Chapter 1 Background and Motivation
1.3 Thesis Organization
This dissertation is organized as follows. In Chapter 2, we give an extensive review of the doubly selective fading channels including the statistical characterization, discrete time model and generating approaches for computer simulations. With the develop-ment of system model of OFDM systems over doubly selective fading channels, a novel ICI indicator is introduced and the statistical analysis as well as possible applications of the ICI indicator are given. Chapter 3 discusses the ICI cancellation and introduces the PSA framework. The PSA PB-ZF and PB-MMSE ICI equalizers that consider the standard constraints and are suitable for implementation are presented. In Chapter 4, the asymptotic and practical diversity orders of BICM-OFDM systems over doubly-selective fading channels are theoretically analyzed. The analysis can be extend to MIMO cases and two techniques, CDD and PRD, are discussed. Finally, we consider possible directions of future works and conclude the dissertation in Chapter 5.
Chapter 2
Doubly Selective Fading Channels
Wireless communication often encounters environments with rich multipaths because of the atmospheric scattering and refraction, or reflections of surrounding objects. Trav-eling through these paths, the signals arrived at the receiver with random delays and attenuations will be added constructively or destructively and result in envelope fluc-tuations significantly, which is referred to fading. Furthermore, in high data rate wide-band systems, e.g., OFDM, the wide-bandwidth of the transmitted signal is larger than the coherence bandwidth of the fading channel, giving rise to frequency-selectivity of fading channels. It becomes clear that emerging mobile applications will experience channel variations within one symbol time where the channel is said to be time-selective. This is mainly due to the changing atmospheric conditions and relative movements between transmitters and receivers. Consequently, the wireless channel of mobile communi-cations is characterized as a time- and frequency-selective (or doubly-selective) fading channel. As these selectivities affect the system performance critically, the understand-ing and modelunderstand-ing of doubly-selective fadunderstand-ing channels are important tasks for devisunderstand-ing countermeasures.
A precise mathematical description of fading channels, i.e., the physical modeling based on electromagnetic radiation [32], for practical scenarios is either unknown or too complex to be tractable. However, considerable efforts have been devoted to the statistical modeling. Moreover, statistical descriptions could provide insights in many typical issues, e.g., the proper packet duration to avoid fades, the relative severity of experienced ISI/ICI, the appropriate subcarrier spacing for frequency diversity, the
necessary interleaving depth for time diversity, expected BER, etc..
In this chapter, we first consider the statistical characterization of doubly-selective fading channels. Then, the discrete-time channel model will be developed and channel simulators for computer simulations will be introduced. As OFDM has become the de facto transmission scheme in modern wireless communications, we consider the system model for OFDM over doubly-selective fading channels where ICI exists. Based on empirical observations, a measure called the ICI indicator was introduced to indicate the ICI level as well as to estimate the maximum Doppler frequency on each subcar-rier. We also provide a thorough analysis of the ICI indicator to reveal why it works.
Collectively, its PDF also provides valuable information on, for example, the maximum Doppler spread. Some applications of the ICI indicator are given.
2.1 Baseband Equivalent Representation and Sta-tistical Characterization
Assume there exist multiple propagation paths, αn(t) is the attenuation factor for the signal received on the n-th path, and τn(t) is the propagation delay of the n-th path.
The bandpass received signal through discrete multipath channels in the absence of noise can be expressed in the form
y(t) =X
n
αn(t)x(t − τn(t)) (2.1)
where x(t) is the bandpass transmit signal.
Using the complex envelope and expressing x(t) as Re[xbb(t)ej2πfct] where xbb(t) is the baseband equivalent transmit signal and fc is the central carrier frequency, we can carry out analysis and simulation of carrier-modulated (bandpass) signals and systems
at baseband. Equation (2.1) can be re-written as
y(t) = Re[{X
n
αn(t)e−j2πfcτn(t)xbb(t − τn(t))}ej2πfct]. (2.2)
From the above expression it is straightforward to write the baseband equivalent (or lowpass equivalent) signal as
ybb(t) =X
n
αn(t)e−j2πfcτn(t)xbb(t − τn(t)). (2.3)
By transmitting a conceptually ideal impulse, the complex baseband equivalent time-varying channel impulse response (CIR) is obtained as
c(t; τ ) =X
where δ(·) denotes the Dirac delta function.
For another type of channel model, the diffuse multipath channel, the signal is composed of a continuum of unresolvable components and is expressed in the integral form
y(t) = Z ∞
−∞
α(t; τ )x(t − τ )dτ (2.5)
where α(t; τ ) denotes the attenuation of the signal at delay τ and at time instant t.
Following the similar procedures as in the case of discrete multipath channels, the CIR can be expressed by
c(t; τ ) = α(t; τ )e−j2πfcτ. (2.6)
In Equation (2.4), the channel fading is described by time variations in the mag-nitudes and phases of c(t; τ ). These variations appear to be random and usually are treated as random processes. If there are numerous propagation paths, which usually is the case of wireless channels, we can model the CIR c(t; τ ) as a complex-valued
Gaussian random process by the central limit theorem. When the CIR is zero-mean, the envelope r = |c(t; τ )| is Rayleigh-distributed that has the form [33]
fR(r|σ2) = r
σ2e−r2/(2σ2), r ≥ 0 (2.7) and it is called the Rayleigh fading channel. This kind of simplification often applies to the non-light-of-sight (NLOS) case. When there is a direct-link between the transmitter and the receiver, i.e., the LOS case, the CIR cannot be modeled as zero-mean, and the channel is often modeled by Ricean fading following the PDF [33]
fR(r|ν, σ2) = r
σ2e−(r2+ν2)/(2σ2)I0(rν
σ2), r ≥ 0 (2.8)
where I0(·) is the zero-th order modified Bessel function of the first kind. When ν = 0, which means the power of the LOS path is zero, the distribution reduces to a Rayleigh distribution.
As the basic Rayleigh/Ricean model gives the PDF of the channel envelope, we now consider the question of how fast the signal fades in time and frequency. To answer the question, we need further characterize the CIR by its autocorrelation function and power spectral density (PSD); together they form a Fourier transform pair [33].
2.1.1 Channel Autocorrelation Functions and Power Spectra
Autocorrelation Functions
We assume c(t; τ ) is wide-sense-stationary (WSS) and define the autocorrelation func-tion of c(t; τ ) as
Rc(∆t; τ1, τ2) = E[c(t; τ1)c∗(t + ∆t; τ2)]. (2.9) In most cases, the attenuation and phase shift at the delay τ1 path is uncorrelated with that at τ2, which is known as uncorrelated scattering (US). With the WSSUS condition,
(2.9) can be decoupled into
Rc(∆t; τ1, τ2) = Rc(∆t; τ1)δ(τ2− τ1). (2.10)
Notice that Rc(0; τ ) , Rc(τ ) is called the power delay profile (PDP); the range of τ within which Rc(τ ) is non-zero is termed the maximum delay spread of the channel and is denoted as Tm.
PDP describes the average received power as a function of delay and is one of the most important parameters for channel modeling. We will see later that many indus-trial standards specify PDPs in their testing environments. PDP can be measured by probing the channel with a wideband radio-frequency (RF) waveform that is generated by modulating a high-rate pseudo-noise (PN) sequence. By cross correlating the re-ceiver output against delayed versions of the PN sequence and measuring the average value of the correlator output, one can obtain the power versus delay profile. Just like there are may equally valid definitions of bandwidth, other useful measurements of the delay spread are possible. One of them is the root-mean-square (RMS) delay spread, which is defined by
TRMS =
sR τ2Rc(τ )dτ
R Rc(τ )dτ − (R τ Rc(τ )dτ
R Rc(τ )dτ )2. (2.11)
Now consider channel characterizations in the frequency domain. By taking the Fourier transform of c(t; τ ) w.r.t. the variable τ , the time-variant channel frequency response is
C(t; f ) = Z ∞
−∞
c(t; τ )e−j2πf τdτ. (2.12) Similarly, we can define the autocorrelation function of C(t; f ) as
RC(∆t; f1, f2) = E[C(t; f1)C∗(t + ∆t; f2)]. (2.13)
Relating (2.13) to (2.10), it can be shown that
RC(∆t; f1, f2) = Z ∞
−∞
Rc(∆t; τ1)e−j2π∆f τ1dτ1 , RC(∆t; ∆f )
(2.14)
where ∆f = f2 − f1. Equation (2.14) describes the autocorrelation function in the frequency variable. Moreover, the range of ∆f within which the components of RC(∆f ) are highly correlated is defined as the coherence bandwidth of the channel and denoted as (∆f )c. As Rc(∆t; τ1) and RC(∆t; ∆f ) form a Fourier transform pair, a very rough relation is that the coherence bandwidth is reciprocally proportional to the maximum delay spread [33]
(∆f )c≈ 1
Tm. (2.15)
If the signal bandwidth is large compared to the channel’s coherence bandwidth, the signal will be distorted and the channel is called frequency-selective. This is equivalent to the case where the delay spread is larger than the symbol time, which is also termed time-dispersion because transmitting an ideal impulse through the channel will yield a receive signal with several delayed pulses. In this case, the interference among different symbols occur and called inter-symbol interference (ISI).
Power Spectral Density
Now, we consider the time variations of the channel and investigate the Fourier trans-form pair
SC(λ; ∆f ) = Z ∞
−∞
RC(∆t; ∆f )e−j2πλ∆td∆t. (2.16)
If ∆f is set to 0, SC(λ) is called the channel’s Doppler PSD and λ represents the Doppler frequency. The range of λ within which SC(λ) is non-zero is termed the maximum Doppler spread of the channel and is denoted as fD. The maximum Doppler
Table 2.1: Time and Frequency Dispersion
Time Dispersion Frequency Dispersion Time Domain Delay spread Time selective fading (fast fading)
Interpretation ISI Coherence time
Frequency Domain Frequency selective fading Doppler spread
Interpretation Coherence bandwith ICI
frequency can be roughly calculated by
fD = vfc
c (2.17)
with v being the mobile speed and c the speed of light. Similarly, the range of ∆t within which the components of RC(∆t) are highly correlated is defined as the channel’s coherence time and is denoted as (∆t)c. Again, because they form a Fourier transform pair, the maximum Doppler spread and the coherence time are reciprocally related via
(∆t)c≈ 1
fD. (2.18)
Similarly, if the signal duration is large compared to the channel’s coherence time, the channel is called time-selective. This is equivalent to the case where the Doppler spread is large enough, and a pure-tone transmit signal passing through the channel will yield a receive signal with several frequency components; we call this phenomenon frequency-dispersion. The terminologies and their relationships are summarized in the Table 2.1.
To relate the parameters τ , λ, ∆f , and ∆t, we define the scattering function of the channel
Ssc(λ; τ ) = Z ∞
−∞
RC(∆t; ∆f )e−j2πλ∆tej2πτ ∆fd∆t d∆f , (2.19)
which is the double Fourier transform of RC(∆t; ∆f ).
Jakes’ model is widely adopted to describe the time variation of the mobile radio channels with the the corresponding autocorrelation function
RC(∆t) = J0(2πfD∆t) (2.20)
where J0(·) is the zero-th order Bessel function of the first kind. The Dopper PSD is obtained by Fourier transform, that is
SC(f ) =
Maxmium Delay Spread = 80 ns
RMS Delay Spread = 15.6 ns
Figure 2.1: PDP of IEEE 802.11n channel model B.
Here we give some examples of the statistical channel parameters in practical systems. For the Wireless Local Area Network (WLAN), which is usually used in the
indoor, there are five channel models proposed by the IEEE 802.11 a/n standard [34]
where the RMS delay spreads are about 0 ns to 150 ns depending on the scenarios, and the Doppler power spectrum is Bell-shaped, specified by
SC(f ) = 1 1 + A(ff
D)2. (2.22)
Fig. 2.1 shows the PDP of the IEEE 802.11n channel B whose the maximum delay spread is 80 ns and the RMS delay spread is 15.6 ns. According to (2.15), the co-herence bandwidth is around 12.5 MHz. The channel frequency response is shown in Fig. 2.2 and it can be seen that the signal with 20 MHz bandwidth experience frequency-selectivity while the CFR varies slow within 5 MHz and thus the channel of the signal with 5 MHz bandwidth can be considered as flat-fading. However, since the application is in indoor, the mobility is very small and the channel varies very slowly so we demonstrate the Doppler effect in the following example.
For the wireless Metropolitan Area Network (MAN), e.g., IEEE 802.16e, the Inter-national Telecommunication Union (ITU) channel models are adopted where the RMS delay spreads for outdoor scenarios are around 2 us to 20 us, much longer than that of WLAN scenarios. The Jakes’ U-shape Doppler spectrum is assumed and the typical maximum Doppler spreads at 2.5 GHz carrier frequency are 4.6, 104.2, and 231.5 Hz corresponding to the speed of 2, 45, and 100 km/h, respectively. The corresponding coherence time is 200, 10, and 4 ms, respectively. Fig. 2.3 shows the path gain varia-tions of ITU Vehicular-A channel and it can be seen that higher speeds result in faster channel variations. In IEEE 802.16e, the symbol duration is around 100 µs so that even at 100 km/h the channel still can be considered as linearly time-varying within one symbol.
0 1 2 3 4 5 6 7 8 9 10 x 107 0
1 2 3 4 5 6
frequency (Hz)
Amplitude
5 MHz BW
20 MHz BW
Figure 2.2: Channel frequency response of the PDP described in Fig. 2.1.
Large-scale versus small-scale fading
It is worthwhile to mention that channel fading can also be categorized into large-scale and scale fading [32]. While we have discussed the causes and effects of small-scale fading in the above, the large-small-scale fading is caused by propagation loss over long distance and by shadowing due to obscuring objects that attenuate the received signal strength. Since large-scale fading varies much slower compared to small-scale fading and the induced issues are more related to cell planning and receivers’ sensitivity, in this dissertation we only focus on small-scale fading caused by multipath propagation.
0 1 2 3 4 5 6 7 8 9 10 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
ms
Magnitude
fD = 4.6 Hz fD = 104.2 Hz fD = 231.5 Hz
Figure 2.3: Path gain variations of ITU Vehicular-A channel with speeds at 2, 45, and 100 km/h.
2.2 Discrete Time Channel Model and Simulators
It has been shown that both PDP and Doppler PSD are effective tools to characterize wireless channels, we now consider the problem of how to efficiently generate channels for computer simulations. We focus on Rayleigh fading since the Ricean model can be obtained from the Rayleigh model by adding a non-zero mean.
2.2.1 Discrete Time Model
For most time the input-output relationship can be modeled as a linear time-varying system. The continuous time model often is not suitable for simulations because of
the intensive computational complexity due to very high oversampling. It has been shown that using discrete time representation realized by a tapped delay line model with time-variant tap gains is usually good enough for simulation purposes.
Let pT(t) and pR(t) be the time-invariant impulse response of the transmit filter and the receive filter, respectively, and both are normalized with unit energy. At the receiver, the resulting channel will have the combined impulse response as
h(t; τ ) = pR(τ ) c(t; τ ) pT(τ ) (2.23) where a(τ ) b(t; τ ) = R b(t; τ0)a(τ − τ0)dτ0 denotes the convolution operation. We consider typical digital communication where the transmit sequence s(l) consists of complex symbols with the symbol time equals to Tsym. By replacing (2.4) with (2.23) and using (2.3), the baseband equivalent received signal y(t) can be expressed by
y(t) =X
l
x(l)h(t; t − lTsym) (2.24)
where the subscript bb in (2.3) is omitted for brevity unless stated otherwise since we are concerned with the baseband equivalent model throughout most part of this dissertation.
The sampled version of y(t) with the sampling period Ts is given by y(kTs) = X
l
x(l)h(kTs; kTs− lγTs) (2.25)
where γ is an integer oversampling factor and Ts = Tsym/γ. For considering RF- or analog-related effects, it may require oversampling, e.g., γ = 2, 4, .... If oversampling is used, the data sequence {x(k)} should also be oversampled by inserting (γ − 1) zeros between each symbol x(k). It can be shown that the symbol-spaced model contains sufficient statistics for data detection if the transmit and receiver filters are carefully chosen, i.e., fulfilling the Nyquist condition. Consequently, we consider the the case γ = 1 and, thus, Ts can be dropped for brevity.
2.2.2 Channel Simulators
As the discrete time channel model in Equation (2.25) acts as an FIR time-varying filter, the problem is transformed to generating the tap gains for simulations. We consider two well-known methods: the first is to generate and combine several complex exponentials, and thus is properly termed sum-of-sinusoidal; the second is to generate a Gaussian random sequence and passing the sequence through an FIR filter with the transfer function designated to be the square-root of the required Doppler PSD.
Simulation Methods I: Sum-of-sinusoidals
The sum-of-sinusoidals is proposed by Jakes [35]:
h(k; l) = σl pNpath
Npath
X
n=1
ej(θn−2πλnkTs) (2.26)
where σl denotes the tap gain of the l-th path and θn are i.i.d. random variables with uniform distribution over the range of 0 to 2π. A small number of Npath, say, Npath = 10, will approximate Rayleigh fading quite accurately. The block diagram of the channel tap generation process is shown in Fig. 2.4.
Figure 2.4: Channel tap generation of sum-of-sinusoidals channel simulator.
Simulation Methods II: Filtering
Another approach to simulate the tap gains with desired PDP and Doppler spectrum is [36,37]: first generate a complex white Gaussian random sequence; second, input the generated sequence to an FIR filter with the square root of the desired Doppler PSD as its transfer function; finally, properly scale the corresponding filter tap to follow the desired PDP. The block diagram of the channel tap generation process is shown in Fig.
2.5.
Figure 2.5: Channel tap generation of filtering-based channel simulator.
Both methods have their advantages and disadvantages. The filtering-based ap-proach has accurate statistics but usually requires high computational complexity. Fur-thermore, the Doppler spectrum usually is discontinuous at the maximum Doppler frequencies and it makes the corresponding FIR filter to be very long. The statistics associated with the sum-of-sinusoidal approach may not be what we expect and some techniques have been proposed to improve the accuracy [38–40].
2.3 OFDM Systems over Doubly-Selective Fading Channels
OFDM is a promising transmission technique to achieve high data rates over wireless mobile channels. Besides bandwidth efficiency, OFDM offers many advantages over conventional single carrier systems, for example, robustness against multipath delay
spread, all-digital-FFT implementation, the possibility of adaptive channel allocation and adaptive modulation of the subcarriers for maximizing the capacity [41]. Thanks to cyclic prefix (CP), the ISI can be avoided, and the channel effect can easily be com-pensated by applying one-tap frequency domain equalizers since the frequency selective fading channel is transformed into parallel flat fading channels. However, if there ex-ists channel variations within one OFDM symbol will introduce interferences among subcarriers, called ICI. OFDM is vulnerable to ICI due to tightly spacing between subcarriers.
Consider the baseband equivalent OFDM system model with N subcarriers. By doing N -point IDFT of the data sequence of the OFDM symbol, {S(m)}, the time-domain transmitted signal x(k) at the k-th time instant can be written as
x(k) = 1 N
N −1
X
m=0
S(m)ej2πmkN , k = 0, 1, · · · , N − 1. (2.27)
Assume a CP with length of NCP longer than the maximum delay spread of the channel, denoted as L; the total length of a transmitted OFDM symbol is NS = N + NCP. Let h(k; l) represent the l-th delay path of the time-varying dispersive channel impulse response at the k-th sampling instant. The channel path {h(k; l)} is assumed to follow the WSSUS model, i.e., E[h(k; l)h∗(k; l+l0)] = σl2ξl0 where ξl0denotes the 1-D Kronecker delta and σ2l is the average power of the l-th path. Assuming CP removal and perfect synchronization, the sampled baseband equivalent received signal y(k) of the OFDM symbol can be expressed as [9,10]
y(k) =
L−1
X
l=0
h(k; l)x(k − l) + n(k), 0 ≤ k ≤ N − 1 (2.28)
where n(k) is the additive white Gaussian noise.
Taking discrete Fourier transform of Equation (2.28), the frequency-domain
re-ceived signal on the i-th subcarrier is Note that H(m; k) can be interpreted as the CFR on the m-th subcarrier at the k-th time instant. The ICI on the i-th subcarrier which is caused by the signal on the m-th subcarrier comes through the ICI channel Hi,m.
Assume that the time variation of CIR is linear within one OFDM symbol [13,15].
It is a reasonable assumption in most practical cases when the normalized Doppler frequency (normalized to subcarrier spacing) is smaller than 0.1, which corresponds to 500 km/h in WiMAX standard. The CIR of the p-th OFDM symbol can be written as
h(pNS− N + k; l) = h(pNS −N − 1
2 ; l) +k − N −12
NS δ(pNS; l) (2.32) where NS denotes the length of CP plus N . h(pNS−N −12 ; l) represents the center point of the p-th OFDM symbol and δ(pNS; l) is the slope of the l-th channel path at the p-th OFDM symbol. Note that with the linear variation assumption the channel at the center point is also the averaged channel in one OFDM symbol.
From (2.27) and (2.28), the received signal can be compactly expressed in vector