序列之轉換域產生法及其在多輸入多輸出正交分頻多工系統之應用

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୯!ҥ!Ҭ!೯!ε!Ꮲ!

!!!!!!!

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ႝߞπำᏢسᅺγ੤!

ᅺγፕЎ!

ׇӈϐᙯඤୱౢғݤϷځӧ

ӭᒡΕӭᒡр҅Ҭϩᓎӭπس಍ϐᔈҔ

!

Transform Domain Approach for Sequence Design

and

Its Applications to MIMO-OFDM Systems

ࣴ ز ғǺጰໜ౰ Student: Lung-Sheng Tsai

ࡰᏤ௲௤Ǻ᝵ػቺ റγ Advisor: Dr. Yu Ted Su

ׇӈϐᙯඤୱౢғݤϷځӧ

ӭᒡΕӭᒡр҅Ҭϩᓎӭπس಍ϐᔈҔ

Transform Domain Approach for Sequence Design and

Its Applications to MIMO-OFDM Systems

ࣴ ز ғǺጰໜ౰ Student: Lung-Sheng Tsai

ࡰᏤ௲௤Ǻ᝵ػቺ 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

ӭᒡΕӭᒡр҅Ҭϩᓎӭπس಍ϐᔈҔ

ࣴزғǺጰໜ౰!! ! ! ! ! ࡰᏤ௲௤Ǻ᝵ػቺറγ!

!

୯ҥҬ೯εᏢႝߞπำᏢسᅺγ੤!

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!

ύЎᄔा!

!

ӭᒡΕӭᒡрس಍)NJNP*མଛ҅Ҭϩᓎӭπ)PGEN*מೌڀԖ࣬྽ε

ޑወΚёаၲډ׳ଯޑ໺ᒡ৒ໆǴӧჴሞس಍Бय़ςԖ࣬྽ӭޑࣴزԋ

݀ǴՠϝԖ΋٤ᜢᗖ܄ޑ᝼ᚒࡑլܺǶҁፕЎख़ᗺӧܭ MIMO-OFDM

س಍܌ሡޑ߻࿼ૻဦ೛ीǶ

ӧҁፕЎύǴךॺගр΋س಍ϯޑᙯඤୱׇӈౢғݤǹԜБݤёа

ౢғ΋ಔڀԖؼӳԾ࣬ᜢ(autocorrelation)ᆶϕ࣬ᜢ(cross-correlation)܄

፦ޑׇӈǶ೭٤ׇӈΨёҗԖज़ૻဦဂ໣ᗺ(finite constellation points)࿶

җϸӛᚆණഡճယᙯඤٰౢғǶ΋٤ςԖޑׇӈёаҔךॺ܌ගрޑБ

ԄౢғǴฅԶךॺගрޑׇӈၨᙑԖޣڙၨϿޑज़ڋǶ୷ܭךॺ܌ගр

ޑཷۺǴ೭٤ࡌҥ΋ᆢׇӈޑБݤёаᇸܰӦۯ՜ډӭᆢତӈׇӈ

)multi-dimensional array sequences*ޑࡌᄬǶ

ճҔҁЎ܌ϟಏޑཥׇӈǴךॺගрΑёҔܭ MIMO-OFDM س಍ޑ

߻࿼ߞဦ่ᄬǹךॺΨ૸ፕΑ୷ܭԜ่ᄬჹᔈޑᓎ౗ᅆ౽՗ෳаϷ೯ၰ

՗ෳᄽᆉݤǶҗႝတኳᔕޑ่݀ёޕךॺ܌ගрޑբݤޑዴၲډΑന٫

ޑਏૈ߄౜Ƕ!

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_ci,ci(λ)’s for 0 < i < K are simply frequency-shifting functions of Θ_c0,c0(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_b2 = 16, and

N_s= KN_b2 = 64) . . . 18 3.1 3-IDFT of b_i. The elements of the matrix G, g₀ ∼ g₈, 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; (K₁ = 2, K₂ = 2, N₁ = 4, and N₂= 5.) . . . . 31 4.2 Magnitude plot for the two-dimensional periodic AC function of proposed

array sequences, |R_C(i)|. (K₁ = 2, K₂= 2, N₁ = 4, and N₂ = 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 2m_{−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_iv_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_a defined by

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,  θ2_c N  + N − 1 N (K− 1)  θ_a2 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_a(n) = 0. If we use these sequences 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

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_Nm)−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_c 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 se-quence is a good candidate that meets our need. Denote the FZC code of length N_c by the N_c-by-1 vector x and let c₀, c₁, . . . , c_K−1 be the sequences to be generated which have 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_iW_Nkn_c, W_Nc = 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_i. The AC function of the new sequence c_i is given by

θ(c_i)(n) = N −n−1
l=0 c_l+n,ic∗_l,i+ N −1
l=N −n c_l+n−N,ic∗_l,i = N W_Nniδ(R(n, N_c)) (2.11)

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_ci,ci(λ) = C_i(λ)C_i∗(λ), 0≤ λ < N. (2.14) Substituting (2.10) into (2.14), Θ_ci,ci(λ) can be expressed as:

Θ_ci,ci(λ) = 

K2X(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_cδ(n). (2.16) Taking N_c-point DFT on (2.16), we have

Θ(x, x)(k) = X(k)X∗(k) = N_c. (2.17)

Comparing (2.15) and (2.17), we find out that Θ_c0,c0(λ) is an rate-expanded version of Θ(x, x)(k), and K is the expanding rate. That is,

Θ_c0,c0(λ) = 

K2Θ(x, x)(k) = K2N_c ; λ = Kk,

Hence we can expect that the AC function of the newly generated sequence c₀be periodic with period N_c, i.e.,

θ_c0,c0(n) = N δ(R(n, N_c)), (2.19)

where R(·) is defined in (2.5). The other Θ_ci,ci(λ)’s for 0 < i < K are simply frequency-shifting functions of Θ_c0,c0(λ), i.e.,

Θ_ci,ci(λ) = Θ_c0,c0(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_ci,ci(n) = W_Nniθ_c0,c0(n) = N W_Nniδ(R(n, N_c)), (2.21) 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_c properly. Fig. 2.3 is a typical plot for the AC function of the new sequences. In this example, the sequence length is

N = KN_c = 2× 32.

2. Crosscorrelation function:

Let us denote two new sequences as c_i and c_j. The CC function of the two sequences is 0 if i= j.

Proof :

From (2.15), we have∀λ,

Θ_ci,ci(λ)Θ_cj,cj(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_ci,cj(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_cof 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_b2. The value K determines how many different sequences having excellent CC function we can used. The value N_b2 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.

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_i’s are all located on the unit circle in the complex plane. We first generate an orthogonal sequence from

b_i’s. Simply take{W_N0_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_b× N_b as

G = F(Nb,−m)_D({b

i}). (2.25)

The basic orthogonal sequence {g_p} of length N_b2 is defined by [8]

g_p = G_{Q(p,Nb),R(p,Nb)} (2.26) or equivalently, if g = [g₀, g₁, . . . , g_N2

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
_b2−1 p=0 g_pδ(i− k − pK), (2.29) N_s= KN_b2, 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_b2× K) is defined as

C = [c_l,k] = 1

N_bF

(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,kc∗_l,k+ N
s−1 l=Ns−τ c_{l+τ −Ns,k}c∗_l,k = N_sW_Nτ k_sδ(R(τ, N_b2)). (2.31) The AC function has a nonzero value only when R(τ, N_b2) = 0; i.e., τ = IN_b2, where I is an integer. We can control the interval or period by properly choosing the value of N_b2. 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.

2. Crosscorrelation function:

Let us denote two PS sequences as {c_l,kI} and {c_l,kII}. 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_b2 with perfect AC function is generated. This step is similar to generating a FZC sequence of length N_b2in 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

function as Fig. 2.3, since 32= N_b2, for any natural number N_b. 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].

O

O

O

O

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_ci,ci(λ)’s for 0 < i < K are simply frequency-shifting functions of Θ_c0,c0(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_b2 = 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_b× N_b as

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)



b₀ b₁ b₂ , (3.3)

where b₀ = [b₀ 0 0]T_{, b1 = [0 b1 0]}T_{, and b2 = [0 0 b2]}T_{. The ith column vector of the}

matrix G, g_i , is equal to the N_b-IDFT of b_i. Since there is only one non-zero element in the vector b_i and the elements in F are all of unit magnitude, the elements of g_i must have identical magnitude. In order to make the explanation clear, we denote

G = [g₀ g₁g₂] = ⎡ ⎣ gg0₁ gg3₄ gg6₇ g₂ g₅ g₈ ⎤ ⎦ . (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, g₀ ∼ g₈, are complex number with unit magnitude.

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

g₀, g₁, and g₂ by inserting some zeros with an expanding rate of N_b = 3, we obtain the expanded vectors (N_b2× 1)  g_e0 = [ g₀ 0 0 g₁ 0 0 g₂ 0 0 ]T (3.5)  g_e1 = [ g₃ 0 0 g₄ 0 0 g₅ 0 0 ]T (3.6)  g_e2 = [ g₆ 0 0 g₇ 0 0 g₈ 0 0 ]T. (3.7)

After taking N_b2-DFT on the expanded vectors(sequences), the corresponding frequency-domain sequences will repeat periodically. Then we do some time-shifting on g_e1 and



g_e2 such that there is no overlapping if we sum g_e0, g_e1, and g_e2 together. Denote the time-shifted vectors as g_es0, g_es1, and g_es2, and then

 g_es0 = [ g₀ 0 0 g₁ 0 0 g₂ 0 0 ]T (3.8)  g_es1 = [ 0 g₃ 0 0 g₄ 0 0 g₅ 0 ]T (3.9)  g_es2 = [ 0 0 g₆ 0 0 g₇ 0 0 g₈ ]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_es0, g_es1, and g_es2 together, we obtain



g = g_es0+ g_es1+ g_es2 = vec(GT). (3.11) We plot the corresponding result in Fig. 3.3. The corresponding N_b2-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_cδ(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_cejφk_{, i.e., all frequency components have the same magnitude, then (3.13) can be} achieved. The constant √N_c can be ignored since it does not affect the AC and CC properties. Notice that there is no constraint on the choice of {φ_k}. We can conclude

that 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 bG b > @ 1 0 1 0 T bG b > @ 2 0 0 2 T bG 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_iδ(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_b− 1)]T, (3.14) where x(m)_i (n) = N
b−1 k=0 X_i(k)W_Nkmn_b = b_iW_Nimn_b . (3.15) For the case that m = 1, x(1)_i (n) = x_i(n) = b_iW_Nin_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_iδ(k− i) =⇒ x(m)_i (n) DF T−→ X_i[(k− i(m − 1))_Nb], (3.16) and X_i[(k− i(m − 1))_Nb] = b_iδ[(k− im)_Nb]. For convenience, we will use the notation (n)_Nb to denote (n modulo N_b). The term b_iδ[(k − im)_Nb] means that the effect of

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}) = [b₀, . . . ,b_Nb−1], such 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, W₄ = ej2π/4, and b ={W₄1, W₄2, W₄3, W₄4}. 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, W₄ = ej2π/4_{, and b =}{W1

4, W42, W43, W44}.

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,N2−1 a_1,0 a_1,1 · · · a_1,N2−1 · · · · · · · · · · · · a_N1−1,0 a_N1−1,1 · · · a_{N1−1,N2−1} ⎤ ⎥ ⎥ ⎦ . (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_p,qb∗_p+φ,q+ω. (4.2) An array is called perfect array if its periodic AC function satisfies

R_A,A(φ, ω) = R_A(φ, ω) = 

E, (φ, ω) = (0, 0)

where E =N1−1

p=0 N2−1

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 = N₁N₂. Then

the array {a_m,n} defined by

a_m,n= b_l, m = l mod N₁, n = l mod N₂ (4.4)

is perfect if gcd(N₁, N₂) = 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

N₁N₂, where gcd(N₁, N₂) = 1, and then we get an N₁× N₂ 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,qW_N−pu₁ W_N−qv₂ . (4.5) Suppose that the new arrays C(i)’s are represented by K₁N₁× K₂N₂ matrices, and their corresponding two-dimensional DFT’s are F(i)(U, V ) defined by

F(i)(U, V ) = K1
N1−1 p=0 K2
N2−1 q=0 c_p,q(i)W_K−pU_1N1W_K−qV_2N2, i = 0, . . . , (K₁K₂− 1). (4.6) We assign F(i)(U, V ) according to

F(i)(U, V ) = 

K₁K₂F (u, v) ; U = K₁u + α, V = K₂v + β

0 ; otherwise , (4.7)

where i = K₂α + β, 0≤ α < K₁, and 0≤ β < K₂.

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 K₁N₁K₂N₂ K1
N1−1 U =0 K2
N2−1 V =0 F(i)(U, V )W_KmU_1N1W_KnV_2N2. (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; (K₁ = 2, K₂= 2, N₁ = 4, and N₂ = 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 K₁N₁× K₂N₂. The AC function|R_C(i)(φ, ω)| is periodic in both arguments–the period in φ is N₁ while the

period in ω is N₂. The CC function between any two arrays of C(i)’s is exactly zero and we can have a family of K₁× K₂ such array sequences.

Example 3 Suppose we have a perfect array of dimension N₁× N₂ already. We can

generate K₁K₂ PS-like arrays of dimension K₁N₁× K₂N₂. Here we set N₁ = 4, N₂ = 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

⎡ ⎢ ⎢ ⎣ b₀ b₁₆ b₁₂ b₈ b₄ b₅ b₁ b₁₇ b₁₃ b₉ b₁₀ b₆ b₂ b₁₈ b₁₄ b₁₅ b₁₁ b₇ b₃ b₁₉ ⎤ ⎥ ⎥ ⎦ . (4.9)

By performing the procedure in the previous section, we can have K₁K₂ = 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)|. (K₁ = 2, K₂ = 2, N₁ = 4, and N₂ = 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_p1,p2,...,pn} of dimension N₁ × N₂ × · · · × N_n. Taking n-dimensional DFT on this basis array, we have

F (p₁, p₂, . . . , p_n) = F (p) = N
1−1 p1=0 N
2−1 p2=0 · · · N
n−1 pn=0 a_p1,p2,...,pnW_N−p₁1u1W_N−p₂2u2· · · W_N−p_nnun.(4.10)

Suppose that the new arrays C(i)’s are K₁N₁× K₂N₂× · · · × K_nN_n matrices, and their corresponding n-dimensional DFT’s are F(i)(P₁, P₂, . . . , P_n):

F(i)(P₁, P₂, . . . , P_n) = F(i)( P ) = K1
N1−1 p0=0 K2
N2−1 p1=0 · · · Kn
Nn−1 pn=0 c(i)_p1,p2,...,pnW_K−p_1N1P₁1W_K−p_2N2P₂2· · · W_K−p_nNnP_nn, (4.11) where i = 0, . . . , (K₁K₂· · · K_n− 1). Then we assign F(i)( P ) by the following rule

F(i)( P ) =



K₁K₂· · · K_nF (p₁, p₂, . . . , 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)(p₁, . . . , p_n)

= 1 K₁N₁K₂N₂· · · K_nN_n K1
N1−1 P1=0 K2
N2−1 P2=0 · · · Kn
Nn−1 Pn=0 F(i)( P )W_Kp1₁P_N1₁W_Kp2₂P_N2₂· · · W_Kpn_nP_Nn_n.(4.13)

By applying n-dimensional IDFT on F(i)( P ), we obtain an array sequence C(i) of dimension K₁N₁× K₂N₂× . . . × K_nN_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.

S₁ Tx1 Tx2 0 Ns 2Ns time (samples) 0 3Ns 4Ns time (samples) S₁ 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.)

S₁ S₁ 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_cejφk_{. 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_t+ jβ_t, (5.2) where α_t = N  0,1 2σ 2 t  , (5.3) β_t = N  0,1 2σ 2 t  , t = 0, 1, 2,· · · (5.4)

σ2_t = σ2₀e−TmaxtTs , σ2₀ = 1− e−TmaxTs . (5.5)

T_sis 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.