Volume 2008, Article ID 840237,9pages doi:10.1155/2008/840237
Research Article
Joint Effects of Synchronization Errors of OFDM Systems in
Doubly-Selective Fading Channels
Wen-Long Chin1and Sau-Gee Chen (EURASIP Member)2
1Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
2Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 30050, Taiwan Correspondence should be addressed to Wen-Long Chin,[email protected]
Received 26 July 2008; Revised 8 November 2008; Accepted 3 December 2008 Recommended by George Tombras
The majority of existing analyses on synchronization errors consider only partial synchronization error factors. In contrast, this work simultaneously analyzes joint effects of major synchronization errors, including the symbol time offset (STO), carrier frequency offset (CFO), and sampling clock frequency offset (SCFO) of orthogonal frequency-division multiplexing (OFDM) systems in doubly-selective fading channels. Those errors are generally coexisting so that the combined error will seriously degrade the performance of an OFDM receiver by introducing intercarrier interference (ICI) and intersymbol interference (ISI). To assist the design of OFDM receivers, we formulate the theoretical signal-to-interference-and-noise ratio (SINR) due to the combined error effect. As such, by knowing the required SINR of a specific application, all combinations of allowable errors can be derived, and cost-effective algorithms can be easily characterized. By doing so, it is unnecessary to run the time-consuming Monte Carlo simulations, commonly adopted by many conventional designs of synchronization algorithms, in order to know those combined error effects.
Copyright © 2008 W.-L. Chin and S.-G. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. INTRODUCTION
Orthogonal frequency-division multiplexing (OFDM) is a promising technology for broadband transmission due to its
high spectrum efficiency, and its robustness to the effects
of multipath fading channels and impulse noises. However, OFDM systems are sensitive to synchronization errors.
There are three major synchronization errors, including the symbol time offset (STO), carrier frequency offset (CFO), and sampling clock frequency offset (SCFO) in OFDM systems. When the symbol time (ST) is not located in the intersymbol interference (ISI) free region, ISI is introduced. The time-selective channel, CFO, and SCFO will introduce additional intercarrier interference (ICI).
The effects of the synchronization errors had been
studied in literature, for example, in [1–15]. For conciseness,
we only discuss some representative works here. In [1],
the signal-to-interference-and-noise ratio (SINR) is analyzed
considering the effect of the CFO in time-selective channels;
the work in [2] analyzes the effect of the STO in
frequency-selective channels; the work in [3] also analyzes the effect
of the STO without considering ISI; the work in [4]
analyzes the effect of the Doppler spread; the work in [5]
analyzes the effect of the CFO and SCFO in time-selective
channels; the work in [6] only analyzes the effect of the
STO in doubly-selective channels; the works in [7–9] analyze
the synchronization errors separately in frequency-selective
channels; the works in [10–12] analyze the effect of the
CFO in frequency-selective channels; and the works in [13–
15] consider the combined effects of the STO and CFO in
frequency-selective channels.
Some works [1–6,8–15] only consider partial
synchro-nization errors; some works [1–9, 12, 13, 15] separately
consider synchronization errors; and some works [2,6–14]
only consider frequency-selective channels. In addition, the
works in [6,8,10,11,13,14] consider the STO, while
assum-ing that the STO is small; therefore, nonnegligible ISI was often neglected. In summary, the current analyses mostly do not consider joint effects of the combined synchronization errors due to nonideal synchronization process in the
envi-ronments of mobility (causing time-selective channel effect)
Table 1: Comparison of synchronization errors analyses.
Reference Consider ISI Consider STO Consider CFO Consider SCFO Fast fading channel Combined analysis
[1] No No Yes No Yes No
[2] Yes Yes No No No No
[3] No No No No Yes No
[4] No No Yes Yes Yes No
[5] Yes Yes No No Yes No
[6] No Yes Yes Yes No No
[7] Yes Yes Yes Yes No No
[8] No Yes Yes Yes No No
[9] No No Yes No No No
[10,11] No Yes Yes No No Yes
[12] No No Yes No No No
[13] No Yes No No No No
[14] No Yes Yes No No Yes
[15] No No Yes No No No
This work Yes Yes Yes Yes Yes Yes
Data source Signal mapper Xl,k N-point IFFT CP insertion P/S DAC 1/TS Data sink Signal demap-per Equaliz er Xl,k N-point FFT CP removal S/P ADC nδ 1/TS=(1 +εt)/TS e− j2π(1−εf)fct ej2π fct AWGN Channel
Figure 1: A simplified OFDM system model.
effect). For clarity,Table 1summarizes the concerned errors
and conditions, of some key representative works and the proposed work, on the synchronization error analyses.
The main contribution of this paper is that for better characterizations of synchronization errors under a practical communication environment, that is, in doubly-selective
fading channels, we analyze joint effects of the mentioned
three major synchronization errors, without the assumption of small STO. Another contribution is that compact forms can be derived from our work to gain further insights on the
synchronization error effects. To this end, we first analyze
the signal model of the combined synchronization errors in time-selective and frequency-selective fading channels by
extending the works in [1–15]. Next, based on this model,
the theoretical SINR is formulated. The derived SINR can be exploited to obtain all possible combinations of syn-chronization errors that meet the required SINR constraint, knowing that the allowable synchronization errors could help design suitable synchronization algorithms and shorten the design cycle. To gain further insights, some compact
results are deduced from the derived SINR formulation. In
addition, the works in [1,2] are found to be special cases
of this work; and our work is more accurate than that in [10].
The rest of this paper is organized as follows. The notations used in this work are summarized in the Notaions section at the end of the paper. The OFDM system model in
doubly-selective fading channels is introduced inSection 2.
The signal model with synchronization errors is analyzed,
and its theoretical SINR is formulated in Section 3. Some
compact results are given in Section 4. Numerical and
simulation results are provided in Section 5. Finally, we
conclude our work inSection 6.
2. SYSTEM AND CHANNEL MODELS
2.1. System model
In the following discussion, all the quantities indexed with
model is shown in Figure 1. In this figure, Xl,k/Xl,k is
the transmitted/received frequency-domain data at the kth
subcarrier;nδis the STO; 1/TSis the transmitter’s sampling
frequency; 1/TS = (1 + εt)/TS is the receiver’s sampling
frequency, whereεt is the SCFO normalized by 1/TS;εf is
the CFO normalized by the subcarrier spacing; and fc is
the carrier frequency. On the transmitter side, N complex data symbols are modulated onto N subcarriers by using
the inverse fast Fourier transform (IFFT). The last NGIFFT
output samples are copied to form the CP which is inserted at the beginning of each OFDM symbol. By inserting the CP, a guard interval is created so that ISI can be avoided and the orthogonality among subcarriers can be sustained. The receiver uses the fast Fourier transform (FFT) to demodulate received data.
2.2. Channel model
In this work, hl(n, τ) denotes the τth channel tap of the
discrete time-selective CIR at time n of the lth symbol. Furthermore, the following two assumptions regarding the channels are made: (a) the channels are wide-sense stationary and uncorrelated scattering (WSSUS), and (b) the Doppler
spectrum follows Jakes’ model [16]. Based on these
assump-tions, the cross-correlation of the CIR can be obtained by
Ehl n1,τ1 h∗l n2,τ2 =Ehl n1,τ1 h∗l n2,τ2 δτ1−τ2 =J0 βΔn σh2ττ=τ1=τ2, 0≤τ≤τd, (1)
where δ(·) is the Dirac delta function; J0(·) is the
zeroth-order Bessel function of the first kind;Δn n1−n2; σh2τ =
E[|hl(τ)|2] is the power of the τth channel tap; τd is the
maximum delay spread of the channel; andβ = 2π fdT/N
where fd represents the maximum Doppler shift, fdT is the
normalized Doppler frequency (NDF), N is the number of
subcarriers, andT=NTSis the symbol duration.
3. ANALYSIS OF RECEIVED FREQUENCY-DOMAIN DATA AND SINR
For convenience, let us define the start of the lth symbol
(excluding the CP with length NG) at the time origin zero
in the time coordinate. The estimated ST can be found to be located in one of the following three regions of an OFDM symbol: the Bad-ST1 region, the Good-ST region (also known as the ISI-free region), and the Bad-ST2 region
in which the STOs are confined within the ranges of−NG ≤
nδ≤ −NG+τd−1,−NG+τd≤nδ≤0, and 1≤nδ≤N−1,
respectively. Note that the first two regions are in the guard interval. Moreover, the transmitted signal of the lth symbol can be represented as xl(t)= 1 N m Xl,mej2πmt/NTS, −NGTS≤t < NTS, (2)
where m is the transmitter subcarrier index. It is assumed that the symbol index l is the same for both the receiver
and the transmitter sides due to the ST and/or SCFO compensations. Consequently, after undergoing a multipath fading channel, the received signal can be determined as
xl(t)= τd τ=0 xl t−τTS hl(t, τ), −NGTS≤t < N + τd TS, (3)
where hl(t, τ) is the continuous time-selective CIR
experi-enced by the lth symbol. Then the overall received baseband signal, with the impairment of the CFO, can be written in the following summation form:
x(t)= l xl t−lNSTS +w(t), (4)
whereNS =N + NGis the OFDM symbol length including
the CP, xlt−lNSTS xl t−lNSTS ej2πεft/NTS, w(t) w(t)ej2πεft/NTS, (5)
andw(t) is AWGN. In (4), the summation form can clearly describe the ISI effect between two consecutively received symbols when the ST is located in the Bad-ST regions.
The desired signal and interference (due to synchroniza-tion errors and time-selective channels) in the three different ST regions are separately analyzed as follows.
3.1. Estimated ST located in good ST region
In the Good-ST region,nδis within the range of−NG+τd≤
nδ ≤ 0. The received frequency-domain data, at the kth
subcarrier, are the FFT of the received time-domain data as written below Xl,k,0=FFT xl,n+wngNn−nδ, (6) where xl,n= xlt−lNSTSt=(lNS+n)TS (7)
is the receivednth sample of the lth symbol, FFT{·}is the
FFT operation, gN(n)= ⎧ ⎨ ⎩ 1, 0≤n < N 0, otherwise, (8)
is the rectangular window function, and wn
w(t)|t=(lNS+n)TS is the discrete-time AWGN. Note that
the subscript 0 in (6) denotes the Good-ST region. With
(2)–(5) and (7), the received frequency-domain data in (6),
after some manipulations, can be found to be
Xl,k,0= Xl,k,0dsr + Nk,0, (9) where Xl,k,0dsr = Hk,0Xl,kW [lNs(kεt−εf)−knδ] N (10)
is the desired signal, and Nk,0 m /=k Xl,m 1 N N−1 n=0 Hl(n,m)Wn φm,k N WlNS(mεt−εf)−knδ N +vk (11) is the combined ICI and AWGN caused by the CFO, SCFO,
AWGN, and doubly-selective channels. Note that in (10), the
following notations are used:WN e−j2π/N, and
Hk,0 1 N N−1 n=0 Hl(n,k)W −n(εf−kεt) N (12)
is the time-averaged time-selective frequency response of the channel where Hl(n,k) τd τ=0 hl(n,τ)WNkτ (13)
is the time-selective frequency response of the channel. Also note that in (11),vk FFT{wn}, and
φm,k−m
1−εt
+k−εf (14)
is the normalized phase rotation which contains the CFO and
SCFO effects. With (10) and (1), the desired signal power is
derived inAppendix Aand rewritten here for convenience:
E Xl,k,0dsr 2 =CσX2 N−1 Δn=1−N N−ΔnJ0 βΔn WΔnφk,k N , (15) whereσ2
Xis the signal power andC
τd τ=0σh2τ/N
2 =σ2
H/N2
is the total channel power normalized by N2. Similarly, with
(11) and (1), the power of combined ICI and AWGN can be
shown to be ENk,02 =CσX2 m /=k N−1 Δn=1−N N−ΔnJ0 βΔn WΔnφm,k N +σv2, (16) whereσ2
vis the AWGN power.
3.2. Estimated ST located in bad-ST1 region
In the Bad-ST1 region,nδis within the range of−NG≤nδ≤
−NG+τd−1. Under this condition, the firstN1 = −NG+
τd−nδsamples received for the FFT operation are corrupted
with the ISI incurred from the (l−1)th symbol. Similar to (6),
the received signal on the kth subcarrier can be determined
by separating the following N-point FFT into three different
parts as
Xl,k,1= Xl,k,1 +Xl,k,1 +Xl,k,1 +vk. (17)
Note that the subscript 1 denotes the Bad-ST1 region. The first part of (17) Xl,k,1 = N1−1 n=0 τd τ=NG+n+nδ+1 1 N m Xl−1,mW −m[ψl,n,nδ−(lNS+τ)] N ×hl−1 NS+nδ+n,τ W−εfψl,n,nδ N Wkn N (18) is the N-point discrete Fourier transform (DFT) operated
on the last N1 output samples contributed by the linear
convolution of the (l−1)th symbol and the channel which
results in ISI, where×denotes multiplication and
ψl,n,nδ (lNS+n
+n
δ)TS
TS .
(19)
The second part (which contributes to ICI)
Xl,k,1 = N1−1 n=0 NG+n+nδ τ=0 1 N m Xl,mW −m[ψl,n,nδ−(lNS+τ)] N ×hl n+nδ,τ W−εfψl,n,nδ N Wkn N (20)
is the N-point DFT operated on the first N1 samples,
extracted bygN(n), from the linear convolution result of the
lth transmitted symbol’s firstτd samples with the CIR. The
third part Xl,k,1 = N−1 n=N1 τd τ=0 1 N m Xl,mW −m[ψl,n,nδ−(lNS+τ)] N ×hl n+nδ,τ WN−εfψl,n,nδ WNkn (21)
is the N-point DFT operated on the remaining N −N1
samples from the circular convolution result of the lth
transmitted symbol’sN−N1samples (i.e., from the (−NG+
τd)th to the (N−nδ −1)th samples) with the CIR. The
remaining derivation is detailed in Appendix B, and final
results are rewritten here:
E Xdsr l,k,1 2 =Cσ2 X N−N1−1 Δn=−(N−N1−1) N−N1−ΔnJ0 βΔn Wφk,kΔn N (22)
is the desired signal power, and E Nk,1 2 =Cσ2 X m /=k N−N1−1 Δn=−(N−N1−1) N−N1−ΔnJ0 βΔn Wφm,kΔn N +CσX2 m N1−1 Δn=−(N1−1) N1−ΔnJ0 βΔn Wφm,kΔn N + 2σ 2 X N2 m /=k N−1 n1=N1 N1−1 n2=0 J0 βΔn Wφm,kΔn N nδ+NG+n2 τ=0 σ2 hτ+σ 2 v (23) is the power of the combined interference (including ISI and ICI) and AWGN.
3.3. Estimated ST located in bad-ST2 region
In the Bad-ST2 region,nδis within the range of 1≤nδ≤N−
1. Since the derivation is similar toSection 3.2, it is omitted
here. The desired signal power can be found to be
E Xl,k,2dsr 2 =CσX2 N−nδ−1 Δn=−(N−nδ−1) N−nδ−ΔnJ0 βΔn Wφk,kΔn N , (24) E Nk,2 2 =CσX2 m /=k N−nδ−1 Δn=−(N−nδ−1) N−nδ−ΔnJ0 βΔn Wφm,kΔn N +CσX2 m nδ−1 Δn=−(nδ−1) nδ−Δn)J0 βΔn Wφm,kΔn N + 2 σ2 X N2 × m /=k N−nδ−1 n1=0 N−1 n2=N−nδ J0 βΔn Wφm,kΔn N τd τ=−N+nδ+n2+1 σ2 hτ+σ 2 v (25) is the power of the combined interference and AWGN.
3.4. SINR analysis
Finally, based on the results in Sections3.1to3.3, the SINR
can be written as ηk,r= E Xdsr l,k,r 2 E Nk,r 2, (26)
wherer=0, 1, 2 denotes those three different ST regions.
As shown inAppendix C, an interesting observation is
that the ICI powers (16) are approximately the same when
fdT=
√
2εf. It can be easily verified that this is also true for
the desired signal power in all of the three ST regions. So are the SINRs.
4. MORE COMPACT RESULTS
By utilizing the fact that
m /=k W−Δn(m−k) N = ⎧ ⎨ ⎩ −1, Δn=/0 N−1, Δn=0, (27)
and both (24) and (25) are even functions ofΔn, given that
the SCFO is negligible, one can reduce (24) and (25) to a
more simpler form as
E Xl,k,rdsr 2 =Cσ2 X N−nδ + 2Cσ2 X N−nδ−1 Δn=1 N−nδ−Δn ×J0 βΔn cos 2πΔ nεf N , E Nk,r 2 =CσX2 N(N−1) +nδ −2CσX2 × N−nδ−1 Δn=1 N−nδ−Δn J0 βΔn cos 2πΔ nεf N −2σ 2 X N2 N−nδ−1 n1=0 N−1 n2=N−nδ J0 βΔn W−Δnεf N τd τ=−N+nδ+n2+1 σh2τ+σ 2 v. (28) It is shown that both compact forms are independent of the subcarrier index. By contrast, the SINR depends on the subcarrier index under the influence of the SCFO. Note that
this result can be applied to the cases ofr = 0 (by setting
nδ=0) andr=2.
To gain further insight into (28), the SIR ρ, under the
influence of STO alone, and the influence of combined CFO and NDF, can be respectively reduced to
ρSTO= (N−nδ)2 (2N−nδ)nδ−2((N−nδ)/σH2)X , fdT=εf =0, (29) whereX denotesNn2−=1N−nδ τd τ=−N+nδ+n2+1σ 2 hτ, and ρCFO&NDF= N + 2Y N(N−1)−2Y, nδ=0, (30)
whereY denotesNΔn−=11(N−Δn)J0(βΔn) cos(2πΔnεf/N).
Note that based on our derivation, the result in [1,
Equation (17)] can be further reduced to a more concise
form as (30), and the result in [2, Equation (2)] is the same
as (29).
With (30) and Taylor’s series of the cosine function, after
some manipulations and the fact that N2 1, the SINR
under the influence of the CFO can be shown to be
ηCFO≈ 6−2π2(ε f)2 π2(ε f)2+ 6/γ , fdT=nδ=0, (31) whereγ is SNR.
0 5 10 15 20 25 30 SINR (dB) This work, SNR=23 dB This work, SNR=29 dB Sim., SNR=23 dB Sim., SNR=29 dB The work in [8], SNR=23 dB The work in [8], SNR=29 dB 0 0.05 0.1 0.15 0.2 0.25 CFO
Figure 2: SINR plotted against CFO, under SNR=23 and 29 dBs.
14 15 16 17 18 19 20 21 22 23 SIR (dB) −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 STO Anal.fdT=0.06 Anal.fdT=0.07 Anal.fdT=0.08 Anal.εf =0.0424 Anal.εf =0.0495 Anal.εf =0.0566 Sim.fdT=0.06 Sim.fdT=0.07 Sim.fdT=0.08 Sim.εf =0.0424 Sim.εf =0.0495 Sim.εf =0.0566
Figure 3: SIR plotted against STO, under the influences of the CFO and NDF.
To verify the concise result of (31), the SINR as a
function of the CFO is shown in Figure 2. The result
in [10, Equation (15)] and the simulation result are also
included for validation, assuming quadrature phase-shift
keying (QPSK) modulation,N = 256, γ = 23 and 29 dBs.
As can be seen, the derived result (31) is more accurate than
that in [10, Equation (15)]. 10 20 30 40 50 60 70 80 SIR (dB) 0 0.5 1 1.5 2 2.5 3 3.5 4 STO N=512,εt=10 ppm N=512,εt=15 ppm N=512,εt=20 ppm N=32,εt=10 ppm N=32,εt=15 ppm N=32,εt=20 ppm Figure 4: SIR plotted against STO under the influence of the SCFO. NDF=CFO=0. Subcarrier index=6.
5. NUMERICAL AND SIMULATION RESULTS
In the simulation, an OFDM system, with N = 256
sub-carriers and a guard interval of NG = N/8 = 32
samples, is considered. The adopted modulation scheme is QPSK. The signal bandwidth is 2.5 MHz, and the radio frequency is 2.4 GHz. The subcarrier spacing is 8.68 kHz. The
OFDM symbol duration is 115.2μs. The maximum delay
spread τd of the channel is 24 samples. The channel taps
are randomly generated by independent zero-mean
unit-variance complex Gaussian variables withτE{|hl(τ)|2} =
1 for each simulation run. In each simulation run, 10 000 OFDM symbols are tested. The same channels are used for both the numerical and simulation analyses. All the results are obtained by averaging over 2000 independent channel realizations.
The following example demonstrates some design
con-straints to achieve the typical condition of SIR> 20 dB. The
SIR curves under the joint effects of the STO, NDF, and
CFO are shown inFigure 3. As shown, for the condition of
SIR > 20 dB to be satisfied, the NDF should be less than 8%
as observed in [1], and the CFO should be less than 6%. This
figure also shows that the SIRs are the same when fdT =
√
2εf. To achieve SIR > 20 dB, the STO, when fdT = 0.00,
should be less than 8 samples.
As can be seen in Figure 3, the SIRs are 22.2 dB and
20.9 dB due to the single error of NDF= 0.06 and STO = 6
(samples), respectively. However, when both errors of
NDF= 0.06 and STO = 6 coexist, the SIR drops to 18.5 dB.
The degradation due to the combined synchronization errors is 3.7 dB more than the single error of NDF, while 2.4 dB more than the single error of STO. Therefore, the degrada-tion of the SINR due to the combined synchronizadegrada-tion errors may be much more severe than a single synchronization error.
The SIR curves under the joint effects of the STO
and SCFO are shown in Figure 4. When the STO= 0, and
under the same SCFO condition, the SIR deteriorates as N
increases; on the contrary, when the STO /=0, the SIR also
decreases as N decreases, because there are less numbers of subcarriers. In other words, the impact on performance due to the STO is more apparent for a smaller N than a larger N.
It can also be seen that the SCFO has a very minor effect on
the SIR. Moreover, effect of the STO is much more significant
than that of the SCFO.
6. CONCLUSION
The impacts of the combined synchronization errors have been analyzed. It has been found that the NDF and CFO
have the same impacts on the SIR when fdT =
√
2εf.
Due to impairments of the synchronization algorithms, the tolerance regarding those synchronization errors should be taken into consideration, especially in a mobile environment. In addition, it has also been found that the effect of the combined synchronization errors on the SINR may be much more severe than a single synchronization error. Therefore, it is beneficial to study the effects of combined synchronization errors. The derived results can be used as design guidelines for devising suitable synchronization algorithms in doubly-selective fading channels.
APPENDICES
A. DERIVATION OF THE SIGNAL POWER OF (15) FOR
THE GOOD-ST REGION INSECTION 3.1
Since the channel fading characteristic is independent of the
transmitted data, the signal power (15) can be found to be
E Xl,k,0dsr 2 = 1 N2E Xl,k2 N−1 n1=0 N−1 n2=0 EHl n1,k Hl n2,k ∗ Wφk,k(n1−n2) N . (A.1)
With (13) and (1), the correlation of the time-selective
transfer function of the channel in (A.1) can be found to be
EHl n1,k Hl n2,k ∗= J0 βn1−n2 τd τ=0 σ2 hτ. (A.2)
Finally, by inserting (A.2) into (A.1), and knowing thatΔn=
n1−n2, the signal power can be shown to be
E Xl,k,0dsr 2 =Cσ2 X N−1 Δn=1−N N−ΔnJ0 βΔn Wφk,kΔn N , (A.3)
whereσX2 is the transmitted signal power andC
τd τ=0σh2τ/
N2=σ2
H/N2is the total channel power normalized byN2.
B. DETAILED DERIVATION OF SIGNAL AND INTERFERENCE POWERS FOR THE BAD-ST1 REGION INSECTION 3.2
From (17), we can separate the desired signal, and the
combined interference and AWGN as
Xl,k,1= Xl,k,1dsr +Nk,1, (B.1) where Xl,k,1dsr = Hk,1Xl,kW [lNs(kεt−εf)−knδ] N (B.2)
is the desired data,
Hk,1 1 N N−1 n=N1 Hl n+nδ,k W−n(εf−kεt) N (B.3)
is the time-averaged time-selective transfer function of the channel, and
Nk,1= Xl,k,1 +Xl,k,1 + Xl,k,1 − Xl,k,1dsr
+vk (B.4)
is the combined interference (caused by the STO, CFO,
SCFO, and time-selective channels) and AWGN. With (B.2),
(B.3), (13), and (1), it can be shown that
E Xl,k,1dsr 2 =Cσ2 X N−N1−1 Δn=−(N−N1−1) N−N1−ΔnJ0 βΔn Wφk,kΔn N . (B.5) Since transmitted data of different symbols are independent, the power of the combined interference and AWGN can be determined as E Nk,1 2 =E Xl,k,1 2 +E Xl,k,1 − Xl,k,1dsr 2 + 2EXl,k,1 Xl,k,1 − Xl,k,1dsr ∗ +E Xl,k,1 2 +σ2 v. (B.6) After some manipulations, it can be shown that
E Xl,k,1 2+E Xl,k,1 2 =Cσ2 X m N1−1 Δn=−(N1−1) N1−ΔnJ0 βΔn Wφm,kΔn N , E Xl,k,1 − Xl,k,1dsr 2 =CσX2 m /=k N−N1−1 Δn=−(N−N1−1) N−N1−ΔnJ0 βΔn Wφm,kΔn N , EXl,k,1 Xl,k,1 − Xl,k,1dsr ∗ = σX2 N2 m /=k N−1 n1=N1 N1−1 n2=0 J0 βΔn Wφm,kΔn N nδ+NG+n2 τ=0 σ2 hτ. (B.7)
Finally, by inserting (B.7) into (B.6), the power of the combined interference and AWGN can be written as
E Nk,1 2 =Cσ2 X m /=k N−N1−1 Δn=−(N−N1−1) N−N1−ΔnJ0 βΔn Wφm,kΔn N +CσX2 m N1−1 Δn=−(N1−1) N1−ΔnJ0 βΔn Wφm,kΔn N + 2σ 2 X N2 m /=k N−1 n1=N1 N1−1 n2=0 J0 βΔn Wφm,k N nδ+NG+n2 τ=0 σ2 hτ+σ 2 v. (B.8)
C. THE RELATIONSHIP OF THE NDF AND CFO THAT EXHIBITS THE SAME ICI POWER IN (16)
In the following, we will find the condition when NDF has the same impact on the ICI power with the CFO.
With (16), the ICI powers under the influence of the NDF
(without the CFO) and CFO (without the NDF) are
Cσ2 X N−1 Δn=−(N−1) N−ΔnJ0 βΔn W−Δn[m(1−εt)−k] N , (C.1) Cσ2 X N−1 Δn=−(N−1) N−ΔnW− Δnεf N WN−Δn[m(1−εt)−k], (C.2)
respectively. When (C.1) equals (C.2), the zeroth-order
Bessel function of the first kind J0(·) has the same value
with the complex exponential function WN = e−j2π/N. In
addition, the Taylor series of the zeroth-order Bessel function of the first kind and the complex exponential function are
J0 x1 =1− x1/2 2 (1!)2 + x1/2 4 (2!)2 − x1/2 6 (3!)2 +· · ·, (C.3) e(x2)=1 +x2 1! + x2 2 2! + x3 2 3! +· · ·, (C.4)
respectively, wherex1 =2π fdTΔn/N and x2 = j2πΔnεf/N.
Since Δn ranges from−(N −1) to N−1, the odd power
terms (and pure imaginary) of (C.4) will be cancelled when
they are inserted into (C.2). Furthermore, since fdT and
εf are typically less than 10−1, (C.3) and (C.4) can be
well approximated by the first two terms. As a result, the
condition of (C.1)= (C.2) implies that
1− x1/2 2 (1!)2 =1 + x2 2 2! (C.5)
which leads to the result of fdT=
√
2εf. SUMMARY OF NOTATIONS
Since there are so many notations used in this work, for clarity, the notations are collectively defined and summarized
in this section. Please note that subscriptsl, r, k (or m), n
denote the lth symbol, rth ST region, kth (or mth) subcarrier, and nth sample, respectively.
δ(·): Dirac delta function
ηk,r: SINR
ρ: Signal-to-interference ratio (SIR)
γ: Signal-to-noise ratio (SNR)
σ2
hτ: Power of theτth channel tap
σ2
v: Additive white Gaussian noise (AWGN) power
σX2: Transmitted signal power
Δn: Time difference
τd: Maximum delay spread of the channel
τ: Path delay of the channel
β: 2π fdT/N
φm,k: Normalized phase rotation which contains the
CFO and SCFO effects
(·)N: Modulo N operation
εf: CFO εt: SCFO
cos(·): Cosine function
C: τd τ=0σh2τ/N
2 = σ2
H/N2, total channel power
normalized by N2
fc: Carrier frequency
fd: Maximum Doppler shift in Hertz
FFT{·}: Fast Fourier transform (FFT) operation
gN(n): Rectangular window function
hl(n, τ): τth channel tap of the discrete time-variant
channel impulse responses (CIR)
hl(t, τ): τth channel tap of the continuous-time
time-variant channel impulse responses (CIR)
Hk,r: Time-averaged time-variant transfer function of
the channel
Hl(n, m): Time-variant transfer function of the channel
J0(·): Zeroth-order Bessel function of the first kind
nδ: STO
N: Number of subcarriers
NG: Cyclic prefix (CP) length
NS: OFDM symbol length including the CP
Nk,r: Combined interference and AWGN
N1: Length of corrupted samples when the symbol
time is located in the Bad-ST1 region (please see
Section 3.2)
T: Symbol duration including the CP
1/TS: Transmitter’s sampling frequency
1/TS: Receiver’s sampling frequency
vk: AWGN at the kth subcarrier
w(t): Continuous-time AWGN
w(t): AWGN affected by the CFO
wn: Discrete-time AWGN
WN: e−j2π/N, twiddle factor
xl(t): Transmitted time-domain signal
xl(t): Time-domain signal under the influence of the
channel
xl(t): Time-domain signal under the influence of CFO
x(t): Overall received baseband signal
xl,n: Received time-domain data
Xl,k: Transmitted frequency-domain data
Xl,k,r: Received frequency-domain data
ACKNOWLEDGMENT
The authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions in improving the quality of this paper.
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