generation of the new set of sequences . . . 8

2.2.4 Properties of the new set of sequences . . . 9

2.2.5 Summary of the new set of sequences . . . 12

2.3 PS Sequences . . . 12

2.3.1 Generation of the PS sequence . . . 13

2.3.2 Properties of the PS sequences . . . 14

2.4 Comparison . . . 14

**3** **The Basis Sequence of the PS Sequences** **19**
3.1 Preliminary . . . 19

3.2 Generating basis sequences . . . 20

**4** **Multi-dimensional Arrays** **29**
4.1 Array correlation functions . . . 29

4.2 New 2D arrays . . . 30

4.3 Properties of the new proposed 2D arrays . . . 31

4.4 Extension to multi-dimensional arrays . . . 32

**5** **Preamble Structure for MIMO-OFDM WLAN Systems** **34**
5.1 Backgrounds . . . 34

5.1.1 MIMO-OFDM WLAN systems . . . 34

5.2 Proposed preamble structure . . . 34

5.2.1 Cyclic preﬁx . . . 36

5.2.2 Length of the training sequence . . . 36

5.2.3 Constraints on the constellation of training symbols . . . 37

5.3 Simulation environment . . . 38

**6** **Fine Frequency Oﬀset Estimation and Channel Estimation** **40**
6.1 Timing and frequency synchronization for SISO OFDM systems . . . 40

6.1.1 Schmidl and Cox’s algorithms . . . 41

6.2 Frequency synchronization schemes for MIMO-OFDM systems . . . 43

6.3 Channel Estimation . . . 44

6.3.1 System model . . . 44

6.3.2 A least square error channel estimator . . . 46

6.3.3 Numerical results . . . 47

**7** **Conclusion** **51**

**List of Figures**

2.1 *Construct the C _{i}(λ)’s from the DFT points of the basis sequence. . . . .* 16
2.2 Θ

_{c}_{i}_{,c}_{i}(λ)’s for 0 < i < K are simply frequency-shifting functions of Θ_{c}_{0}

_{,c}_{0}

*(λ). 17*2.3

*The autocorrelation function of the new sequence.(K = 2, N*

_{c}*= 32, and*

*N = KN _{c}* = 64) . . . 17
2.4

*The autocorrelation function of the PS sequence.(K = 4, N*2

_{b}*= 16, and*

*N _{s}= KN_{b}*2 = 64) . . . 18
3.1

*3-IDFT of b*

_{i}. The elements of the matrix G, g_{0}

*∼ g*

_{8}, are complex number

with unit magnitude. . . 20 3.2 Rate-expanding and time-shifting. The time-domain sequences and their

corresponding frequency-domain sequences are showed. . . 22 3.3 Generate the new sequence with perfect periodic AC property. All

fre-quency components are of equal magnitude. . . 23
3.4 *g.c.d.(m, N _{b}) = 1. Magnitude plot for the DFT of each column vector of*

*G(m)*. . . 27
3.5 *g.c.d.(m, N _{b}) = 1. AC function of the sequence of m = 3. . . .* 27
3.6

*g.c.d.(m, N*)

_{b}*= 1. Magnitude plot for the DFT of each column vector of*

*G(m)*. . . 28
3.7 *g.c.d.(m, N _{b}*)

*= 1. AC function of the sequence of m = 2. . . 28*

4.1 *(a) Construct the F*(0)*(U, V ) from the two-dimensional DFT points of*
the basis array. (b) Diﬀerent symbols represent the non-zero positions of

*F(i)(U, V ) for diﬀerent i’s; (K*_{1} *= 2, K*_{2} *= 2, N*_{1} *= 4, and N*_{2}= 5.) . . . . 31
4.2 Magnitude plot for the two-dimensional periodic AC function of proposed

array sequences, *|R _{C}(i)|. (K*

_{1}

*= 2, K*

_{2}

*= 2, N*

_{1}

*= 4, and N*

_{2}= 5.) . . . 32

5.1 OFDM training structure in IEEE 802.11a standard. (We will redesign the long training sequences.) . . . 35 5.2 A time orthogonal preamble for a MIMO conﬁguration with 2 transmit

antennas. (The guard intervals are not shown in this ﬁgure.) . . . 36 5.3 A coded orthogonal preamble for a MIMO conﬁguration with 2 transmit

antennas. . . 36 5.4 Suggested long training symbol structure for a MIMO conﬁguration with

2 transmit antennas. . . 37 5.5 Transmitter block diagram for the OFDM PHY. . . 37

6.1 A typical result of Schmidl and Cox’s synchronization algorithm. The es-timated frequency oﬀset is in unit of the subcarrier spacing. (Parameters: r.m.s. delay spread = 50ns, frequency oﬀset = 0.3 subcarrier spacing, and SNR = 10dB.) . . . 42 6.2 MSE of the frequency oﬀset estimator in diﬀerent multi-path

environ-ments (2Txs and 1 Rx). Frequency oﬀset = 0.3 subcarrier spacing. . . 43 6.3 MSE of the frequency oﬀset estimator for systems of diﬀerent number of

transmit antennas. (Frequency oﬀset=0.3 subcarrier spacing and r.m.s. delay spread=50ns.) . . . 44 6.4 A system with two transmit antennas and one receive antenna. . . 45 6.5 Mean squared channel estimation error. Noise variance can be reduced

6.6 Mean squared channel estimation error for diﬀerent channel delay spread.(Guard Interval = 16 samples.) . . . 50

**Chapter 1**

**Introduction**

It has been shown that the capacity of a wireless communication system can be greatly increased by employing multiple transmit and receive antennas. The Orthogonal Frequency Division Multiplexing (OFDM) scheme, because of its proven superiority over other wideband transmission alternatives, is a natural candidate choice for use in conjunction with the Multiple-Input Multiple-Output (MIMO) technique. Although the data rate of a MIMO-OFDM system can be increased drastically, the number of system parameters that need to be estimated in either the initial link set-up stage or the regular transmission stage increases as well.

Eﬀorts to extend some existing OFDM based standards or products like the Local Area Network (WLAN) (IEEE 802.11a) and Direct Video Broadcasting (DVB) such that they are compatible with a MIMO scenario have been actively pursued recently. The main purpose of this thesis is to design a preamble structure that is optimal for MIMO-OFDM based WLAN systems. The main utility of a preamble is to facilitate the receiver’s synchronization and channel estimation tasks (data-aided synchronization and channel estimation) with as little overhead as possible. An ideal preamble wave-form should therefore has high time and frequency resolutions while render little or no ambiguity, be it resulted from multiple propagation paths or interference bearing simi-lar structures. These desired properties usually entail a preamble that has a Dirac-like auto-correlation (AC) and, if a family of preambles is in question, zero cross-correlation

(CC). Practical considerations also require that the length of the preamble be arbitrary and the family size be as large as possible while maintaining the above two properties. Unfortunately, as far as a family of preamble sequences is concerned, one can not have all the nice properties simultaneously. The trade-oﬀs are discussed in Chapter 2.

It is instructive to consider a multiple-transmit-antenna signal as sum of diﬀerent
user signals in a multiple access systems. Our investigation begins with the consideration
of some signature sequences that are being used in CDMA systems. Gold code [1] is a
popular choice but it is not appropriate for use as a training sequence of an OFDM system
for the following two reasons. Firstly, the length of the Gold code is limited to 2*m _{−1 while}*

an OFDM frame size is often a power of 2 because of FFT implementation. Secondly,
the mutual interference caused by cross-correlation is not as low as desired especially
when short period code is used. Scholtz and Welch presented a class of sequences based
*on [2] Group Characters. These sequences have similar autocorrelation properties as*

*m-sequences, and their cross-correlations can be smaller than those of Gold codes. But*

the sequence duration must be a prime number and the cross-correlation is still not low enough when the sequence duration is short. Walsh-Hadamard sequences are orthogonal (zero cross-correlation) in a frame-synchronous condition but do not possess the desired Dirac-like autocorrelation property, suﬀering from severe self-interference when multiple propagation paths exist.

As one cannot have both the ideal auto-correlation and cross-correlation, an
alter-nate design philosophy is to sacriﬁce some of the desired properties that do not bring
in either self interference or interference from other system users. For example, the
autocorrelation at time-lags larger than the transmission channel’s multipath delay do
not contribute to self-interference (inter-symbol interference, ISI). A set of sequences,
*called the PS sequences does have some desirable properties that can be applied to be*
the preamble design for the case of interest to us. In the ensuing chapter we present a
new family of preamble sequences that have the desired AC and CC properties similar to

those of the PS sequences but have less constraint on the sequence durations. We pro-pose a transform (frequency) domain approach such that the AC and CC requirements are directly transformed into the frequency domain identities. Although this approach has been suggested to generate complex sequences with Dirac-like aperiodic AC property and study some combinatorial design problems, as far as we know, it has never applied to the design of preamble sequences with predetermined periodic AC and CC functions. Our approach also has the beneﬁt of interpreting the PS sequences from the fre-quency (transform) domain’s viewpoint. The sequences proposed in [8] have perfect AC properties and were used to generate the PS sequences. In Chapter 3, we explain why the sequences of [8] can have perfect AC properties by using similar concepts of chapter 2. In Chapter 4, We extend the concepts developed for one-dimensional sequences to multi-dimensional array sequences. In Chapter 5, we propose a preamble structure for MIMO-OFDM WLAN systems, and show why our new sequence has some ﬂexibility when applied in OFDM systems. Chapter 6 presents algorithms for frequency oﬀset estimation and channel estimation. We show that the proposed preamble structure can achieve the lower bound of the channel estimation mean square error.

**Chapter 2**

**Orthogonal Sequences and Related**

**Properties**

*Park et al. [3] invented a set of sequences (called PS sequences) that has excellent*
periodic autocorrelation and cross-correlation properties. The periodic AC of the
se-quence is 0 except at periodic intervals and the CC function between properly selected
sequences is identical to 0 everywhere. Because of the AC and CC properties, they
propose to use the PS sequences as the signature sequences for a cellular CDMA
sys-tem that operates in an environment whose delay spread is less than the period of the
sequences. Without the bandlimiting eﬀect, such a system is free from ISI and multiple
access interference (MAI) that limit the system capacity.

Our main interest is to ﬁnd proper training sequences for use in a preamble so that a MIMO-OFDM receiver can easily accomplish the link setup process within the pream-ble period. The setup process includes at least package detection, frame and frequency synchronization and channel estimation. Such a synchronization procedure involves the detection and estimation of some signal and channel parameters in a multiple antenna scenario. Conventional maximum likelihood (ML) paradigm solves this data-aided es-timation and detection problem by an estimator-correlator type receiver structure and necessitates the ideal AC and CC properties on the part of the training sequences. As will become clear in the next section that one can not have both ideal AC and CC properties. The next best thing one can have is something similar to the PS sequences,

assuming a known delay spread channel. In that situation we can employ the PS se-quences, using diﬀerent members of the family in diﬀerent transmit antennas during the preamble period. To remove the constraint on the PS sequence length, we propose a new family of sequences which have similar AC and CC behavior. Instead of using the ad hoc approach of [3], we interpret the AC and CC requirements in the transform domain and provide a simpler and more natural derivation. We oﬀer a new simple derivation of the PS family and show that the new family of sequences is a generalization of the PS sequences.

**2.1**

**Welch bound (Sarwate bound)**

Sets of periodic sequences with good correlation properties are desired in many communication applications. Oftentimes we hope to have a set of sequences whose AC function has a single peak at the zero delay and whose CC values are identically zero. Such sequences can be used to avoid or minimize the interference from other antennas (or other users) and eliminate the ISI due to a multi-path channel. However, it is observed that a set of sequences having good AC properties, e.g., PN sequences and Gold sequences, does not have good CC properties. On the other hand, the ideal AC requirement can not be met if the set has good CC properties. Walsh-Hadamard code is a typical example. In fact the bounds on CC and AC of sequences derived in [4] and [5] indicate that there is a tradeoﬀ between AC and CC when designing sequences. For convenience of reference we present these bounds in the followings.

**Theorem 1 Let X denote a set of K complex-valued sequences of period N , i.e., for**

*every sequence u∈ X, u _{i}= u_{i+N}, for all i∈ Z, Z being the set of integers. The periodic*

*CC function θ(u, v)(·) for sequences u, v ∈ X is deﬁned by*

*θ(u, v)(l) =*
*N −1*

*i=0*

*u _{i}v_{i+l}∗*

*,*

*l*

*∈ Z,*(2.1)

*The periodic AC function θ(u)(l) for the sequence u is just θ(u, u)(l). We assume that*

*θ(u)(0) = N for all u∈ X then it is obvious that |θ(u)(l)| ≤ N and |θ(u, v)(l)| ≤ N for*

*all u, v* *∈ X. For the set X, the maximum periodic CC magnitude θ _{c}*, and the maximum

*out-of-phase periodic AC magnitude θ*deﬁned by

_{a}*θ _{c}* = max

*{|θ(u, v)(l)| : u, v ∈ X, u = v, 0 ≤ l ≤ N − 1}*

*θ _{a}* = max

*{|θ(u)(l)| : u ∈ X, 0 < l ≤ N − 1}*must satisfy

**Theorem 2 For any set X of K sequences of period N satisfying θ(u)(0) = N for all**

*u∈ X,*
*θ*2_{c}*N*
+ *N* *− 1*
*N (K− 1)*
*θ _{a}*2

*N*

*≥ 1.*(2.2)

The proof was given in [5]. Invoking this theorem, we assign properly-selected sequences
*with period N (N* *≥ 2) to diﬀerent transmit antennas. For a MIMO receive it is necessary*
to separate signals emitting from diﬀerent transmitting antennas, or equivalently, it
should have the capability to resolve and estimate the impulse response of each
sub-channels between any pair of transmit-receive antennas. One way to achieve
near-optimal channel estimation is to use pilot sequences that have perfect CC properties,
*i.e., θ _{c}*

*= 0. If there are K transmit antennas, we need at least K diﬀerent preamble*

*sequences. The above theorem implies that θ*

_{a}≥ N

*K−1*

*N −1* and, for a MIMO system with

*two transmit antennas (K = 2), we have θ _{a}*

*≥*

*√N*

*N −1*

*≥*

*√*

*N . Thus even for a set of*

*only two sequences of length N with perfect CC properties, i.e., θ _{c}(l) = 0 for all l, it is*

*impossible for them to yield the ideal AC function θ*as the training signals in the system, the received signal will be interfered by their delay versions in a multi-path environment. The above discussion convince us that instead of trying to ﬁnd a set of sequences with ideal correlation properties, we might as well focus our attention on ﬁnding some set of sequences that have nonzero AC values at

_{a}(n) = 0. If we use these sequences*some desired positions n’s whose corresponding AC values θ(u)(n) = n _{l}*

*= 0 meet our*requirements.

**2.2**

**A new set of orthogonal sequences**

In this section, we present a new set of sequences having some desired periodic AC and CC properties. This new family of sequences is a generalized version of what had been referred to as the PS sequences [3]. Notations and deﬁnitions are given ﬁrst and then a class of sequences to be used to generate the new family is introduced before presenting the derivation of the new sequences and the associated properties.

**2.2.1**

**Notation and deﬁnitions**

**Deﬁnition 1 Let us deﬁne the N***× N DFT matrix with index m as*

*F(N,m)(k, l) = [W _{N}−klm] = (W_{N}m*)

*−kl,*(2.3)

*where m is a natural number, k, l = 0, 1, . . . , N* *− 1, W _{N}*

*= ej2π/N*

*and j =√−1.*

**Deﬁnition 2 The diagonalized matrix D(**{x_{l}}) associated with the sequence {x_{l}} is

*de-ﬁned as*

*D({x _{l}}) = diag({x_{l}}).* (2.4)

**Deﬁnition 3 The quotient and residual functions Q and R corresponding to the devisee**

*and divisor (α, β) are deﬁned as*

*Q(α, β) = q, R(α, β) = r,* (2.5)

*where α and q are integers, β is a natural number, and α = qβ +r with r = 0, 1, . . . , β−1.*

**2.2.2**

**FZC sequences**

The well-known complex sequences, Frank-Zadoﬀ-Chu (FZC) sequences [6], [7]
ren-der a Dirac-like periodic AC functions whose values are zeros for all non-zero lags. More
speciﬁcally, a FZC sequence *{a _{k}} of length N has entries of unity-modulus complex*

*numbers, i.e., a*

_{k}*= ejαk*

_{, k = 0, . . . , N}− 1. When N is even, they are given by*a _{k}* = exp

*jM πk*2

*N*

*,*(2.6)

*where M is an integer prime to N , while if N is odd,*

*a _{k}* = exp

*jM πk(k + 1)*

*N*

*,*(2.7)

*where M is also an integer prime to N . It is proved that for any such length N sequence,*

*θ(a)(n) = N δ(n), for n = 0, 1, . . . , N− 1. The single maximum of magnitude N occurs*

*at n = 0.*

**2.2.3**

**Generation of the new set of sequences**

*We now introduce a procedure to generate a family of sequences of length N = KN _{c}*

*based on the FZC code. To begin with, we need a length-N*sequence having perfect AC property, i.e., the AC function of this sequence is zero for all non-zero delays. This sequence will be refer to as the basis or the generating sequence henceforth. FZC

_{c}*se-quence is a good candidate that meets our need. Denote the FZC code of length N*by

_{c}*the N*

_{c}-by-1 vector x and let c_{0}

*, c*

_{1}

*, . . . , c*be the sequences to be generated which have

_{K−1}*the desired AC and CC properties with the vector c*

_{i}*being the ith sequence of length*

*N .*

*Taking the N _{c}-point DFT of the vector x, we obtain*

*X(k) =*

*N**c−1*

*n=0*

*x _{i}W_{N}kn_{c}, W_{N}_{c}*

*= e−j2π/Nc*

_{, and 0}≤ k < N*c.* (2.8)

*The N -DFT of the vector c _{i}* is

*C _{i}(λ) =*

*N −1*
*n=0*

*Using following assignments on C _{i}(λ) and λ*

*C _{i}(λ) =*

*KX(k) ; λ = Kk + i,*

0 *; otherwise,* (2.10)

*and taking the N -point IDFT on C _{i}(λ), we obtain a set of sequences of length N . This*
procedure is illustrated in Fig. 2.1.

**2.2.4**

**Properties of the new set of sequences**

*1. Autocorrelation function:*

*Let c _{l,i}*

*represents the lth element in the sequence vector c*. The AC function of the

_{i}*new sequence c*is given by

_{i}*θ(c _{i})(n) =*

*N −n−1*

*l=0*

*c*+

_{l+n,i}c∗_{l,i}*N −1*

*l=N −n*

*c*

_{l+n−N,i}c∗_{l,i}*= N W*)) (2.11)

_{N}niδ(R(n, N_{c}Before proving the AC function above, we need introduce some simple lemmas:

**Lemma 1 The periodic autocorrelation function of x(n), θ(x, x)(n), is equivalent to the**

*circular convolution function between x(n) and x∗*(*−n).*

*Proof :*

*The periodic autocorrelation function of a sequence of length N ,* *{x(n)}, is deﬁned as*

*θ(x, x)(n)*
*N −1*

*τ =0*

*x(n + τ )x∗(τ ). The circular convolution function between x(n) and*

*x∗*(*−n) is*
*x(n) x∗*(*−n) =*
*N −1*
*τ =0*
*x(n− τ)x∗*(*−τ)*
=
*N −1*
*τ =0*
*x(n + τ )x∗(τ )*
*= θ(x, x)(n)* (2.12)

*Using the same argument, we conclude that the cross-correlation function θ(x, y)(n) is*
*equivalent to x(n) y∗*(*−n).*

**Lemma 2 For any two sequences x(n), y(n) with perfect CC property, i.e., θ(x, y)(n) =**

*0 for all integer n, their DFT’s must satisfy*

*X[k]Y [k] = 0,* *∀ k.*
*Proof :*

Perfect CC function implies *θ(x, y)(n) = 0 for all n, or equivalently, as implied by*

*Lemma 1, x(n) y∗*(*−n) = 0. Taking DFT on both sides, we have X[k]Y∗[k] = 0.*

*Proof of (2.11)*

*Lemma 1 implies that the AC function for c _{i}* is given by

*θ(c _{i}, c_{i})(n) = c_{i}(n) c∗_{i}*(

*−n).*(2.13)

*Taking N -point DFT on both side, we have*

Θ* _{c}_{i}_{,c}_{i}(λ) = C_{i}(λ)C_{i}∗(λ),* 0

*≤ λ < N.*(2.14) Substituting (2.10) into (2.14), Θ

_{c}_{i}_{,c}_{i}(λ) can be expressed as:Θ_{c}_{i}_{,c}_{i}(λ) =

*K*2*X(k)X∗(k) ; λ = Kk + i,*

0 *; otherwise,* (2.15)

*where X(k)’s ares the N _{c}-point DFTs of the FZC sequence x which is of length N_{c}*.

*Since x is an FZC sequence, the AC function of x is*

*θ(x, x)(n) = x(n) x∗*(*−n) = N _{c}δ(n).* (2.16)

*Taking N*-point DFT on (2.16), we have

_{c}*Θ(x, x)(k) = X(k)X∗(k) = N _{c}.* (2.17)

Comparing (2.15) and (2.17), we ﬁnd out that Θ_{c}_{0}_{,c}_{0}*(λ) is an rate-expanded version of*
*Θ(x, x)(k), and K is the expanding rate. That is,*

Θ_{c}_{0}_{,c}_{0}*(λ) =*

*K*2*Θ(x, x)(k) = K*2*N _{c}*

*; λ = Kk,*

*Hence we can expect that the AC function of the newly generated sequence c*_{0}be periodic
*with period N _{c}*, i.e.,

*θ _{c}*

_{0}

_{,c}_{0}

*(n) = N δ(R(n, N*(2.19)

_{c})),*where R(·) is deﬁned in (2.5). The other Θ _{c}_{i}_{,c}_{i}(λ)’s for 0 < i < K are simply *
frequency-shifting functions of Θ

_{c}_{0}

_{,c}_{0}

*(λ), i.e.,*

Θ_{c}_{i}_{,c}_{i}(λ) = Θ_{c}_{0}_{,c}_{0}*(λ− i).* (2.20)
Fig. 2.2 gives a graphic explanation on this relation. The frequency-shifting operation
induces a phase rotation in time-domain. This means that

*θ _{c}_{i}_{,c}_{i}(n) = W_{N}niθ_{c}*

_{0}

_{,c}_{0}

*(n) = N W*(2.21)

_{N}niδ(R(n, N_{c})),*where W*

_{N}*is deﬁned by ej2π/N*as before. The AC function has a nonzero value only

*when R(n, N*

_{c}) = 0; i.e., n = IN_{c}, where I is an integer. One can control the interval*(period of the AC function) by choosing the value of N*properly. Fig. 2.3 is a typical plot for the AC function of the new sequences. In this example, the sequence length is

_{c}*N = KN _{c}* = 2

*× 32.*

*2. Crosscorrelation function:*

*Let us denote two new sequences as c _{i}*

*and c*. The CC function of the two sequences

_{j}*is 0 if i= j.*

*Proof :*

From (2.15), we have*∀λ,*

Θ* _{c}_{i}_{,c}_{i}(λ)Θ_{c}_{j}_{,c}_{j}(λ) = 0, for i= j, and 0 ≤ i, j < K.* (2.22)
Substituting (2.14) into the equation above, we have:

*C _{i}(λ)C_{i}∗(λ)C_{j}(λ)C_{j}∗(λ) = 0, for i= j, and 0 ≤ i, j < K.* (2.23)
It implies that

Hence Θ_{c}_{i}_{,c}_{j}(λ) = 0 for all λ. By applying Lemma 1 and Lemma 2, we conclude that,*for i= j, θ(c _{i}, c_{j})(n) = 0 for all n. Hence they do have perfect CC properties.*

The new set of sequences can be used as the training sequences for MIMO-OFDM
*systems. The parameters K and N _{c}*of the sequences are to be determined by the number
of transmit antennas and the length of the maximum delay spread.

**2.2.5**

**Summary of the new set of sequences**

The major attributes of the proposed family of sequences are : 1. Sequence are composed of complex numbers with unity magnitude.

2. The basis sequence determines the AC function in one period, and the CC properties between sequences are determined by their DFTs.

*3. Given a sequence of length N _{c}* with the perfect periodic AC property, we generate

*K diﬀerent sequences of length KN _{c}*; all of them have periodic impulse-like AC function.

**2.3**

**PS Sequences**

*In [3], a new class of polyphase sequences (called PS sequences) for CDMA systems*
and its generation method are suggested. *The PS sequence is of length N _{s}*, where

*N _{s}= KN_{b}*2

*. The value K determines how many diﬀerent sequences having excellent CC*

*function we can used. The value N*2 determines the period of the AC function of the sequences. The PS sequences have perfect CC function so we can completely reject the interference from other users in a multiuser system with these sequences. However, the PS sequence still has some unwanted peaks in its AC function. It degrades the system performance when the system is operated in a multi-path environment. Hence there is a restriction in using the PS sequence in CDMA systems. The delay spread of the reverse-link channels must be small. In this section, we will introduce the PS sequence and its generation methods.

_{b}**2.3.1**

**Generation of the PS sequence**

*Deﬁne the basic symbols as N _{b}(not necessarily distinct) symbols b_{i}, i = 0, . . . , N_{b}−1,*

*all with equal magnitude. Without loss of generality, we assume b*’s are all located on the unit circle in the complex plane. We ﬁrst generate an orthogonal sequence from

_{i}*b _{i}*’s. Simply take

*{W*0

_{N}

_{b}, . . . , WNb−1*Nb* *}(uniformly distributed on the unit circle) as a set of*
basic symbols. For a set of basic symbols *{b _{i}} and 1 ≤ m ≤ N_{b}− 1, we deﬁne the basic*

*orthogonal sequence matrix G of size N*as

_{b}× N_{b}*G = F(Nb,−m) _{D(}_{{b}*

*i}).* (2.25)

The basic orthogonal sequence *{g _{p}} of length N_{b}*2 is deﬁned by [8]

*g _{p}*

*= G*

_{Q(p,N}_{b}_{),R(p,N}_{b}_{)}(2.26)

*or equivalently, if g = [g*

_{0}

*, g*

_{1}

*, . . . , g*2

_{N}*b−1*]

*T*_{,}

*g = vec(GT),* (2.27)
where vec(*·) denotes the stacking operator.*

Using the basic orthogonal sequence*{g _{p}}, we form the N_{s}× K matrix H as*

*H = [h _{i,k}],* (2.28)
where

*h*=

_{i,k}*N*

_{}

*2*

_{b}*−1*

*p=0*

*g*(2.29)

_{p}δ(i− k − pK),*N*2

_{s}= KN_{b}*, δ(n) =*

*1, n = 0*

*0, n= 0* *, and K is a natural number.*

*The PS sequence matrix C of size N _{s}× K (or KN_{b}*2

*× K) is deﬁned as*

*C = [c _{l,k}*] = 1

*N _{b}F*

*(Ns,−1) _{H.}*

_{(2.30)}

*Each column vector of C forms a sequence* *{c _{l,k}, l = 0, 1, . . . , N_{s}− 1} which is called a*

*PS sequence.*

**2.3.2**

**Properties of the PS sequences**

*1. Autocorrelation function:*

The AC function of the PS sequence is given by

*θ(c)(τ ) =*
*Ns**−τ−1*
*l=0*
*c _{l+τ,k}c∗_{l,k}*+

*N*

*s−1*

*l=Ns−τ*

*c*

_{l+τ −N}_{s}_{,k}c∗_{l,k}*= N*2

_{s}W_{N}τ k_{s}δ(R(τ, N_{b}*)).*(2.31)

*The AC function has a nonzero value only when R(τ, N*2

_{b}*) = 0; i.e., τ = IN*2

_{b}*, where I is*

*an integer. We can control the interval or period by properly choosing the value of N*2. On the contrary, the PN sequence has nonzero values of the AC function at all intervals. The PS sequence has better CC properties than the PN sequence. Fig. 2.4 is a typical plot for the AC function of the PS sequences.

_{b}*2. Crosscorrelation function:*

Let us denote two PS sequences as *{c _{l,k}I} and {c_{l,k}II}. The CC function of the two*

*sequences is 0 if kI*

*= kII*.

**2.4**

**Comparison**

If we compare the proposed sequences with PS sequences, we can ﬁnd that the main
diﬀerence between them is the choice of the basic orthogonal sequence. From (2.25)
*to (2.27), a sequence of length N _{b}*2 with perfect AC function is generated. This step is

*similar to generating a FZC sequence of length N*2in our procedure. (This sequence may be not the same with the sequence generated by (2.25), (2.26), and (2.27).) In fact, the only constraint on the basis sequence is its AC property only. We can generate a family of PS-like sequences as soon as we ﬁnd a sequence with perfect AC function. Comparing to the PS sequences, our new sequences have less constraint on sequence length. Fig. 2.3 represents a typical plot for the AC function of the new sequences whose length is

_{b}function as Fig. 2.3, since 32*= N _{b}*2

*, for any natural number N*. We can also apply the new concept discussed in section 2.2.4 to show why the sequence generated from (2.25), (2.26), and (2.27) must be an orthogonal sequence. In chapter 3, we will explain it in another point of view, which is diﬀerent from what had been discussed in [8]. And from (2.28) to (2.30), these steps are equivalent to the realization described in Fig. 2.2, such that perfect CC properties can be obtained. However we give a much easier proof for CC properties instead of the proof in the [3].

_{b}### O

˃ ˞ ˅˞ ʻˡ_{˶}ˀ˄ʼ˞ ˃ ˄ ˅ ʻˡ

_{˶}ˀ˄ʼ ˾ ˫ʻ˾ʼ

### O

˄ ˞ʾ˄ ˅˞ʾ˄ ʻˡ_{˶}ˀ˄ʼ˞ʾ˄ 0( )

*C*O 1( )

*C*

### O

### O

˞ˀ˄ ˅˞ˀ˄ ˆ˞ˀ˄ ˡ_{˶}˞ˀ˄ 1( )

*K*

*C*

_{}

### O

O
˃ ˞ ˅˞ ʻˡ˶ˀ˄ʼ˞
0,0( )
*c c*
T O
O
˄ ˞ʾ˄ ˅˞ʾ˄ ʻˡ˶ˀ˄ʼ˞ʾ˄
1,1( )
*c c*
T O
1, 1( )
*K* *K*
*c* *c*
T _{} _{} O
O
˞ˀ˄ ˅˞ˀ˄ ˆ˞ˀ˄ ˡ˶˞ˀ˄

Figure 2.2: Θ_{c}_{i}_{,c}_{i}(λ)’s for 0 < i < K are simply frequency-shifting functions of Θ_{c}_{0}_{,c}_{0}*(λ).*

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 AutoCorrelation function |θ(c)(n)| n

*Figure 2.3: The autocorrelation function of the new sequence.(K = 2, N _{c}*

*= 32, and*

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Auto−Correlation |θ(c)(τ)| τ

*Figure 2.4: The autocorrelation function of the PS sequence.(K = 4, N _{b}*2

*= 16, and*

**Chapter 3**

**The Basis Sequence of the PS**

**Sequences**

**3.1**

**Preliminary**

In the last chapter, we have shown that the PS sequences are generated by a basis sequence with perfect AC property. One of the candidate basis sequences was that presented in [8]. We will demonstrate that one can use the technique similar to that discussed in Chapter 2 to explain why the sequences of [8] have perfect AC function.

For a set of basic symbols*{b _{i}} located on the unit circle (|b_{i}| = 1), 0 ≤ m ≤ N_{b}− 1,*

*we deﬁne the basic orthogonal sequence matrix G of size N*as

_{b}× N_{b}*G = F(Nb,−m) _{D(}_{{b}*

*i}),* (3.1)

*where F(N,m)(k, l) = [W _{N}−klm] = (Wm*

*N*)*−kl*, 0 *≤ k, l < N, and WN* *= ej2π/N*. Hence
*F(N,1)x is equivalent to the DFT of the time-domain vector x, and F(N,−1)y is equivalent*

*to the IDFT of the frequency-domain vector y if the constant factor 1/N is omitted.*

*In the following discussion, we will omit the 1/N factor since it does not aﬀect the AC*
property. The basis sequence used in PS sequences is given by:

*g = vec(GT),* (3.2)
where vec(*·) denotes the stacking operator. Here we give a more meaningful *
interpreta-tion about that why the sequence generated in this way will have perfect periodic AC
function under some conditions.

**3.2**

**Generating basis sequences**

*We take the case for N _{b}*

*= 3 as an example without loss of generality. Assume m = 1,*then:

*G = F(Nb,−1) _{D(}_{{b}*

*i}) = F(Nb,−1)*

*b*_{0} *b*_{1} *b*_{2} *,* (3.3)

*where b*_{0} *= [b*_{0} 0 0]*T _{, b}*

_{1}

_{= [0 b}_{1}

_{0]}

*T*

_{, and b}_{2}

_{= [0 0 b}_{2}

_{]}

*T*

_{. The ith column vector of the}*matrix G, g _{i}*

*, is equal to the N*. Since there is only one non-zero element in

_{b}-IDFT of b_{i}*the vector b*

_{i}*and the elements in F are all of unit magnitude, the elements of g*must have identical magnitude. In order to make the explanation clear, we denote

_{i}*G = [g*_{0} *g*_{1}*g*_{2}] =
⎡
⎣ *gg*0_{1} *gg*3_{4} *gg*6_{7}
*g*_{2} *g*_{5} *g*_{8}
⎤
*⎦ .* (3.4)

*The relation between b _{i}’s and g_{i}*’s is showed in Fig. 3.1. We proceed by invoking a

˃
˵˃
˅ ˹
˄ ˃
˵_{˄}
˅
˄ ˹ ˃ ˄ ˅ ˹
˵˅
˃ ˄ ˅ ̇ ˃ ˄ ˅ ˃ ˄ ˅ ̇
̇
˺˄ ˺˅ ˺ˆ ˺ˇ ˺ˈ ˺ˉ ˺ˊ
˺˃ ˺ˋ
ˆˀ˜˗˙˧

*Figure 3.1: 3-IDFT of b _{i}. The elements of the matrix G, g*

_{0}

*∼ g*

_{8}, are complex number with unit magnitude.

technique similar to (2.15). Performing the rate expanding on the time-domain vectors

*g*_{0}*, g*_{1}*, and g*_{2} *by inserting some zeros with an expanding rate of N _{b}* = 3, we obtain the

*expanded vectors (N*2

_{b}*× 1)*

*g*

_{e}_{0}

*= [ g*

_{0}

*0 0 g*

_{1}

*0 0 g*

_{2}0 0 ]

*T*(3.5)

*g*

_{e}_{1}

*= [ g*

_{3}

*0 0 g*

_{4}

*0 0 g*

_{5}0 0 ]

*T*(3.6)

*g*

_{e}_{2}

*= [ g*

_{6}

*0 0 g*

_{7}

*0 0 g*

_{8}0 0 ]

*T.*(3.7)

*After taking N _{b}*2-DFT on the expanded vectors(sequences), the corresponding

*frequency-domain sequences will repeat periodically. Then we do some time-shifting on g*

_{e}_{1}and

*g _{e}*

_{2}

*such that there is no overlapping if we sum g*

_{e}_{0}

*, g*

_{e}_{1}

*, and g*

_{e}_{2}together. Denote the

*time-shifted vectors as g*

_{es}_{0}

*, g*

_{es}_{1}

*, and g*

_{es}_{2}, and then

*g _{es}*

_{0}

*= [ g*

_{0}

*0 0 g*

_{1}

*0 0 g*

_{2}0 0 ]

*T*(3.8)

*g*

_{es}_{1}

*= [ 0 g*

_{3}

*0 0 g*

_{4}

*0 0 g*

_{5}0 ]

*T*(3.9)

*g*

_{es}_{2}

*= [ 0 0 g*

_{6}

*0 0 g*

_{7}

*0 0 g*

_{8}]

*T.*(3.10)

This shifting operation will cause some phase rotation in frequency-domain. In fact the phase rotation is not important, and we will show the reason later. We depict these steps in Fig. 3.2 in which the phase rotation eﬀects are also shown.

*Summing the vectors g _{es}*

_{0}

*, g*

_{es}_{1}

*, and g*

_{es}_{2}together, we obtain

*g = g _{es}*

_{0}

*+ g*

_{es}_{1}

*+ g*

_{es}_{2}

*= vec(GT).*(3.11)

*We plot the corresponding result in Fig. 3.3. The corresponding N*2

_{b}*-DFT points of g*

are all with the same magnitude.

In order to explain the eﬀects of the phase rotation, we need to recall some properties
*introduced in chapter 2. Denote the AC function of a sequence x(n) as θ(x, x)(n) and*
*its N _{c}-DFT as Θ(x, x)(k), we hope that*

*θ(x, x)(n) = N _{c}δ(n),* (3.12)
and

*Θ(x, x)(k) = X(k)X∗(k) = N _{c}.* (3.13)

*These two equations had been mentioned in (2.16) and (2.17). Hence if we set X(k) =*

*√*

*N _{c}ejφk*

_{, i.e., all frequency components have the same magnitude, then (3.13) can be}achieved. The constant

*√N*can be ignored since it does not aﬀect the AC and CC properties. Notice that there is no constraint on the choice of

_{c}*{φ*

_{k}}. We can concludethat a sequence has perfect periodic AC property if all of its frequency components
*have the same magnitude. Hence the new sequence g has perfect AC property. This*

statement also explains why the phase rotation induced from the time-shifting operation is not important. From Figs. 3.2 and 3.3, we also can get the idea why the transpose and the stacking operations in (3.2) are needed. Up to now, we have concentrate our

˃
˵˃
˅ ˹
˄
˃ ˄ ˅ ̇
˺˄
˺˃ ˺˅
ˆˀ˜˗˙˧
˅ ̇
˺˅
˺˄
˃
˺˃
˄ ˆ ˇ ˈ ˉ ˊ ˋ
˅
˵˃
˵˃
˃
˵˃
˄ ˆ ˇ ˈ ˉ ˊ ˋ
˹
˥˴̇˸ˀ˸̋̃˴́˷˼́˺
˃
˵˅
˅ ˹
˄
˃ ˄ ˅ ̇
˺ˉ ˺ˊ ˺ˋ
ˆˀ˜˗˙˧
˅ ̇
˃ ˄ ˆ ˇ ˈ ˉ ˊ ˋ
˅
˵˅˪ˡ˄ˉ
˵˅˪ˡ˄˃
˃
˵˅˪ˡˇ
˄ ˆ ˇ ˈ ˉ ˊ ˋ
˹
˥˴̇˸ˀ˸̋̃˴́˷˼́˺
˴́˷
̇˼̀˸ˀ̆˻˼˹̇˼́˺
ʻ̅˼˺˻̇ʳ̆˻˼˹̇ʳ˵̌ʳ˅ʼ
˥˸̃˸˴̇ʳ̃˸̅˼̂˷˼˶˴˿˿̌
˵̈̇ʳ
̊˼̇˻ʳ̃˻˴̆˸ʳ̅̂̇˴̇˼̂́
˃
˵˄
˅ ˹
˄
˃ ˄ ˅ ̇
˺ˆ ˺ˇ ˺ˈ
˅ ̇
˺ˈ
˺ˇ
˃
˺ˆ
˄ ˆ ˇ ˈ ˉ ˊ ˋ
˅
˵˄˪ˡˊ
˵˄˪ˡˇ
˃
˵˄˪ˡ˄
˄ ˆ ˇ ˈ ˉ ˊ ˋ
˹
˥˴̇˸ˀ˸̋̃˴́˷˼́˺
˴́˷
̇˼̀˸ˀ̆˻˼˹̇˼́˺
ʻ̅˼˺˻̇ʳ̆˻˼˹̇ʳ˵̌ʳ˄ʼ
ˆˀ˜˗˙˧ ˺ˉ ˺ˊ ˺ˋ
˥˸̃˸˴̇ʳ̃˸̅˼̂˷˼˶˴˿˿̌
˵̈̇ʳ
̊˼̇˻ʳ̃˻˴̆˸ʳ̅̂̇˴̇˼̂́
˥˸̃˸˴̇ʳ̃˸̅˼̂˷˼˶˴˿˿̌
ˆ˅_{ˀ˗˙˧}
ˆ˅_{ˀ˗˙˧}
ˆ˅_{ˀ˗˙˧}
> @
0 0 0 0
*T*
*b*G *b*
> @
1 0 1 0
*T*
*b*G *b*
> @
2 0 0 2
*T*
*b*G *b*

Figure 3.2: Rate-expanding and time-shifting. The time-domain sequences and their corresponding frequency-domain sequences are showed.

̇
˺ˈ
˺ˇ
˺ˆ
˅
˵˃
˵˃
˃
˵˃
˄ ˆ ˇ ˈ ˉ ˊ ˋ
˹
˵˅˪ˡ˄ˉ
˵˅˪ˡ˄˃
˵˅˪ˡˇ
˺ˋ
˺ˊ
˺ˉ
˵˄˪ˡˊ
˵˄˪ˡˇ
˵˄˪ˡ˄
˺˄ ˺˅
ˆ˅_{ˀ˗˙˧}
˺˃
˅
˃ ˄ ˆ ˇ ˈ ˉ ˊ ˋ

Figure 3.3: Generate the new sequence with perfect periodic AC property. All frequency components are of equal magnitude.

*discussion on the case m = 1 only. The case m= 1 is the subject of our next discourse.*
We need to re-deﬁne some notations.

*Let X _{i}(k) = b_{i}δ(k− i), then [X_{i}(0), X_{i}(1), . . . , X_{i}(N_{b}− 1)]T*

_{= b}*i. Taking Nb*-point

*IDFT on X(k)’s, we get a sequence x _{i}(n), for 0≤ n ≤ N_{b}−1, and [x_{i}(0), x_{i}(1), . . . , x_{i}(N_{b}−*
1)]

*T*

_{= }_{g}(m=1)*i* * g*

(1)

*i* *. This is equivalent to the case discussed before for m = 1. The ith*

*column vector of the matrix G(m)* *with m= 1 can be written as:*

*g _{i}(m)= [x(m)_{i}*

*(0), x*

_{i}(m)(1), . . . , x(m)_{i}*(N*(3.14) where

_{b}− 1)]T,*x(m)*

_{i}*(n) =*

*N*

*b−1*

*k=0*

*X*

_{i}(k)W_{N}kmn_{b}*= b*

_{i}W_{N}imn_{b}*.*(3.15)

*For the case that m = 1, x*(1)

_{i}*(n) = x*

_{i}(n) = b_{i}W_{N}in_{b}. We can observe that x(m)_{i}*(n) is a*

*phase-rotated version of x*

_{i}(n). Hence we have following relations:*x _{i}(n)DF T−→ X_{i}(k) = b_{i}δ(k− i) =⇒*

*x(m)*

_{i}*(n)*

*DF T−→ X*(3.16)

_{i}[(k− i(m − 1))_{N}_{b}],*and X*]. For convenience, we will use the notation

_{i}[(k− i(m − 1))_{N}_{b}] = b_{i}δ[(k− im)_{N}_{b}*(n)*

_{N}_{b}*to denote (n modulo N*

_{b}). The term b_{i}δ[(k*− im)*] means that the eﬀect of

_{N}_{b}*diﬀerent values of m is simply the step size of index shifting in frequency-domain. If we*
can keep the shifted tones still non-overlapping, the sequence vec*{G(m)T} can preserve*
perfect periodic AC property. To change another words, if there are some values of

*m such that the sequence vec{G(m)T} has perfect AC function, then we can ﬁnd a*

*column-reordered matrix P from the diagonal matrix D({b _{i}}) = [b*

_{0}

*, . . . ,b*], such

_{N}_{b}_{−1}*that F(Nb,−m)*

_{D(}_{{b}*i}) = F(Nb,−1)P. The RHS is the case that m = 1, which has been*

*discussed. Now we check what the values of m should be such that the shifted tones are*
*still non-overlapping. We consider two cases separately, depending on if m is prime to*

*N _{b}* or not.

* Case I g.c.d.(m, N_{b}*) = 1

If there is one tone lapped over another after being shifted, then

*im mod N _{b}= jm mod N_{b}, for i= j, 0 ≤ i, j < N_{b}.*

*⇒ (i − j)m mod Nb*= 0

*⇒ (i − j)m = pNb, p∈ Z* (3.17)

*The assumption that g.c.d.(m, N _{b}) = 1 implies that N_{b}|(i − j), ∴ i = j. We thus*

*conclude that if there are two tones with i= j, they will not overlap after shifting.*

**Example 1 Consider the case, N**_{b}*= 4, m = 3, W*_{4} *= ej2π/4, and b ={W*_{4}1*, W*_{4}2*, W*_{4}3*, W*_{4}4*}.*
*In this case we have g.c.d.(m, N _{b}) = 1. The basic orthogonal sequence matrix G is *

*de-ﬁned by F(Nb,−m) _{D(}{b*

*i}). Taking DFT on each column vector of G(m), we get the column*
*vectors of P . The result is shown in Fig. 3.4, and the AC function of the sequence*
*vec(G(m)T) is plotted in Fig. 3.5.*

*Let m = hd, N _{b}*

*= kd, and g.c.d.(h, k) = 1.*

*im mod N*

_{b}*= jm mod N*

_{b}*⇒ (i − j)hd mod kd = 0*

*⇒ (i − j)h mod k = 0*

*⇒ (i − j)h = qk, q ∈ Z*(3.18)

*g.c.d.(h, k) = 1*

*⇒ k|(i − j) ∴ i = j mod k.*

*Hence there will be d tones lapped together. We can denote* *{j, j + k, j + 2k, . . .} as a*
*coset. There will be d elements in this coset and one can ﬁnd N/d diﬀerent cosets.*

From the above discussion we conclude that the sequence vec*{G(m)T} has perfect*
*periodic AC property if g.c.d.(m, N _{b}*) = 1.

**Example 2 Consider the case–N**_{b}*= 4, m = 2, W*_{4} *= ej2π/4 _{, and b =}{W*1

4*, W*42*, W*43*, W*44*}.*

*In this case we have g.c.d.(m, N _{b}) = 2. The basic orthogonal sequence matrix G(m)*

*is*

*deﬁned by F(Nb,−m)*

_{D(}_{{b}*i}). Taking DFT on each column vector of vec(G(m)*

*T*

*), we obtain*

*the desired result as shown in Fig. 3.6. We notice that if we sum up all column vectors,*
*there must be some tones overlapping each other. Hence the AC function will not be*
*perfect; see Fig. 3.7.*

*The above discussion indicates that the sequence vec(G(m)T*) has perfect AC function if

*g.c.d.(m, N _{b}) = 1. Furthermore, if the b_{i}*’s are of the same magnitude, the generated
sequence will be composed of complex numbers with the same magnitude. Observing
Fig. 3.3, we can notice that the elements of the sequence have the same magnitude both
in time-domain and frequency-domain. As mentioned before, a sequence has perfect
AC function if all of its frequency components have the same magnitude. Hence if we
exchange the roles of the “time-domain” sequence and the “frequency-domain” sequence,
the AC property still can be maintained. Both the sequence generated in this chapter
and the FZC sequence have this property. With this property, the step of (2.10) can be

modiﬁed as

*C _{i}(λ) =*

*Kx _{k}*

*; λ = Kk + i,*

0 *; otherwise.* (3.19)

0 0.5 1 1.5 2 2.5 3 0

2 4

The case for m=3

0 0.5 1 1.5 2 2.5 3 0 2 4 0 0.5 1 1.5 2 2.5 3 0 2 4 0 0.5 1 1.5 2 2.5 3 0 2 4

*Figure 3.4: g.c.d.(m, N _{b}) = 1. Magnitude plot for the DFT of each column vector of*

*G(m)*.
0 5 10 15
0
2
4
6
8
10
12
14
16

AutoCorrelation function for the basis sequence with m=3

0 0.5 1 1.5 2 2.5 3 0

2 4

The case for m=2

0 0.5 1 1.5 2 2.5 3 0 2 4 0 0.5 1 1.5 2 2.5 3 0 2 4 0 0.5 1 1.5 2 2.5 3 0 2 4

*Figure 3.6: g.c.d.(m, N _{b}*)

*= 1. Magnitude plot for the DFT of each column vector of*

*G(m)*.
0 5 10 15
0
2
4
6
8
10
12
14
16

AutoCorrelation function for the basis sequence with m=2

**Chapter 4**

**Multi-dimensional Arrays**

Like the one dimensional (1D) case, two dimensional (2D) arrays that possess some desired AC or CC properties are useful in sonar/radar and multimedia applications. Similarly, higher dimensional array signal are needed in some cognitive radio and com-puter graphics. In this chapter, we extend the concepts developed for one-dimensional sequences to two or higher dimensions cases. The notations and deﬁnitions used here follow those of [9].

**4.1**

**Array correlation functions**

*Let an array sequence A = a _{i,j}* be denoted by

*A =*
⎡
⎢
⎢
⎣
*a _{0,0}*

*a*

_{0,1}*· · ·*

*a*

_{0,N}_{2}

_{−1}*a*

_{1,0}*a*

_{1,1}*· · ·*

*a*

_{1,N}_{2}

_{−1}*· · ·*

*· · ·*

*· · ·*

*· · ·*

*a*

_{N}_{1}

_{−1,0}*a*

_{N}_{1}

_{−1,1}*· · · a*

_{N}_{1}

_{−1,N}_{2}

*⎤ ⎥ ⎥*

_{−1}*⎦ .*(4.1)

*The two-dimensional periodic AC function between two array sequences A and B having*
the same dimensions is deﬁned as

*R _{A,B}(φ, ω) =*

*N*1

*−1*

*p=0*

*N*2

*−1*

*q=0*

*a*(4.2) An array is called perfect array if its periodic AC function satisﬁes

_{p,q}b∗_{p+φ,q+ω}.*R _{A,A}(φ, ω) = R_{A}(φ, ω) =*

*E, (φ, ω) = (0, 0)*

*where E =N*1*−1*

*p=0*
*N*2*−1*

*q=0* *|ap,q|*

2_{.}

There are many earlier works on the syntheses of perfect arrays. We will apply one of the synthesis methods introduced in [9] to obtain a perfect array. This method is based on

**Theorem 3 (Folding method) Let b**_{l}*be a perfect sequence of length N = N*_{1}*N*_{2}*. Then*

*the array* *{a _{m,n}} deﬁned by*

*a _{m,n}= b_{l}, m = l mod N*

_{1}

*, n = l mod N*

_{2}(4.4)

*is perfect if gcd(N*_{1}*, N*_{2}*) = 1.*

**4.2**

**New 2D arrays**

To begin with, we need a perfect array sequence. This sequence will be referred to as the basis array. We apply the folding method to the FZC sequence of length

*N*_{1}*N*_{2}*, where gcd(N*_{1}*, N*_{2}*) = 1, and then we get an N*_{1}*× N*_{2} perfect array. Taking the
two-dimensional DFT on this basis array, we obtain

*F (u, v) =*
*N*1*−1*
*p=0*
*N*2*−1*
*q=0*
*a _{p,q}W_{N}−pu*

_{1}

*W*

_{N}−qv_{2}

*.*(4.5)

*Suppose that the new arrays C(i)’s are represented by K*

_{1}

*N*

_{1}

*× K*

_{2}

*N*

_{2}matrices, and their

*corresponding two-dimensional DFT’s are F(i)(U, V ) deﬁned by*

*F(i)(U, V ) =*
*K*1*N*1*−1*
*p=0*
*K*2*N*2*−1*
*q=0*
*c _{p,q}(i)W_{K}−pU*

_{1}

_{N}_{1}

*W*

_{K}−qV_{2}

_{N}_{2}

*, i = 0, . . . , (K*

_{1}

*K*

_{2}

*− 1).*(4.6)

*We assign F(i)(U, V ) according to*

*F(i)(U, V ) =*

*K*_{1}*K*_{2}*F (u, v)* *; U = K*_{1}*u + α, V = K*_{2}*v + β*

0 ; otherwise *,* (4.7)

*where i = K*_{2}*α + β, 0≤ α < K*_{1}*, and 0≤ β < K*_{2}*.*

This assignment is illustrated in Fig. 4.1. Taking the two-dimensional IDFT on

two-dimensional IDFT is deﬁned by
*C(i)(m, n) =* 1
*K*_{1}*N*_{1}*K*_{2}*N*_{2}
*K*1*N*1*−1*
*U =0*
*K*2*N*2*−1*
*V =0*
*F(i)(U, V )W _{K}mU*

_{1}

_{N}_{1}

*W*

_{K}nV_{2}

_{N}_{2}

*.*(4.8) ˃ ˅ ˇ ˉ ˃ ˅ ˇ ˉ ˋ ʻ˴ʼ ʻ˵ʼ

*Figure 4.1: (a) Construct the F*(0)*(U, V ) from the two-dimensional DFT points of the*
*basis array. (b) Diﬀerent symbols represent the non-zero positions of F(i)(U, V ) for*
*diﬀerent i’s; (K*_{1} *= 2, K*_{2}*= 2, N*_{1} *= 4, and N*_{2} = 5.)

**4.3**

**Properties of the new proposed 2D arrays**

The new array sequences possess some desired properties similar to those in
*one-dimensional case. The new array sequences C(i)’s are of dimension K*_{1}*N*_{1}*× K*_{2}*N*_{2}. The
AC function*|R _{C}(i)(φ, ω)| is periodic in both arguments–the period in φ is N*

_{1}while the

*period in ω is N*_{2}*. The CC function between any two arrays of C(i)*’s is exactly zero and
*we can have a family of K*_{1}*× K*_{2} such array sequences.

**Example 3 Suppose we have a perfect array of dimension N**_{1}*× N*_{2} *already. We can*

*generate K*_{1}*K*_{2} *PS-like arrays of dimension K*_{1}*N*_{1}*× K*_{2}*N*_{2}*. Here we set N*_{1} *= 4, N*_{2} *= 5,*

*we have a 4× 5 perfect array. Denote the FZC sequence as {b _{l}}, l = 0, . . . , 19. The*

*corresponding perfect array*

*{a*

_{m,n}} will be⎡
⎢
⎢
⎣
*b*_{0} *b*_{16} *b*_{12} *b*_{8} *b*_{4}
*b*_{5} *b*_{1} *b*_{17} *b*_{13} *b*_{9}
*b*_{10} *b*_{6} *b*_{2} *b*_{18} *b*_{14}
*b*_{15} *b*_{11} *b*_{7} *b*_{3} *b*_{19}
⎤
⎥
⎥
*⎦ .* (4.9)

*By performing the procedure in the previous section, we can have K*_{1}*K*_{2} *= 4 diﬀerent *

*PS-like arrays. The magnitude plot of the AC function of these arrays,* *|R _{C}(i)|, i = 0, . . . , 3,*

*is shown in Fig. 4.2. The magnitude of the AC function* *|R _{C}(i)(φ, ω)| is periodic in both*

*two axes. The period along φ-axis is 4, and the period along ω-axis is 5.*

−10 −5 0 5 10 −10 −5 0 5 10 0 10 20 30 40 50 60 70 80 90 ω φ |R C ( φ , ω )|

Figure 4.2: Magnitude plot for the two-dimensional periodic AC function of proposed
array sequences, *|R _{C}(i)|. (K*

_{1}

*= 2, K*

_{2}

*= 2, N*

_{1}

*= 4, and N*

_{2}= 5.)

**4.4**

**Extension to multi-dimensional arrays**

One of the key step in generalizing the technique of section 4.2 to synthesizing multi-dimensional arrays is to ﬁnd a multi-dimensional perfect array.

Suppose we have an perfect array *{a _{p}*

_{1}

_{,p}_{2}

_{,...,p}_{n}} of dimension N_{1}

*× N*

_{2}

*× · · · × N*.

_{n}*Taking n-dimensional DFT on this basis array, we have*

*F (p*_{1}*, p*_{2}*, . . . , p _{n}) = F (p) =*

*N*1

*−1*

*p*1=0

*N*2

*−1*

*p*2=0

*· · ·*

*N*

*n−1*

*pn*=0

*a*

_{p}_{1}

_{,p}_{2}

_{,...,p}_{n}W_{N}−p_{1}1

*u*1

*W*

_{N}−p_{2}2

*u*2

*· · · W*

_{N}−p_{n}nun.(4.10)*Suppose that the new arrays C(i)’s are K*_{1}*N*_{1}*× K*_{2}*N*_{2}*× · · · × K _{n}N_{n}* matrices, and their

*corresponding n-dimensional DFT’s are F(i)(P*

_{1}

*, P*

_{2}

*, . . . , P*):

_{n}*F(i)(P*_{1}*, P*_{2}*, . . . , P _{n}) = F(i)( P )*
=

*K*1

*N*1

*−1*

*p*0=0

*K*2

*N*2

*−1*

*p*1=0

*· · ·*

*Kn*

*Nn−1*

*pn*=0

*c(i)*

_{p}_{1}

_{,p}_{2}

_{,...,p}_{n}W_{K}−p_{1}

*1*

_{N}*P*

_{1}1

*W*

_{K}−p_{2}

*2*

_{N}*P*

_{2}2

*· · · W*(4.11)

_{K}−p_{n}_{N}nP_{n}n,*where i = 0, . . . , (K*

_{1}

*K*

_{2}

*· · · K*

_{n}− 1). Then we assign F(i)( P ) by the following rule*F(i)( P ) =*

*K*_{1}*K*_{2}*· · · K _{n}F (p*

_{1}

*, p*

_{2}

*, . . . , p*)

_{n}*; P = f (p, i)*

0 ; otherwise *,* (4.12)

*where f (p, i) deﬁnes the non-zero positions in transform domain for the ith new *

*gener-ated array. (similar to Fig. 4.1(b) in 2D case.) For a certain i, the non-zero positions in*
transform domain are equally spaced along all axes.

*The n-dimensional IDFT is deﬁned by C(i)(p*_{1}*, . . . , p _{n}*)

= 1
*K*_{1}*N*_{1}*K*_{2}*N*_{2}*· · · K _{n}N_{n}*

*K*1

*N*1

*−1*

*P*1=0

*K*2

*N*2

*−1*

*P*2=0

*· · ·*

*Kn*

*Nn−1*

*Pn*=0

*F(i)( P )W*1

_{K}p_{1}

*P*1

_{N}_{1}

*W*2

_{K}p_{2}

*P*2

_{N}_{2}

*· · · W*

_{K}pn_{n}P_{N}n_{n}.(4.13)*By applying n-dimensional IDFT on F(i)( P ), we obtain an array sequence C(i)* of
*dimension K*_{1}*N*_{1}*× K*_{2}*N*_{2}*× . . . × K _{n}N_{n}*. The CC function between any two generated
array sequences is exactly zero.

**Chapter 5**

**Preamble Structure for**

**MIMO-OFDM WLAN Systems**

**5.1**

**Backgrounds**

**5.1.1**

**MIMO-OFDM WLAN systems**

In [10], a TDMA-like preamble structure was suggested for MIMO-OFDM system. In this structure, conventional algorithms for synchronization, channel estimation, etc. in SISO-OFDM system can be extended directly since the receiver can distinguish the signals from diﬀerent transmit antennas separately. However, the total length of the proposed preamble grows linearly with the number of the transmit antennas. It is not highly eﬃcient because of the increased overhead. Moreover, when one transmit antenna is idling, the receiver cannot get any information about the idling transmitter(ex.: chan-nel information) during this period. Hence we hope to ﬁnd a more eﬃcient preamble structure.

**5.2**

**Proposed preamble structure**

The preamble structure proposed here is based on the training symbol structure in IEEE 802.11a standard[13], which is showed in Fig. 5.1. We will focus on the long training symbol design. Channel estimation and ﬁne frequency oﬀset estimation are the main tasks during the long training symbols. In conventional OFDM systems, several

algorithms based on long preamble symbols are presented to work jointly to attain syn-chronization tasks and channel estimation. We are going to apply the new sequences that we had been discussed to be the training sequences in the MIMO-OFDM WLAN systems.

t1 t2t3 t4 t5t6t7t8 t9 GI2 T1 T2 GI SIGNAL GI Data 1 GI Data 2

8 + 8 = 16 µs

10× 0.8 = 8 µs 2 × 0.8 + 2 × 3.2 = 8.0 µs _{0.8 +3.2 = 4.0 µs} 0.8 + 3.2 = 4.0 µs 0.8 + 3.2 = 4.0 µs

Signal Detect, AGC, Diversity

Coarse Freq.

Offset EstimationChannel and Fine Frequency RATE SERVICE + DATA DATA

t10

Selection Timing Synchronize Offset Estimation

LENGTH

Figure 5.1: OFDM training structure in IEEE 802.11a standard. (We will redesign the long training sequences.)

*In the IEEE 802.11a standard, the guard interval is of length L, and it needs to be*
larger than the maximum delay spread. Hence we should generate the PS sequences or
our new sequences according to the length of the guard interval. Since the maximum
delay spread is bounded, we can choose a suitable sequence length such that the
un-wanted peak values of the AC function can be avoided. By Choosing the new proposed
*sequences of length N = KN _{c}*

*= 16K, where K is related to the number of transmit*antennas we use, the unwanted peak values of the AC function can be avoided. Consid-ering the system with 2 transmit antennas, we need at least two new proposed sequences

*with perfect CC properties. In this case, K = 2 and hence the period of the sequences*

*need to be N = 32. To suit the long training symbol length in 802.11a standard, we*

*simply set K = 4. This means this set of sequences can at most support 4 transmit*antennas. Fig. 5.2 shows the structure adopted in [10]. Fig. 5.3 shows the structure we adopt. The overhead is highly reduced.

S_{1}
Tx1
Tx2
0 Ns 2Ns time (samples)
0 3Ns 4Ns time (samples)
S_{1}
S
2 S2

Figure 5.2: A time orthogonal preamble for a MIMO conﬁguration with 2 transmit antennas. (The guard intervals are not shown in this ﬁgure.)

S_{1} S_{1}
S
2 S2
Tx1
Tx2
0 Ns 2Ns time (samples)
0 Ns 2Ns time (samples)

Figure 5.3: A coded orthogonal preamble for a MIMO conﬁguration with 2 transmit antennas.

**5.2.1**

**Cyclic preﬁx**

The cyclic preﬁx parts in conventional OFDM system are still necessary for some consideration. For time-domain channel estimation algorithms, which will be introduced in chapter 6, the added cyclic preﬁx can help us to preserve the periodic AC and CC properties of the adopted sequences, even though we remove the cyclic preﬁx at the channel estimation stage. Fig. 5.4 is the suggested preamble structure for a MIMO conﬁguration with 2 transmit antennas.

**5.2.2**

**Length of the training sequence**

We had introduced the PS sequences and a new set of orthogonal sequences in chapter 2. Both of them have periodic AC and excellent CC properties. However, our new sequences are more ﬂexible to the sequences length. In diﬀerent systems, the deﬁned training symbol length may be not the same. For example, the length of the training

˦_{˄} ˦_{˄}
˦_{˅} ˦_{˅}
˧̋˄
˧̋˅
˃ ˡ̆ ˅ˡ̆ ̇˼̀˸ʳʻ̆˴̀̃˿˸̆ʼ
˃ ˡ̆ ˅ˡ̆ ̇˼̀˸ʳʻ̆˴̀̃˿˸̆ʼ
˚˜ʻ˄ʼ
˚˜ʻ˅ʼ
ˀˡ˺
ˀˡ˺
˖̂̃̌

Figure 5.4: Suggested long training symbol structure for a MIMO conﬁguration with 2
transmit antennas.
FEC
Coder
Interleaving+ _{IFFT} GI
Addition
Symbol
Wave
Shaping
IQ
Mod.
HPA
Mapping

Figure 5.5: Transmitter block diagram for the OFDM PHY.

symbol deﬁned in 802.11a and 802.16 are diﬀerent. Hence a set of sequences with less constraint on the sequences length is important for preamble signal design.

**5.2.3**

**Constraints on the constellation of training symbols**

In OFDM systems [13], the OFDM subcarriers shall be modulated by using BPSK, QPSK, 16-QAM, or 64-QAM modulation depending on the transmission rate requested. The encoded and interleaved are divided into groups to form symbols and then converted into complex numbers representing BPSK, QPSK, 16-QAM, or 64-QAM constellation points. The modulation symbols are mapped to the inputs of the IDFT block. These operations are showed in Fig. 5.5. In previous discussion, we generate the training symbols in time domain. If we want to generate them from frequency domain, which is the case in IEEE 802.11a, we need to put the DFT values of these training sequences as

the input of IDFT block. The DFT values of the sequences we adopted usually do not fall on the constellations selected. Hence we propose another way to generate desired training sequence from frequency domain.

Similar to the steps introduced in section 2.2.3, we need a basis sequence of length

*N _{c}, x(n), with perfect AC function ﬁrst. We hope that this sequence can be generated*

*from its IDFT, X[k], and X[k]’s are BPSK or QPSK constellation points. By using the*concepts discussed in chapter 3, we know that if all frequency components of a sequence have the same magnitude, then perfect AC property can be achieved. Hence we simply

*set X(k) =√N*

_{c}ejφk_{. There is no constraint on the phase of each frequency component.}Therefore, we can limit the choice of

*{φ*

_{k}} to ﬁnite M-ary constellation points, i.e.,*φ _{k}* = exp

*j2πnk*

*M*

*, n*

_{k}*∈ integer.*(5.1)

Moreover, if the peak-to-average power ratio(PAPR) problem is further considered, the complementary sequences can be applied[14].

**5.3**

**Simulation environment**

Subsequent discussion addresses the issues of synchronization, channel estimation based on the proposed preamble structure. Our proposed algorithms are to be tested through computer simulation of transmission over real-world wireless channels. Table 6.1 lists some parameters adopted in the IEEE 802.11a standard. Exponentially decayed Rayleigh fading channels are used in our simulation with the impulse response given by

*h _{t}*

*= α*(5.2) where

_{t}+ jβ_{t},*α*

_{t}*= N*

*0,*1 2

*σ*2

*t*

*,*(5.3)

*β*

_{t}*= N*

*0,*1 2

*σ*2

*t*

*, t = 0, 1, 2,· · ·*(5.4)

*σ*2_{t}*= σ*2_{0}*e−TmaxtTs* *, σ*2_{0} = 1*− e−TmaxTs* *.* (5.5)

*T _{s}is the sampling period, T_{RM S}is the root mean squared delay, and T_{max}* is the maximum
delay spread.

There are lots of works in literatures that discuss the channel capacity of the MIMO systems, and we know that the channel capacity can be maximized if the sub-channels from diﬀerent transmit antennas to every receive antenna are independent. Hence we generate the channel responses independently in our simulation.