Exhibit 2029
`
`Exhibit 2029
`
`

`

`1422
`
`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006
`
`Performance Degradation of OFDM Systems
`Due to Doppler Spreading
`
`Tiejun (Ronald) Wang, Student Member, IEEE, John G. Proakis, Life Fellow, IEEE, Elias Masry, Fellow, IEEE,
`and James R. Zeidler, Fellow, IEEE
`
`Abstract— The focus of this paper is on the performance of
`orthogonal frequency division multiplexing (OFDM) signals in
`mobile radio applications, such as 802.11a and digital video
`broadcasting (DVB) systems, e.g., DVB-CS2. The paper considers
`the evaluation of the error probability of an OFDM system
`transmitting over channels characterized by frequency selectivity
`and Rayleigh fading. The time variations of the channel during
`one OFDM symbol
`interval destroy the orthogonality of the
`different subcarriers and generate power leakage among the
`subcarriers, known as Inter-Carrier Interference (ICI). For con-
`ventional modulation methods such as phase-shift keying (PSK)
`and quadrature-amplitude modulation (QAM), the bivariate
`probability density function (pdf) of the ICI is shown to be a
`weighted Gaussian mixture. The large computational complexity
`involved in using the weighted Gaussian mixture pdf to evaluate
`the error probability serves as the motivation for developing a
`two-dimensional Gram-Charlier representation for the bivariate
`pdf of the ICI. It is demonstrated that its truncated version
`of order 4 or 6 provides a very good approximation in the
`evaluation of the error probability for PSK and QAM in the
`presence of ICI. Based on Jakes’ model for the Doppler effects,
`and an exponential multipath intensity profile, numerical results
`for the error probability are illustrated for several mobile speeds.
`
`Index Terms— OFDM, Doppler spreading, ICI, C/I ratio,
`Gaussian mixture, two-dimensional Gram-Charlier series.
`
`that the ICI distribution is Gaussian by invoking the central
`limit theorem. In other related papers [5]-[9], efforts have
`been made to evaluate the effect of ICI by calculating its
`average power and comparing it with the power of the desired
`signal. In [5][6], the carrier to interference (C/I) ratio has
`been introduced to demonstrate the effect of the ICI under
`various maximum Doppler spreads and different Doppler
`spectra. Through numerical evaluations of the C/I ratios, it is
`reported in [7] that an OFDM system is robust to frequency
`selectivity but quite sensitive to time varying fading channels.
`Li and Cimini [8] provide universal bounds on the ICI in
`an OFDM system over Doppler fading channels, which are
`easier to evaluate and can provide useful insights compared
`with the exact ICI expression. Furthermore, the closed-form
`expression of ICI power is derived and evaluated in [9], where
`the normalized ICI power is represented as a function of the
`normalized Doppler spread. However, all these papers do not
`attempt to determine the underlying probability distribution
`function (pdf) of the ICI.
`In this paper we focus on providing a statistical analysis
`for the ICI in an OFDM system that employs conventional
`PSK and QAM signal modulation in a frequency selective,
`Rayleigh fading time-varying channel. The channel, which
`is assumed to be wide-sense stationary with uncorrelated
`scattering (WSSUS), is modeled by a two-dimensional corre-
`lation function in time and frequency, representing the time
`variations and frequency selectivity of the channel. Each
`subchannel is assumed to be frequency flat and, based on
`the power series model developed by Bello [10], a two-term
`Taylor series expansion is used to model the time variations
`in an OFDM symbol. Jakes’ model [14] is used as the model
`for the Doppler power spectrum and an exponential multipath
`intensity profile is the model adopted for the multipath effects.
`A cyclic prefix is assumed to remove the effects of inter-
`symbol interference. Based on this channel model, the ICI is
`expressed as the summation of leakage terms into each of the
`subcarriers and its pdf is shown to be characterized statistically
`by a bivariate pdf that is a weighted sum of Gaussian pdfs.
`In deriving the probability of error for the OFDM system
`in the presence of ICI, the use of the weighted Gaussian pdf
`proves to be computationally intensive. This difficulty serves
`as the motivation to develop a two-dimensional Gram-Charlier
`series to represent the pdf of the ICI. A truncated version of the
`Gram-Charlier is used in the evaluation of the error probability
`for PSK and QAM signal modulations in an OFDM system.
`The paper is organized as follows: In Section II we describe
`the model for the OFDM system. In Section III we describe
`1536-1276/06$20.00 c(cid:2) 2006 IEEE
`VIS EXHIBIT 2029
`
`I. INTRODUCTION
`
`I N OFDM systems, a serial data stream is split into parallel
`
`streams that modulate a group of orthogonal sub-carriers.
`Compared to single carrier modulation, OFDM symbols have
`a relatively long time duration, but a narrow bandwidth. Con-
`sequently, OFDM is robust to channel multipath dispersion
`and results in a decrease in the complexity of equalizers for
`high delay spread channels or high data rates. However, the
`increased symbol duration makes an OFDM system more
`sensitive to the time variations of mobile radio channels.
`In particular, the effect of Doppler spreading destroys the
`orthogonality of the sub-carriers, resulting in inter-carrier
`interference (ICI) due to power leakage among subcarriers.
`In several previous publications [1]-[4], the system per-
`formance for OFDM was analyzed based on the assumption
`
`Manuscript received April 15, 2004; revised November 21, 2004; accepted
`April 26, 2005. The associate editor coordinating the review of this paper
`and approving it for publication was A. Molisch. This work was supported
`by the Center for Wireless Communications under the CoRe research grant
`core 00-10071 and 03-10148.
`The authors are with the Center for Wireless Communications, Uni-
`versity of California, San Diego, La Jolla, CA 92093-0407 USA (e-
`mail:
`ronald@cwc.ucsd.edu;
`jproakis@ucsd.edu; emasry@ucsd.edu; zei-
`dler@ucsd.edu).
`Digital Object Identifier 10.1109/TWC.2006.04223
`
`Page 1 of 11
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`

`

`WANG et al.: PERFORMANCE DEGRADATION OF OFDM SYSTEM DUE TO DOPPLER SPREADING
`
`1423
`
`dk,i are statistically independent, identically distributed and
`E[dk,r] = E[dk,i] = 0 .
`
`III. CHANNEL MODEL
`We consider a frequency selective randomly varying chan-
`nel with impulse response h(t, τ). Within the narrower band-
`width of each sub-carrier, compared with the coherence
`bandwidth of the channel, the sub-channel is modeled as a
`frequency nonselective Rayleigh fading channel. Hence, the
`channel impulse response hk(t, τ) for the kth subchannel is
`denoted as
`
`hk(t, τ) = βk(t)δ(τ)
`(4)
`where the process {βk(t),−∞ < t <∞} is a stationary,
`zero mean complex-valued process described as follows: It is
`assumed that the processes {βk(t),−∞ < t < ∞}, k =
`1, . . . , N, are complex-valued jointly stationary and jointly
`Gaussian with zero means and cross covariance function
`(τ) := E[βk(t + τ)β∗
`l (t)], k, l = 1, . . . , N.
`Rβk,βl
`For each fixed k, the real and imaginary parts of the process
`{βk(t),−∞ < t < ∞} are assumed independent with
`identical covariance function. We further assume that
`the
`(τ) has the following factorable
`correlation function Rβk,βl
`form
`(τ) = R1(τ)R2(k − l)
`literature,
`the
`frequently
`used
`in
`been
`has
`which
`to represent
`the
`e.g.,[4][15][16], and which is sufficient
`frequency selectivity and the time-varying effects of the
`channel. R1(τ) gives the temporal correlation for the process
`{βk(t),−∞ < t <∞} which is seen to be identical
`for all k = 1, . . . , N. R2(k) represents the correlation in
`frequency across subcarriers. We assume in this paper that
`the corresponding spectral density ψ1(f) to R1(τ) is given
`(cid:6)
`by the Doppler power spectrum, modeled as in Jakes [14],
`i.e.,
`|f| ≤ Fd
`otherwise
`
`(5)
`
`(6)
`
`)2
`
`(7)
`
`Rβk,βl
`
`ψ1(f) =
`
`1
`
`πFd·1−( f
`0
`
`Fd
`
`where Fd is the (maximum) Doppler bandwidth. Note that
`R1(τ) = J0(2πFdτ)
`
`(8)
`
`Fig. 1. Base-band OFDM transmisson model with N subcarriers.
`
`the channel model and use a Taylor series expansion for
`the time variations within an OFDM symbol. In Section
`IV an expression for the ICI and its power is presented.
`Section V provides a thorough analysis of the statistics of the
`ICI, its joint probability density, joint moments, and a two-
`dimensional Gram-Charlier representation. In Section VI, the
`error rate performance of BPSK and 16-QAM OFDM systems
`are presented and compared. Finally, concluding remarks are
`given in Section VII.
`
`II. OFDM SYSTEM
`An OFDM system with N subcarriers is represented in Fig.
`1. In an OFDM system that employs M -ary digital modula-
`tion, a block of log2 M input bits is mapped into a symbol
`constellation point dk by a data encoder, and then N symbols
`are transferred by the serial-to-parallel converter (S/P). If 1/T
`is the symbol rate of the input data to be transmitted, the
`symbol interval in the OFDM system is increased to N T ,
`which makes the system more robust against the channel delay
`spread. Each sub-channel, however, transmits at a much lower
`bits/s. The parallel symbols (d1,··· , dN )
`bit rate of log2 M
`N T
`(cid:3)
`(cid:2) N T
`modulate a group of orthogonal subcarriers, which satisfy
`1
`1 i = j
`exp(j2πfit) exp(j2πfjt)dt =
`0 i (cid:2)= j
`N T
`(i = 1, 2,··· , N)
`where fi = i−1
`N T ,
`Consider the system shown in Fig. 1. The baseband trans-
`mitted signal can be represented as
`1√
`skej2πfk t, 0 ≤ t ≤ N T,
`N T
`
`(1)
`
`. (
`
`fk = k − 1
`
`N T
`
`0
`
`s(t) =
`
`N(cid:4)
`
`k=1
`
`2)
`We denote by 2Es the average energy for the complex
`(cid:5)
`baseband symbol sk. Then sk is given by
`sk =
`2Esdk
`(3)
`where dk = dk,r + jdk,i , is the signal constellation point
`(e.g. BPSK, QPSK, QAM, etc.) with normalized variance
`E[|dk|2] = 1 . Square M-QAM signal constellations may be
`√
`M-PAM signals on orthogonal
`viewed as two independent
`carriers. In this case, the real and imaginary parts dk,r and
`
`Page 2 of 11
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`

`

`1424
`
`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006
`
`multipath intensity profile
`
`correlation function in frequency
`
`1
`
`0.9
`
`0.8
`
`0.7
`
`1
`
`0.9
`
`0.8
`
`0.7
`
`0.6
`
`0.5
`
`s(τ)/α
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`0.6
`
`0.5
`
`2(kΔf)|
`|Ψ
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`−2
`
`−1
`
`0
`f/Δ f
`
`1
`
`2
`
`the following form:
`
`2 , 0 ≤ t ≤ N T.
`t0 = N T
`(10)
`Therefore, the impulse response of the kth subchannel is
`expressed as
`
`(t0)(t − t0),
`
`(cid:2) k
`
`βk(t) = βk(t0) + β
`
`hk(t, τ) = βk(t)δ(τ) = [βk(t0) + β
`
`(t0)(t − t0)]δ(τ). (11)
`Since R1(τ) of (8) is infinitely differentiable, all mean-square
`derivatives exist and thus the differentiation above is justified.
`We use this model for the time variations of the channel within
`an OFDM symbol.
`
`(cid:2) k
`
`IV. EXPRESSION FOR THE ICI AND ITS POWER
`Let s(t) be the baseband signal transmitted over the channel
`with impulse response h(t, τ) as modeled above. Then the
`baseband received signal with additive noise may be expressed
`as
`
`N(cid:4)
`
`βk(t)skej2πfkt +n(t)
`
`r(t) = h(t, τ)(cid:8)s(t)+n(t) =
`
`1√
`N T
`
`k=1
`
`(12)
`where (cid:8) denotes convolution and n(t) is the additive noise,
`which is modeled as a Gaussian process with zero mean
`and spectrally flat within the signal bandwidth, with one-
`sided spectral density N0 watts/Hz. By using the Taylor series
`expansion for βk(t) as given in (11), we obtain
`N(cid:4)
`1√
`N T
`N(cid:4)
`1√
`N T
`
`r(t) =
`
`βk(t0)skej2πfkt
`
`k=1
`
`(13)
`
`(t0)(t − t0)skej2πfkt + n(t).
`
`(cid:2) k
`
`β
`
`+
`
`k=1
`
`The received signal in a symbol interval is passed through
`a parallel bank of correlators, where each correlator is tuned
`
`0
`
`0
`
`1
`
`2
`
`3
`
`α τ
`
`4
`
`5
`
`Fig. 2. Multipath delay profile and frequency correlation function.
`
`where J0(τ) is the zero-order Bessel function of the first
`kind. In order to specify the correlation in frequency across
`subcarriers, we adopt an exponential multipath power intensity
`of the form S(τ) = αe−ατ ,
`τ > 0, α > 0 where α is
`a parameter that controls the coherence bandwidth of the
`channel. The Fourier transform of S(τ) yields
`
`ψ2(f) =
`
`α
`α + j2πf
`
`(9)
`
`which provides a measure of the correlation of the fading
`across the subcarriers as illustrated in Fig. 2. Then R2(k) =
`ψ2(Δf k) where Δf = 1
`N T is the frequency separation
`between two adjacent subcarriers. The 3dB bandwidth of
`ψ2(f) is defined as the coherence bandwidth of the channel
`√
`and easily shown to be fcoherent =
`3α
`2π .
`The channel model described above is suitable for modeling
`OFDM signal
`transmission in mobile radio systems, e.g.,
`cellular systems and broadcasting systems. For example, in
`DVB-CS2 with 2000 subcarriers, the symbol duration N T is
`500μs. In contrast, the delay spread of many fading channels
`is much smaller, which make it reasonable to view each
`subcarrier as a flat fading channel. However, compared with
`the entire OFDM system bandwidth W = 1/T , the coherence
`bandwidth fcoherent is usually smaller, fcoherent < W ,
`especially in an outdoor wireless communication environment.
`Hence,
`the channel
`is frequency-selective over the entire
`OFDM bandwidth.
`We now turn our attention to modeling the time variations
`of the channel within an OFDM symbol interval. For most
`practical mobile radio fading channels, the time-varying ef-
`fects in the channel are sufficiently slow, i.e., the coherence
`time is always much larger than the interval of an OFDM
`symbol [17][18]. For such slow fading channels, we use the
`two terms Taylor series expansion, first introduced by Bello
`[10], to represent the time-varying fading response βk(t) as
`
`Page 3 of 11
`
`

`

`WANG et al.: PERFORMANCE DEGRADATION OF OFDM SYSTEM DUE TO DOPPLER SPREADING
`
`1425
`
`from Equation (25)
`Upper bound from Reference [8]
`Lower bound from Reference [8]
`
`21.95
`
`21.9
`
`21.85
`
`21.8
`
`21.75
`
`21.7
`
`21.65
`
`21.6
`
`21.55
`
`C/I (dB)
`
`from Equation (25)
`Reference [9]
`Reference [5],[6]
`
`36
`
`34
`
`32
`
`30
`
`28
`
`26
`
`24
`
`22
`
`C/I (dB)
`
`20
`100
`
`600
`
`21.5
`490
`
`505
`500
`495
`500
`400
`300
`200
`Doppler frequency in Hz
`Doppler frequency in Hz
`(b)
`(a)
`Fig. 3. C/I ratio curves of an OFDM system N = 256 subcarriers, subcarrier distance Δf = 7.81KHz, and carrier frequency fc = 2GHz.
`
`510
`
`where ni is a complex Gaussian noise with zero mean and
`variance N0/Es. Thus we have
`N(cid:4)
`(cid:9)(cid:10)
`k=1
`k(cid:4)=i
`
`(cid:2) k
`
`(t0)dk
`k − i
`(cid:11)
`
`β
`
`(cid:7)di = βi(t0)di
`(cid:8) (cid:9)(cid:10) (cid:11)
`
`desired_signal
`
`+ N T
`2πj
`(cid:8)
`
`+ni .
`
`(19)
`
`ICI
`
`In Section V we establish the statistical properties of the
`ICI term. Here, we obtain the C/I ratio and we compare the
`result with those obtained in [5],[6],[8], and [9], which are
`based on different models for the time variations.
`From Equation (19), the average power of the desired signal
`
`to one of the N subcarriers. The output of the ith correlator
`is
`
`(cid:2) N T
`
`0
`
`1√
`N T
`
`r(t)e−j2πfi tdt.
`
`(14)
`
`(cid:7)di =
`
`Substituting (13) into (14), we obtain
`
`1√
`N T
`
`k=1
`
`(1)
`
`βk(t0)skej2π(fk−fi)tdt
`(cid:11)
`
`1√
`2Es
`(cid:2) N T
`
`0
`
`(cid:7)di =
`
`+
`
`1√
`(cid:8)
`2Es N T
`
`+
`
`N(cid:4)
`(cid:9)(cid:10)
`N(cid:4)
`
`(t0)(t − t0)skej2π(fk−fi)tdt
`(cid:9)(cid:10)
`(cid:11)
`
`(cid:2) k
`
`β
`
`0
`
`k=1
`
`(2)
`
`is
`
`(15)
`
`C = E[|βi(t0)di|2] = E[|βi(t0)|2]E[|di|2] = 1 .
`
`(20)
`
`(τ) = R1(τ) is infinitely differentiable, all
`Since Rβk,βk
`the process {βk(t),−∞ <
`(mean-square) derivatives of
`t < ∞} exist.
`In particular,
`the first-order derivative
`process {β(cid:5)
`k(t),−∞ < t < ∞} is a zero mean complex-
`valued Gaussian process with correlation function E[β(cid:5)
`k(t +
`k(t))∗] = −R(cid:5)(cid:5)
`τ)(β(cid:5)
`1(τ) (identical for all k) with corresponding
`spectral density
`⎧⎨
`⎩
`
`ψ3(f) =
`
`(2πf )2
`πFd·1−( f
`0
`
`Fd
`
`)2
`
`|f| ≤ Fd
`otherwise
`
`(21)
`
`df = 2π2Fd
`
`2.
`
`(22)
`
`)2
`
`ψ3(f)df
`(cid:17)
`(2πf)2
`1 − ( f
`
`πFd
`
`Fd
`
`(cid:2) ∞
`(cid:2) Fd
`
`−∞
`
`−Fd
`
`Then,
`
`(t)|2] =
`
`(cid:2) k
`
`E[|β
`
`=
`
`1√
`(cid:8)
`2Es N T
`(cid:2) N T
`1√
`N T
`(cid:2) N T
`(cid:9)(cid:10)
`
`1√
`(cid:8)
`2Es N T
`
`n(t)e−j2πfi tdt
`(cid:11)
`
`0
`
`(3)
`
`The first term yields
`
`N(cid:4)
`
`k=1
`
`βk(t0)dk
`
`(cid:12)
`
`1
`N T
`
`(cid:13)
`
`ej2π(fk−fi)tdt
`
`= βi(t0) di. (16)
`
`(cid:2) N T
`
`0
`
`(cid:13)
`
`1
`N T
`N(cid:4)
`
`k=1
`k(cid:4)=i
`
`(cid:12)
`
`=
`
`(t0)dk
`
`(cid:2) k
`
`β
`
`N(cid:4)
`
`k=1
`
`The second term yields
`
`(cid:2) N T
`
`0
`
`(t − t0)ej2π(fk−fi)tdt
`N(cid:4)
`
`.
`
`(cid:2) k
`
`(t0)dk
`k − i
`
`β
`
`k=1
`k(cid:4)=i
`
`= N T
`2πj
`
`(cid:2) k
`
`(t0)dk
`β
`j2π(fk − fi)
`
`Finally, the additive noise term is
`1√
`1√
`2Es
`N T
`
`(cid:2) N T
`
`0
`
`ni =
`
`(17)
`
`(18)
`
`n(t)e−j2πfitdt
`
`Page 4 of 11
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`

`

`1426
`
`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006
`
`from Equation (25)
`Reference [9]
`Lower bound from Reference [8]
`Reference [5],[6]
`
`Fd = 5KHz −−> Fd*NT = 0.64
`Fd = 10KHz −−> Fd*NT = 1.28
`Fd = 40KHz −−> Fd*NT = 5.12
`
`0.5
`
`1
`
`2.5
`2
`1.5
`Doppler frequency in Hz
`
`3
`
`3.5
`
`4
`x 104
`
`20
`
`15
`
`10
`
`5
`
`0
`
`−5
`
`−10
`
`−15
`
`−20
`
`0
`
`C/I (dB)
`
`Fig. 4. C/I ratio curves of an OFDM system N = 256 subcarriers, subcarrier
`distance Δf = 7.81KHz, and carrier frequency fc = 2GHz.
`
`very good approximation of the time variations encountered
`in many physical channels.
`
`V. THE DISTRIBUTION AND MOMENTS OF THE ICI
`In this section we derive expressions for the bivariate
`probability density and joint moments of the ICI complex ran-
`dom variable Z including a two-dimensional Gram-Charlier
`expansion. We carry out this analysis in a fairly general setting
`so that the results can be applied to situations where the
`channel model and the signal constellation are different from
`those assumed in the previous sections. Let
`
`Z :=
`
`akXkdk
`
`(26)
`
`k=1
`where the ak’s are real-valued constants and the sets of
`random variables {Xk}N
`k=1 and {dk}Nk=1 are independent.
`
`The random vector ˜X := (X1, . . . , XN)T is complex-valued
`circular Gaussian with zero mean and N × N complex-valued
`covariance matrix (cid:28)Σ = E[ ˜X ˜X
`∗T ]. We allow (cid:28)Σ to be an
`arbitrary covariance matrix. It is readily evident from (26) that,
`given the {dk}, Z is conditionally complex Gaussian random
`variable. We seek an explicit expression for the the joint
`probability density of its real and imaginary parts. Decompose
`Xk into its real and imaginary parts, Xk = Xk,r + jXk,i, k =
`1, . . . , N, and define the 2N × 1 real-valued Gaussian vector
`X := [X1,r ··· XN,rX1,i ··· XN,i]T . The 2N × 2N real-
`valued covariance matrix Σ = E[XX T ] is given by (see for
`(cid:18)
`(cid:20)
`(cid:29)
`(cid:30)
`example [19])
`(cid:6)[(cid:28)Σ] −(cid:7)[(cid:28)Σ]
`(cid:7)[(cid:28)Σ] (cid:6)[(cid:28)Σ]
`
`=:
`
`Σ11 −Σ12
`Σ12
`Σ22
`
`.
`
`(27)
`
`1 2
`
`Σ =
`
`Next the signal constellation points dk’s are assumed to be
`i.i.d. complex-valued random variables and we set dk =
`dk,r + jdk,i for its real and imaginary parts. Again, in the
`general setting here, we do not assume that dk,r and dk,i are
`independent taking values with equal probabilities. Instead we
`assume that
`P [dk,r = bl, dk,i = cm] = pl,m, l, m = 1, . . . , L
`
`(28)
`
`N(cid:4)
`
`2(cid:20)
`
`(cid:19)(cid:19)(cid:19)(cid:19)(cid:19)
`
`(cid:2) k
`
`β
`
`(t0)dk
`k − i
`
`(cid:18)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19) N T
`N(cid:4)
`(cid:22)2 N(cid:4)
`
`2πj
`
`k=1
`k(cid:4)=i
`N(cid:4)
`
`Thus, the power of the interference (ICI) is
`
`(t0)dl)∗(cid:24)
`(t0)dl)∗(cid:24)
`
`(cid:2) l
`
`(t0)dk)(β
`
`(cid:2) k
`
`(β
`
`(cid:23)
`
`(cid:2) l
`
`(t0)dk)(β
`
`(cid:2) k
`
`(β
`
`1
`(k − i)(l − i) E
`(cid:23)
`
`=: J1 + J2.
`(23)
`(t0)) is independent of (dk, dl). Also, the
`(t0), β
`Note that (β
`dk’s are i.i.d. with zero means. Thus J1 = 0. It then follows
`(cid:21)
`(cid:22)2 N(cid:4)
`that
`
`(cid:2) l
`
`(cid:2) k
`
`I =
`
`=
`
`N T
`2π
`
`k=1
`k(cid:4)=i
`N(cid:4)
`(N T Fd)2
`2
`
`k=1
`k(cid:4)=i
`
`(t0)|2]E[|dk|2]
`(k − i)2
`
`(cid:2) k
`
`E[|β
`
`1
`(k − i)2
`
`.
`
`(24)
`
`Thus the signal to interference ratio (C/I) ratio can be ex-
`pressed as
`
`(25)
`
`1
`(k−i)2
`
`1
`N(cid:27)
`k=1
`k(cid:4)=i
`
`=
`
`(N T Fd)2
`2
`
`C I
`
`The C/I curve of the middle subcarrier, i.e., subcarrier index
`k = N/2, is plotted versus the Doppler frequency Fd in Fig.
`3 (a). The OFDM system is assumed to have N = 256 sub-
`carriers, with subcarrier spacing Δf = 1/N T = 7.81KHz,
`and carrier frequency fc = 2GHz. The C/I curve given
`in Fig. 3 (a) matches very well with the results given in
`[5][6][9], which were obtained without the use of Taylor series
`approximation. Furthermore, using an expanded scale in Fig.
`3 (b), we compare our analytical result with the upper and
`lower bounds on C/I given by Li and Cimini [8].
`In order to evaluate the approximation effect embodied in
`the two-term Taylor series expansion, we also demonstrate
`in Fig. 4 the comparison of the C/I ratio curves of the
`same OFDM system in very high Doppler frequency regimes.
`We can observe from Fig. 4 that the two-term Taylor series
`expansion model well approximates the actual fading channel
`up to a relatively large Doppler frequency 10000Hz, i.e., the
`two-term Taylor series expansion model becomes inaccurate
`only when Fd is larger than 10000Hz, which is equivalent to
`a normalized Doppler frequency of 1.28. This is larger than
`any practical wireless fading channel for OFDM applications.
`For example, if we consider 802.11a with carrier frequency of
`5GHz, the terminal must be moving at a speed of 2160km/hr.
`Overall, the two term Taylor expansion channel model is a
`
`(cid:25) N T
`
`(cid:26)2
`
`k=1
`l=1
`k(cid:4)=i
`l(cid:4)=i
`N(cid:4)
`N(cid:4)
`
`I = E
`(cid:21)
`
`N T
`2π
`
`=
`
`=
`
`2π
`(cid:12)
`
`+
`
`N T
`2π
`
`1
`(k − i)(l − i) E
`(cid:23)|β
`
`l=1
`k=1
`l(cid:4)=i
`k(cid:4)=i
`k(cid:4)=l
`
`(cid:13)2 N(cid:4)
`
`k=1
`k(cid:4)=i
`
`(cid:24)
`
`(t0)dk|2
`
`(cid:2) k
`
`1
`(k − i)2
`
`E
`
`Page 5 of 11
`
`

`

`WANG et al.: PERFORMANCE DEGRADATION OF OFDM SYSTEM DUE TO DOPPLER SPREADING
`
`1427
`
`where bl and cm are real-valued. This allows general signal
`constellations. Finally the constants {ak} appearing in (26)
`are arbitrary; in the special case that will be considered later,
`we set ak = 1/(k − i), for k (cid:2)= i and ak = 0 for k = i. In
`this case, ICI = N T
`2πj Z.
`Decompose the complex-valued random variable Z of (26)
`into its real and imaginary parts,
`N(cid:4)
`ak[Xk,rdk,r − Xk,idk,i]
`N(cid:4)
`
`k=1
`
`k=1
`
`+j
`
`ak[Xk,rdk,i + Xk,idk,r]
`
`(29)
`
`Z := Zr + jZi =
`
`and let fZr,Zi(u, v) be its joint probability density. We first
`seek an expression for this density and its joint moments.
`It can be seen that fZr,Zi(u, v) is a Gaussian mixture and
`(cid:12)
`(cid:13)
`L(cid:4)
`L(cid:4)
`L(cid:4)
`L(cid:4)
`N(cid:31)
`one can write
`fY1,Y2(u, v)
`
`pln,mn
`
`fZr,Zi(u, v)=
`. . .
`m1=1
`l1=1
`lN =1
`
`mN =1
`
`n=1
`
`(30)
`where fY1,Y2(u, v) is a bivariate Gaussian density with zero
`means and 2× 2 dimensional covariance matrix whose entries
`depend on (bl1 , . . . , blN ; cm1, . . . , cmN ) (i.e., on the values of
`the signal constellation points {dk}N
`k=1). Moreover, one can
`verify that under our assumptions we have cov{Y1, Y2} = 0
`whereas the variance of Y1 and Y2 are identical and given by
`Y (bl1 , . . . , blN ; cm1 , . . . , cmN )
`σ2
`N(cid:4)
`N(cid:4)
`
`Note that when the real and imaginary parts of the signal
`constellation points are independent and equally probable,
`there is a significant simplification in (32) and (35) since then
`pl,m is a constant. Computationally, finding fZr,Zi(u, v) from
`(32) is very intensive for large N and L since the dependence
`on the values of the signal constellation points appear in the
`exponent of the Gaussian density fY (u) via its variance σ2
`Y .
`On the other hand, computing the joint moments E[Z k1
`] is
`r Z k2
`i
`considerably simpler as the dependence on the values of the
`constellation points appear directly in σY and only in term
`of sums of products of two values as seen from (31). We
`therefore plan to use the joint moments E[Z k1
`] to obtain
`r Z k2
`i
`an approximation of the joint probability density fZr ,Zi(u, v).
`This is carried out below using a two-dimensional Gram-
`Charlier expansion.
`The one-dimensional Gram-Charlier expansion and its rate
`of convergence are as follows [20] [21]: Let φ(x) be the
`standard Gaussian probability density with zero mean and
`unit variance. The nth Hermite polynomial Hn(x) is defined
`by Hn(x) := (−1)nex2/2Dn[e−x2/2] where D stands for
`derivative. Hn(x) is given explicitly as
`[n/2](cid:4)
`(−1)kxn−2k
`k! 2k (n − 2k)! .
`
`Hn(x) = n!
`
`k=0
`
`(36)
`
`The first few Hermite polynomials are given by
`H0(x) = 1, H1(x) = x, H2(x) = x2 − 1,
`H3(x) = x3 − 3x, H4(x) = x4 − 6x2 + 3.
`(cid:2) ∞
`The Hermite polynomials satisfy the orthogonality relationship
`
`(37)
`
`Hn(x)Hk(x)φ(x)dx = n! δn,k
`
`(38)
`
`−∞
`where δn,k is the Kronecker delta. Let g(x) be a probability
`density function of a real-valued random variable X. It can
`be expanded in a Gram-Charlier series
`∞(cid:4)
`
`g(x) =
`
`θnHn(x)φ(x)
`
`n=0
`where the coefficient θn is given by
`
`1 n
`
`g(x)Hn(x)dx =
`
`1 n
`
`!
`
`θn =
`
`(cid:2) ∞
`
`(−1)kE[X n−2k]
`k! 2k (n − 2k)! .
`
`θn =
`
`k=0
`
`! E[Hn(X)]
`−∞
`and by (36), can be expressed in terms of the moments of X
`up to order n:
`[n/2](cid:4)
`
`(39)
`
`(40)
`
`(41)
`
`K(cid:4)
`
`The convergence properties of the Gram-Charlier series are
`presented in [21]: Set
`
`gK(x) :=
`
`θnHn(x)φ(x)
`
`(42)
`
`n=0
`
`then if g(x) has s continuous derivatives satisfying certain
`integrability conditions ([21], Theorem II and Corollary I),
`we have
`|g(x) − gK(x)| ≤ constant
`K s/2
`
`(43)
`
`=
`
`+ cmicmk
`
`]
`
`aiakσ11(i, k)[bliblk
`+ σ12(i, k)[−cmiblk
`]
`+ blicmk
`(31)
`where σ11(i, k) is the (i, k)th entry of the matrix Σ11 and
`σ12(i, k) is the (i, k)th entry of the matrix Σ12 defined in
`(cid:12)
`(cid:13)
`(27). It follows that
`L(cid:4)
`L(cid:4)
`L(cid:4)
`L(cid:4)
`N(cid:31)
`fY1(u)fY2(v)
`pln,mn
`
`i=1
`
`k=1
`
`fZr,Zi(u, v)=
`. . .
`m1=1
`l1=1
`lN =1
`
`mN =1
`
`n=1
`
`(32)
`where fY (u) is a one-dimensional Gaussian density with zero
`mean and variance given by (31). Equation (32) allows the
`computation of the joint moments of (Zr, Zi). We have
`(cid:12)
`(cid:13)
`]
`E[Z k1
`r Z k2
`L(cid:4)
`L(cid:4)
`L(cid:4)
`L(cid:4)
`N(cid:31)
`i
`
`=
`
`. . .
`
`pln,mn
`
`E[Y k1
`
`
`1 ]E[Y k22 ](33)
`
`lN =1
`
`mN =1
`
`n=1
`
`m1=1
`l1=1
`and since
`
`E[Y k] =
`
`(cid:3)
`
`1 × 3 . . . × (k − 1)σk
`Y , k even
`0,
`k odd
`
`(34)
`
`it follows that
`E[Z k1
`] =
`r Z k2
`L(cid:27)
`L(cid:27)
`i
`
`⎧⎪⎪⎨
`⎪⎪⎩
`
`l1=1
`
`0,
`
`(cid:22)
`N!
`L(cid:27)
`L(cid:27)
`[1 · 3 ··· (k1 − 1)]·
`. . .
`pln,mn
`m1=1
`mN =1
`lN =1
`×[1 · 3 ··· (k2 − 1)] · σk1+k2
`k1&k2 even
`,
`otherwise
`
`(cid:21)
`
`n=1
`
`Y
`
`(35)
`
`Page 6 of 11
`
`

`

`1428
`
`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006
`
`uniformly in x. Thus, the smoother g(x) is, the better the ap-
`proximation of g(x) by gK(x). In two-dimensions, the Gram-
`Charlier series takes the following form (see for example
`[22]): Let (X, Y ) be jointly distributed random variables with
`probability density function g(x, y). We can then represent
`g(x, y) in a two-dimensional Gram-Charlier series
`∞(cid:4)
`∞(cid:4)
`
`g(x, y) =
`
`and the representation is truncated at n + m ≤ K.
`We now specialize the above general results to the case
`introduced in the previous sections. For the covariance matrix
`of the vector X, set
`Xk,r := (cid:6)[β(cid:5)k(t0)], Xk,i := (cid:7)[β(cid:5)k(t0)].
`
`
`(52)
`It then follows from (6) that the (i, k)th component of the
`covariance matrix Σ11 is given by
`σ11(i, k) := E[Xi,rXk,r]
`= − 1
`1 (0)(cid:6)[R2(i − k)], i, k = 1, . . . , N.
`2 R(cid:5)(cid:5)
`(53)
`Similarly, it is seen that the (i, k)th component of the covari-
`ance matrix Σ12 is given by
`1 (0)(cid:7)[R2(l − k)].
`R(cid:5)(cid:5)
`σ1,2(l, k) := E[Xl,rXk,i] =
`(54)
`Note that this vanishes for l = k. Next we specify the values
`and corresponding probabilities of the signal constellation
`points. Unlike the general formulation earlier, we now assume
`that dk,r and dk,i are independent taking values from a square
`M-point QAM signal constellation with equal probabilities
`P [dk,r = ld] = P [dk,i = ld]
`√
`1√
`M − 1).
`, l = ±1,±3, . . . ,±(
`=
`M
`√
`This corresponds to setting pl,m = 1√
`M with L =
`M in our
`general formulation. Finally, we set ak = 1/(k − i), for k (cid:2)= i
`and ak = 0 for k = i. Thus for the special case, the expression
`for σ2
`Y of (31) remains applicable with ak as specified above
`and bk = kd,
`cm = md and the values of σ11(i, k) and
`σ12(i, k) are specified in (53) and (54) respectively. The
`expression for fZr,Zi(u, v) of (32) simplifies to
`1
`M N−1
`
`1 2
`
`(cid:4) m
`(cid:4) l
`
`N
`
`N
`
`. . .
`
`(cid:4) m
`(cid:4) l
`
`1
`
`1
`
`fZr,Zi(u, v) =
`
`(55)
`
`fY1(u)fY2(v)
`
`(56)
`
`(57)
`
`.
`
`where we use the compact notation
`M−1(cid:4)
`√
`√
`l=−(
`M−1)
`l odd
`
`(cid:4) l
`
`:=
`
`] simplify to
`
`×
`
`0,
`
`l1
`
`m1
`
`lN
`
`mN
`
`Y
`
`The joint moments E[Z k1
`r Z k2
`i
`⎧⎨
`(cid:27)
`(cid:27)
`(cid:27)
`(cid:27)
`E[Z k1
`] = 1
`r Z k2
`M N−1
`i
`[1 × 3 . . . × (k1 − 1)]·
`. . .
`⎩
`×[1 × 3 . . . × (k2 − 1)]σk1+k2
`,
`k1 and k2 even
`otherwise
`(58)
`The truncated two-dimensional Gram-Charlier representation
`is again given by (49) with coefficients θn,m given by (48)
`involving the joint moments E[Z k1
`] now given by (58).
`r Z k2
`i
`Thus, we only need to obtain an expression for σ2
`Y . Straight-
`(cid:22)2 N(cid:4)
`(cid:21)
`forward calculations using (31) (53) and (54) show that
`N(cid:4)
`(cid:6)[ψ2(Δf(r − s))](lrls + mrms)
`N T Fd d
`(r − i)(s − i)
`2
`r=1
`s=1
`r(cid:4)=i
`s(cid:4)=i
`(cid:7)[ψ2(Δf(r − s))](−mrls + lrms)
`(r − i)(s − i)
`
`Y =
`σ2
`
`+
`
`.
`
`(59)
`
`θn,mHn(x)Hm(y)φ(x)φ(y)
`(cid:2) ∞
`(cid:2) ∞
`
`Hn(x)Hm(y)g(x, y)dxdy
`
`(44)
`
`(45)
`
`(46)
`
`θn,m =
`
`=
`
`θn,m =
`
`n=0
`m=0
`where the coefficient θn,m is given by
`1
`n! m!
`−∞
`−∞
`1
`n! m! E[Hn(X)Hm(Y )].
`In view of (36), it is seen that θn,m can be expressed in terms
`of the joint moments of (X, Y ):
`[n/2](cid:4)
`[m/2](cid:4)
`(−1)k+lE[X n−2kY m−2l]
`k! l! 2k+l (n − 2k)! (m − 2l)! .
`
`k=0
`l=0
`Applying the representation (44) to the bivariate density
`fZr,Zi(u, v) we have
`∞(cid:4)
`
`∞(cid:4)
`
`fZr,Zi(u, v) =
`
`θn,mHn(u)Hm(v)φ(u)φ(v)
`
`(47)
`
`θn,m =
`
`r
`
`i
`
`(48)
`
`n=0
`m=0
`where the coefficient θn,m is now given by
`[n/2](cid:4)
`[m/2](cid:4)
`(−1)k+lE[Z n−2k
`Z m−2l
`]
`k! l! 2k+l (n − 2k)! (m − 2l)! .
`k=0
`l=0
`Note that by (35) θn,m is nonzero only if both n and m are
`even. In practice the two-dimensional Gram-Charlier series
`(47) is truncated at a total order K, i.e.,
`∞(cid:4)
`∞(cid:4)
`
`f (K)
`Zr,Zi
`
`(u, v) =
`
`θn,mHn(u)Hm(v)φ(u)φ(v).
`
`(49)
`
`n=0
`m=0
`n+m≤K
`Thus if, say, K = 4, we only need to compute the marginal
`moments E[Z 4
`
`
`r ] and the joint moment E[Z 2r Z 2i ] in (35); in
`turn this only requires the computation of σ2
`Y of (31).
`It should be noted that the standard Gram-Charlier expan-
`sion is generated by a normalized Gaussian density φ(x) with
`zero mean and unit variance. Clearly, if the variables Zr and
`Zi in (49) have variances substantially different from one, we
`are likely to need large truncation level K for an acceptable
`approximation. This problem could be easily remedied as
`
`
`follows: Let s21 and s22 be the variances of the random
`variables Zr and Zi respectively. We can generate a two-
`dimensional Gram-Charlier expansion starting from Gaussian
`
`
`density functions with zero means and variances s21 and s22. It
`∞(cid:4)
`∞(cid:4)
`is easy to see that instead of (47) we now have
`(cid:25) u
`(cid:26)
`(cid:25) v
`(cid:26)
`(cid:25) u
`(cid:26)
`(cid:25) v
`(cid:26)
`1
`¯θn,mHn
`s1s2
`
`fZr ,Zi(u, v)=
`
`
`
`n=0m=0
`
`Hm
`
`s1
`
`s2
`
`φ
`
`s1
`
`φ
`
`s2
`(50)
`
`with coefficients
`
`[n/2](cid:4)
`
`[m/2](cid:4)
`
`¯θn,m =
`
`k=0
`
`l=0
`
`(−1)k+lE[Z n−2k
`Z m−2l
`]
`(n − 2k)! (m − 2l)!
`k! l! 2k+l sn−2k
`sm−2l
`(51)
`
`r
`
`i
`
`1
`
`2
`
`Page 7 of 11
`
`

`

`WANG et al.: PERFORMANCE DEGRADATION OF OFDM SYSTEM DUE TO DOPPLER SPREADING
`
`1429
`
`Similar computations hold for the scaled Gram-Charlier ex-
`pansion (50).
`The above Gram-Charlier approximation method can be
`easily applied to Rician fading statistics. The only difference
`in applying the above proposed approximation approach is to
`adjust the corresponding ICI joint moments. In this paper, our
`results are limited to Rayleigh fading statistics.
`
`VI. EVALUATION OF OFDM SYSTEM PERFORMANCE
`
`We now evaluate the error probability for an M -QAM
`system with coherent detection. In view of (19) and (26) we
`can write
`
`(cid:7)di = βi(t0)di + N T
`
`2πj
`
`Z + ni
`
`(60)
`
`.
`
`(61)
`
`2
`
`2
`
`.
`
`.
`
`(63)
`A final decision is made by comparing the location of random
`
`2
`
`2
`
`where Z represents the ICI contribution whose statistics were
`thoroughly studied in Section V. We assume that we have
`perfect knowledge of βi(t0) in each sub-channel and we form
`the decision variable
`(cid:7)di βi
`(cid:7)Di =
`∗(t0)
`+ ni β∗
`Z β∗
`i (t0)
`i (t0)
`= di + N T
`|βi(t0)|2
`|βi(t0)|2
`|βi(t0)|2
`2πj
`Set a := N T /(2π), Z = Zr + jZi, βi(t0) = W1 + jW2, and
`ni = ni,1 + jni,2 for their real and imaginary parts. Then
`(cid:6)[(cid:7)Di] = (cid:6)[di] + a(ZiW1 − ZrW2)
`+ ni,1W1 + ni,2W2
`1 + W 2
`1 + W 2
`W 2
`W 2
`(62)
`(cid:7)[(cid:7)Di] = (cid:7)[di] − a(ZrW1 + ZiW2)
`+ ni,2W1 − ni,2W2
`1 + W 2
`1 + W 2
`W 2
`W 2
`variable (cid:7)Di with the M -QAM constellation points and select-
`
`ing the signal point that is nearest to di. The decision of the
`detector is based on (61): Assuming a symmetric rectangular
`QAM signal constellation, the detector performs independent
`decisions on the real and imaginary parts conditioned on a
`particular channel realization βi(t0) and ICI realization Z. A
`symbol error occurs if either the real or imaginary components
`are in error. Let Ai be the event of making an error in the real
`(or