有近似週期性導引之TDD-OFDMA信號之頻率同步

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國立交通大學

電信工程學系碩士班

碩士論文

有近似週期性導引之 TDD-OFDMA 信號之頻率同步

Frequency-Synchronization for TDD-OFDMA Systems

with Almost-Periodic Preamble

研究生：楊士賢 Student: Shih-Hsien Yang

指導教授：蘇育德博士 Advisor:

Dr. Yu Ted Su

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有近似週期性導引之 TDD-OFDMA 信號之頻率同步

Frequency-Synchronization for TDD-OFDMA Systems with

Almost-Periodic Preamble

研究生：楊士賢

Student : Shih-Hsien Yang

指導教授：蘇育德博士

Advisor

:

Dr.

Yu

T. Su

國立交通大學

電信工程學系碩士班

碩士論文

A Thesis Submitted to

The 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

July 2006

Hsinchu, Taiwan, Repubic of China

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有近似週期性導引之 TDD-OFDMA 信號之頻率同步

研究生：楊士賢

指導教授：蘇育德博士

國立交通大學電信工程學系碩士班

中文摘要

本文旨在研究 OFDMA 系統的頻率同步相關課題。跟一般 OFDM 系統不同的是來從事頻率同步的導引信號只具備近似的週期性。缺乏理想週期性的原因是由於離散傅立葉轉換大小為 2 的冪次方，無法被奇數整除。我們若仍然運用傳統基於週期性導引信號的相關性來做頻率估計的方法，其性能將可能很不理想。我們利用內插法從接收的信號樣本中重建具完整週期性的導引信號，使得應用時域相關性的頻率偏移估測方法可以適當的使用而不至造成性能損失。我們會比較線性和基於 sinc 函數（即理想低通）的內插濾波器。如同我們所預期的，電腦模擬的結果顯示我們所提出的解決方法在 AWGN 和複徑衰退的通道下皆有性能的改善，尤以後者更為顯著。我們也觀察到性能增益與 OFDM 符號(或者 FFT ) 長度的有很大的關係。這個長度決定了非週期性的程度也直接影響了內插（或性能）增益的多寡。

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Frequency-Synchronization for TDD-OFDMA

Systems with Almost-Periodic Preamble

Student : Shih-Hsien Yang Advisor : Dr. Yu Ted Su

Department of Communications Engineering National Chiao Tung University

Abstract

This thesis deals with the frequency synchronization issue of an OFDMA system. We consider the situation that the preamble used for synchronization and channel estimation has an almost-periodic structure. The lack of ideal periodicity is due to the fact that the discrete Fourier transform size, being a power of two, is not divisible by three. This non-periodicity causes the frequency synchronizer that assumes a periodic pilot structure to suffer from serious performance loss.

We propose an interpolation approach to reconstruct the received preamble samples in a periodic format so that conventional time-domain correlation-based frequency off-set estimation algorithms become applicable. We examine both linear and sinc-based interpolation filters. Simulation results indicate that, as expected, the proposed solution brings about performance improvement in both AWGN and multipath fading channels. Significant performance enhancement is obtained in the latter case. We also observe that the performance gain is a function of the OFDM symbol (or FFT) size which determines the extent of non-periodicity and therefore the interpolation gain.

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List of Figures

2.1 Block diagram of an OFDM transmitter. . . 6

2.2 Insertion of Cyclic Prefix (CP). . . 6

2.3 OFDMA Sub-Carrier Structure. . . 8

2.4 Basic structure of OFDMA downlink transmission. . . 8

2.5 Frequency-domain structure of OFDMA downlink preamble for 128-point FFT size. . . 11

2.6 Time-domain preamble of TDD-OFDMA WiMAX system. . . 11

2.7 Almost-periodic preamble sequences in time-domain. . . 12

2.8 Illustration of the almost-periodic preamble samples for IDcell 0 preamble in segment 0. . . 13

2.9 Block diagram of algorithm 1. . . 14

2.13 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 0 without interpolation. . . 19

2.14 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 1 without interpolation. . . . 20

2.15 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 2 without interpolation. . . . 20

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2.16 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

fading channel using IDcell 0 preamble in segment 0 without interpolation. 21 2.17 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

fading channel using IDcell 0 preamble in segment 1 without interpolation. 21 2.18 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

fading channel using IDcell 0 preamble in segment 2 without interpolation. 22

3.1 Preamble sequences via oversampling in time-domain. . . 24

3.2 Three perfect preamble sequences via oversampling for IDcell 0 in segment 0. . . 25

3.3 A factor-of-L interpolator. . . . 26

3.4 Illustration of the linear interpolation method. . . 27

3.5 IDcell 0 preamble sequences in segment 0 via linear interpolation. . . 28

3.6 The impulse response of hsinc(m) with L = 5. . . . 29

3.7 Illustration of the sinc interpolation method. . . 30

3.8 IDcell 0 preamble sequences in segment 0 via sinc interpolation with side-lobe length = 5. . . 31

3.9 MSE between any two parts of the three repetition via sinc interpolation. 31 4.1 Structure of a interpolated preamble symbol, where samples connected together by the same arrow-line are almost identical. . . 33

4.2 Block diagram of modified algorithm 1 via interpolations. . . 35

4.5 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 0 via Algorithm 1. . . 39

4.6 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath fading channel using IDcell 0 preamble in segment 0 via Algorithm 1. . . 40

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4.7 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

channel using IDcell 0 preamble in segment 0 via Algorithm 2. . . 40 4.8 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

fading channel using IDcell 0 preamble in segment 0 via Algorithm 2. . . 41 4.9 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

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4.20 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

channel using IDcell 0 preamble in segment 0 via maximum-likelihood estimation. . . 60 5.2 MSE of normalized CFO ˆε estimate as a function of the SNR in

multi-path fading channel using IDcell 0 preamble in segment 0 via maximum-likelihood estimation. . . 61 5.3 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

channel using IDcell 0 preamble in segment 1 via maximum-likelihood estimation. . . 61

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5.4 MSE of normalized CFO ˆε estimate as a function of the SNR in

multi-path fading channel using IDcell 0 preamble in segment 1 via maximum-likelihood estimation. . . 62 5.5 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

channel using IDcell 0 preamble in segment 2 via maximum-likelihood estimation. . . 62 5.6 MSE of normalized CFO ˆε estimate as a function of the SNR in

multi-path fading channel using IDcell 0 preamble in segment 2 via maximum-likelihood estimation. . . 63 6.1 Discrete-time model of the baseband OFDMA system. . . 65 6.2 Frequency-domain subcarriers allocation for each active user. . . 66 6.3 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

channel via Algorithm 1. . . 67 6.4 MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

fading channel via Algorithm 1. . . 68 6.5 MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

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A.1 DL Frequency Diverse Sub-Channel . . . 74 A.2 Tile Structure for UL PUSC . . . 75 A.3 WiMAX OFDMA Frame Structure . . . 78

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List of Tables

2.1 Parameter setting of OFDMA downlink preamble. . . 9 3.1 MSE between any two parts of the three repetitions by linear interpolation. 28 A.1 OFDMA Scalability Parameters . . . 76 A.2 Preamble modulation series per segment and IDcell for the 128 FFT mode 79 A.3 Preamble modulation series per segment and IDcell for the 128 FFT mode

(continued) . . . . 80 A.4 Preamble modulation series per segment and IDcell for the 128 FFT mode

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Chapter 1 Introduction

The Orthogonal Frequency Division Multiplexing (OFDM) technique enjoys several distinct advantages–multipath immunity, bandwidth efficiency, and resistance to nar-rowband interference–that make it a very attractive transmission scheme for high rate wireless communications.

It has been adopted in several international communication standards, e.g., digital audio broadcasting (DAB), digital video broadcasting-terrestrial (DVB-T), high per-formance local area networks (HIPERLAN/2) [7], IEEE 802.11a/g wireless local area networks (WLAN) [6], and very-high-speed digital subscriber line (VDSL). Recently, it has been used or is being considered in IEEE 802.11n , IEEE 802.16 [1]-[4], and IEEE 802.20. Some of these standards are likely to include the Multiple Input Multiple Output (MIMO) option for capacity enhancement.

The main difference between traditional frequency division multiplexing (FDM) and OFDM is that in OFDM, the spectra of the individual carriers do overlap. The overlap-ping property can increase the frequency efficiency in an OFDM system. Nevertheless, the OFDM carriers exhibit orthogonality over a symbol interval only if they are spaced in frequency domain exactly at the reciprocal of the symbol interval, which can be accom-plished by using the discrete Fourier transform (DFT). Hence, a set of equally spaced subcarriers is used for parallel data transmission in OFDM systems.

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com-pletely eliminated by inserting between symbols a small time interval known as a guard interval (GI). The length of the GI is made equal to or greater than the maximum delay spread of the class of multipath fading channels of concern. If the symbol signal wave-form is extended periodically into the GI (normally referred to as cyclic prefix, CP), orthogonality of the carrier is maintained over the symbol period, thus eliminating ICI. ISI is also eliminated because successive symbols do not overlap due to the CP. Hence the receiver needs only to perform one-tap equalization, greatly reducing its complexity. Such an arrangement enables an OFDM-based system to overcome frequency selective fading in broadband wireless transmission and is the major reason for its current popu-larity.

With all its merits, OFDM systems, unfortunately, are far more sensitive to synchro-nization errors, especially the carrier frequency offset (CFO), than single-carrier systems. The CFO is caused by the misalignment in subcarrier frequencies due to fluctuations in transmitter and receiver radio frequency (RF) oscillators or the Doppler shift induced by the time-varying channel effect. This frequency offset can destroy the subcarrier orthogonality and introduces ICI. Hence, if the CFO is not properly estimated and com-pensated for, the ICI would cause significantly degradation of the bit-error-rate (BER) performance.

Therefore, the main purpose of frequency synchronizer is to ensure inter-carrier or-thogonality. Various approaches have been proposed to estimate the CFO, either blindly or with the aid of pilot symbols and training sequences. Blind estimation algorithms achieve higher spectrum efficiency at the cost of increased complexity and slow conver-gence rate whence are not suitable for tracking time-varying CFO. An alternative design technique is data-aided estimation which inserts pilot tones in some or all subcarriers. It is a simpler method to obtain reliable estimate at the cost of lower effectively data rate [13].

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division multiple access is employed then the resulting transmission technique is referred to as OFDMA. This scheme has been adopted by IEEE 802.16e [1]-[4]; it is proposed as a candidate air interface for the next generation broadband wireless networks [8]. In an OFDMA system, each user is assigned an exclusive set of orthogonal subcarriers for transmission. There are two major subcarrier-assignment schemes, namely, subband based and interleaved. The former scheme divides the whole bandwidth into small continuous subbands, and each user is assigned to one or several subbands. For the interleaved subcarrier assignment scheme, subcarriers assigned to different users are interleaved over the whole bandwidth. Both schemes, however, inherit from OFDM the weakness of being sensitive to frequency error. In an OFDMA system, CFO will further cause multiple-access interference (MAI) which might become the dominant factor that limits the system performance.

In this thesis, we use IEEE 802.16e standard (WiMAX) uplink as a model system for it has an almost-periodic pilot format that we are interested in. For the broadcast link (downlink) of the WiMAX system, CFO estimation is relatively simple since relative orthogonality among different users’ assigned subcarriers can be maintained.

The remainder of this thesis is organized as follows. In Chapter 2, the system model and a time-domain correlation-based pilot-aided fractional CFO estimation method are introduced. Chapter 3 then presents two interpolations to generate periodic time-domain preamble for CFO estimation application. Based on the repetitive structure of the interpolated preamble, we examine the performance of the correlation-based algorithms in Chapter 4 in details. In Chapter 5, we further investigate the performance of the optimal maximum-likelihood frequency estimator and its simplified version. We extend our discussion to frequency synchronization of a subband-based OFDMA system in Chapter 6, assuming multiple preambles are available. A guard band between subbands is inserted so that signals from different users can be separated by filter banks. Existing CFO estimation algorithms can then be applied after each user signal is filtered. Finally,

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Chapter 2 Frequency Synchronization with

Almost-Periodic Preamble for

TDD-OFDMA System

For practical OFDM(A) applications, data transmission is organized in frames, and training blocks (carrying known symbols) are located at the beginning of each frame for synchronization purposes. In this chapter, we will explore the downlink fractional CFO estimation based on the received preamble symbols.

2.1 OFDM System Fundamentals

OFDM is a multi-carrier transmission technique that subdivides the whole band-width into multiple frequency sub-carriers as shown in Fig. 2.1. In an OFDM system, the input data stream is divided into several parallel sub-streams of reduced data rate (thus increased symbol duration) and each sub-stream is modulated and transmitted on a separate orthogonal sub-carrier. The increased symbol duration improves the robust-ness of OFDM against the channel delay spread. Furthermore, the introduction of the CP can completely eliminate ISI as long as the CP duration is longer than the chan-nel delay spread. The CP is typically a repetition of the last samples of data portion of the block that is appended to the beginning of the data payload as shown in Fig. 2.2. Hence, each time-domain OFDM symbol waveform contains a “useful interval” of

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length Tu and CP with duration Tg. The value of Tg/Tu might be 1/4, 1/8, 1/16 and

1/32, depending on the transmission environment. The CP prevents block inter-ference and makes the channel appear circular and permits low-complexity frequency domain equalization. A perceived drawback of CP is that it introduces overhead, which effectively reduces bandwidth efficiency.

Data Source FEC Encoder Digital Modulator S/P Conversion IFFT 0 X 1 X 1 N X P/S D/A Converter x x x x x x x x x

Figure 2.1: Block diagram of an OFDM transmitter.

C yclic

Prefix Data Payload s T u T Total Sy mbol Period

U seful Sym bol Period

g

T

g

T

Figure 2.2: Insertion of Cyclic Prefix (CP).

A typical OFDM transmitter takes an N-point inverse discrete fourier transform (IDFT) on every block of N complex datas {Xk} drawn from a QAM or PSK

constel-lation before making parallel-to-serial conversion on the resulting time-domain block. An OFDM symbol (block) is then preceded by an Ng-sample cyclic prefix that is longer

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than the maximum channel delay spread to form an “extended” symbol so that ISI can be eliminated at the receiving end by simply discarding the prefix part. The received (time-domain) OFDM signal, y(n), are given by

y(n) = √1 N N −1_X k=0 XkHke j2πn(k+ε) N + w(n) = x(n)ej2πnεN + w(n) n = −Ng, −(Ng − 1), ..., 0, 1, 2, ..., N − 1 (2.1)

where the subscripts, n and k denote the nth OFDM sample in one OFDM block and the kth subcarrier, respectively. Xk is the transmitted frequency-domain complex data. Hkis the complex transfer function of the channel at the frequency of the kth subcarrier. x(n) is the time-domain complex signal after passing Xk through a multipath channel

without both CFO and AWGN effect. ε is the CFO normalized to the subcarrier spacing (assume the sampling interval is Ts(or Tu/N), the subcarrier spacing is 1/NTs(or 1/Tu))

and w(n) denotes the samples of the complex envelop of AWGN.

2.2 OFDMA System Fundamentals

OFDMA is a multiple-access and multiplexing scheme that provides multiplexing operation of data streams from multiple users through the downlink sub-channels and uplink multiple access by means of uplink sub-channels. The OFDMA symbol structure consists of three types of sub-carriers as shown in Fig. 2.3.

• Data sub-carriers for data transmission

• Pilot sub-carriers for channel estimation and synchronization purposes

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G uard Band Sub-carriers Pilot Sub-carriers Data Sub-carriers D C Sub-carrier . . . . . .

. . .

Figure 2.3: OFDMA Sub-Carrier Structure.

2.3 Downlink Structure

In IEEE 802.16e system, the downlink can be divided into a three segment structure and includes a preamble which begins the transmission. This preamble subcarriers are divided into three carrier-sets. There are three possible groups consisting of a carrier-set. Each of them may be used by any segment. A typical downlink period is illustrated in Fig. 2.4.

P

re

am

b

le

D

at

a

S

y

m

bo

l

D

at

a

S

y

m

bo

l

D

at

a

S

y

m

bo

l

D

at

a

S

y

m

bo

l

D

at

a

S

y

m

bo

l

D

at

a

S

y

m

bo

l

Figure 2.4: Basic structure of OFDMA downlink transmission.

2.3.1 Preamble Structure

The first symbol of the downlink transmission is the preamble. There are three types of preamble carrier-sets, those are defined by allocation of different subcarriers for each one of them. Those subcarriers are modulated using a boosted BPSK modulation with a specific Pseudo-Noise (PN) code. 114 preambles forms sequentially indexed with

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k Guard Band Length (GB) Nu

FFT size (N) = 128 -18 ∼ 17 10 54 FFT size (N) = 512 -72 ∼ 71 40 216 FFT size (N) = 1024 -142 ∼ 141 86 426 FFT size (N) = 2048 -284 ∼ 283 172 852

Table 2.1: Parameter setting of OFDMA downlink preamble.

{0, 1, ..., 113}. The 114 preamble forms can be divided into three segments indexed

from segment 0 to segment 2 and every segment has 38 preamble forms. The preamble modulation series in Hexadecimal format can be found in Appendix A.

The preamble carrier-sets in frequency-domain are defined in (2.2),

P reambleCarrierSetn= n + 3 · k (2.2)

where

• P reambleCarrierSetn specifies all subcarriers allocated to the specific preamble,

• n is the segment of the preamble carrier-set indexed 0...2,

• k is a running index defined as in Tab. 2.1.

Each segment uses one type of preamble composed of a carrier-set out of the three available carrier-sets in the following manner: (In the case of segment 0, the DC carrier will not be modulated at all and the appropriate PN will be discarded; therefore, DC carrier shall always be zeroed. For the preamble symbol, there will be GB guard band subcarriers defined as in Tab. 2.1 on the left side and the right side of the spectrum.)

• Segment 0 uses preamble carrier-set 0

• Segment 1 uses preamble carrier-set 1

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From above definition, (2.1) can be rewritten as y(n) = √1 N X k∈Di XkHke j2πn(k+ε) N + w(n) = x(n)ej2πnεN + w(n) n = −Ng, −(Ng − 1), ..., 0, 1, 2, ..., N − 1 (2.3)

where the subscripts, i denotes the ith segment, Di = {−Nu+i, −Nu+i+3, ..., −3+i, 0+ i, 3+i, ..., Nu−3+i} is the set of modulated subcarrier indices and Nu is defined in Tab.

2.1. As an example, Fig. 2.5 depicts the frequency-domain preamble structure of each segment for 128-point FFT size. The subcarrier locations of preamble in segment 1 and segment 2 are only circular shifting in segment 0. From circular frequency-shifting theorem [27], the inverse DFT of the circularly frequency-shifted DFT X[k] =

G[hk − k0iN], with k0 an integer, is given by x[n] = WN−k0ng[n], where g[n] is the IDFT

of G[k] and WN is defined as e j2π

N ; that is, W−k0n

N g[n] ⇔ G[hk − k0iN] (2.4)

Hence, each segment has the same structure in time-domain but with some different phase rotation (or called linear phase shift).

From above, this preamble structure can be visualized as upsampling in frequency-domain [26]. A factor-of-L sampling rate expansion thus leads to a L-fold repetition of the original time-domain waveform, indicating that the inverse Fourier transform is compressed by a factor of L. Hence, in IEEE 802.16e preamble structure, every third subcarrier is used, preamble consists of three identical parts within an OFDM symbol. Figure 2.6 shows the structure of preamble in time domain, where N is the FFT length. Due to the periodic property, we can utilize the correlation between any two parts of the received (time-domain) preamble sequences to perform frequency synchronization.

Unfortunately, these three repetitions are not exactly the same, since the FFT size (N=128, 512, 1024, 2048) is not the multiple of 3. Hence, the N samples obtained from

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-54 -5 1 -5 3 -50 -5 2 -49 0 0 0 51 52 53 f f f Segment 0 Segment 1 Segment 2 3 -3 -2 1 -1 2

Figure 2.5: Frequency-domain structure of OFDMA downlink preamble for 128-point FFT size.

A A A

N

Figure 2.6: Time-domain preamble of TDD-OFDMA WiMAX system.

the A/D converter can not be divided by 3 exactly and will lead to three incomplete repetitions. The reason is owing to the samples obtained from the A/D converter is not complete. Therefore, this imperfectly repetitive structure in time-domain will lead to fault frequency synchronization. This almost-periodic property can be depicted in Fig. 2.7. The blue solid curve denotes the continuous waveform before the A/D converter and the red circles denote the discrete samples after the A/D converter. Hence, we can easily know that there is no alignment of the samples between any two parts of the three repetitions in Fig. 2.7.

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0 2 4 6 8 10 12 14 16 18 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 2.7: Almost-periodic preamble sequences in time-domain.

the three truncated repetition for demonstration. Because 128 is not multiple of 3 and the remainder of 128 divided by 3 is equal to 2, we discard two points (the 43-th point and the 86-th point) to keep the three incomplete repetition and utilize the three parts to implement correlation-based CFO estimation. Hence, repetition 1 is indexed from sample 1 to sample 42, repetition 2 is indexed from sample 44 to sample 85 and repetition 3 is indexed from sample 87 to sample 128.

2.4 Time-Domain Correlation-Based Fractional CFO

Estimation by Using Almost-Periodic Preamble

After the division of preamble sequences as shown in Fig. 2.8, each part of the three sections can be considered as a repetition. Then, we will present four correlation-based algorithms for fractional CFO estimation. In the following, we still take 128-point FFT size (N = 128) as an applicable example and the other cases (N = 512, 1024, 2048) can also be implemented by the same way.

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5 10 15 20 25 30 35 40 −4 −3 −2 −1 0 1 2 3 4 5 sample Phase

Phase distribution for IDcell 0 in segment 0 (N=128)

Repetition 1 Repetition 2 Repetition 3

Figure 2.8: Illustration of the almost-periodic preamble samples for IDcell 0 preamble in segment 0.

2.4.1 Algorithm 1

First, we consider the correlation between the first two incomplete repetitions as shown in Fig. 2.9. According to (2.3), the sample-by-sample correlation function ignoring the noise effect can be written as

C1 = 41 X n=0 y(n)y(n + 43)∗ = 41 X n=0 x(n)ej2πnε/N _{· [x(n + 43)e}j2π(n+43)ε/N_]∗ = 41 X n=0 x(n)x(n + 43)∗_ej2πnε/N_e−j2π(n+43)ε/N = e−j2π·43·ε/N 41 X n=0 x(n) · x(n + 43)∗ = e−j86πε/N _{· Φ} 1 (2.5)

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where Φ1 = P₄₁ n=0x(n) · x(n + 43)∗ and N = 128. CP 0~41 42 43~84 85 86~127 conj angle -64/43

Figure 2.9: Block diagram of algorithm 1.

If we ignore the Φ1 effect arises from the incomplete repetitions, the normalized CFO

ε estimation can be obtained by using the phase of the correlation function C1 and is

given by ˆ ε1 = − 64 43π∠C1. (2.6)

2.4.2 Algorithm 2

Then, we consider the correlation between the first and the third repetitions as shown in Fig. 2.10. According to (2.3), the sample-by-sample correlation function ignoring the noise effect can be written as

C2 = 41 X n=0 y(n)y(n + 86)∗ = 41 X n=0 x(n)ej2πnε/N _{· [x(n + 86)e}j2π(n+86)ε/N_]∗ = 41 X n=0 x(n)x(n + 86)∗_ej2πnε/N_e−j2π(n+86)ε/N = e−j2π·86·ε/N 41 X n=0 x(n)x(n + 86)∗ = e−j172πε/N _{· Φ} 2 (2.7)

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where Φ2 = P₄₁ n=0x(n) · x(n + 86)∗ and N = 128.

CP

0~41

42 43~84

85 86~127

conj

angle

-32/43

given by ˆ ε2 = − 32 43π∠C2. (2.8)

2.4.3 Algorithm 3

The algorithm 3 is a combination of algorithm 1 and algorithm 2 as shown in Fig. 2.11. Hence, the sample-by-sample correlation function can be obtained from the multiplication of C1 and C2 and is given by

C3 = C1 × C2 = e−j86πε/N 41 X n=0 x(n) · x(n + 43)∗× e−j172πε/N 41 X n=0 x(n)x(n + 86)∗ = e−j258πε/N 41 X n=0 x(n) · x(n + 43)∗_× 41 X n=0 x(n)x(n + 86)∗ = e−j258πε/NΦ1× Φ2 = e−j258πε/N_Φ 3 (2.9) where Φ3 = Φ1× Φ2 and N = 128.

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CP 0~41 42 43~84 85 86~127

conj

angle - 64/129

given by

ˆ

ε3 = −

64

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2.4.4 Algorithm 4

Finally, the algorithm 4 is an extension of algorithm 1 by extending the length of the correlation window and is depict in Fig. 2.12.

C4 = 41 X n=0 y(n)y(n + 43)∗× 41 X n=0 y(n + 43)y(n + 86)∗ = 41 X n=0 x(n)ej2πnε/N · [x(n + 43)ej2π(n+43)ε/N]∗× { 41 X n=0 x(n + 43)ej2π(n+43)ε/N _{· [x(n + 86)e}j2π(n+86)ε/N_]∗_} = 41 X n=0 x(n)x(n + 43)∗_ej2πnε/N_e−j2π(n+43)ε/N_× 41 X n=0 x(n + 43)x(n + 86)∗ej2π(n+43)ε/Ne−j2π(n+86)ε/N = e−j2π86ε/N 41 X n=0 x(n)x(n + 43)∗ _× 41 X n=0 x(n + 43)x(n + 86)∗ = e−j172πε/NΦ4 (2.11) where Φ4 = P₄₁ n=0x(n)x(n + 43)∗× P₄₁ n=0x(n + 43)x(n + 86)∗ and N = 128.

given by

ˆ

ε4 = −

32

43π∠C4. (2.12)

2.5 Numerical Results and Discussion

Numerical results are provided in this section to demonstrate the performance of the four algorithms. The MSE is defined as

MSE = 1 Mc Mc X k=1 (ε − ˆε(k))2 (2.13) where Mc is the total Monte Carlo runs, and ˆε(k) is the normalized CFO estimate of

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C P 0~41 42 43~84 85 86~127

conj conj

angle -32/43

give good MSE performance in AWGN channel even if the three repetitions are not incomplete.

The frequency-selective fading channel is modelled as a linear FIR filter with impulse response given by

h(k) = CXL−1

n=0

αne−jΦδ(k − n) (2.14)

where Φ is uniformly distributed in [0, 2π) and αn is Rayleigh distributed with an

expo-nential power profile

¯

α2

n = (1 − e−Ts/Trms)e−nTs/Trms (2.15)

with CL= 16, Trms = 30ns and Ts= 50ns. We use Jakes channel model with maximum

Doppler shift of 500 Hz to simulate time-correlated Rayleigh fading αn.

While multipath fading channel exists, the four proposed estimators give bad MSE performance if we do not do any things to deal with the three incomplete repetitions first; see Fig. 2.16, Fig. 2.17 and Fig. 2.18. Even if the SNR is very high, the MSE of each algorithm is bounded at 10−2_.

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Hence, in order to reduce the MSE of the four proposed estimators when multipath exists, we will introduce the interpolations in next chapter. The interpolations can be used to construct the losing samples from the existing samples by some proper weighting factor. Then, the three repetitions are nearly identical and the MSE performance will be better. 0 5 10 15 20 25 30 35 40 10−5 10−4 10−3 10−2 10−1 SNR (dB)

MSE of the Carrier Frequency Offset

FFT size = 128, for segment 0

Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4

Figure 2.13: MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN

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0 5 10 15 20 25 30 35 40 10−4 10−3 10−2 10−1 SNR (dB)

channel using IDcell 0 preamble in segment 1 without interpolation.

0 5 10 15 20 25 30 35 40 10−5 10−4 10−3 10−2 10−1 SNR (dB)

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0 5 10 15 20 25 30 35 40 10−3

10−2 10−1

SNR (dB)

Figure 2.16: MSE of normalized CFO ˆε estimate as a function of the SNR in multipath

fading channel using IDcell 0 preamble in segment 0 without interpolation.

0 5 10 15 20 25 30 35 40

10−3 10−2 10−1

SNR (dB)

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0 5 10 15 20 25 30 35 40 10−3

10−2 10−1

SNR (dB)

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Chapter 3 Interpolation Methods for

Almost-Periodic Preamble in

Time-Domain

In chapter 2, we have provided some general preamble-based fractional CFO estima-tion schemes for TDD OFDMA downlink system. However, the estimaestima-tion performances are not very good especially when the multipath exists. The reason is that there is no alignment of the preamble samples between any two parts of the three incomplete rep-etitions as shown in Fig. 2.7. This property leads to fault correlation-based frequency synchronization. Hence, this chapter will develop two interpolation methods to recon-struct the complete preamble. Based on the interpolation methods, the correlation-based CFO estimation can be improved as well.

3.1 Oversampling

From Fig. 2.7, if we increase the sampling rate of the A/D converter by an integer factor of 3, we will have three completely identical parts. Then, the samples of the complex envelope of the received OFDM signal is given by

yos(m) = 1 √ N N −1_X k=0 XkHke j2πm(k+ε) N + w(m) = xos(m)e j2πmε N + w(m) m = −Ng, −Ng+ 1 3, −Ng+ 2 3, −(Ng− 1), ..., 0, 1 3, 2 3, 1, 1 1 3, ..., N − 1 3 (3.1)

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where we assume that the oversampling interval is Tu/3N, Tu is the useful duration of

one OFDMA block.

Then, the three perfectly repetitive structures in time domain can be obtained by oversampling if we ignore the channel, CFO, and noise effect; see Fig. 3.1. In Fig. 3.1, the red circles denote the originally discrete samples after the A/D converter without oversampling and the green squares denote the new samples obtained from oversampling. A practical example via oversampling for 128-point FFT size is shown in Fig. 3.2

0 2 4 6 8 10 12 14 16 18 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 3.1: Preamble sequences via oversampling in time-domain.

In practical systems, however, to increase the sampling rate is almost impossible. Therefore, we should provide some alternative methods such as interpolations instead of oversampling. In order to obtain the green squares in Fig. 3.1 without oversampling, we can utilize the original samples such as the red circles in Fig. 3.1 to interpolate by taking weighted average. In the remaining of this chapter, some interpolations (either linear or sinc) will be presented. In other words, all these methods are based on the originally discrete samples (red circles) to generate the new samples (green squares). The detail description of the two interpolations will be introduced in Sec. 3.2 and Sec. 3.3 respectively.

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20 40 60 80 100 120 −4 −3 −2 −1 0 1 2 3 4 5 sample Phase

Figure 3.2: Three perfect preamble sequences via oversampling for IDcell 0 in segment 0.

3.2 Linear Interpolation

The first method is the linear interpolation that is often employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence. The linear interpolation is implemented by first passing the input sequence y(n) to be interpolated through an up-sampler whose output yu(m) is then passed through a second

discrete-time system that “fills in” the zero-valued samples inserted by the up-sampler with values obtained by a linear interpolation of the pair of input samples surrounding the zero-valued samples, as indicated in Fig. 3.3. In our case, we develop the input-output relation of a linear factor-of-3 interpolator. Hence, the output of the up-sampler can be

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expressed as yu(m) =    y(n), m = 3n 0, m = 3n + 1 0, m = 3n + 2 0 ≤ n ≤ N − 1 (3.2) where y(n) is the received preamble sequences after the A/D converter and is defined in (2.3). L D is c rete-time sy stem

)

(n

y

yu(m) y_int(m)

Figure 3.3: A factor-of-L interpolator.

The overall input-output relationship at the linear interpolator output can be written as ylinear(m) =    y(n), m = 3n 2 3y(n) + 13y(n + 1), m = 3n + 1 1 3y(n) + 2 3y(n + 1), m = 3n + 2 0 ≤ n ≤ N − 1 (3.3) where ylinear(m) is the output of the linear interpolator, and (1₃,2₃) are the weighting

factors.

Here, if yu(m) is two zero-valued samples inserted between a pair of input samples,

it is replaced with the average of the four original input samples, yu(m − 2), yu(m − 1), yu(m + 1) and yu(m + 2) : ylinear(m) = yu(m) + 2 3[yu(m − 1) + yu(m + 1)] + 1 3[yu(m − 2) + yu(m + 2)]. (3.4) On the other hand, if yu(m) is one of the original input samples, its neighbors, yu(m−2), yu(m − 1), yu(m + 1) and yu(m + 2) are all equal to 0. Hence, (3.4) can be expressed in

convolution sum as

ylinear(m) = yu(m) ∗ hlinear(m) 0 ≤ m ≤ 3N − 1 (3.5)

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given by hlinear(m) =            1 3, m = −2 2 3, m = −1 1, m = 0 2 3, m = 1 1 3, m = 2 (3.6)

The interpolated samples thus lie on a straight line joining the pair of input samples, as illustrated in Fig. 3.4 for a factor-of-3 linear interpolation. The green triangles denote the result of the linear interpolation and the orange dotted curve denotes the continuous waveform obtained from linear interpolation. From practical example, the repetitive property of the time-domain preamble sequences by linear interpolation can be shown in Fig. 3.5. Hence, linear interpolation is a feasible method to solve the foregoing problem. 0 5 10 15 20 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 3.4: Illustration of the linear interpolation method.

3.2.1 Analysis of The Three Repetitive Structures via Linear

Interpolation

In this subsection, we will compare the difference between any two parts of the three identical structures. A convenient way is to find the MSE. This result is shown in Tab.

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20 40 60 80 100 120 −4 −3 −2 −1 0 1 2 3 4 5 sample Phase

Figure 3.5: IDcell 0 preamble sequences in segment 0 via linear interpolation.

FFT size = 128 & IDcell = 0 in segment 0 MSE repetition 1 & 2 0.1612 × 10−3

repetition 2 & 3 0.1608 × 10−3

repetition 1 & 3 0.1601 × 10−3

Table 3.1: MSE between any two parts of the three repetitions by linear interpolation. 3.1. Although the MSE is quite small, there is little amount of information since each reconstructed sample is interpolated by only two neighboring samples. Therefore, this method is suboptimal and then we will present a better method in the next section.

3.3 Sinc Interpolation

The second method is sinc interpolation which is derived from the sampling theo-rem. We can utilize the sinc function to reconstruct the new samples from old samples. Likewise, it can be easily shown that for a factor-of-3 sinc interpolator, the discrete-time

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system following the factor-of-3 up-sampler as illustrated in Fig. 3.3 is characterized by the input-output relation written as

ysinc(m) = yu(m) ∗ hsinc(m) 0 ≤ m ≤ 3N − 1 (3.7)

where hsinc(m) is the impulse response of the discrete-time system in Fig. 3.3 and is

given by

hsinc(m) = sinc( m

3) − 3L ≤ m ≤ 3L (3.8)

with L is the truncated sidelobe length of the sinc function. Fig. 3.6 illustrates the impulse response of hsinc(m) with L = 5.

−5 0 5 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 3.6: The impulse response of hsinc(m) with L = 5.

In Fig. 3.7, the black triangles denote the result of sinc interpolation. From prac-tical examples, the repetitive property of the time-domain preamble sequences by sinc interpolation can be shown in Fig. 3.8. This result reveals that the sinc interpolation is more accurate than linear interpolation but with higher computational complexity.

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0 2 4 6 8 10 12 14 16 18 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 3.7: Illustration of the sinc interpolation method.

3.3.1 Analysis of The Three Repetitive Structures via Sinc

In-terpolation

Likewise, we still compare the difference between any two parts of the three repetitive structures. The MSE as a function of the sidelobe length is shown in Fig. 3.9. Hence, in order to achieve the MSE of 10−4_{, the sidelobe length of 5 (L = 5) is enough.}

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20 40 60 80 100 120 −4 −3 −2 −1 0 1 2 3 4 5 sample Phase

Figure 3.8: IDcell 0 preamble sequences in segment 0 via sinc interpolation with side-lobe length = 5. 5 10 15 20 25 30 1 10−7 10−6 10−5 10−4 10−3 sidelobe length MSE

The MSE analysis of the DL preamble by sinc interpolation (N=128 for segment 0)

difference between 1 & 2

difference between 2 & 3 difference between 3 & 1

difference between 3 & 1 (discard some points) difference between 2 & 3 (discard some points)

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Chapter 4 Downlink Pilot-Assisted Frequency

Synchronization via Interpolation

After the interpolations (either linear or sinc) introduced in chapter 3, we will have three repetitive structures. Based on the three repetitive structures, the correlation between the three repetitive samples in time-domain can be used to perform CFO esti-mation. This chapter still presents the same correlation-based algorithms as introduced in chapter 2 for CFO estimation but with interpolation. Hence, the four algorithms should be modified properly. Besides, the performance of the identical algorithms with-out interpolation is also illustrated. Finally, the simulation results verify the superior performance of the proposed methods (interpolations) with regard to estimation accu-racy.

4.1 Repetitive Structure of Interpolated Preamble

The structure of the factor-of-3 interpolated preamble sequences with index set

{0, 1, ..., 3N − 1} is illustrated in Fig. 4.1, where samples connected together by the same arrow-line are almost identical. After the factor-of-3 interpolation, in addi-tion to the original N preamble samples, we will have addiaddi-tional 2N samples obtained from interpolations. In Fig. 4.1, the N original preamble samples are indexed with

{0, 3, ..., 3(N − 1)}, and the interpolated samples are indexed with {1, 2, ..., 3N − 2, 3N −

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when the sample-by-sample correlation is implemented. Consequently, repetition 1 is indexed with {0, 1, ..., N − 1}, repetition 2 is indexed with {N, N + 1, ..., 2N − 1}, and repetition 3 is indexed with {2N, 2N + 1, ..., 3N − 1}.

C P 0 1 · · · N-2 N-1 N N+1 · · · 2N- 2 2N- 1 2N 2N +1 · · · 3N-2 3N-1

Figure 4.1: Structure of a interpolated preamble symbol, where samples connected to-gether by the same arrow-line are almost identical.

From above descriptions, the four algorithms introduced in chapter 2 can be per-formed well without discarding some samples and only require some modifications when interpolation exists. The modification of each algorithm is described in detail in the next section.

4.2 Time-Domain Correlation-based Fractional CFO

Estimation by Using Almost-Periodic Preamble

with Interpolations

If we neglect the inaccuracy of the interpolated samples and AWGN effect, THREE conditions are satisfied in a preamble symbol

• Condition 1 :

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• Condition 2 :

yint(m + N) = yint(m + 2N) 0 ≤ m ≤ N − 1 (4.2)

• Condition 3 :

yint(m) = yint(m + 2N) 0 ≤ m ≤ N − 1 (4.3)

where yint(m) is the output of the interpolator either form linear interpolation ylinear(m)

or from sinc interpolation ysinc(m).

Because we neglect the inaccuracy of the interpolated samples, the output of the interpolator yint(m) is equal to the oversampling sample yos(m) as defined in (3.1).

Consequently, we can utilize (3.1) to modify the four algorithms as introduced in chapter 2 in the following. For derivation purpose, (3.1) can be rewritten as

yos(m) = 1 √ N N −1 X k=0 XkHke j2πm(k+ε) 3N + w(m) = xos(m)e j2πmε 3N + w(m) 0 ≤ m ≤ 3N − 1 (4.4) where m is an integer. Hence, we can replace yos(m) with yint(m) and replace xos(m)

with xint(m) respectively.

4.2.1 Modified Algorithm 1 via Interpolations

First, we consider the correlation between the first two repetitions satisfing condition 1 as shown in Fig. 4.2. If we ignore the noise effect, the sample-by-sample correlation

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function can be written as C1 =

N −1_X m=0

yint(m)yint(m + N)∗

=

N −1_X m=0

xint(m)ej2πmε/3N · [xint(m + N)ej2π(m+N )ε/3N]∗

= N −1_X m=0 xint(m)xint(m + N)∗ej2πmε/3Ne−j2π(m+N )ε/3N = e−j2πN ε/3N N −1 X m=0 kxint(m)k2 = e−j2πε/3 N −1_X m=0 kxint(m)k2 (4.5)

The normalized CFO ε is estimated using the phase of the correlation function and is given by ˆ ε1 = − 3 2π∠C1. (4.6)

CP

IFFT

conj

IFFT

angle

-3/2

Figure 4.2: Block diagram of modified algorithm 1 via interpolations.

4.2.2 Modified Algorithm 2 via Interpolations

Then, we consider the correlation between the first and the third repetitions satisfy-ing condition 3 as shown in Fig. 4.3. If we ignore the noise effect, the sample-by-sample

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correlation function can be written as C2 =

N −1_X m=0

yint(m)yint(m + 2N)∗

=

N −1_X m=0

xint(m)ej2πmε/3N · [xint(m + 2N)ej2π(m+2N )ε/3N]∗

= N −1_X m=0 xint(m)xint(m + 2N)∗ej2πmε/3Ne−j2π(m+2N )ε/3N = e−j2π2N ε/3N N −1_X m=0 kxint(m)k2 = e−j4πε/3 N −1_X m=0 kxint(m)k2 (4.7)

The normalized CFO ε is estimated using the phase of the correlation function and is given by

ˆ

ε2 = −

3

4π∠C2. (4.8)

CP

IFFT

conj

angle

-3/4

4.2.3 Modified Algorithm 3 via Interpolations

Algorithm 3 is a combination of algorithm 1 and algorithm 2, and is shown in Fig. 4.4. Hence, the sample-by-sample correlation function can be obtained from the

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multiplication of C1 and C2 and is given by C3 = C1 × C2 = e−j2πε/3 N −1 X m=0 kxint(m)k2 × e−j4πε/3 N −1_X m=0 kxint(m)k2 = e−j2πε_{ N −1_X m=0 kxint(m)k2}2 (4.9)

ˆ

ε3 = −

1

2π∠C3. (4.10)

CP

IFFT

conj

angle

-1/2

4.2.4 Modified Algorithm 4 via Interpolations

Finally, the modified algorithm 4 is an extension of modified algorithm 1 by extend-ing the length of the correlation window from N samples to 2N samples. If we ignore

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the noise effect, the sample-by-sample correlation function can be written as C4 =

2N −1_X

m=0

yint(m)yint(m + N)∗

=

2N −1_X

m=0

xint(m)ej2πmε/3N · [xint(m + N)ej2π(m+N )ε/3N]∗

= 2N −1_X m=0 xint(m)xint(m + N)∗ej2πmε/3Ne−j2π(m+N )ε/3N = e−j2πN ε/3N 2N −1_X m=0 kxint(m)k2 = e−j2πε/3 2N −1_X m=0 kxint(m)k2 (4.11)

ˆ

ε4 = −

3

2π∠C4. (4.12)

4.3 Simulation Results and Discussions

Performance improvement due to linear or sinc interpolation can be obtained in the proposed correlation-based algorithms in both the AWGN channel and the multipath fading channel. We discuss each algorithm respectively and compare the performance improvement between the cases with and without interpolations (linear or sinc).

For AWGN channel, performance improvement of algorithm 1 is about 2 dB at low SNR if the interpolation is present. When SNR increases, the performance advantages of the proposed scheme will increase; see Fig. 4.5, Fig. 4.13 and Fig. 4.21. Significant performance improvement can be obtained in the multipath fading channel by using the proposed interpolations; see Fig. 4.6, Fig. 4.14 and Fig. 4.22. The performance improvement of algorithm 2 with interpolations in AWGN channel is about 5 dB at low

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SNR. When SNR increases, the performance advantages of the proposed scheme will increase; see Fig. 4.7, Fig. 4.15 and Fig. 4.23. When multipath exists, significant performance improvement is obtained under the interpolations; see Fig. 4.8, Fig. 4.16 and Fig. 4.24. For the other two algorithms, we have similar simulation results.