在快速時變且多重路徑通道下使用白化處理於殘留載波間干擾和通道雜訊的正交分頻多工訊號接收

國 立

交 通 大

學

電 子 工 程 學 系 電 子 研 究 所

博 士 論 文

在快速

_{時變且多重路徑通道下使用白化處理於殘留載波}

間

_{干擾和通道雜訊的正交分頻多工訊號接收}

OFDM Signal Detection in Doubly Selective Channels with

Whitening of Residual Intercarrier Interference and Noise

研 究

生： 王海薇

指

_{導教授： 林大衛博士}

桑梓賢

_博士

在快速

_{時變且多重路徑通道下使用白化處理於殘留載波間干擾和通}

道

_{雜訊的正交分頻多工訊號接收}

OFDM Signal Detection in Doubly Selective Channels with

Whitening of Residual Intercarrier Interference and Noise

研 究

生 ： 王海薇

_{Student: Hai-wei Wang}

指

_{導 教 授 ： 林大衛博士}

_{Advisors: Dr. David W. Lin}

桑梓賢

_博士

_{Dr. Tzu-Hsien Sang}

國 立

交 通 大

學

電子工程學系電子研究所

博 士 論 文

A Dissertation

Submitted to Department of Electronics Engineering and

Institute of Electronics

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in

Electronics Engineering

July 2012

Hsinchu, Taiwan, Republic of China

中華民國一百零一年七月

在快速

_{時變且多重路徑通道下使用白化處理於殘留載波間干擾和通}

道

_{雜訊的正交分頻多工訊號接收}

學生： 王海薇

指

_{導 教 授 ： 林大衛博士}

桑梓賢

_博士

國立交通大學

_{電子工程學系電子研究所}

摘

要

數下_{是不變的， 該系統參數包含最大督普勒頻率位移(maximum Doppler shift)、}

多 重 路 徑 通 道 數 據(multipath channel profile) 、 功 率 頻 譜 密 度(power spectral density)、正交分頻多工系統之取樣週期(sampling period)、離散傅利葉轉換之長 度(DFT size)、 正交性分頻多工系統之符號週期(symbol period)以及平均傳送符 號能(average transmitted symbol energy)。 以上的發現說明了該相鄰次載波之殘

留的載波間干擾(residual ICI)高度的正規化相關性和容易估計的特性非常適合應

用於實_{際通訊系統接收。}

白化處理的一接受器可以得到非常低的雜訊底(noise floor)， 進而使通訊系統具有 很好的傳輸性能，例如電腦模擬顯示採用最大可能序列估計(maximum-likelihood

sequence estimation, MLSE)的接收器 並且考慮上述白化處理用於該相鄰載波之殘

留的載波間_{干擾(residual ICI)可以降低位元誤差率(BER)之誤差底(error floor) 數}

個級數(order)， 可明顯看出利用該統計特性結合傳統接收偵測方法對於提升通訊 系統之接收效_{能有很大的貢獻。} 更近一步，本論文提供一個考慮上述白化程序用於 “相鄰載波所殘留的載波 間_{干擾”的線性最小均方誤差(LMMSE)和遞迴線性最小均方誤差接收器。 相對於} 最大可能_{序列估計，這個方法在良好的偵測性能和低複雜度間提供另一折衷選} 擇。

OFDM Signal Detection in Doubly Selective Channels with

Whitening of Residual Intercarrier Interference and Noise

Student: Hai-wei Wang

Advisors: Dr. David W. Lin

Dr. Tzu-Hsien Sang

Department of Electronics Engineering

& Institute of Electronics

National Chiao Tung University

ABSTRACT

Orthogonal frequency-division multiplexing (OFDM) is a popular broadband wire-less transmission technique, but its performance can suffer severely from the inter-carrier interference (ICI) induced by fast channel variation arising from high-speed motion. Existing ICI countermeasures usually address a few dominant ICI terms only and treat the residual similar to white noise.

We show that the residual ICI has high normalized autocorrelation and that this normalized autocorrelation is insensitive to the maximum Doppler frequency and the multipath channel profile, the OFDM sample period, the discrete Fourier transform (DFT) size, the OFDM symbol time, the transmitted symbol energy. Consequently, the residual ICI plus noise can be whitened in a nearly channel-independent manner, leading to significantly improved detection performance. Simulation results confirm the theoretical analysis. As a result, a whitening transform for the residual ICI plus noise can be obtained based solely on the ICI-to-noise ratio. Such a transform can be used in association with many different signal detection schemes to significantly improve the detection performance.

In particular, they show that the proposed technique can significantly lower the ICI-induced error floor by several orders of magnitude in maximum-likelihood sequence estimation (MLSE) designed to address a few dominant ICI terms. For QPSK, the proposed method can lower the error floor induced by ICI to under 10−6

channel state information (CSI).

Furthermore, we consider linear minimum mean-square error (LMMSE) and iterative LMMSE detection with the above partial whitening of additive distur-bance, together with soft decision feedback. The method is shown to provide good performance-complexity tradeoff compared to other ICI countermeasures.

誌

誌

誌

謝

謝

謝

Contents

書名頁_{. . . . i}

中文摘要 _{. . . . ii}

Abstract . . . iv

誌謝 _{. . . . vi}

Table of Contents . . . vii

List of Tables . . . x

List of Figures . . . xi

1 Thesis Introduction . . . 1

1.1 System Model . . . 4

1.2 Thesis Organization and Contributions . . . 5

2 Wireless Channel Characterization . . . 8

2.1 Wireless Channel . . . 8

2.2 Multipath Fading . . . 9

2.2.1 Statistical Characterization of Multipath Channels . . . 10

2.2.2 Doubly Selective Channel Model . . . 10

2.4 Statistical Characterization: The WSSUS Model . . . 12

2.5 The Time-Varying Channel . . . 13

2.5.1 Jakes Doppler Spectrum . . . 13

2.5.2 Doppler Spreading Simulation . . . 14

3 Autocorrelation of Residual Intercarrier Interference . . . 16

3.1 Derivation of Autocorrelation of Residual ICI . . . 17

3.2 Numerical Examples . . . 21

3.3 Derivation of (3.6) and Some Related Comments . . . 30

3.4 Summary of Results . . . 32

4 MLSE Detection with Whitening of Residual ICI Plus Noise . . . 33

4.1 MLSE Detection with Whitening of Residual ICI Plus Noise . . . 34

4.2 Complexity Analysis . . . 37

4.3 Simulation Results on Detection Performance . . . 38

4.4 Dependence of Detection Performance on Parameter Setting . . . 45

4.5 Summary of Results . . . 48

5 Low Complexity Detection with Whitening of Residual ICI Plus Noise . . . 49

5.1 LMMSE Signal Detection with Whitening of Residual ICI Plus Noise 50 5.2 Simulation Results . . . 53

5.3 Summary of Results . . . 57

6 Thesis Conclusions and Potential Future Topics . . . 58

6.1 Thesis Conclusions . . . 58

Appendix A: The Whiteners of Residual ICI Plus Noise . . . 61

簡歷 _{. . . 72} 著作目錄 _{. . . 73}

List of Tables

4.1 Two Channel Power-Delay Profiles Used in This Study, Where TU6 Corresponds to the COST 207 6-Tap Typical Urban Channel And SUI4 the SUI-4 3-Tap Channel . . . 37

List of Figures

1.1 OFDM system model. . . 2

2.1 Frequency Selective Fading Channel Simulators. . . 11

3.1 Normalized autocorrelation of residual ICI over multipath Rayleigh fading channel at K = 0, with N = 128 and Tsa = 714 ns. The

first-order approximation (3.11)–(3.13) yields 0.6079 for r = 1 and 0.1520 for r = 2, which are quite accurate at low fd values. . . 22

3.2 Normalized autocorrelation of residual ICI over multipath Rayleigh fading channel at K = 1, with N = 128 and Tsa = 714 ns. The

first-order approximation (3.11)–(3.13) yields 0.7753, 0.6461, 0.5599, 0.3036, 0.1912, and 0.1317, for r = 1–6, respectively, which are quite accurate. . . 23

3.3 Normalized autocorrelation of residual ICI over multipath Rayleigh fading channel at K = 2, with N = 128 and Tsa = 714 ns. The

first-order approximation (3.11)–(3.13) yields 0.8440, 0.7358, 0.6612, 0.6014, and 0.5534, for r = 1–5, respectively, which are quite accurate. 24

3.4 Normalized autocorrelation of residual ICI over one-Doppler-line chan-nel at K = 0, with N = 128 and Tsa= 714 ns. . . 25

3.5 Normalized autocorrelation of residual ICI over one-Doppler-line chan-nel at K = 1, with N = 128 and Tsa= 714 ns. . . 26

3.6 Normalized autocorrelation of residual ICI over one-Doppler-line chan-nel at K = 2, with N = 128 and Tsa= 714 ns. . . 27

4.1 Trellis structure for MLSE-based detection using the Viterbi algo-rithm, under QPSK modulation and with p = 1, where numerals 0–3 represent the QPSK constellation points. . . 36

4.2 Error performance in TU6 channel of the conventional OFDM signal detection method and ICI-whitening MLSE (the proposed method) with K = 0 and p = q = 1 in noise-free condition. . . 41

4.3 Comparison of proposed technique in TU6 and SUI4 channels with that treating residual ICI as white; SNR = ∞. . . 42

4.4 Performance of proposed technique versus Doppler spread in the TU6 channel with p = q = K = 1, at N = 128 and Tsa= 714 ns and under

QPSK subcarrier modulation. . . 43

4.5 Performance versus Eb/N0 of different methods in the TU6 channel,

with N = 128, Tsa = 714 ns, fd = 1500 Hz (normalized peak Doppler

frequency fdTsaN = 0.1371) and QPSK subcarrier modulation.

(Re-sults with N = 1024 are very close.) . . . 44

4.6 SINR performance of different methods in the TU6 channel, with N = 128 and Tsa= 714 ns and assuming perfect CSI. . . 47

5.1 Bit error rate of different detection methods in the TU6 channel, with N = 128, Tsa = 714 ns, fd = 1500 Hz (normalized peak Doppler

frequency fdTsaN = 0.1371) and QPSK subcarrier modulation. . . 54

5.2 Bit error rate floor versus Doppler spread of different detection meth-ods in the TU6 channel with N = 128, Tsa = 714 ns, and QPSK

5.3 Bit error rate floor versus Doppler spread of different detection meth-ods in the TU6 channel with N = 128, Tsa = 714 ns, and 16QAM

subcarrier modulation. . . 56

A.1 Performance of proposed MLSE p = q = K = 1 and MMSE p = q = 1, K = 2, with imperfect whitener in the TU6 channel, at N = 128 and Tsa = 714 ns fdTsaN = 0.137 and under QPSK subcarrier

Chapter 1

Thesis Introduction

Orthogonal frequency-division multiplexing (OFDM) is widely adopted in broad-band wireless signal transmission due to its high spectral efficiency. However, its performance can suffer severely from the intercarrier interference (ICI) induced by fast channel variation resulting from high-speed motion. Such an effect is sometimes referred to as loss of subcarriers orthogonality. The problem becomes increasingly acute as the carrier frequency or the speed of motion increases. For instance, with a 500 km/h mobile speed and a 6 GHz carrier frequency, the peak Doppler frequency can be as high as about 2800 Hz, which translates to over 0.25 times the 10.94 kHz subcarrier spacing in the Mobile WiMAX standard [1]. The signal detection performance can become intolerable without proper countermeasures.

Consider the typical OFDM system illustrated in Fig.1.1. In a system without ICI, the channel frequency response matrix that relates the inputs of the inverse discrete Fourier transform (IDFT) and the outputs of the DFT is diagonal. Fast channel variation introduces sizable off-diagonal elements in the matrix, thus result-ing in ICI.

The direct minimum mean square error and zero-forcing equalizers for OFDM symbols requires a large matrix inversion. Several algorithms [14–16] were developed to reduce the complexity of this direct matrix inverse for OFDM symbols .

..

.

..

.

Figure 1.1: OFDM system model.

Choi et al. [14] proposed a MMSE equalizer for OFDM symbols incorporat-ing with successive interference cancellation. In [15], Cai and Giannakis derived recursive algorithms for calculation of the matrix inversion by combining the meth-ods [2,15].

The above equalizers [14,15] still require ≥ O(N2_{) complexity, where N is the}

number of subcarriers. Hsu and Wu [16] proposed a successive detection combined with Newton’s iterative matrix inversion, requiring O(NlogN) complexity. However, as the subcarrier numbers in one OFDM symbol increases, the direct implementation of a traditional MMSE or ZF equalizer should be avoided.

In theory, an optimal signal detector should take all ICI terms into account. But for reasons of complexity and robustness, usually only the dominant terms are compensated for. As these dominant terms are normally concentrated (circulantly) around the diagonal, the channel matrix shows a (circulant) band structure [2–4,22]. Several frequency-domain equalization techniques based on band approxima-tion to channel matrices have been proposed, including blockwise zero-forcing linear equalization [2], linear minimum mean-square error (LMMSE) equalization [3,22,23], and maximum-likelihood sequence estimation (MLSE) [4].

An interested reader may refer to [16] for additional introduction to various ICI mitigiation studies.

Jeon et al. [2] consider the situation where the normalized peak Doppler fre-quency (i.e., peak Doppler frefre-quency expressed in units of frefre-quency spacing of subcarriers) is on the order of 0.1 or less. In this situation, the channel variation over one OFDM symbol time is approximately linear. A frequency-domain equalizer that exploits the ensuing band channel matrix structure is proposed. Schniter [22] considers substantially higher normalized peak Doppler frequencies, under which the ICI is more widespread. Time-domain windowing is used to partially counter-act the effect of channel variation and shrink the bandwidth of the channel matrix. An iterative minimum mean-square error (MMSE) equalizer is then used to detect the signal. Rugini et al. [3] employ block-type linear MMSE equalization, wherein the band channel matrix structure is exploited (via triangular factorization of the autocorrelation matrix) to reduce the equalizer complexity. Ohno [4] addresses the ICI via maximum-likelihood sequence estimation (MLSE) in the frequency domain, where the band channel matrix structure is utilized to limit the trellis size.

The consideration of only the dominant ICI terms results in an irreducible error floor in time-varying channels [2–4,22]. Moreover, while the uncompensated residual ICI is colored [5,6,24], for various reasons it is often treated as white [4–7,24].

In principle, the error performance floor can be reduced by whitening. Although whitening of “I+N” (i.e., sum of ICI and additive channel noise) can lead to improved signal detection performance, it requires knowing the autocorrelation function of I+N, which remains a key problem awaiting solution [6,24]. Without knowing the autocorrelation function, one can only resort to less sophisticated techniques, such as simple differencing of the received signals at neighboring subcarriers [8]. Authors [8] further point out that the noise and channel statistics is a challenging and interesting problem under investigation.

In this thesis, we attempt to characterize this autocorrelation function of resid-ual ICI pluse noise for the benefit of signal detection.

1.1

System Model

Fig.1.1shows the discrete-time baseband equivalent model of the considered OFDM system. The input-output relation of the channel is given by

yn= L−1

X

l=0

hn,lxn−l+ wn (1.1)

where xn and yn are, respectively, the channel input and output at time n, L is

the number of multipaths, hn,l is the complex gain of the lth path (or tap) at time

n, and wn is the complex additive white Gaussian noise (AWGN) at time n. We

assume that the length of the cyclic prefix (CP) is sufficient to cover the length of the channel impulse response (CIR) (L − 1)Tsa, where Tsa denotes the sampling

period.

One common way of expressing the received signal in the DFT domain is Ym = N −1_X k=0 L−1 X l=0 XkH_l(m−k)e−j2πlk/N+ Wm, 0≤m≤N − 1, (1.2)

where Xk and Ym are, respectively, the channel input and output in the frequency

domain (see Fig.1.1), N denotes the size of DFT, Wm denotes the DFT of wm, and

H_l(k) is the frequency spreading function of the lth path given by H_l(k)= 1

N

N −1_X n=0

hn,le−j2πnk/N. (1.3)

Another way of expressing it is

y = Hx + w (1.4) where y = [Y0, ..., YN −1]′, x = [X0, ..., XN −1]′, w = [W0, ..., WN −1]′, and H =         a0,0 a0,1 · · · a0,N −1 a1,0 a1,1 · · · a1,N −1 ... ... . .. ... aN −1,0 aN −1,1 · · · aN −1,N −1         , (1.5)

with ′ _{denoting transpose and}

am,k = L−1

X

l=0

The quantity am,k is the “ICI coefficient” from subcarrier k to subcarrier m. For

a time-invariant channel, H_l(k) vanishes ∀k 6= 0 and H becomes diagonal, implying absence of ICI.

As mentioned, a band approximation to H that retains only the dominant terms about the diagonal may ease receiver design and operation, but also results in an irreducible error floor. Consider a symmetric approximation with one-side bandwidth K, that is, am,k = 0 for |(m − k)%N| > K where K is a nonnegative

integer and % denotes modulo operation. Then the ICI at each subcarrier consists of contributions from at most 2K nearest (circularly) subcarriers. In this chapter, we exploit the correlation of the residual ICI outside the band to attain a significantly enhanced signal detection performance. For convenience, in the following we omit explicit indication of modulo-N in indexing a length-N sequence, understanding an index, say n, to mean n%N.

Let the channel be wide-sense stationary uncorrelated scattering (WSSUS) [10] with

E[hn,lh∗n−q,l−m] = σ2lrl(q)δ(m) (1.7)

where E[·] denotes expectation, σ2

l denotes the variance of the lth tap gain, rl(q)

denotes the normalized tap autocorrelation (where rl(0) = 1), and δ(m) is the

Kronecker delta function. For convenience, assumeP_lσ2

l = 1. Let Pl(f ) denote the

Doppler power spectral density (PSD) of path l and thus rl(q) = Z fd −fd Pl(f )ej2πf τdf   τ =Tsaq , (1.8)

where fd denotes the peak Doppler frequency of the channel. We assume that the

paths may be subject to arbitrary, different fading so that Pl(f ) may be asymmetric

about f = 0 and different for different l.

1.2

Thesis Organization and Contributions

The content of Chap. 3,4 has been published in [9,25] and the content of Chap. 5

The contribution of the present thesis is twofold.

First, we explore the correlation property of ICI outside the band and derive an approximate mathematical expression for it. The expression applies not only to classical multipath Rayleigh fading, but also to arbitrary Doppler spectrum shapes in general. It is found that the correlation values are based solely on the ICI-to-noise ratio. Moreover, the correlation values are very high for the residual ICI beyond the few dominant terms.

Secondly, to capitalize on the above high correlation to improve signal recep-tion over fast varying channels, we consider performing simple blockwise whitening of the residual I+N before signal detection (i.e., equalization), where the whitener makes use of the ICI characteristics as found. Numerical results show that substan-tial gains can be achieved with this approach.

This chapter describes the system model and introduced this thesis organiza-tion.

In Chap. 2, we introduce some mobile channel characterization.

In Chap. 3,we find that, in a mobile time-varying channel, the residual ICI beyond several dominant terms had high normalized autocorrelation. We derive a rather precise closed-form approximation for the (unnormalized) autocorrelation function. As a result, a whitening transform for the residual ICI plus noise can be obtained based solely on the ICI-to-noise ratio.

In Chap. 4, we consider MLSE-type signal detection in ICI with blockwise whitening of the residual ICI plus noise. Simulations and SINR numerical analysis are provided.

In Chap. 5, we consider LMMSE signal detection with blockwise whitening of residual ICI plus noise. We present some simulation results based on 3 × 3 block whitening and three-sample equalization. The results show that a good tradeoff between complexity and performance could be achieved.

Chapter 2

Wireless Channel Characterization

2.1

Wireless Channel

In general, “channel” can be used to mean everything between the source and the There may be more than one path over which the signal can travel between the transmitter and receiver over the air. Various signals are sent from the transmitter antennas and the all paths before it reaches the receiver antennas are referred as channel. The wireless users communicate over the air and then there is significant interference over channels. The wireless channel could be a simple straight line (Line of Sight, LOS). It also may be interfered by other factors, such as multi-path effects, which are due to atmospheric scattering and reflections from buildings and other objects.

Before arriving at the receiving antenna, the transmitted signal follows many different paths, and these paths constitute the multipath radio propagation channel. The resulting signal strength will undergo large fluctuations. How to deal with fading and with interference over channel is a key issue for the design of communication systems. The time variation of the channel strengths due to the small-scale effect of

multipath effects, as well as larger-scale effects such as shadowing by obstacles and path loss by distance attenuation. Shadow fading reveals itself as an attenuation of the average signal power. Shadow fading is induced by obstacles (buildings, hills, etc.) between transmitter and receiver. The wireless users communicating over the air often encounters both types of fading: multipath fading superimposed on the slower fading. The channel impulse response in the complex-lowpass equivalent form is composed of two components,

h(τ, t) = s(t) × ˜c(τ, t) (2.1)

where s(t) denotes the shadow fading component and ˜c(τ, t) denotes the multipath component. (2.1) means the multipath fading is superimposed on the shadow fading . It turns out that channels gains vary over multiple scales. At a fast time-scale, channels vary due to the multipath effects. At a slow time-time-scale, channels vary due to large-scale fading effects such as shadowing and path loss by distance attenuation. The duration of a shadow fade lasts for multiple seconds or minutes, and hence occurs at a much slower time-scale compared to multipath fading. Since the shadow fading is slow and is often compensated by power control, it may be regarded as quasi-static. Large-scale shadowing fading is often relevant to issues such as cell-site planning. Small-scale multipath fading is often relevant to the wireless communication systems design. For a given shadow fading component, the signal envelope is conditionally Rayleigh or Ricean distribution. If there is no LOS signal contribution to the receiver, the signal follows a Rayleigh distribution. If there is a LOS signal contribution to the receiver, the signal follows a Ricean distribution.

2.2

Multipath Fading

In a multipath channel, the transmitted signals arriving along different paths can have different attenuations and delays and they might be superimposed either con-structively or decon-structively at the receiver. This is the phenomenon of multipath fading.

channels [31]. Many realistic channels contain both diffuse and discrete properties. Those two properties often are separated for the purpose of channel modeling.

1. Diffuse multipath channel: The multipath signal paths are generated by a large number of unresolvable reflections. The Diffuse multipath fading might occur in an urban or a mountainous area. The signal envelope generated by lots unresolvable reflections is Rayleigh or Ricean distribution.

2. Discrete multipath channel: The multipath paths are made up of a few identifiable and resolvable components, which are reflected by hills or structures in open or rural areas. This results in a channel model with a finite number of multipath components.

2.2.1

Statistical Characterization of Multipath Channels

The multipath channels for both the diffuse and discrete effects have the following statistical characterization [31]:

1. Time spreading of the symbol duration in τ , which can be modeled as a set of discrete resolvable multipath components [31]: The channels effect is equivalent to filtering and band-limiting. A popular model for discrete multipath channels is the tapped-delay-line (TDL) channel model [33,34].

2. A time-variant channel behavior in t due to the motion of the receiver or the changing environment such as movements of reflectors or scatters: A popular channel model describing a time-variant behavior is the Jakes Doppler Spectrum.

2.2.2

Doubly Selective Channel Model

The doubly selective channel actually means the multipath channel with the time-variant behavior. Different time-time-variant and frequency selective fading channels may be simulated, depending on the settings of gain and time delay. They are shown in Fig.2.1.

1 n t -1 t 0 t 1 n

s

s

s

å

˗˸˿˴̌̆

˚˴˼́̆

g

g

g

g

g

g

-˙˴˷˼́˺ʳ̆˼̀̈˿˴̇̂̅̆

Figure 2.1: Frequency Selective Fading Channel Simulators.

2.3

Statistical Characterization of the

Time-Variant Behavior

The components of the multipath fading received signal can be modeled by treating ˜

c(τ, t) as a random process in t. Since ˜c(τ, t) arises from a large number of reflections and scattering, then by the central limit theorem, it can be modeled as a complex Gaussian process. In radio communications, the most common model describing flat fading in urban/suburban environments is Clarke’s model [35].

At any time t, the probability density functions of the real and imaginary parts of ˜c(τ, t) are Gaussian. If ˜c(τ, t) has a zero mean, then the envelope |˜c(τ, t)| = r can be shown [36] to be Rayleigh-distributed, i.e. with probability density function (pdf):

p(r) = r σ2e

where σ2 _{is the time-average power of the received signal before envelope}

detec-tion If ˜c(τ, t) has a nonzero mean, which implies there is a significant line-of-sight component present, can then be shown [36] to be Rician-distributed, i.e. with pdf:

p(r) = r σ2e

−(r2+A2) 2σ2 I₀(Ar

σ2) (2.3)

where A is the nonzero mean of and I0(.)is the zero-order modified Bessel function of

the first kind. In such a situation, random multipath components arriving at differ-ent angles are superimposed on a stationary dominant signal. A ratio K = A2_/(2σ2₎

is an indicator of the relative power in the faded and unfaded components.K is termed the Rician K-factor and completely specifies the Ricean distribution. As K >> 1, and as the dominant path fades away, the Ricean distribution degenerates to a Rayleigh distribution.

2.4

Statistical Characterization: The WSSUS

Model

A frequency-flat fading channel simulator needs to reproduce the Doppler spread-ing only, while a frequency-selective fadspread-ing channel simulator should emulate both Doppler spreading and time spreading. In general, the time spreading and Doppler spreading are mutually related. However, most channel simulators treat the two spreading processes independently for simplicity.

Such simulators are said to follow the Wide-Sense Stationary Uncorrelated Scat-tering (WSSUS) assumption in [32]. In the sections below, common approaches are reviewed for separately simulating the Doppler spreading process and the time spreading process. A model for the multipath channel that includes both the vari-ations in t and τ was introduced by Bello [32]. The time-varying channel ˜c(τ, t) is modeled as a wide-sense stationary (WSS) random process in t with an autocorre-lation function

Rc˜(τ1, τ2, ∆t) = E[˜c∗(τ1, t)˜c(τ2, t + ∆t)] (2.4)

delays may be uncorrelated. This is the uncorrelated scattering (US) assumption, which leads to

R˜c(τ1, τ2, ∆t) = Rc˜(τ1, ∆t)δ(τ2 − τ1) (2.5)

The most important class of stochastic time-variant linear channel models is repre-sented by models belonging to the WSS models as well as to the US models. These channel models with both the WSS and US assumptions are called WSSUS mod-els (WSSUS, wide-sense stationary uncorrelated scattering). This autocorrelation function is denoted by R˜c(∆τ, ∆t), and

R˜c(∆τ, ∆t) = E[˜c(τ, t)˜c(τ + ∆τ, t + ∆t)] (2.6)

Due to their simplicity, they are of great practical importance and are nowadays almost exclusively employed for modeling frequency-selective mobile radio channels.

2.5

The Time-Varying Channel

For mobile radio applications, the channel is time-varying because the motion be-tween the transmitter and receiver results in propagation paths change. It should be noted that since the channel characteristics are dependent on the relative posi-tions of the transmitter and receiver, time variance is equivalent to space variance. As mentioned previously, the time variation of the channel is characterized by the Doppler power spectrum. Although Doppler power spectrums apply to any time-variant model, for the sake of simplicity we present the commonly used Jakes model.

2.5.1

Jakes Doppler Spectrum

Jakes Doppler spectrum applies to time-varying channels. The so-called ”Jakes” Doppler power spectrum model is due to Gans [37]. Gans analyzed the Doppler spectrum of time-varying channels by Clarke’s model [35], which is also called the ”classical model”. Jakes Doppler spectrum follows the following assumptions [38,39]:

2. The ray arrival angles at receivers are uniformly distributed over [−π, π] . 3. The receiver’s antenna is omnidirectional. The normalized Jakes Doppler spectrum is given by Sj(f ) = 1 πfd q 1 − (f /fd)2 , |f | ≥ fd (2.7)

where fdis the maximum Doppler shift. And the corresponding autocorrelation

is then:

Rj(τ ) = J0(2πfdτ ) (2.8)

where J0(z)is the Bessel function of the first kind of order 0. We will have the

amplitude of the frequency response as |Hj(f )| =

q

Sj(f ) (2.9)

2.5.2

Doppler Spreading Simulation

The Rayleigh or Rician fading simulators designed to ensure that the following two properties are approximately verified, Due to the Doppler spreading, its power spectrum is given by the Clarke model, or by any other specified spectrum. For simulated fading process, its envelope should be Rayleigh or Rician-distributed. Two popular methods are sum-of-sinusoids (SoS) simulators and filtered Gaussian noise (FGN) simulators [39].

1. Simulators by Summing of Sinusoids

Like Clarke’s model, many sum of sinusoids simulators for fading channel have been proposed over the past three decades. Simulators by summing of sinusoids create the fading process by superposing several waves, each one being characterized by random amplitude, angle of arrival, and phase. As mention above, the resulting process of fading tends towards a Gaussian distribution due to the central limit theorem.

2. Simulators Filtering Gaussian Process with The Doppler Filters

A straightforward method of constructing simulators is to filter two independent white Gaussian noise with low-pass filters (Doppler filter). The Doppler filters H(f ) (impluse response) are to approximate the desired Doppler spectrum by eq(2.9). A complex Gaussian fading process with desired spectrums can be obtained by filtering with a Doppler filter. Both finite impulse response (FIR) filters and infinite impulse response (IIR) filters have been proposed as the Doppler filters. The filtering operation can be carried out in either the time domain or the frequency domain. A simple example was shown in [31] p.575.

Chapter 3

Autocorrelation of Residual

Intercarrier Interference

In this chapter, we try to characterize this autocorrelation function of residual ICI pluse noise from the viewpoint of signal detection. We derived a rather precise closed-form approximation for the (unnormalized) autocorrelation function. It is found that the correlation values are based solely on the ICI-to-noise ratio. More-over, the correlation values are very high for the residual ICI beyond the few domi-nant terms.

The remainder of this chapter is organized as follows. Sec. 3.1 analyzes the correlation property of ICI. Sec.3.2, we verify some key results above by considering multipath Rayleigh fading and simple Doppler frequency shift. Finally, Sec.3.4gives a summary.

3.1

Derivation of Autocorrelation of Residual

ICI

Assume a signal detector (equalizer) able to handle 2K terms of nearest-neighbor ICI. We may partition the summation over k in (1.2) into an in-band and an out-of-band term as Ym = m+K_X k=m−K L−1 X l=0 H_l(m−k)e−j2πlk/N_X k+ X k /∈[m−K,m+K] L−1 X l=0 H_l(m−k)e−j2πlk/N_X k | {z } ,cm,K +Wm, (3.1) where cm,K is the out-of-band term, i.e., residual ICI. Alternatively, using the

nota-tion of (1.6), Ym = m+K_X k=m−K am,kXk+ cm,K + Wm (3.2) where cm,K = X k /∈[m−K,m+K] am,kXk. (3.3)

For large enough N, the residual ICI may be modeled as Gaussian by the central limit theorem.

It turns out that the analysis can be more conveniently carried out by way of the frequency spreading functions of the propagation paths than by way of am,k.

Hence consider (3.1). From it, the autocorrelation of cm,K at lag r is given by

E[cm,Kc∗m+r,K] = Es× X k /∈[m−K,m+K] ∪[m+r−K,m+r+K] L−1 X l=0 E[H_l(m−k)H_l(m+r−k)∗] = Es× X k /∈[−K,K]∪[−K−r,K−r] L−1 X l=0 E[H_l(k)H_l(k+r)∗] (3.4) where Es is the average transmitted symbol energy and we have assumed that Xk

is white. Invoking (1.3) and (1.7), we get E[cm,Kc∗m+r,K] = Es N2 L−1 X l=0 N −1 X n=0 N −1 X n′₌₀ X k /∈[−K,+K] ∪[−K−r,K−r] σ2_lrl(n − n′)ej2π[n ′_{(k+r)−nk]/N} . (3.5)

We show in the Sec. 3.3 that E[cm,Kc∗m+r,K] ≈ 4π2Tsa2Es L−1 X l=0 σ_l2σD2l ! ρ(K, r, N) (3.6)

where σD2l is the mean-square Doppler spread of path l given by σD2l =

Rfd −fdPl(f )f 2_{df and} ρ(K, r, N) = X k /∈[−K,K]∪[−K−r,K−r] 1 (1 − e−j2πk/N_{)(1 − e}j2π(k+r)/N₎. (3.7) Note that ρ(K, r, N) = X k∈[0,N −1]\{0,−r} 1 (1 − e−j2πk/N_{)(1 − e}j2π(k+r)/N₎ | {z } ,ρ0(r,N ) − X k∈[−K,K]∪[−K−r,K−r]\{0,−r} 1 (1 − e−j2πk/N_{)(1 − e}j2π(k+r)/N₎ | {z } ,ρ1(K,r,N ) ,(3.8)

where the exclusion of 0 and −r from both ranges of summation is to skip over the points of singularity where the summands are null anyway. Note further that −1/(1 − e−j2πk/N_{) and −1/(1 − e}−j2π(k+r)/N_{) (as sequences in k) are the DFTs of}

[n − (N − 1)/2]/N and e−j2πrn/N_{[n − (N − 1)/2]/N (as sequences in n), respectively.}

Hence, with Parseval’s theorem we get

ρ0(r, N) = 1 N N −1_X n=0  n −N − 1 2 2 ej2πrn/N =    N2₋₁ 12 , r = 0, −2 (1−ej2πr/N₎2, r 6= 0. (3.9) For ρ1(K, r, N), we have ρ1(K, r, N) = ρ∗1(K, −r, N), (3.10)

i.e., it is conjugate symmetric in r. Moreover, the summands in the last summation in (3.8) are symmetric over the range of summation. But the range of summation does not allow us to obtain a compact expression for ρ1(K, r, N) as that for ρ0(r, N).

As mentioned, the proposed receiver will whiten the residual I+N before equal-ization. Here we make some observations of the properties of the normalized autocor-relation of residual ICI, i.e., E[cm,Kc∗m+r,K]/E[|cm,K|2], that are relevant to whitener

depends only on K and N through ρ(K, r, N); the other factors cancel out. Thus this normalized autocorrelation is independent of the average transmitted symbol energy Es and the sample period Tsa. More interestingly, it is also independent of

the power-delay profile (PDP) of the channel (i.e., σ2

l vs. l) and the Doppler PSD

Pl(f ) of each path. While the independence of the normalized autocorrelation on the

average transmitted symbol energy may be intuitively expected, its independence of the sample period, the PDP, and the Doppler PSDs of channel paths appears somewhat surprising.

Moreover, the normalized autocorrelation is also substantially independent of the DFT size N. To see this, note that for complexity reason, in a practical receiver both the whitener and the equalizer are likely short. A short equalizer implies a small K and a short whitener implies a small range of r over which the normalized autocorrelation needs to be computed. Hence, when N is large, the exponential functions in the above summations for ρ0(r, N) and ρ1(K, r, N) can all be well

ap-proximated with the first two terms of their respective power series expansion (i.e., ex _{≈ 1 + x when |x| ≪ 1). As a result, we have}

ρ(K, r, N) = ρ0(r, N) − ρ1(K, r, N) (3.11) where ρ0(r, N) ≈    N2 12, r = 0, N2 2π2_r2, r 6= 0, (3.12) ρ1(K, r, N) ≈ X k∈[−K,K]∪[−K−r,K−r]\{0,−r} N2 4π2_{k(k + r)}. (3.13)

Thus the normalized autocorrelation, being essentially given by ρ(K, r, N)/ρ(K, 0, N), is substantially independent of the DFT size N.

The rules are given below. Property 1

Assume a receiver partition the frequency channel matrix with band width K into an in-band and an out-of-band term and is out-of-band term at m-th subcarrier.

The normalized autocorrelation of cm,K at lag r is given by E[cm,Kc∗m+r,K] E[|cm,K|2] ≈ 1/r2₋ P k∈[−K,K]∪[−K−r,K−r]\{0,−r} 1/2k(k + r) π2_{/6 −} PK k=1 1/k2 , (3.14)

that will approximate to a constant on condition that K, r are given.

Although the above observations concern ICI only, it is straightforward to extend them to the sum of ICI and AWGN channel noise.

Property 2

The normalized autocorrelation of Zm (i.e. cm,K + Wm) at lag r is given by

E[ZmZm+r∗] E[|Zm|2] ≈ 1/r2₋ P k∈[−K,K]∪[−K−r,K−r]\{0,−r} 1/2k(k + r) π2_{/6 −} PK k=1 1/k2 1 1 + E[|Wm|2] E[|cm,K|2] (3.15)

that will approximate to a function only depends on E[|Wm|2]

E[|cm,K|2] with K, r given.

In particular, the resulting whitening filter and its performance can also disregard a variety of system parameters and channel conditions, including the DFT size, the sample period, the system bandwidth (which is approximately proportional to the inverse of the sample period), the OFDM symbol period NTsa, the channel PDP,

and the Doppler PSDs of the channel paths. They only depend on the ICI-to-noise power ratio (INR) at the receiver.

As a result, a whitener parameterized on receiver INR can be designed for all operating conditions, which is advantageous for practical system implementation. (The estimation of ICI and noise powers is outside the scope of the present work. Some applicable methods have been proposed in the literature, e.g., [11] for ICI power and [12] for noise power.)

The whitener performance can be understood to a substantial extent by examin-ing the above approximation to the normalized autocorrelation E[cm,Kc∗m+r,K]/E[|cm,K|2].

We leave a detailed study along this vein to potential future work. For now, we shall be content with a first-order understanding by a look at its value at lag r = 1. A

large value indicates that whitening can effectively lower the residual ICI. For this, we see from the above approximation (after some straightforward algebra) that

E[cm,Kc∗m+1,K] E[|cm,K|2] ≈ ρ(K, 1, N) ρ(K, 0, N) ≈ 1 −PK_k=11/[k(k + 1)] π2_{/6 −}PK k=11/k2 = 1/(K + 1) π2_{/6 −}PK k=11/k2 . (3.16) For example, its values for K = 0–3 are, respectively, 0.6079, 0.7753, 0.8440, and 0.8808, which are substantial indeed.

As a side remark that will be of use later, we note the following properties from (3.6) and (3.11).

Property 3

The total ICI power E[|cm,0|2] can be approximated as

σ_c02 , E[|cm,0|2] ≈ 4π2Tsa2Es L−1 X l=0 σ_l2σD2l ! ρ(0, 0, N) ≈ Es 12(2πTsaN) 2 L−1 X l=0 σ2_lσD2l ! , (3.17)

which is in essence the upper bound derived in [11]. Moreover, we have an approxi-mation to the partial ICI power beyond the 2K central terms.

Property 4

The total ICI power E[|cm,K|2] can be approximated as

σ_cK2 , E[|cm,K|2] ≈ 4π2Tsa2Es L−1 X l=0 σ_l2σD2l ! ρ(K, 0, N) ≈ σ2_c0 1 − 6 π2 K X k=1 1 k2 ! . (3.18)

In the following section, we provide some numerical examples to verify the above results on ICI correlation. Then, in the next section, we consider how to incorporate a whitener for residual ICI plus noise in the receiver.

3.2

Numerical Examples

In this section, we verify some key results above by considering two very different channel conditions: multipath Rayleigh fading and simple Doppler frequency shift.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Peak Doppler Frequency fd (Hz)

E [cm ,K c ∗ m + r ,K ]/ E [| cm ,K | 2 ] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32 Normalized Peak Doppler Frequency (fdTsaN)

r=1, theory r=1, simul. r=2, theory r=2, simul.

Figure 3.1: Normalized autocorrelation of residual ICI over multipath Rayleigh fad-ing channel at K = 0, with N = 128 and Tsa = 714 ns. The first-order

approxi-mation (3.11)–(3.13) yields 0.6079 for r = 1 and 0.1520 for r = 2, which are quite accurate at low fd values.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Peak Doppler Frequency fd (Hz)

E [cm ,K c ∗ m + r ,K ]/ E [| cm ,K | 2 ] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32 Normalized Peak Doppler Frequency (fdTsaN)

r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul. r=6, theory r=6, simul.

Figure 3.2: Normalized autocorrelation of residual ICI over multipath Rayleigh fad-ing channel at K = 1, with N = 128 and Tsa = 714 ns. The first-order

approxi-mation (3.11)–(3.13) yields 0.7753, 0.6461, 0.5599, 0.3036, 0.1912, and 0.1317, for r = 1–6, respectively, which are quite accurate.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Peak Doppler Frequency fd (Hz)

E [cm ,K c ∗ m + r ,K ]/ E [| cm ,K | 2 ] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32 Normalized Peak Doppler Frequency (fdTsaN)

r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul.

Figure 3.3: Normalized autocorrelation of residual ICI over multipath Rayleigh fad-ing channel at K = 2, with N = 128 and Tsa = 714 ns. The first-order

approxi-mation (3.11)–(3.13) yields 0.8440, 0.7358, 0.6612, 0.6014, and 0.5534, for r = 1–5, respectively, which are quite accurate.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Peak Doppler Frequency fd (Hz)

E [cm ,K c ∗ m + r ,K ]/ E [| cm ,K | 2 ] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32

Normalized Peak Doppler Frequency (fdTsaN)

r=1, theory r=1, simul. r=2, theory r=2, simul.

Figure 3.4: Normalized autocorrelation of residual ICI over one-Doppler-line channel at K = 0, with N = 128 and Tsa= 714 ns.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Peak Doppler Frequency fd (Hz)

E [cm ,K c ∗ m + r ,K ]/ E [| cm ,K | 2 ] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32 Normalized Peak Doppler Frequency (fdTsaN)

r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul. r=6, theory r=6, simul.

Figure 3.5: Normalized autocorrelation of residual ICI over one-Doppler-line channel at K = 1, with N = 128 and Tsa= 714 ns.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Peak Doppler Frequency fd (Hz)

E [cm ,K c ∗ m + r ,K ]/ E [| cm ,K | 2 ] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32

Normalized Peak Doppler Frequency (fdTsaN)

r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul.

Figure 3.6: Normalized autocorrelation of residual ICI over one-Doppler-line channel at K = 2, with N = 128 and Tsa= 714 ns.

First, consider a multipath channel having the COST 207 6-tap Typical Urban (TU6) PDP as shown in Table4.1 [13, p. 94]. Let the paths be subject to Rayleigh fading with the same peak Doppler frequency fd, so that rl(q) = J0(2πfdTsaq) for

all l, where J0(·) denotes the zeroth-order Bessel function of the first kind [10].

Let the OFDM system have N = 128, subcarrier spacing fs = 10.94 kHz, and

sampling period Tsa = 1/(Nfs) = 714 ns, which are some of the Mobile WiMAX

parameters [1].

Figs. 3.1–3.3 illustrate the normalized autocorrelation of the residual ICI for K = 0–2, respectively, where the theoretical values are calculated using (3.5). As points of reference, note that a peak Doppler frequency of 1 kHz corresponds to a 180 km/h mobile speed at a 6 GHz carrier frequency, or a 540 km/h mobile speed at a 2 GHz carrier frequency. Figs.3.1–3.3 show that the theory and the simulation results agree well up to very large Doppler spreads. In addition, they also show that, for given lag r, the normalized autocorrelation increases with K. The last fact can be understood by examining (3.3): as K increases, the residual ICI cm,K is

composed of the sum of increasingly fewer terms with generally smaller magnitudes, which naturally leads to higher normalized autocorrelation.

Next, consider a channel with a one-line Doppler PSD equal to δ(f − fd); in

other words, the channel simply effects a frequency offset of fd. The temporal

autocorrelation of the CIR is given by rl(q) = exp(j2πfdTsaq). It turns out that the

normalized autocorrelation of residual ICI is very similar to that obtained for the previous example, as the theory predicts. Figs.3.4–3.6 illustrate the corresponding normalized autocorrelation of the residual ICI for K = 0–2, respectively. They are very similar to Figs. 3.1–3.3, as the theory predicts.

Looking backwards from the one-Doppler-line example to the earlier analysis in this Section 3.1, we find that this example also provides an alternative way of interpreting the earlier analytical results. Specifically, an arbitrary Doppler PSD can be considered as composed of a (possibly infinite) number of line PSDs. Hence the autocorrelation of residual ICI associated with an arbitrary Doppler PSD may be obtained as a linear combination of the autocorrelation associated with a line

PSD as

E[cm,Kc∗m+r,K]|any shape= L−1 X l=0 σ_l2 Z fd −fd Pl(f )E[cm,Kc∗m+r,K]|line,fdf (3.19)

where E[cm,Kc∗m+r,K]|any shape denotes the autocorrelation of residual ICI associated

with a multipath channel of arbitrary Doppler PSD and E[cm,Kc∗m+r,K]|line,f that

associated with a line Doppler PSD corresponding to a Doppler frequency f . As we have verified now (through Figs.3.4–3.6, for example) that

E[cm,Kc∗m+r,K]|line,fd

E[cm,Kc∗m,K]|line,fd

≈ ρ(K, r, N)

ρ(K, 0, N), (3.20)

substituting it into (3.19) yields E[cm,Kc∗m+r,K]|any shape ≈

ρ(K, r, N) ρ(K, 0, N) × L−1 X l=0 σ_l2 Z fd −fd Pl(f )E[cm,Kc∗m,K]|line,fdf = ρ(K, r, N) ρ(K, 0, N) × E[cm,Kc ∗ m,K]|any shape. (3.21)

In other words, since the single-Doppler-line channel shows substantial invariance of the normalized residual ICI autocorrelation over a large range of operating conditions (as we have seen in the last example), it follows that a channel with any Doppler PSD has a similar property.

In summary, we have confirmed that the normalized autocorrelation of the residual ICI is quite insensitive to various system parameters and channel condi-tions. To lower the error floor, therefore, a whitening filter for the residual ICI plus noise can be designed without regard to these system parameters and channel con-ditions. Such a fixed design can lead to low implementation complexity and robust performance.

3.3

Derivation of (

3.6

) and Some Related

Comments

Substituting the inverse Fourier transform relation in (1.8) into the right-hand side (RHS) of (3.5), we get E[cm,Kc∗m+r,K] = Es N2 L−1 X l=0 N −1_X n=0 N −1_X n′₌₀ X k /∈[−K,+K]∪[−K−r,K−r] σ_l2 · Z fd −fd Pl(f ){cos[2πf Tsa(n − n′)] + j sin[2πf Tsa(n − n′)]}df · ej2π[n ′_{(k+r)−nk]/N} . (3.22) Let ξ denote the quantity that collects all the terms associated with sin[2πf Tsa(n −

n′_{)]. That is,} ξ = Es N2 L−1 X l=0 σ_l2 Z fd −fd df Pl(f ) N −1_X n=0 N −1_X n′₌₀ X k /∈[−K,+K] ∪[−K−r,K−r] j sin[2πf Tsa(n−n′)]ej2π[n ′_{(k+r)−nk]/N} . (3.23) Consider the inner triple sum and denote it by χ. By substituting the variables n, n′_{, and k with ν}′_{, ν, and −(κ + r), respectively, we get, after some straightforward}

algebra, χ = N −1_X ν=0 N −1_X ν′₌₀ X κ /∈[−K,+K]∪[−K−r,K−r] −j sin[2πf Tsa(ν − ν′)]ej2π[ν ′_{(κ+r)−νκ]/N} . (3.24)

A comparison with the inner triple sum in (3.23) shows that χ = −χ, which implies χ = 0 and thus ξ = 0. Therefore, only the cosine terms remain in E[cm,Kc∗m+r,K].

only up to the second-order term as cos x ≈ 1 − x2_{/2, we get} E[cm,Kc∗m+r,K] ≈ Es N2 L−1 X l=0 σ2 l Z fd −fd Pl(f )df X k /∈[−K,K]∪[−K−r,K−r] N −1_X n=0 e−j2πnk/N | {z } =0 N −1_X n′₌₀ ej2πn′_(k+r)/N | {z } =0 − Es 2N2 L−1 X l=0 σ_l2 Z fd −fd Pl(f )(2πf Tsa)2df X k /∈[−K,K] ∪[−K−r,K−r]      N −1_X n=0 n2e−j2πnk/N N −1_X n′₌₀ ej2πn′(k+r)/N | {z } =0 + N −1_X n=0 e−j2πnk/N | {z } =0 N −1_X n′₌₀ n′2ej2πn′(k+r)/N − 2 N −1_X n=0 ne−j2πnk/N N −1_X n′₌₀ n′ej2πn′(k+r)/N      = 4π2T_sa2Es L−1 X l=0 σ_l2 Z fd −fd Pl(f )f2df X k /∈[−K,K] ∪[−K−r,K−r] 1 (1 − e−j2πk/N_{)(1 − e}j2π(k+r)/N₎. (3.25)

In fact, the above second-order approximation to cosine function is tantamount to assuming linearly time-varying paths in the CIR. To see it, let hl(t) denote the

continuous-time waveform of the lth path of the CIR (of which hn,l is a sampled

version) and let h′

l(t) be its time-derivative. Then by a well-known relation between

the time-derivative of a stochastic process and its PSD, we have 4π2σ_l2R Pl(f )f2df =

E [|h′

l(t)|2] [20, Table 7.5-1]. Therefore, if we approximate the channel by one whose

lth path response varies linearly with time in some period with its slope equal to h

|h′ l(t)|2

i1/2

in magnitude (where the overline in the brackets denotes time average over this period), then the autocorrelation of residual ICI of the approximating channel would be exactly that obtained above, without approximation. In this sense, the second-order approximation to cosine function above is tantamount to assuming linearly time-varying paths in the CIR.

Numerical examples in Section 3.1 show that the ensuing approximation to the autocorrelation of the residual ICI is rather accurate even under a relatively large peak Doppler shift.

3.4

Summary of Results

We found that, in a mobile time-varying channel, the residual ICI beyond several dominant terms had high normalized autocorrelation. We derived a rather precise closed-form approximation for the (unnormalized) autocorrelation function. It turns out that, up to a rather high peak Doppler frequency, the normalized autocorrelation was not sensitive to a variety of system parameters and channel conditions, includ-ing the DFT size, the sample period, the system bandwidth, the OFDM symbol period, the average transmitted symbol energy, the multipath channel profile, and the Doppler PSDs of the channel paths. As a result, a whitening transform for the residual ICI plus noise can be obtained based solely on the ICI-to-noise ratio. Such a transform can be used in association with many different signal detection schemes to significantly improve the detection performance. That it depends only on the ICI-to-noise ratio but no other quantities also implies simplicity and robustness.

Chapter 4

MLSE Detection with Whitening

of Residual ICI Plus Noise

In Sec. 4.1, we considered MLSE-type signal detection with blockwise whitening of the residual ICI plus noise. Simulations showed that the proposed technique could lower the ICI induced error floor by several orders of magnitude in MLSE that addressed a few dominant ICI terms.

To capitalize on the above high correlation to improve signal reception over fast varying channels, In Sec.4.4, we consider performing simple blockwise whitening of the residual I+N before signal detection (i.e., equalization), where the whitener makes use of the ICI characteristics as found. Numerical analysis of SINR also confirms that substantial gains can be achieved with this approach. The chapter is organized as follows.

The remainder of this chapter is organized as follows. Sec. 4.2, we presents complexity analysis of proposed method. Sec. 4.3 presents some simulation results on signal detection performance. Sec.4.4explores how signal detection performance depends on whitener parameter setting. Finally, Sec. 4.5 gives a summary.

4.1

MLSE Detection with Whitening of Residual

ICI Plus Noise

As indicated, we propose to whiten the residual ICI plus noise in signal detection. This can be applied to many detection methods, including MMSE, iterative MMSE, decision-feedback equalization (DFE), MLSE, etc., providing a wide range of tradeoff between complexity and performance. In this chapter, we consider an MLSE-based technique both to illustrate how such whitening can be carried out and to demon-strate its benefit. For simplicity, rather than performing whitening over a complete sequence, we do blockwise whitening over windows of size 2q + 1 where q may or may not be equal to K. The details are as follows.

Consider a vector of 2q + 1 frequency-domain signal samples centered at sample m:

ym = [Ym−q · · · Ym · · · Ym+q]′ = Hmxm+ zm (4.1)

where xm = [Xm−p · · · Xm · · · Xm+p]′ for some integer p, Hm is a (2q + 1) ×

(2p + 1) submatrix of H of bandwidth K, and zm collects all the right-hand-side

(RHS) terms in (1.2) (or (1.4)) associated with Yk, m − q ≤ k ≤ m + q, that do

not appear in Hmxm. The elements of zm include both residual ICI and channel

noise. To avoid clogging the mathematical expressions with details, we have omitted explicit indexing of various quantities in (4.1) with the parameters K, p, and q, understanding that their dimensions and contents depend on these parameters. As an example, with the set of parameters {K = 1, q = 1, p = 2} we have

Hm =      am−1,m−2 am−1,m−1 am−1,m 0 0 0 am,m−1 am,m am,m+1 0 0 0 am+1,m am+1,m+1 am+1,m+2      (4.2) whereas with {K = 1, q = 1, p = 1}, Hm =      am−1,m−1 am−1,m 0 am,m−1 am,m am,m+1 0 am+1,m am+1,m+1     . (4.3)

Let Kz = E[zmzHm], i.e., the covariance matrix of zm, where superscript H

stands for Hermitian transpose. The aforesaid blockwise whitening of residual ICI plus noise zm is given by

e ym , K −1₂ z ym = K −1₂ z Hm | {z } , eHm xm+ K −1₂ z zm | {z } ,ezm (4.4) where K−12

z may be defined in more than one way. One choice is to let K −1₂ z =

UΛ−12UH where U is the matrix of orthonormal eigenvectors of K_z and Λ is the

diagonal matrix of corresponding eigenvalues of Kz.

If block-by-block signal detection were desired, then the ML criterion would result in the detection rulex_bm = arg minxmkyem− eHmxmk2. As stated, we consider

MLSE-based detection in this chapter.

In developing the MLSE-based detection method, we treat_ezm, m = 0, . . . , N −1,

as if they were mutually independent, even though this may at best be only nearly so. Then the probability density function of the received sequence conditioned on the transmitted sequence would be

f (y_e0,ey1, . . . ,eyN −1|x0, x1, . . . , xN −1) = f (ez0,ez1, . . . ,ezN −1) = N −1_Y

n=0

f (_ezn). (4.5)

As a result, the recursive progression of the log-likelihood values, i.e.,

Λk , log f(ez0,ez1, . . . ,ezk) = Λk−1+ log f (yek− eHkxk), k = 1, . . . , N − 1, (4.6)

leads to a standard Viterbi algorithm. Disregarding some common terms that do not affect sequence detection, in the Viterbi algorithm we may use k_eyk − eHkxkk2

as the branch metric instead of log f (y_ek − eHkxk). Fig. 4.1 illustrates the trellis

structure of the MLSE detector for p = 1 under QPSK modulation. A tradeoff between complexity and performance can be achieved by different choices of the three parameters K, q, and p, where p determines the number of states in each trellis stage and the three parameters jointly affect the branch metric structure in the trellis and the autocorrelation structure of the residual ICI (and thereby the whitener behavior).

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−

y

H

x

Э

Э

−

y

H x

Э

Э

  

  

  

  

  

  

  

  

  

Figure 4.1: Trellis structure for MLSE-based detection using the Viterbi algorithm, under QPSK modulation and with p = 1, where numerals 0–3 represent the QPSK constellation points.

Table 4.1: Two Channel Power-Delay Profiles Used in This Study, Where TU6 Corresponds to the COST 207 6-Tap Typical Urban Channel And SUI4 the SUI-4 3-Tap Channel Tap Index 1 2 3 4 5 6 TU6 Delay (µs) 0.0 0.2 0.5 1.6 2.3 5.0 Power (%) 19 38 24 9 6 4 Tap Index 1 2 3 – – – SUI4 Delay (µs) 0.0 1.5 4.0 – – – Power (%) 64 26 10 – – –

4.2

Complexity Analysis

Concerning complexity, let NA denote the signal constellation size at each

subcar-rier. Then, for each subcarrier, the nonwhitening MLSE requires O[(2K + 1)N_A2K+1] complex multiplications and additions (CMAs) to build the trellis and O(N_A2K+1) CMAs to conduct the Viterbi search [4]. In contrast, the proposed method requires O[(2K + 1)N_A2p+1 + (2q + 1)2_N2p+1

A ] CMAs to build the trellis, wherein O[(2K +

1)N_A2p+1] are for computing Hmxm and O[(2q + 1)2N_A2p+1] are for multiplying with

K−12

z . Then the Viterbi search requires O[(2q +1)N_A2p+1] CMAs. The computation of

K−12

z requires estimation of the ICI power and the AWGN power, but the complexity

is far lower than building the trellis or performing the Viterbi search and is thus neglected. From the above, the proposed method may seem to require much higher complexity than nonwhitened MLSE. But, to the contrary, the reduced residual I+N through whitening may facilitate using a smaller ICI bandwidth K in the MLSE, culminating in a complexity gain rather than loss. This will be demonstrated in the simulation results below.

4.3

Simulation Results on Detection

Performance

We present some simulation results on signal detection performance in this section. As in Sec.3.2, we let subcarrier spacing fs = 10.94 kHz and sample period Tsa= 714

ns. The subcarriers are QPSK-modulated with Gray-coded bit-to-symbol mapping. There is no channel coding. The channels are multipath Rayleigh-faded WSSUS channels having the PDPs shown in Table4.1.

Unless otherwise noted, we let N = 128 and assume that the receiver has perfect knowledge of the channel state information (CSI), which includes the channel matrix within band K and the covariance matrix Kz of the residual ICI plus noise.

To start, consider the extreme case of K = 0 in absence of channel noise. Through this we look at the limit imposed by the ICI to the performance of the conventional detection method. We also look at the possible gain from blockwise whitening of the full ICI followed by MLSE with p = q = 1, at infinite signal-to-noise ratio (SNR). The ICI covariance matrix in this case is given by

Kz =      1 0.6 0.15 0.6 1 0.6 0.15 0.6 1     σ 2 c0 (4.7)

where recall that σ2

c0 = E[|cm,0|2] is the total ICI power. Fig. 4.2 shows some

simu-lation results for the TU6 channel. The numerical performance for the SUI4 channel is very similar. These results show that ICI-whitening detection (the proposed tech-nique) yields some advantage over conventional detection: the error probability is reduced by about 2.2 times.

Significantly higher gain can be obtained by ICI-whitening MLSE with K = 1. In Fig. 4.3 we compare the corresponding performance of the proposed technique with that of MLSE which treats the residual ICI as white [4], over TU6 and SUI4 channels in the noise-free condition (i.e., SNR = ∞). For the proposed technique, two parameter settings are considered, viz. {q = 1, p = 2} and {q = 1, p = 1}, for

which the covariance matrices Kz of residual ICI are given by, respectively,      1 0.775 0.645 0.775 1 0.775 0.645 0.775 1     σ 2 c1,      1.785 1.16 1.16 1.16 1 1.16 1.16 1.16 1.785     σ 2 c1, (4.8)

where recall that σ2

cK = E[|cm,K|2] is the residual ICI power outside band K.

Consider the case p = q = 1 first. In this case, the proposed method shows a remarkable gain of roughly three to four orders of magnitude in error performance compared to treating residual ICI as white. The error floor induced by the residual ICI can be driven to below 10−5 _{even at the very high normalized peak Doppler}

frequency of 0.32.

Very interestingly, Fig. 4.3 also shows that the setting {q = 1, p = 2} yields a worse performance than p = q = 1, even though the former setting may seem more natural in its associated band channel matrix structure (compare (4.2) with (4.3)), which captures all the ICI terms within the modeling range (K = 1). Moreover, its corresponding trellis has more states than the latter setting (45 _{vs. 4}3_{). The reason}

will be explored in the next section. For now, we note that the above results appear to indicate the suitability of setting p = q = K = 1 in practical system design. It yields good performance without undue complexity. With this observation, we now present some more simulation results under this setting. The aims are to examine the proposed technique’s performance at finite SNR and to compare it with a benchmarking upper bound. For this, we first consider how it varies with Doppler spread and then how it varies with SNR.

Fig. 4.4 shows some results for the TU6 channel with p = q = K = 1 at several SNR values. The results for SUI4 show similar characteristics and are omitted. We compare the performance of the proposed method with a benchmark: the matched-filter bound (MFB), i.e., signal detection with perfect knowledge of the interfering symbols. To make the MFB a more-or-less absolute lower bound, it is obtained with the residual ICI outside band K fully cancelled. Other than these, the same MLSE as in the proposed technique is used. For all three finite SNR values shown, note that the MFB drops monotonically with increasing fd, i.e., with increasing time-variation

of the channel. This is in line with the fact that faster channel variation yields greater time diversity, as various researchers have observed [15,17,18]. However, such time diversity can show clearly only when ICI is sufficiently small (e.g., after ICI cancellation). For the proposed technique, its error performance at Eb/N0 = 15

and 28 dB tracks that of the MFB reasonably closely, deviating by less than a multiplicative factor of three for normalized peak Doppler frequencies up to 0.18 (fd ≤ 2000 Hz). At Eb/N0 = 45 dB, the performance improves with fd until fd

reaches about 1500 Hz (normalized peak Doppler frequency ≈ 0.14). Afterwards, the residual ICI dominates in determining the performance, as can be seen by the closeness between the corresponding curves for Eb/N0 = 45 dB and ∞.

Next, consider how the performance of the proposed method varies with SNR. The solid lines in Fig. 4.5 show results at fd = 1500 Hz (normalized peak Doppler

frequency ≈ 0.14) under perfect CSI. It is seen that the proposed method at K = 1 can yield a substantial performance gain compared to nonwhitening MLSE [4] at K = 2. The dash-dot lines in Fig. 4.5 depict some results under imperfect CSI. Limited by space, we cannot elaborate on the many possible channel estimation methods and their performance. Hence the results shown pertain to a typical con-dition only. For this, we note that the mean-square channel estimation error is typically proportional to the variance of the unestimatable channel disturbance, with the proportionality constant inversely dependent on the sophistication of the channel estimation method [19]. In our case, the unestimatable channel distur-bance includes residual ICI (mostly that beyond K = 1) and additive channel noise (AWGN). At a normalized peak Doppler frequency of 0.14 (fd = 1500 Hz), the first

term is approximately 20 dB below the received signal power. The proportionality constant is set to 1/8. The channel estimation error limits the performance of all detection methods and the residual ICI-free bound in the form of error floors. The floor of the proposed method at K = 1 is seen to be lower than that of nonwhitening MLSE at K = 2 and is relatively close to the bound. We further note that, while Fig.4.5has been obtained with N = 128, the results obtained with N = 1024 (eight times the bandwidth) are very close.

200 400 600 800 1000 1200 1400 1600 10−4

10−3 10−2 10−1

Peak Doppler Frequency fd (Hz)

B it E rr o r R a te 0.0183 0.0366 0.0549 0.0731 0.0914 0.1097 0.128 0.1463 Normalized Peak Doppler Frequency (fdTsaN)

Proposed method, K=0

Conventional OFDM detection method

Figure 4.2: Error performance in TU6 channel of the conventional OFDM signal detection method and ICI-whitening MLSE (the proposed method) with K = 0 and p = q = 1 in noise-free condition.