On the Diversity Order of BICM-OFDM in Doubly Selective Fading Channels
Hsin-De Lin, Student Member, IEEE, and Tzu-Hsien Sang, Member, IEEE
Abstract—Bit-interleaved coded modulation with orthogonal
frequency-division multiplexing (BICM-OFDM) is an attractive approach to achieve time and frequency diversity. Remarkable diversity gain can be ob-tained when the channel is doubly selective fading. In this paper, the asymptotic diversity orders of BICM-OFDM systems in doubly se-lective fading channels for both single-input–single-output (SISO) and multiple-input–multiple-output (MIMO) cases are derived. In addition, the system bit-error-rate (BER) behavior in practical situations with mod-erate signal-to-noise ratios (SNRs) is also investigated. In the SISO case, the diversity order depends on the rank of the channel correlation matrix. Therefore, the channel variations induced by fast fading contribute to improving diversity. In the MIMO case, the diversity order can be further increased when factors such as cyclic delays or phase rolls are introduced. Simulations are provided to verify the analysis.
Index Terms—Bit-interleaved coded modulation (BICM), diversity
or-der, doubly selective fading channel, orthogonal frequency-division multi-plexing (OFDM).
I. INTRODUCTION
Wireless communication faces the ever-present demand for higher data rates with more efficient spectrum usage while maintaining good quality of service in high motion speeds. Two transmission tech-niques that are very popular in that regard are bit-interleaved coded modulation (BICM) and orthogonal frequency-division multiplexing (OFDM). BICM counters fading channels by spreading code-word bits in time to exploit the time diversity available in the time-varying channel response [1], [2], whereas OFDM claims high bandwidth efficiency and simplicity in receiver design.
Doubly selective fading channels in many situations have been viewed as the cause of severe channel impairments, e.g., the inter-carrier interference (ICI), and many techniques, such as ICI-canceling equalizers, have been developed to mitigate the problem [3]. However, it is also considered as a potential source of time and frequency diver-sity, which may enhance the system performance [4]–[6]. Pioneering work by Ma and Giannakis [7] considers the maximum diversity order over doubly selective channels for general block transmission systems without forward error correction (FEC). Recent papers have started investigating time diversity for coded systems. Huang et al. [8] and Liu and Fitz [9] have reported that, in simulations, the performance of BICM-OFDM systems is improved when the channel is fast fading. In [10], the diversity order of BICM-OFDM systems in frequency selective, but not doubly selective, fading channels is analyzed.
This paper aims to extend these pioneering works and provide an analytic study of the diversity order of BICM-OFDM systems
Manuscript received April 30, 2011; revised May 3, 2012, September 17, 2012, and November 26, 2012; accepted December 2, 2012. Date of publication December 19, 2012; date of current version May 8, 2013. This work was supported by the National Science Council of the Republic of China under Grant NSC 101-2220-E-009-003, Grant NSC 101-2220-E-009-058, and Grant NSC 101-2219-E-009-021. The review of this paper was coordinated by Prof. M. Uysal.
The authors are with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: good.ee92g@nctu.edu.tw; tzuhsien54120@faculty.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2012.2235091
Fig. 1. Block diagram of the transmitter and the receiver of the considered system. Soft-input–soft-output demodulator consists of the ICI equalization such as the feedback canceler [8] and the demapper.
over doubly selective fading channels. First, we derive the asymptotic diversity order of BICM-OFDM systems by studying the role of the channel autocorrelation matrix in rank analysis of the typical derivation of diversity order. Second, the effect of significant eigen-values of the channel autocorrelation matrix and the diversity order in realistic situations with moderate SNRs are examined by studying the channel autocorrelation function (ACF) and its Fourier dual. Finally, our analysis framework is extended to multiple-input–multiple-output (MIMO) cases while incorporating more diversity techniques, such as cyclic delay diversity (CDD) and phase-roll diversity (PRD) [11]. This MIMO extension can also be applied in a distributed fashion, for example, in cooperative communications [12].
II. DIVERSITYORDERANALYSIS A. System Model
A block diagram of the BICM-OFDM system under study is shown in Fig. 1. Information bits inare first encoded by a channel encoder,
and the coded bits cn∈ C are passed to interleaver Π, where C is the
set of all possible code words. The interleaved bits cnare divided into
P blocks and modulated using quadrature-amplitude modulation or phase-shift keying (PSK), where γ bits are mapped into one of the total 2γ constellation points. Finally, an N -point inverse discrete Fourier
transform generates the transmitted OFDM signal xn. The size of the
interleaver is γN P bits. Note that the encoded bits are interleaved across several OFDM symbols as in [8] (which calls it time–frequency interleaving). Considering the time-varying dispersive channels, the sampled baseband equivalent received signal yp(k) of the pth OFDM
symbol is yp(k) = L−1 l=0 hp(k; l)xp(k− l) + wp(k) (1)
where hp(k; l) represents the lth delay path of the multipath Rayleigh
fading channel at the kth sampling instant of the pth OFDM sym-bol with E[Ll=0−1|hp(k; l)|2] = 1, L is the number of multipaths,
xp(k) is the transmitted signal, and wp(k) is the additive white
Gaussian noise. The channel {hp(k; l)} is assumed to be
wide-sense stationary uncorrelated scattering. The autocorrelation function (ACF) of the channel in time for the lth path is defined as r(m) = E[hp(k; l)h∗p+m(k; l)].
Assume that the cyclic prefix (CP) is long enough to prevent intersymbol interference and that the synchronization is perfect. The received symbol in the frequency domain after CP removal is
yp= F HpFHxp+ wp
Δ
= Gpxp+ wp, p = 1, 2, . . . , P, (2)
where yp= [Y (Np), Y (Np+ 1), . . . , Y (Np+ N− 1)]T ∈ CN×1is
the frequency-domain signal after discrete Fourier transform (DFT); Np= (p− 1)N marks the starting point of the pth OFDM symbol;
[F ]k,n= (1/
√
N )e−j2πnk/Nis the (k, n)th element in the DFT ma-trix F ; xp= [X(Np), X(Np+ 1), . . . , X(Np+ N− 1)]T ∈ CN×1
is the transmitted coded signal in the frequency domain with the average power normalized to 1; wp= [W (Np), W (Np+
1), . . . , W (Np+ N− 1)]T∈ CN×1represents the noise; and Hp∈
CN×N is the channel impulse response (CIR) matrix given by
[Hp]i, j= hp(i; (i− j) mod N) , 1, j = 0, 1, . . . , N − 1. (3)
Note that the CIR hp(k; l) is zero when l≤ 0 or l ≥ L − 1. The
off-diagonal terms in Gpare the ICI coefficients, where the (r, s)th
element can be expressed as
[Gp]r, s= 1 N N−1 k=0 L−1 l=0 hp(k; l) exp (j2πk(s− r)/N) × exp(−j2πsl/N). (4) The channel matrix Gp can be decomposed as Gp= Gdp+ G
o p,
where Gd
pcontains the diagonal terms of Gp, and Gopconsists all the
off-diagonal terms of Gp. Note that Gdpcorresponds to the DFT of
˜
hp= [˜hp(0), . . . , ˜hp(L− 1)]T, the averaged CIR in the pth OFDM
symbol, i.e., ˜hp(l) = (1/N )
N−1
k=0 hp(k, l). Meanwhile G o p
con-tributes to ICI terms. We can rewrite (2) as
yp= Gdpxp+ Gopxp+ wp= diag(xp)FN×Lh˜p+ zp (5)
where FN×Lis an N× L matrix composed of the first L columns of
F , and zpdenotes the interference-plus-noise terms.
The variation of doubly selective fading channels within one or across several OFDM symbols can provide some benefits. The in-trasymbol channel variation also causes ICI. It has been observed that the diversity order slightly increases if ICI is utilized at the receiver [4]–[6], [9], but the high complexity remains a concern. In this paper, we focus on the much more pronounced benefit provided by the intersymbol channel variation, assuming that ICI is handled by simple techniques such as the decision-feedback canceler [8] with FEC to improve the decision accuracy. In particular, we assume that any residual ICI will not ruin the possible diversity gain provided by intersymbol channel variation. This assumption is reasonable even if the normalized Doppler frequency fdTS (TS is one OFDM symbol
duration) is as high as 0.1, about which many ICI mitigation methods can reach around 30 dB of signal-to-interference ratio without the help of FEC [13], [14]. As a result, the error floor due to residual ICI will not kick in until a very high SNR.
With this focus in mind, the stacked received signal y = [yT
1, yT2, . . . , yT P] Tcan be given as y = X(IP⊗ FN×L)h + z Δ = Xheq+ z (6) in which heq Δ = (IP⊗ FN×L)h, IP is a P× P identity matrix; ⊗
denotes the Kronecker product; h is a length-P L vector defined as h = [˜hT1, ˜h
T
2, . . . , ˜h
T
P]T; X is a P N× P N diagonal data signal
ma-trix given by X = diag(X(0), . . . , X(N− 1), X(N), . . . , X(2N − 1), . . . , X(P N− 1)); and z is a P N × 1 noise vector representing the residual ICI plus noise.
B. Asymptotic Analysis
Here, we derive the asymptotic diversity order of BICM-OFDM by first bounding the pairwise error probability (PEP). Let X be the coded transmit signal corresponding to code word c and ˆX be the detected signal corresponding to code word ˆc, where c= ˆc. To gain diversity, we adopt the assumption in [10] that the interleaver can disperse d consecutive coded bits to different symbols and onto differ-ent OFDM subcarriers, whereas d bits cover the span of consecutive trellis branches on which dfreedistinct bits of any two code words occur. The assumption can be satisfied by practical interleavers, for instance, the block interleaver of IEEE 802.11a [15]. Assume that z is complex Gaussian distributed with zero mean and variance N0. Evoke the maximum-likelihood (ML) decision rule, and the conditional error probability P (c→ ˆc|h) is P ( z 2≥ z + (X − ˆX)h
eq 2|h). It can be expressed by the Q-function as
Q ⎛ ⎝ (X− ˆX)heq 2 2N0(X− ˆX)heq 2 ⎞ ⎠= Q 1 2N0 (X− ˆX)heq (7) where Q(x) = (1/√2π) ∞ x e −(y2/2) dy. The PEP is obtained by ensemble averaging, i.e.,
P (c→ ˆc) = E Q 1 2N0 (X− ˆX)heq = E Q SNR· hHeqD H Dheq 2 (8)
where SNR = 1/N0since the average power of channel and transmit-ted signal are normalized to 1, and D = X− ˆX is the coded symbol difference matrix. According to the assumption of the interleaver, at least dfreenonzero terms exist in D [10]. In the following, we consider the worst case that only dfreeterms are nonzero in D, and each of these terms has Euclidean distance dmin. Note that the parameters of BICM-OFDM are usually chosen as N≥ L and N ≥ dfree.
A typical PEP analysis such as those in [10] and [16] assumes that the vector heqhas independent elements and the rank analysis is focused on the term DHD. Here, however, the correlation of channels
over several OFDM symbols needs to be considered. We adopt the approach in [7] to extract the statistical independent components in heq, which are considered as the source of diversity. After this extraction, the usual PEP analysis can proceed.
Assume the Kronecker model [11], [17], [18] for the channel, i.e., the channel autocorrelation matrix can be decoupled as
R = E[hhH] = Φ
T⊗ ΦL (9)
where ΦT is the P× P time autocorrelation matrix constructed by
stacking the windowed ACFs, i.e., the kth row [ΦT]k,1:P = [r(1− k),
r(2− k), . . . , r(P − k)]. ΦL is the autocorrelation matrix of the
L× L path gain and constructed as a diagonal matrix diag(σ2 0, σ12,
. . . , σ2
L−1) due to the uncorrelated scattering assumption, where σl2
denotes the power of the lth path.
The rank of R can be evaluated through [19, Fact 7.4.20] rank(R) = rank(ΦT⊗ ΦL)
= rank(ΦT)× rank(ΦL) = rT× L (10)
where rT and L are the ranks of ΦT and ΦL, respectively. Note
that rT× L is bounded by P × L. The autocorrelation matrix of heq is Req= E[heqhHeq] = (IP⊗ FN×L)E[hhH](IP⊗ FN×L)H=
matrix (IP⊗ FN×L) also has full rank P L. Therefore, by the rank
property of matrix products [19, Prop. 2.6.2],1the rank of R eqis rank(Req) = rank (IP⊗ FN×L)R(IP⊗ FN×L)H = rank(R) = rT× L. (11)
The eigenvalue decomposition is used to extract the statistically independent components in heq. Consider Req= V ΣhVH, where
V is a P N× rTL matrix satisfying VHV = IrTL and Σh= diag(ξ2
1, . . . , ξr2TL). In the following analysis, heqis substituted by V Σ1/2h h¯eq, where ¯heqis the rTL× 1 normalized equivalent channel
vector containing independent and identically distributed (i.i.d.) zero-mean complex Gaussian random variables with unit variance. It can be shown that the PEP is not affected by this substitution since heq and V Σ1/2h h¯eq have identical distributions, which is known as the isotropy property of the standard Gaussian random vector.
The term hHeqDHDheqin (8) now becomes ¯h
H
eq(Σ 1/2
h )
HVHDH
DV Σ1/2h h¯eq in which the central part (Σ 1/2
h )
HVHDHDV Σ1/2
h
is a Hermitian matrix and can be diagonalized as (Σ1/2h )HVHDH
DV Σ1/2h = U ΛUH, where U is unitary, r
T V is its rank, and Λ =
diag{λ1, . . . , λrT V} contains the eigenvalues. Moreover, define ˘h = UHh¯
eq, and notice that ˘h is i.i.d. complex Gaussian with zero mean and unit variance since U is unitary. Equation (8) becomes
P (c→ ˆc) = E ⎡ ⎣Q ⎛ ⎝ SNR· ˘hHΛ˘h 2 ⎞ ⎠ ⎤ ⎦ = E ⎡ ⎣Q ⎛ ⎝ SNR 2 rT V n=1 λn|˘hn|2 ⎞ ⎠ ⎤ ⎦ (12)
where ˘hnis the nth element in ˘h, and|˘hn| is Rayleigh distributed. The
PEP is bounded by [16] P (c→ ˆc) ≤ E exp −SNR 4 rT V n=1 λn|˘hn|2 =rT V 1 n=1[1 + (λnSNR/4)] . (13)
When SNR is large enough, the bound in (13) can be further simplified to P (c→ ˆc) ≤ r T V n=1 λn −1 SNR 4 −rT V . (14)
In addition, it follows that the asymptotic diversity order is rT V. As for
the value of rT V, since V Σ
1/2
h has rank rTL and the diagonal matrix
D has rank dfree, it follows that rT V ≤ min(rTL, dfree), i.e., the max-imum asymptotic diversity order is no larger than min(rTL, dfree).
1Reference [19, Prop. 2.6.2] states the following: Let A∈ Cn×m, and let C∈ Ck×n be left invertible (or equivalently, rank(C) = n) and B∈ Cm×lbe right invertible (or equivalently, rank(B) = m); then, rank(A) = rank(CA) = rank(AB). It is straightforward that rank(CAB) = rank(A). Since rank(IP⊗ FN×L) = rank((IP⊗ FN×L)H) = P L, (11) can be obtained by designating C = (IP⊗ FN×L), A = R, and B = (IP⊗
FN×L)H.
Fig. 2. BER performance with 8-PSK modulation under two-multipath (L = 2) Rayleigh block fading channels with two different path-gain distri-butions. The DFT size is 64, P = 2, and a rate-1/2 convolutional code with the generator polynomial [133; 171] (dfree= 10) is adopted. 105 channel
realizations are simulated. C. Practical Diversity Gains
Next, we consider the slope of the BER curve that is observed in more realistic situations. The full slope of rT V = min(rTL, dfree) from asymptotic analysis is achieved when SNR approaches infinity. In practical situations, smaller eigenvalues of Reqmay not have the opportunity to contribute to steepening the slope before the effect of other impairments, such as residual ICI, kicks in. As a result, the actual slope that is observed at moderate SNR would not be as large as the asymptotic slope. A similar phenomenon caused by dom-inant eigenvalues in the context of channel estimation has also been reported [20].
Simulations are conducted to demonstrate the effect of the relative sizes of eigenvalues on the observed slope. Consider a BICM-OFDM system over a block fading channel. Here, we did not use doubly selective fading channels because the magnitudes of eigenvalues need to be controlled to show their effects, and it is very difficult to do so for a doubly selective fading channel, whereas in a block fading channel, it can be easily done by setting the path gains. Fig. 2 shows two BER curves under two-multipath (L = 2) Rayleigh block fading channel with two different path gain settings. The system uses 8-PSK modulation, the DFT size is 64, P = 2, and a rate-1/2 convolutional code with the generator polynomial [133;171] (dfree= 10) is adopted such that the diversity order is not limited by dfree. 105 channel realizations are simulated. Two settings of eigenvalues of ΦLare used:
The first consists of (0.995, 0.05), and the second consists of (0.5, 0.5). The full slope is−4 for both cases, but for the first setting, we expect the slope will only reach−2 when the SNR is moderate since only one eigenvalue is significant. As shown in Fig. 2, in the SNR range from 10 to 20 dB, the slopes are−2 and −4, respectively, and the first setting’s slope does not reach−4 until around SNR = 25 dB.
The relation between the practical diversity order and the channel ACF can be further expounded. Consider the number of significant eigenvalues of Req in (11). Since Req can be approximated to be circulant when N is large, its eigenvalues can be obtained by ap-plying DFT to it. It is essentially time-windowing the channel ACF, doing Fourier transform, and sampling the result. Therefore, the faster the channel changes, the narrower the ACF becomes relative to the observation window (the code block length), and a wider Fourier dual
Fig. 3. Eigenvalues are obtained by sampling the convolution of the Doppler spectrum and the sinc function in the frequency domain, which can be recog-nized as the observed Doppler spectrum. The curves show the effect of different window lengths.
and more significant samples follow. Consequently, the diversity order gets larger. Two contrasting examples are shown in Fig. 3, where the observed Doppler spectra are obtained by convolving the Doppler spectra with a sinc function, corresponding to the Fourier transform of time-windowed channel ACFs. After obtaining the observed Doppler spectra, the eigenvalues can be obtained by sampling them.
D. Simulation Results
Back to doubly selective fading channels, Fig. 4 shows the BER performance of a BICM-OFDM system under time-varying chan-nels with two equal-gain paths (L = 2). The experiments are con-ducted with three Doppler spectra and different normalized Doppler frequencies. The system uses 8-PSK modulation, the DFT size is 64, P = 10, and a rate-1/2 convolutional code with the generator polynomial [133; 171] (dfree= 10) is adopted such that the system performance is not limited by dfree. 105 channel realizations are simulated. Fig. 4 verifies our predictions on the diversity orders and the effectiveness of the BICM-OFDM systems. For example, by considering the case of Jakes’ model with fdTS= 0.1, the six largest
eigenvalues of time ACF are 3.96, 3.44, 2.23, 0.349, 0.0162, and 0.0004. The first three eigenvalues contain 99.67% of total channel power. As can be expected, the practical diversity order is limited by rT× L = 3 × 2 = 6. When fdTS is 0.05, the practical diversity
order is further reduced to rT× L = 2 × 2 = 4 since the number of
significant eigenvalues of ΦTis cut to 2. The results confirm that larger
channel variations lead to higher diversity orders. The case of carrier frequency offset (CFO) is also shown to demonstrate channel variation introduced that does not contribute to diversity. The ACF of this case has [ΦT]k,1:P= [ej2π(1−k)fdTS, ej2π(2−k)fdTS, . . . , ej2π(P−k)fdTS]
at the kth row. Alternatively, ΦT can be rewritten as ([1, e−j2πfdTS,
. . . , e−j2π(P −1)fdTS]T⊗ [1, ej2πfdTS, . . . , ej2π(P−1)fdTS]), which results in a rank-one matrix, and no time diversity gain is available.
Note that, in doing the given simulations, relatively simple channel estimation is required in the sense that only the time-averaged channel within each OFDM symbol needs to be estimated for the receiver to function. The study of channel estimation of OFDM over doubly selective channels has been very active, and many efficient methods are available [4], [13], [20]. Subsequently, efficient ICI cancelation can be done with many methods found in, e.g., [8], [13], and [14].
Fig. 4. Comparison of the diversity gain provided by time-varying channels with three kinds of Doppler power spectral density (PSD): Jakes’ model, uni-form PSD, and CFO. The normalized Doppler frequencies of 0.01, 0.05, and 0.1 are simulated. The path gains hp(k; l) for different l values are assumed independent. The time variation of the channel E[hp(k; l)h∗p(m; l)] is J0(2πfd(k− m)T ) · δ(l − l) for Jakes’ model, sin(2πfd(k− m)T )/ (π(k− m)T ) · δ(l − l) for uniform PSD, and exp(j2πfd(k− m)T ) · δ(l− l) for CFO, where fd is the maximum Doppler frequency, T is the OFDM sampling time, J0(·) is the zeroth-order Bessel function of the first
kind, and δ(·) is the Kronecker delta function. Notice that the considered diversity here is the effective diversity order based on the dominant eigenvalues.
III. EXTENSION TO THE MULTIPLE-INPUT–MULTIPLE-OUTPUTCASE
The transmission scheme can be extended to MIMO scenarios to benefit from additional gain from spatial diversity. As has been shown, BICM-OFDM can effectively capture time and frequency diversity; to further take advantage of spatial diversity in MIMO situations, a common way is to transform the spatial diversity to time or frequency diversity [11]. In the following, we consider two techniques that achieve this goal with little modifications and overhead cost with respect to BICM-OFDM design for SISO cases. We choose a different approach from the obvious choice of combining BICM-OFDM with space–time coding (STC), which potentially gives an even larger diversity order of NT× min{L, dfree} [10], out of the considerations that the simple receivers often used for Alamouti-like schemes suffer greatly in double selective fading environments and the ML decoders for general STC usually carry prohibitively high cost [21]–[23].
A. CDD
The first example is the CDD method used in OFDM systems [24] to transform spatial diversity to multipath diversity (or recognized as frequency diversity), which is done by inserting different cyclic delays to the signal at each transmit antenna. The diversity order is analyzed via the framework established in Section II-B; in essence, the asymptotic diversity order is determined by the rank of Req. For simplicity, the case of two transmit antennas and one receive antenna is considered, and the results can be extended to more general cases. Starting from the SISO signal model of (2), inserting an intentional delay Δ is equivalent to multiplying a circular shift matrix P =
0 IΔ
IN−Δ 0
be F HpP FHxp, and we can combine P and Hpinto an equivalent
channel. Without loss of generality, assume that an intentional delay Δ is inserted at the second antenna, and (6) is modified for the MIMO case as follows:
y = X(IP⊗ FN×2L) (h1+ h2) !" #
h
+z = Xheq+ z (15)
where L= L + Δ and h1and h2are, respectively, assembled by h1= $˜ h11(0), . . . , ˜h 1 1(L− 1), 01×Δ, ˜h12(0), . . . , ˜h 1 2(L− 1), 01×Δ, . . . , ˜h1P(0), . . . , ˜h 1 P(L− 1), 01×Δ %T P L×1 h2= $ 01×Δ, ˜h21(0), . . . , ˜h 2 1(L− 1), 01×Δ, ˜h22(0), . . . , ˜ h2 2(L− 1), . . . , 01×Δ, ˜h2P(0), . . . , ˜h 2 P(L− 1) %T P L×1(16)
where superscript is used to denote the transmit antenna index. Assume that the time and path ACFs of h1 and h2 satisfy the Kronecker model in (9) individually, and denote the spatial correlation matrix between two transmit antennas as ΦS=
1 ρ12
ρ21 1
. Further assume that L≤ Δ ≤ Ncp− L, where Ncp is the CP length; then, the taps of h1 and h2 will not overlap in the h in (15), and the length of h of each OFDM symbol is still within Ncp. As a result, the autocorrelation matrix of h will satisfy the MIMO Kronecker model as follows:
R = E[hhH] = Φ
T⊗ ΦS⊗ ΦL, (17)
and thus
rank(Req) = rT× NT× L. (18)
Following the approach in Section II-B, it is straightforward to see from (18) that the maximum achievable diversity order is min{rT×
NT× L, dfree}. Fig. 5 shows the BER performance of a MIMO BICM-OFDM system employing CDD with the normalized Doppler frequency set to 0.05. (Thus, rT is 2.) The slopes coincide with the
analysis results, and the diversity order increases with the path and antenna counts.
B. PRD
The second example is to generalize the phase-roll scheme [11], in which the correlation function of the equivalent channel h[k] = h1+ h2ej2πkθ has zeros at certain delays, i.e., Rk[Δk] = (1/2)
E{h[k]h∗[k + Δk]}=(1/2)(1+ej2πΔkθ) = 0 at Δkθ = (1/2), (3/2),
(5/2), . . .. Zero correlation, as demonstrated in Section II-B, in turn implies independent channel conditions and opportunities to exploit diversity. One interesting scenario happens when multiple CFOs exist among collaborating transmitters in cooperative communications. The scenario can be fitted into a PRD model with unintentional phase differences induced by CFOs. We have reported simulation results of similar schemes in [12] in which the effectiveness of PRD is clearly demonstrated.
Similarly as in the CDD case, by combining the phase rotation matrix E = diag(1, ej(2πε/N ), . . . , ej(2πε(N−1)/N)) and H
pinto an
equivalent channel, (6) can be rewritten as (when there are two transmit antennas)
y = X(IP⊗ FN×L)(E1h1+ E2h2) + z = Xheq+ z (19)
Fig. 5. BER comparison of the MIMO BICM-OFDM employing CDD, PRD, and STBC over doubly selective fading channels. The DFT size is 64, P = 10, and a rate-1/2 convolutional code with the generator polynomial [133; 171] (dfree= 10) is adopted. Notice that the considered diversity here is the
effective diversity order based on the dominant eigenvalues. The channel is the equal-gain dual path at l = 0 and l = 1, and the introduced cyclic delay Δ is 5. The parameters of PRD are chosen as ε1= 0.05 and ε2=−0.05.
where Eα= diag([1, 1, . . . , 1]⊗[ej(2πεα(N−1)/2/N), ej(2πεα2(N−1)/2/N),
. . . , ej(2πεαP (N−1)/2/N)]), α∈ {1, 2}, is a P L × P L diagonal
ma-trix representing phase offsets; and hαis defined as
hα= $˜ hα 1(0), . . . , ˜hα1(L− 1), ˜hα2(0), . . . , ˜hα2(L− 1) . . . , ˜hα P(0), . . . , ˜hαP(L− 1) %T P L×1. (20)
Assume that we correct the phase offset from the first transmit antenna; thus, E1becomes an identity matrix. The rank of Reqcan be bounded by [19, Fact 2.10.7] rank(Req) = rank R + E2REH2 ≤ rank(R) + rankE2REH2 = rT× L × 2. (21)
The last equation follows that rank(R) = rank(E2REH2) = rT× L.
The maximum diversity order is increased by NT due to space
diversity if dfreeis not the limiting factor. Although the analysis does not find the condition when the bound can be achieved, simulation results, such as those shown in Fig. 5, indicate that it is achievable.
For slow fading channels, it is worth noting that, in [17], the diversity gain provided by multiple transmit antennas appears in the same form as our result. However, it is not surprising since both methods transform spatial diversity into other types of diversity for easier harvest. Finally, the simulation results show the weakness of the Alamouti scheme over doubly selective fading channels. Consider the example of space–time Alamouti code where the signal matrix X in (6) becomes diag(XST
1 , . . . , X ST P /2), where X ST p = X2(p−1)+1 X2(p−1)+2 −XH 2(p−1)+2 XH2(p−1)+1 with Xp= diag(X((p− 1)N),
X((p− 1)N + 1), . . . , X((p − 1)N + N − 1)). One would expect the diversity order to be P × min{L, dfree}; however, the performance degrades severely, and an error floor occurs if a typical Alamouti receiver is deployed. The degradation is caused by channel variations within an Alamouti code word and the induced destruction of the Alamouti structure.
IV. CONCLUSION
In this paper, the diversity orders of BICM-OFDM over doubly selective fading channels have been examined for both the asymptotic condition of very high SNR and the realistic situation with moderate SNRs. It is shown by simulation that the later diversity order is practically governed by dominant eigenvalues of the channel ACF. The same analysis framework is extended to show that BICM-OFDM, together with MIMO techniques, can provide even higher diversity gains in, e.g., in asynchronous cooperative communications. The results provide insights into the design/tradeoff of design parameters of BICM-OFDM systems, e.g., the achievable time diversity order versus the length of blocks.
ACKNOWLEDGMENT
The authors thank Prof. D. W. Lin of NCTU and the anonymous reviewers for their careful reading and helpful guidance.
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On the Design of Amplify-and-Forward MIMO-OFDM Relay Systems With QoS Requirements Specified
as Schur-Convex Functions of the MSEs Luca Sanguinetti, Member, IEEE, Antonio A. D’Amico, and
Yue Rong, Senior Member, IEEE
Abstract—In this paper, we focus on the design of linear and
nonlin-ear architectures in amplify-and-forward multiple-input–multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) relay net-works in which different types of services are supported. The goal is to jointly optimize the processing matrices to minimize the total power consumption while satisfying the quality-of-service (QoS) requirements of each service specified as Schur-convex functions of the mean square errors (MSEs) over all assigned subcarriers. It turns out that the optimal solution leads to the diagonalization of the source–relay–destination channel up to a unitary matrix, depending on the specific Schur-convex function.
Index Terms—Amplify-and-forward, multiple-input multiple-output
(MIMO), nonregenerative relay, orthogonal frequency-division multiplex-ing (OFDM), power minimization, quality-of-service (QoS) requirements, Schur-convex functions, transceiver design.
I. INTRODUCTION
Over the past few years, the ever-increasing demand for high-speed ubiquitous wireless communications has motivated an intense research activity toward the development of transmission technologies characterized by high spectral efficiency and high reliability. The most promising solutions in this direction rely on orthogonal frequency-division multiplexing (OFDM) techniques,
multiple-input–multiple-Manuscript received July 23, 2012; accepted December 18, 2012. Date of publication December 25, 2012; date of current version May 8, 2013. The review of this paper was coordinated by Dr. T. Jiang.
L. Sanguinetti and A. A. D’Amico are with the Department of Infor-mation Engineering, University of Pisa, 56126 Pisa, Italy (e-mail: luca. sanguinetti@iet.unipi.it; a.damico@iet.unipi.it).
Y. Rong is with the Department of Electrical and Computer Engineer-ing, Curtin University of Technology, Bentley, WA 6102, Australia (e-mail: y.rong@curtin.edu.au).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2012.2236370 0018-9545/$31.00 © 2012 IEEE
![Fig. 1. Block diagram of the transmitter and the receiver of the considered system. Soft-input–soft-output demodulator consists of the ICI equalization such as the feedback canceler [8] and the demapper.](https://thumb-ap.123doks.com/thumbv2/9libinfo/7706192.145466/1.891.467.829.103.364/transmitter-receiver-considered-demodulator-consists-equalization-feedback-demapper.webp)
