OFDM Signal Detection in Doubly Selective
Channels with Blockwise Whitening of Residual
Intercarrier Interference and Noise
Hai-wei Wang, David W. Lin, Senior Member, IEEE, and Tzu-Hsien Sang, Member, IEEE
Abstract—Orthogonal frequency-division multiplexing (OFDM) is a popular broadband wireless transmission technique, but its performance can suffer severely from the intercarrier interference (ICI) induced by fast channel variation arising from high-speed motion. Existing ICI countermeasures usually address a few dominant ICI terms only and treat the residual similar to white noise. We show that the residual ICI has high normalized autocorrelation and that this normalized autocorrelation is insensitive to the multipath channel profile as well as a variety of other system and channel conditions. Consequently, the residual ICI plus noise can be whitened in a nearly channel-independent manner, leading to significantly improved detection performance. Simulation results confirm the theoretical analysis. In particular, they show that the proposed technique can significantly lower the ICI-induced error floor in maximum-likelihood sequence estimation (MLSE) designed to address a few dominant ICI terms.
Index Terms—Doppler spread, intercarrier interference (ICI), maximum-likelihood sequence estimation (MLSE), orthogonal frequency-division multiplexing (OFDM), time-varying channels.
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) is widely adopted in broadband wireless signal transmission due to its high spectral efficiency. However, its performance can suffer severely from the intercarrier interference (ICI) induced by fast channel variation resulting from high-speed motion. Such an effect is sometimes referred to as loss of subcarrier orthogonality. The problem becomes increasingly acute as the carrier frequency or the speed of motion increases. For instance, with a 500 km/h mobile speed and a 6 GHz carrier frequency, the peak Doppler frequency can be as high as about 2800 Hz, which translates to over 0.25 times the 10.94 kHz subcarrier spacing in the Mobile WiMAX standard [1]. The signal detection performance can become intolerable without proper countermeasures.Consider the typical OFDM system illustrated in Fig. 1. In a system without ICI, the channel frequency response Manuscript received 29 April 2011; revised 24 October 2011. This work was supported in part by the National Science Council of R.O.C. under Grants NSC 99-2219-E-009-009 and 99-2219-E-009-010. Part of this work was published in [11], IEEE Vehicular Technology Conference, May 2010, Taipei, Taiwan.
The authors are with the Department of Electronics Engineering and Insti-tute of Electronics, National Chiao Tung University, Hsinchu, Taiwan 30010, R.O.C. (e-mails: c93jo6@gmail.com, dwlin@mail.nctu.edu.tw, tzuhsien54120 @faculty.nctu.edu.tw).
Digital Object Identifier 10.1109/JSAC.2012.120503.
matrix that relates the inputs of the inverse discrete Fourier transform (IDFT) and the outputs of the DFT is diagonal. Fast channel variation introduces sizable off-diagonal elements in the matrix, thus resulting in ICI. In theory, an optimal signal detector should take all ICI terms into account. But for reasons of complexity and robustness, usually only the dominant terms are compensated for. As these dominant terms are normally concentrated (circulantly) around the diagonal, the channel matrix shows a (circulant) band structure [2]–[5].
Jeon et al. [2] consider the situation where the normalized peak Doppler frequency (i.e., peak Doppler frequency ex-pressed in units of frequency spacing of subcarriers) is on the order of 0.1 or less. In this situation, the channel variation over one OFDM symbol time is approximately linear. A frequency-domain equalizer that exploits the ensuing band channel ma-trix structure is proposed. Schniter [3] considers substantially higher normalized peak Doppler frequencies, under which the ICI is more widespread. Time-domain windowing is used to partially counteract the effect of channel variation and shrink the bandwidth of the channel matrix. An iterative minimum mean-square error (MMSE) equalizer is then used to detect the signal. Rugini et al. [4] employ block-type linear MMSE equalization, wherein the band channel matrix structure is exploited (via triangular factorization of the autocorrelation matrix) to reduce the equalizer complexity. Ohno [5] addresses the ICI via maximum-likelihood sequence estimation (MLSE) in the frequency domain, where the band channel matrix structure is utilized to limit the trellis size.
The consideration of only the dominant ICI terms results in an irreducible error floor in time-varying channels [2]–[5]. Moreover, while the uncompensated residual ICI is colored [6]–[8], for various reasons it is often treated as white [5]– [9]. Although whitening of “I+N” (i.e., sum of ICI and additive channel noise) can lead to improved signal detection performance, it requires knowing the autocorrelation function of I+N, which remains a key problem awaiting solution [7], [8]. Without knowing the autocorrelation function, one can only resort to less sophisticated techniques, such as simple differencing of the received signals at neighboring subcarriers [10]. An attempt to characterize this autocorrelation function for the benefit of signal detection is reported in [11], but a comprehensive understanding of it remains lacking.
The contribution of the present work is twofold. First,
we explore the correlation property of ICI and derive an
..
.
m X yn xn..
.
Ym Symbol Parallel Converter Serial−to− +CP and IDFT Channel Generator Time−Variant and DFT −CP Converter Serial Parallel−to− Detector Symbol Output InputFig. 1. OFDM system model.
approximate mathematical expression for it. The expression applies not only to classical multipath Rayleigh fading, but also to arbitrary Doppler spectrum shapes in general. It is found that the correlation values are insensitive to various system parameters and channel conditions. Moreover, the correlation values are very high for the residual ICI beyond the few dominant terms. Secondly, to capitalize on the above high correlation to improve signal reception over fast varying channels, we consider performing simple blockwise whitening of the residual I+N before signal detection (i.e., equalization), where the whitener makes use of the ICI characteristics as found. Numerical results show that substantial gains can be achieved with this approach.
The remainder of this paper is organized as follows. Sec. II describes the system model. Sec. III analyzes the correlation property of ICI. Sec. IV introduces the proposed detection method that utilizes the residual ICI’s high correlation. It also presents some simulation results. Finally, Sec. V gives a conclusion.
II. SYSTEMMODEL
Fig. 1 shows the discrete-time baseband equivalent model of the considered OFDM system. The input-output relation of the channel is given by
yn =
L−1
l=0
hn,lxn−l+ wn (1)
where xn and yn are, respectively, the channel input and
output at time n, L is the number of multipaths, hn,l is the
complex gain of the lth path (or tap) at time n, and wn is the
complex additive white Gaussian noise (AWGN) at time n. We assume that the length of the cyclic prefix (CP) is sufficient to cover the length of the channel impulse response (CIR) (L − 1)Tsa, where Tsa denotes the sampling period.
One common way of expressing the received signal in the DFT domain is Ym= N −1 k=0 L−1 l=0 XkHl(m−k)e−j2πlk/N + Wm, 0≤m≤N − 1, (2) where Xk and Ym are, respectively, the channel input and
output in the frequency domain (see Fig. 1), N denotes the size of DFT, Wm denotes the DFT of wm, and Hl(k) is the
frequency spreading function of the lth path given by
Hl(k)= 1 N N −1 n=0 hn,le−j2πnk/N. (3)
Another way of expressing it is
y = Hx + w (4) where y = [Y0, ..., YN −1], x = [X0, ..., XN −1], w = [W0, ..., WN −1], and H = ⎡ ⎢ ⎢ ⎢ ⎣ a0,0 a0,1 · · · a0,N−1 a1,0 a1,1 · · · a1,N−1 .. . ... . .. ... aN −1,0 aN −1,1 · · · aN −1,N −1 ⎤ ⎥ ⎥ ⎥ ⎦, (5) with denoting transpose and
am,k=
L−1 l=0
Hl(m−k)e−j2πkl/N. (6)
The quantity am,k is the “ICI coefficient” from subcarrier k
to subcarrier m. For a time-invariant channel, Hl(k) vanishes
∀k = 0 and H becomes diagonal, implying absence of ICI.
As mentioned, a band approximation toH that retains only the dominant terms about the diagonal may ease receiver design and operation, but also results in an irreducible error floor. Consider a symmetric approximation with one-side bandwidth K, that is, am,k= 0 for |(m − k)%N| > K where
K is a nonnegative integer and % denotes modulo operation.
Then the ICI at each subcarrier consists of contributions from at most 2K nearest (circularly) subcarriers. In this work, we exploit the correlation of the residual ICI outside the band to attain a significantly enhanced signal detection performance. For convenience, in the following we omit explicit indication of modulo-N in indexing a length-N sequence, understanding an index, say n, to mean n%N .
Let the channel be wide-sense stationary uncorrelated scat-tering (WSSUS) [12] with
E[hn,lh∗n−q,l−m] = σl2rl(q)δ(m) (7)
where E[·] denotes expectation, σ2l denotes the variance of the
lth tap gain, rl(q) denotes the normalized tap autocorrelation
(where rl(0) = 1), and δ(m) is the Kronecker delta function.
For convenience, assume lσl2 = 1. Let Pl(f) denote the
Doppler power spectral density (PSD) of path l and thus
rl(q) = fd −fd Pl(f)ej2πf τdf τ =Tsaq , (8)
where fd denotes the peak Doppler frequency of the channel.
We assume that the paths may be subject to arbitrary, different fading so that Pl(f) may be asymmetric about f = 0 and
different for different l.
III. AUTOCORRELATION OFRESIDUALICI
Assume a signal detector (equalizer) able to handle 2K terms of nearest-neighbor ICI. We may partition the summa-tion over k in (2) into an in-band and an out-of-band term
as Ym= m+K k=m−K L−1 l=0 Hl(m−k)e−j2πlk/NXk + k /∈[m−K,m+K] L−1 l=0 Hl(m−k)e−j2πlk/NXk cm,K +Wm, (9)
where cm,K is the out-of-band term, i.e., residual ICI.
Alter-natively, using the notation of (6),
Ym= m+K k=m−K am,kXk+ cm,K+ Wm (10) where cm,K = k /∈[m−K,m+K] am,kXk. (11)
For large enough N , the residual ICI may be modeled as Gaussian by the central limit theorem.
It turns out that the analysis can be more conveniently carried out by way of the frequency spreading functions of the propagation paths than by way of am,k. Hence consider
(9). From it, the autocorrelation of cm,K at lag r is given by
E[cm,Kc∗m+r,K] = Es× k /∈[m−K,m+K] ∪[m+r−K,m+r+K] L−1 l=0 E[Hl(m−k)Hl(m+r−k)∗] = Es× k /∈[−K,K]∪[−K−r,K−r] L−1 l=0 E[Hl(k)H (k+r)∗ l ] (12)
where Es is the average transmitted symbol energy and we
have assumed that Xk is white. Invoking (3) and (7), we get
E[cm,Kc∗m+r,K] = Es N2 L−1 l=0 N −1 n=0 N −1 n=0 k /∈[−K,+K] ∪[−K−r,K−r] σl2rl(n − n)ej2π[n (k+r)−nk]/N . (13)
We show in the Appendix that
E[cm,Kc∗m+r,K] ≈ 4π2Tsa2Es L−1 l=0 σ2lσD2l ρ(K, r, N ) (14) where σD2l is the mean-square Doppler spread of path l given
by σD2l = fd −fdPl(f)f 2df and ρ(K, r, N ) = k /∈[−K,K] ∪[−K−r,K−r] 1 (1 − e−j2πk/N)(1 − ej2π(k+r)/N) . (15) Note that ρ(K, r, N ) = k∈[0,N −1]\{0,−r} 1 (1 − e−j2πk/N)(1 − ej2π(k+r)/N) ρ0(r,N) − k∈[−K,K] ∪[−K−r,K−r] \{0,−r} 1 (1 − e−j2πk/N)(1 − ej2π(k+r)/N) ρ1(K,r,N) , (16)
where the exclusion of 0 and−r from both ranges of summa-tion is to skip over the points of singularity where the sum-mands are null anyway. Note further that−1/(1 − e−j2πk/N)
and−1/(1 − e−j2π(k+r)/N) (as sequences in k) are the DFTs
of [n − (N − 1)/2]/N and e−j2πrn/N[n − (N − 1)/2]/N (as sequences in n), respectively. Hence, with Parseval’s theorem we get ρ0(r, N) = 1 N N −1 n=0 n −N − 12 2 ej2πrn/N = N2−1 12 −2, r = 0, (1−ej2πr/N)2, r = 0. (17) For ρ1(K, r, N), we have ρ1(K, r, N) = ρ∗1(K, −r, N), (18) i.e., it is conjugate symmetric in r. Moreover, the summands in the last summation in (16) are symmetric over the range of summation. But the range of summation does not allow us to obtain a compact expression for ρ1(K, r, N) as that for
ρ0(r, N).
As mentioned, the proposed receiver will whiten the resid-ual I+N before eqresid-ualization. Here we make some observations of the properties of the normalized autocorrelation of residual ICI, i.e., E[cm,Kc∗m+r,K]/E[|cm,K|2], that are relevant to
whitener design and performance. For this, note from (14) that E[cm,Kc∗m+r,K]/E[|cm,K|2] depends only on K and
N through ρ(K, r, N ); the other factors cancel out. Thus
this normalized autocorrelation is independent of the average transmitted symbol energy Es and the sample period Tsa.
More interestingly, it is also independent of the power-delay profile (PDP) of the channel (i.e., σl2 vs. l) and the Doppler PSD Pl(f) of each path. While the independence of the
normalized autocorrelation on the average transmitted symbol energy may be intuitively expected, its independence of the sample period, the PDP, and the Doppler PSDs of channel paths appears somewhat surprising.
Moreover, the normalized autocorrelation is also substan-tially independent of the DFT size N . To see this, note that for complexity reason, in a practical receiver both the whitener and the equalizer are likely short. A short equalizer implies a small K and a short whitener implies a small range of r over which the normalized autocorrelation needs to be computed. Hence, when N is large, the exponential functions in the above summations for ρ0(r, N) and ρ1(K, r, N) can all be well approximated with the first two terms of their respective
power series expansion (i.e., ex≈ 1 + x when |x| 1). As a result, we have ρ(K, r, N ) = ρ0(r, N) − ρ1(K, r, N) (19) where ρ0(r, N) ≈ N2 12, r = 0, N2 2π2r2, r = 0, (20) ρ1(K, r, N) ≈ k∈[−K,K]∪[−K−r,K−r]\{0,−r} N2 4π2k(k + r). (21) Thus the normalized autocorrelation, being essentially given by ρ(K, r, N )/ρ(K, 0, N ), is substantially independent of the DFT size N .
Although the above observations concern ICI only, it is straightforward to extend them to the sum of ICI and AWGN channel noise. In particular, the resulting whitening filter and its performance are also independent of a variety of system parameters and channel conditions, including the DFT size, the sample period, the system bandwidth (which is approximately proportional to the inverse of the sample period), the OFDM symbol period N Tsa, the channel PDP, and the Doppler PSDs
of the channel paths. They only depend on the ICI-to-noise power ratio (INR) at the receiver. As a result, a whitener parameterized on receiver INR can be designed for all op-erating conditions, which is advantageous for practical system implementation. (The estimation of ICI and noise powers is outside the scope of the present work. Some applicable methods have been proposed in the literature, e.g., [13] for ICI power and [14] for noise power.)
The whitener performance can be understood to a sub-stantial extent by examining the above approximation to the normalized autocorrelation E[cm,Kc∗m+r,K]/E[|cm,K|2]. We
leave a detailed study along this vein to potential future work. For now, we shall be content with a first-order understanding by a look at its value at lag r = 1. A large value indicates that whitening can effectively lower the residual ICI. For this, we see from the above approximation (after some straightforward algebra) that E[cm,Kc∗m+1,K] E[|cm,K|2] ≈ ρ(K, 1, N ) ρ(K, 0, N ) ≈ 1 − K k=11/[k(k + 1)] π2/6 − Kk=11/k2 = 1/(K + 1) π2/6 − Kk=11/k2. (22)
For example, its values for K = 0–3 are, respectively, 0.6079, 0.7753, 0.8440, and 0.8808, which are substantial indeed.
As a side remark that will be of use later, we note that from (14) and (19), the total ICI power E[|cm,0|2] can be
approximated as σ2c0 E[|cm,0|2] ≈ 4π2T2 saEs L−1 l=0 σ2lσD2l ρ(0, 0, N ) ≈ Es 12(2πTsaN )2 L−1 l=0 σ2lσD2l , (23) TABLE I
TWOCHANNELPOWER-DELAYPROFILESUSED INTHISSTUDY, WHERE TU6 CORRESPONDS TO THECOST 207 6-TAPTYPICALURBAN
CHANNELANDSUI4THESUI-4 3-TAPCHANNEL
Tap Index 1 2 3 4 5 6 TU6 Delay (µs) 0.0 0.2 0.5 1.6 2.3 5.0 Power (%) 19 38 24 9 6 4 Tap Index 1 2 3 – – – SUI4 Delay (µs) 0.0 1.5 4.0 – – – Power (%) 64 26 10 – – –
which is in essence the upper bound derived in [13]. Moreover, we have an approximation to the partial ICI power beyond the 2K central terms as σcK2 E[|cm,K|2] ≈ 4π2T2 saEs L−1 l=0 σl2σD2l ρ(K, 0, N ) ≈ σ2 c0 1 − 6 π2 K k=1 1 k2 . (24)
In the following subsection, we provide some numerical examples to verify the above results on ICI correlation. Then, in the next section, we consider how to incorporate a whitener for residual ICI plus noise in the receiver.
A. Numerical Examples
In this subsection, we verify some key results above by considering two very different channel conditions: multipath Rayleigh fading and simple Doppler frequency shift.
First, consider a multipath channel having the COST 207 6-tap Typical Urban (TU6) PDP as shown in Table I [15, p. 94]. Let the paths be subject to Rayleigh fading with the same peak Doppler frequency fd, so that rl(q) = J0(2πfdTsaq) for
all l, where J0(·) denotes the zeroth-order Bessel function of the first kind [12]. Let the OFDM system have N = 128, subcarrier spacing fs = 10.94 kHz, and sampling period
Tsa = 1/(Nfs) = 714 ns, which are some of the Mobile
WiMAX parameters [1].
Figs. 2–4 illustrate the normalized autocorrelation of the residual ICI for K = 0–2, respectively, where the theoretical values are calculated using (13). As points of reference, note that a peak Doppler frequency of 1 kHz corresponds to a 180 km/h mobile speed at a 6 GHz carrier frequency, or a 540 km/h mobile speed at a 2 GHz carrier frequency. Figs. 2–4 show that the theory and the simulation results agree well up to very large Doppler spreads. In addition, they also show that, for given lag r, the normalized autocorrelation increases with
K. The last fact can be understood by examining (11): as K
increases, the residual ICI cm,K is composed of the sum of
increasingly fewer terms with generally smaller magnitudes, which naturally leads to higher normalized autocorrelation.
Next, consider a channel with a one-line Doppler PSD equal to δ(f − fd); in other words, the channel simply effects a
frequency offset of fd. The temporal autocorrelation of the
CIR is given by rl(q) = exp(j2πfdTsaq). It turns out that the
normalized autocorrelation of residual ICI is very similar to that obtained for the previous example, as the theory predicts.
0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Peak Doppler Frequency fd(Hz)
E [cm, K c ∗ m+ r, K ]/E [| cm, K | 2] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32
Normalized Peak Doppler Frequency (fdTsaN)
r=1, theory r=1, simul. r=2, theory r=2, simul.
Fig. 2. Normalized autocorrelation of residual ICI over multipath Rayleigh fading channel atK = 0, with N = 128 and Tsa= 714 ns. The first-order approximation (19)–(21) yields 0.6079 for r = 1 and 0.1520 for r = 2, which are quite accurate at lowfdvalues.
0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Peak Doppler Frequency fd(Hz)
E [cm, K c ∗ m+ r, K ]/E [| cm, K | 2] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32
Normalized Peak Doppler Frequency (fdTsaN)
r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul. r=6, theory r=6, simul.
Fig. 3. Normalized autocorrelation of residual ICI over multipath Rayleigh fading channel atK = 1, with N = 128 and Tsa= 714 ns. The first-order approximation (19)–(21) yields 0.7753, 0.6461, 0.5599, 0.3036, 0.1912, and 0.1317, forr = 1–6, respectively, which are quite accurate.
For space reason, we only illustrate the numerical data for
K = 1 in Fig. 5, which can be compared with Fig. 3.
Looking backwards from the one-Doppler-line example to the earlier analysis in this Section III, we find that this example also provides an alternative way of interpreting the earlier analytical results. Specifically, an arbitrary Doppler PSD can be considered as composed of a (possibly infinite) number of line PSDs. Hence the autocorrelation of residual ICI associated with an arbitrary Doppler PSD may be obtained as a linear combination of the autocorrelation associated with a line PSD as
E[cm,Kc∗m+r,K]|any shape
0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Peak Doppler Frequency fd(Hz)
E [cm, K c ∗ m+ r, K ]/E [| cm, K | 2] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32
Normalized Peak Doppler Frequency (fdTsaN)
r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul.
Fig. 4. Normalized autocorrelation of residual ICI over multipath Rayleigh fading channel atK = 2, with N = 128 and Tsa= 714 ns. The first-order approximation (19)–(21) yields 0.8440, 0.7358, 0.6612, 0.6014, and 0.5534, forr = 1–5, respectively, which are quite accurate.
0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Peak Doppler Frequency fd(Hz)
E [cm, K c ∗ m+ r, K ]/E [| cm, K | 2] 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32
Normalized Peak Doppler Frequency (fdTsaN)
r=1, theory r=1, simul. r=2, theory r=2, simul. r=3, theory r=3, simul. r=4, theory r=4, simul. r=5, theory r=5, simul. r=6, theory r=6, simul.
Fig. 5. Normalized autocorrelation of residual ICI over one-Doppler-line channel atK = 1, with N = 128 and Tsa= 714 ns.
=L−1 l=0 σ2l fd −fd Pl(f)E[cm,Kc∗m+r,K]|line,fdf (25)
where E[cm,Kc∗m+r,K]|any shape denotes the autocorrelation
of residual ICI associated with a multipath channel of arbitrary Doppler PSD and E[cm,Kc∗m+r,K]|line,f that associated with
a line Doppler PSD corresponding to a Doppler frequency f . As we have verified now (through Fig. 5, for example) that
E[cm,Kc∗m+r,K]|line,fd
E[cm,Kc∗m,K]|line,fd
≈ ρ(K, r, N )
ρ(K, 0, N ), (26)
substituting it into (25) yields
E[cm,Kc∗m+r,K]|any shape
≈ ρ(K, r, N ) ρ(K, 0, N )× L−1 l=0 σl2 fd −fd Pl(f)E[cm,Kc∗m,K]|line,fdf
= ρ(K, r, N)
ρ(K, 0, N )× E[cm,Kc
∗
m,K]|any shape. (27)
In other words, since the single-Doppler-line channel shows substantial invariance of the normalized residual ICI autocor-relation over a large range of operating conditions (as we have seen in the last example), it follows that a channel with any Doppler PSD has a similar property.
In summary, we have confirmed that the normalized auto-correlation of the residual ICI is quite insensitive to various system parameters and channel conditions. To lower the error floor, therefore, a whitening filter for the residual ICI plus noise can be designed without regard to these system param-eters and channel conditions. Such a fixed design can lead to low implementation complexity and robust performance.
IV. SIGNALDETECTION WITHWHITENING OFRESIDUAL ICI PLUSNOISE
As indicated, we propose to whiten the residual ICI plus noise in signal detection. This can be applied to many detec-tion methods, including MMSE, iterative MMSE, decision-feedback equalization (DFE), MLSE, etc., providing a wide range of tradeoff between complexity and performance. In this work, we consider an MLSE-based technique both to illustrate how such whitening can be carried out and to demonstrate its benefit. For simplicity, rather than performing whitening over a complete sequence, we do blockwise whitening over windows of size 2q + 1 where q may or may not be equal to K. The details are as follows.
Consider a vector of 2q + 1 frequency-domain signal sam-ples centered at sample m:
ym= [Ym−q · · · Ym · · · Ym+q]= Hmxm+ zm (28)
where xm = [Xm−p · · · Xm · · · Xm+p] for some integer
p, Hmis a (2q + 1) × (2p + 1) submatrix of H of bandwidth
K, and zmcollects all the right-hand-side (RHS) terms in (2)
(or (4)) associated with Yk, m − q ≤ k ≤ m + q, that do not
appear in Hmxm. The elements of zm include both residual
ICI and channel noise. To avoid clogging the mathematical expressions with details, we have omitted explicit indexing of various quantities in (28) with the parameters K, p, and
q, understanding that their dimensions and contents depend
on these parameters. As examples, with the set of parameters
{K = 1, q = 1, p = 2}, Hmis given by ⎡ ⎣am−1,m−20 am−1,m−1am,m−1 am−1,mam,m am,m+10 00 0 0 am+1,m am+1,m+1 am+1,m+2 ⎤ ⎦ (29) whereas with{K = 1, q = 1, p = 1}, Hmis given by
⎡
⎣am−1,m−1am,m−1 am−1,mam,m am,m+10
0 am+1,m am+1,m+1
⎤
⎦ . (30)
Let Kz = E[zmzHm], i.e., the covariance matrix of zm,
where superscript H stands for Hermitian transpose. The aforesaid blockwise whitening of residual ICI plus noise zm
is given by ym K− 1 2 z ym= K− 1 2 z Hm eHm xm+ K− 1 2 z zm ezm (31) 2 1 1 1 m m m y H x Э Э ЭymH xm mЭ2
Fig. 6. Trellis structure for MLSE-based detection using the Viterbi algorithm, under QPSK modulation and withp = 1, where numerals 0–3 represent the QPSK constellation points.
where K−12
z may be defined in more than one way. One choice is to letK−12
z = UΛ−12UH whereU is the matrix of orthonormal eigenvectors ofKzandΛ is the diagonal matrix of corresponding eigenvalues ofKz. If block-by-block signal detection were desired, then the ML criterion would result in the detection rule xm= arg minxmym− Hmxm2. As
stated, we consider MLSE-based detection in this work. In developing the MLSE-based detection method, we treat zm, m = 0, . . . , N − 1, as if they were mutually independent,
even though this may at best be only nearly so. Then the prob-ability density function of the received sequence conditioned on the transmitted sequence would be
f (y0, y1, . . . , yN −1|x0, x1, . . . , xN −1)
= f(z0, z1, . . . , zN −1) = N −1
n=0
f (zn). (32)
As a result, the recursive progression of the log-likelihood values, i.e.,
Λk log f(z0, z1, . . . , zk) = Λk−1+log f(yk− Hkxk) (33)
(where k = 1, . . . , N − 1), leads to a standard Viterbi algorithm. Disregarding some common terms that do not affect sequence detection, in the Viterbi algorithm we may use
yk− Hkxk2 as the branch metric instead of log f (yk−
Hkxk). Fig. 6 illustrates the trellis structure of the MLSE
detector for p = 1 under QPSK modulation. A tradeoff between complexity and performance can be achieved by different choices of the three parameters K, q, and p, where
p determines the number of states in each trellis stage and the
three parameters jointly affect the branch metric structure in the trellis and the autocorrelation structure of the residual ICI (and thereby the whitener behavior).
A. Complexity Analysis
Concerning complexity, let NA denote the signal
constel-lation size at each subcarrier. Then, for each subcarrier, the nonwhitening MLSE requires O[(2K + 1)NA2K+1] complex multiplications and additions (CMAs) to build the trellis and O(NA2K+1) CMAs to conduct the Viterbi search [5]. In
200 400 600 800 1000 1200 1400 1600
10−4
10−3
10−2
10−1
Peak Doppler Frequency fd(Hz)
B it E rro r R a te 0.0183 0.0366 0.0549 0.0731 0.0914 0.1097 0.128 0.1463
Normalized Peak Doppler Frequency (fdTsaN)
Proposed method, K=0 Conventional OFDM
detection method
Fig. 7. Error performance in TU6 channel of the conventional OFDM signal detection method and ICI-whitening MLSE (the proposed method) withK = 0 and p = q = 1 in noise-free condition.
contrast, the proposed method requires O[(2K + 1)NA2p+1+ (2q + 1)2N2p+1
A ] CMAs to build the trellis, wherein O[(2K +
1)NA2p+1] are for computing Hmxm and O[(2q + 1)2NA2p+1]
are for multiplying withK−12
z . Then the Viterbi search requires
O[(2q + 1)NA2p+1] CMAs. The computation of K
−1 2
z requires estimation of the ICI power and the AWGN power, but the complexity is far lower than building the trellis or performing the Viterbi search and is thus neglected. From the above, the proposed method may seem to require much higher complexity than nonwhitened MLSE. But, to the contrary, the reduced residual I+N through whitening may facilitate using a smaller ICI bandwidth K in the MLSE, culminating in a complexity gain rather than loss. This will be demonstrated in the simulation results below.
B. Simulation Results on Detection Performance
We present some simulation results on signal detection per-formance in this subsection. As in Sec. III-A, we let subcarrier spacing fs = 10.94 kHz and sample period Tsa = 714 ns.
The subcarriers are QPSK-modulated with Gray-coded bit-to-symbol mapping. There is no channel coding. The channels are multipath Rayleigh-faded WSSUS channels having the PDPs shown in Table I. Unless otherwise noted, we let N = 128 and assume that the receiver has perfect knowledge of the channel state information (CSI), which includes the channel matrix within band K and the covariance matrix Kz of the residual ICI plus noise.
To start, consider the extreme case of K = 0 in absence of channel noise. Through this we look at the limit imposed by the ICI to the performance of the conventional detection method. We also look at the possible gain from blockwise whitening of the full ICI followed by MLSE with p = q = 1, at infinite signal-to-noise ratio (SNR). The ICI covariance
500 1000 1500 2000 2500 3000 3500 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
Peak Doppler Frequency fd(Hz)
B it E rro r R a te 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32
Normalized Peak Doppler Frequency (fdTsaN)
TU6 SUI4
MLSE with K=1, residual ICI treated as white
Proposed method, K=1,q=1,p=1 Proposed method, K=1,q=1,p=2
Fig. 8. Comparison of proposed technique in TU6 and SUI4 channels with that treating residual ICI as white; SNR =∞.
matrix in this case is given by
Kz= ⎡ ⎣ 0.61 0.6 0.151 0.6 0.15 0.6 1 ⎤ ⎦ σ2 c0 (34)
where recall that σ2c0 = E[|cm,0|2] is the total ICI power.
Fig. 7 shows some simulation results for the TU6 channel. The numerical performance for the SUI4 channel is very similar. These results show that ICI-whitening detection (the proposed technique) yields some advantage over conventional detection: the error probability is reduced by about 2.2 times.
Significantly higher gain can be obtained by ICI-whitening MLSE with K = 1. In Fig. 8 we compare the corresponding performance of the proposed technique with that of MLSE which treats the residual ICI as white [5], over TU6 and SUI4 channels in the noise-free condition (i.e., SNR =∞). For the proposed technique, two parameter settings are considered,
viz. {q = 1, p = 2} and {q = 1, p = 1}, for which
the covariance matrices Kz of residual ICI are given by, respectively, ⎡ ⎣0.7751 0.775 0.6451 0.775 0.645 0.775 1 ⎤ ⎦ σ2 c1, ⎡ ⎣1.785 1.16 1.161.16 1 1.16 1.16 1.16 1.785 ⎤ ⎦ σ2 c1, (35) where recall that σcK2 = E[|cm,K|2] is the residual ICI power
outside band K.
Consider the case p = q = 1 first. In this case, the proposed method shows a remarkable gain of roughly three to four orders of magnitude in error performance compared to treating residual ICI as white. The error floor induced by the residual ICI can be driven to below 10−5 even at the very high normalized peak Doppler frequency of 0.32.
Very interestingly, Fig. 8 also shows that the setting {q = 1, p = 2} yields a worse performance than p = q = 1, even though the former setting may seem more natural in its asso-ciated band channel matrix structure (compare (29) with (30)), which captures all the ICI terms within the modeling range
0 500 1000 1500 2000 2500 3000 3500 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
Peak Doppler Frequency fd(Hz)
B it E rro r R a te 0 0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32
Normalized Peak Doppler Frequency (fdTsaN)
E b/N0 = 28 dB Proposed method Proposed method Bound E b/N0 = 15 dB Bound E b/N0 = 45 dB Proposed method Bound E b/N0 = ∞ Proposed method
Fig. 9. Performance of proposed technique versus Doppler spread in the TU6 channel withp = q = K = 1, at N = 128 and Tsa = 714 ns and under QPSK subcarrier modulation.
(K = 1). Moreover, its corresponding trellis has more states than the latter setting (45vs. 43). The reason will be explored in the next subsection. For now, we note that the above results appear to indicate the suitability of setting p = q = K = 1 in practical system design. It yields good performance without undue complexity. With this observation, we now present some more simulation results under this setting. The aims are to examine the proposed technique’s performance at finite SNR and to compare it with a benchmarking upper bound. For this, we first consider how it varies with Doppler spread and then how it varies with SNR.
Fig. 9 shows some results for the TU6 channel with
p = q = K = 1 at several SNR values. The results for SUI4
show similar characteristics and are omitted. We compare the performance of the proposed method with a benchmark: the matched-filter bound (MFB), i.e., signal detection with perfect knowledge of the interfering symbols. To make the MFB a more-or-less absolute lower bound, it is obtained with the residual ICI outside band K fully cancelled. Other than these, the same MLSE as in the proposed technique is used. For all three finite SNR values shown, note that the MFB drops monotonically with increasing fd, i.e., with increasing
time-variation of the channel. This is in line with the fact that faster channel variation yields greater time diversity, as various researchers have observed [16]–[18]. However, such time diversity can show clearly only when ICI is sufficiently small (e.g., after ICI cancellation). For the proposed technique, its error performance at Eb/N0 = 15 and 28 dB tracks
that of the MFB reasonably closely, deviating by less than a multiplicative factor of three for normalized peak Doppler frequencies up to 0.18 (fd ≤ 2000 Hz). At Eb/N0= 45 dB, the performance improves with fduntil fdreaches about 1500
Hz (normalized peak Doppler frequency≈ 0.14). Afterwards, the residual ICI dominates in determining the performance, as can be seen by the closeness between the corresponding curves for Eb/N0= 45 dB and ∞.
10 15 20 25 30 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0(dB) B it E rro r R a te
Proposed method, p=q=K=1, perfect CSI Residual ICI−free bound, K=1, perfect CSI Nonwhitening MLSE, K=1, perfect CSI Nonwhitening MLSE, K=2, perfect CSI Proposed method, p=q=K=1, noisy CSI Residual ICI−free bound, K=1, noisy CSI Nonwhitening MLSE, K=1, noisy CSI Nonwhitening MLSE, K=2, noisy CSI
Fig. 10. Performance versusEb/N0of different methods in the TU6 channel, withN = 128, Tsa= 714 ns, fd= 1500 Hz (normalized peak Doppler frequency fdTsaN = 0.1371) and QPSK subcarrier modulation. (Results withN = 1024 are very close.)
Next, consider how the performance of the proposed method varies with SNR. The solid lines in Fig. 10 show results at
fd = 1500 Hz (normalized peak Doppler frequency ≈ 0.14)
under perfect CSI. It is seen that the proposed method at
K = 1 can yield a substantial performance gain compared
to nonwhitening MLSE [5] at K = 2. The dash-dot lines in Fig. 10 depict some results under imperfect CSI. Limited by space, we cannot elaborate on the many possible channel estimation methods and their performance. Hence the results shown pertain to a typical condition only. For this, we note that the mean-square channel estimation error is typically propor-tional to the variance of the unestimatable channel disturbance, with the proportionality constant inversely dependent on the sophistication of the channel estimation method [19]. In our case, the unestimatable channel disturbance includes residual ICI (mostly that beyond K = 1) and additive channel noise (AWGN). At a normalized peak Doppler frequency of 0.14 (fd= 1500 Hz), the first term is approximately 20 dB below
the received signal power. The proportionality constant is set to 1/8. The channel estimation error limits the performance of all detection methods and the residual ICI-free bound in the form of error floors. The floor of the proposed method at
K = 1 is seen to be lower than that of nonwhitening MLSE
at K = 2 and is relatively close to the bound. We further note that, while Fig. 10 has been obtained with N = 128, the results obtained with N = 1024 (eight times the bandwidth) are very close.
C. Dependence of Detection Performance on Parameter Set-ting
As mentioned, we here explore how signal detection perfor-mance depends on whitener parameter setting. In particular, recall that one intriguing phenomenon observed earlier is the worse performance with p = 2 than with p = 1 (both at
q = K = 1), although the former is associated with a
more expanded MLSE trellis. A comprehensive analysis would require examining the distance property of the received signal after the proposed blockwise whitening. However, a crude understanding can be obtained by looking at the signal-to-interference-plus-noise ratio (SINR) after blockwise whiten-ing.
From (28) and (31), the pre- and post-whitening SINRs are given by, respectively,
SINRpre= E[xHmHHmHmxm]/E[zHmzm], (36)
SINRpost= E[xHmHHmK−1z Hmxm]/E[zHmK−1z zm]. (37)
For the power of residual ICI plus noise, we have E[zH
mzm] = tr(E[zmzHm]) = tr(Kz) and
E[zH
mK−1z zm] = tr(E[K−1z zmzHm]) = tr(K−1z Kz) = 2q + 1,
where tr(A) denotes the trace of a matrix A. For the signal power, we have E[xH
mHHmHmxm] = tr(E[HHmHmxmxHm]) =
Es · tr(E[HHmHm]) = Es · tr(E[HmHHm]) and
E[xH
mHHmK−1z Hmxm] = tr(E[HHmK−1z HmxmxHm]) =
Es · tr(E[HHmK−1z Hm]) = Es· tr(K−1z E[HmHHm]), where
Es is as defined previously (the average energy of the
transmitted signal samples) and we have assumed that the transmitted signal is independent and identically distributed (i.i.d.).
Note that the factor E[HmHHm] appears in the signal power
terms of both SNRs. Employing a procedure similar to that for
E[cm,Kc∗m+r,K] in Sec. III, we can derive an expression for
E[HmHHm] in terms of the channel parameters as in the case
of E[cm,Kc∗m+r,K]. However, although such an expression can
provide more precise numerical results, an illuminating insight into the SNR impact of the proposed blockwise whitening technique can already be gathered with a very simple ap-proximation to E[HmHHm], and this insight is sufficient for
the purpose of the present work. Specifically, in the limit of little ICI, Hm approaches a diagonal matrix of the channel
frequency response. In this case, E[HmHHm] ≈ (
L−1 l=0 σ2l)I
where I denotes an identity matrix and recall that we have assumed a unity channel power gain, i.e., lσ2l = 1. Hence
SINRpre≈ (2q + 1)Es/tr(Kz), (38)
SINRpost≈ Es· tr(K−1z )/(2q + 1). (39) As a result, SINRpost SINRpre ≈ tr(K−1z ) · tr(Kz) (2q + 1)2 . (40)
Now let λi, 0≤ i ≤ 2q, denote the eigenvalues of Kz. Then
the eigenvalues ofK−1z are given by λ−1i and we have SINRpost
SINRpre ≈
( 2qi=0λ−1i )( 2qi=0λi)
(2q + 1)2 . (41)
Therefore, the more disparate the eigenvalues of Kz are, the greater gain the proposed blockwise whitening can offer. If the eigenvalues are all equal, then no gain is attained.
As examples, we consider the previously considered cases
1) {K = 0, q = 1, p = 1}, 2) {K = 1, q = 1, p = 1},
and 3) {K = 1, q = 1, p = 2}, all at infinite SNR. The corresponding Kz matrices are given in (34) and (35). For case 1), we obtain the eigenvalues 0.2232σc02, 0.8500σ2c0, and 1.9268σ2
c0; for case 2), 0.0654σ2c1, 0.6250σc12, and 3.8796σc12;
10 15 20 25 30 35 5 10 15 20 25 30 35 40 Eb/N0(dB) SI N R (d B)
Simul., proposed method, K=q=p=1 Theory, proposed method, K=q=p=1 Simul., proposed method, K=q=1, p=2 Theory, proposed method, K=q=1, p=2 Simul., nonwhitening MLSE, K=1 Theory, nonwhitening MLSE, K=1
f
d = 500 Hz
fd = 3500 Hz
Fig. 11. SINR performance of different methods in the TU6 channel, with N = 128 and Tsa= 714 ns and assuming perfect CSI.
and for case 3), 0.1800σ2c1, 0.3550σc12, and 2.4650σ2c1. The resulting post- to pre-SINR ratios are 2.0588, 8.7052, and 2.9258, respectively. They do correspond monotonically to the performance gains shown in Figs. 7 and 8. However, the mathematical relation between SINR and bit error rate (BER) is not straightforward—a point worth remembering when comparing the SINR performance of different detection methods and different parameter settings.
With the above caveat, we show some SINR performance results at finite SNR values in Fig. 11, both to verify the theory derived in this subsection and to further illustrate the performance of different detection methods. In the case of the proposed method, the theoretical SINR values shown in the figure have been obtained using (38) and (39), i.e.,
SINRpost = Es · tr(K−1z )/(2q + 1), whereas in the case
of nonwhitening MLSE, the values of “I” in the theoretical SINR are simply given by σc12, which are calculated using (24) with K = 1. We see that, in the case fd = 500 Hz
(normalized peak Doppler frequency ≈ 0.046), the theory and the simulation results agree almost exactly, whereas in the case fd = 3500 Hz (normalized peak Doppler frequency
≈ 0.32), the theory consistently underestimates the SINR
performance by a fraction of a dB. The latter phenomenon can be understood by the fact that the σ2c0 as given in (23) is a progressively looser upper bound to the actual ICI power as the normalized peak Doppler frequency increases [13]. The figure confirms the earlier observation concerning the superiority of the proposed method with K = q = p = 1, especially in high SNR or high Doppler spread.
V. CONCLUSION
We found that, in a mobile time-varying channel, the resid-ual ICI beyond several dominant terms had high normalized autocorrelation. We derived a rather precise closed-form ap-proximation for the (unnormalized) autocorrelation function. It turns out that, up to a rather high peak Doppler frequency, the normalized autocorrelation was not sensitive to a variety
of system parameters and channel conditions, including the DFT size, the sample period, the system bandwidth, the OFDM symbol period, the average transmitted symbol energy, the multipath channel profile, and the Doppler PSDs of the channel paths. As a result, a whitening transform for the residual ICI plus noise can be obtained based solely on the ICI-to-noise ratio. Such a transform can be used in association with many different signal detection schemes to significantly improve the detection performance. That it depends only on the ICI-to-noise ratio but no other quantities also implies simplicity and robustness.
We considered MLSE-type signal detection in ICI with blockwise whitening of the residual ICI plus noise. Simu-lations showed that the proposed technique could attain a substantially lower ICI-induced error floor than conventional MLSE.
APPENDIX
DERIVATION OF(14)ANDSOMERELATEDCOMMENTS Substituting the inverse Fourier transform relation in (8) into the right-hand side (RHS) of (13), we get
E[cm,Kc∗m+r,K] = Es N2 L−1 l=0 N −1 n=0 N −1 n=0 k /∈[−K,+K]∪[−K−r,K−r] σl2 · fd −fd Pl(f){cos[2πfTsa(n − n)] + j sin[2πfTsa(n − n)]}df · ej2π[n (k+r)−nk]/N . (42)
Let ξ denote the quantity that collects all the terms associated with sin[2πf Tsa(n − n)]. That is,
ξ = Es N2 L−1 l=0 σl2 fd −fd df Pl(f) · N −1 n=0 N −1 n=0 k /∈[−K,+K] ∪[−K−r,K−r] j sin[2πf Tsa(n − n)] · ej2π[n(k+r)−nk]/N. (43)
Consider the inner triple sum and denote it by χ. By substi-tuting the variables n, n, and k with ν, ν, and −(κ + r), respectively, we get, after some straightforward algebra,
χ = N −1 ν=0 N −1 ν=0 κ /∈[−K,+K] ∪[−K−r,K−r] {−j sin[2πfTsa(ν − ν)]} · ej2π[ν(κ+r)−νκ]/N. (44)
A comparison with the inner triple sum in (43) shows that
χ = −χ, which implies χ = 0 and thus ξ = 0. Therefore, only
the cosine terms remain in E[cm,Kc∗m+r,K]. Approximating
the cosine function by taking its power series expansion and retaining only up to the second-order term as cos x ≈ 1−x2/2,
we get E[cm,Kc∗m+r,K] ≈ Es N2 L−1 l=0 σl2 fd −fd Pl(f)df k /∈[−K,K] ∪[−K−r,K−r] N −1 n=0 e−j2πnk/N =0 · N −1 n=0 ej2πn(k+r)/N =0 − Es 2N2 L−1 l=0 σ2l fd −fd Pl(f)(2πfTsa)2df · k /∈[−K,K] ∪[−K−r,K−r] ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ N −1 n=0 n2e−j2πnk/N N −1 n=0 ej2πn(k+r)/N =0 + N −1 n=0 e−j2πnk/N =0 N −1 n=0 n2ej2πn(k+r)/N − 2 N −1 n=0 ne−j2πnk/N N −1 n=0 nej2πn(k+r)/N = 4π2T2 saEs L−1 l=0 σl2 fd −fd Pl(f)f2df · k /∈[−K,K] ∪[−K−r,K−r] 1 (1 − e−j2πk/N)(1 − ej2π(k+r)/N) . (45)
In fact, the above second-order approximation to cosine function is tantamount to assuming linearly time-varying paths in the CIR. To see it, let hl(t) denote the continuous-time
waveform of the lth path of the CIR (of which hn,l is a
sampled version) and let hl(t) be its time-derivative. Then by a well-known relation between the time-derivative of a stochastic process and its PSD, we have 4π2σl2 Pl(f)f2df =
E|hl(t)|2 [20, Table 7.5-1]. Therefore, if we approximate the channel by one whose lth path response varies linearly with time in some period with its slope equal to
|h
l(t)|2
1/2 in magnitude (where the overline in the brackets denotes time average over this period), then the autocorrelation of residual ICI of the approximating channel would be exactly that obtained above, without approximation. In this sense, the second-order approximation to cosine function above is tantamount to assuming linearly time-varying paths in the CIR. Numerical examples in Section III show that the ensuing approximation to the autocorrelation of the residual ICI is rather accurate even under a relatively large peak Doppler shift.
REFERENCES
[1] IEEE Std. 802.16-2009, IEEE Standard for Local and Metropolitan Networks — Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems. New York: IEEE, May 2009.
[2] W. G. Jeon, K. H. Chang, and Y. S. Cho, “An equalization technique for orthogonal frequency-division multiplexing systems in time-variant multipath channels,” IEEE Trans. Commun., vol. 47, no. 1, pp. 27–32, Jan. 1999.
[3] P. Schniter, “Low-complexity equalization of OFDM in doubly selective channels,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 1002–1011, Apr. 2004.
[4] L. Rugini, P. Banelli, and G. Leus, “Simple equalization of time-varying channels for OFDM,” IEEE Commun. Lett., vol. 9, no. 7, pp. 619–621, July 2005.
[5] S. Ohno, “Maximum likelihood inter-carrier interference suppression for wireless OFDM with null subcarriers,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., vol. 3, 2005, pp. 849–852.
[6] M. Russell and G. L. St¨uber, “Interchannel interference analysis of OFDM in a mobile environment,” in IEEE Veh. Technol. Conf., vol. 2, July 1995, pp. 820–824.
[7] A. A. Hutter and R. Hasholzner, “Determination of intercarrier interfer-ence covariance matrices and their application to advanced equalization for mobile OFDM,” in Proc. 5th Int. OFDM Workshop, Hamburg, Germany, Sep. 2000, pp. 33-1–33-5.
[8] A. A. Hutter, J. S. Hammerschmidt, E. de Carvalho, and J. M. Cioffi, “Receive diversity for mobile OFDM systems,” in Proc. IEEE Wirel. Commun. Networking Conf., Sep. 2000, pp. 707-712.
[9] S. Ohno and K. A. D. Teo, “Approximate BER expression of ML equalizer for OFDM over doubly selective channels,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., 2008, pp. 3049–3052. [10] Y.-H. Yeh and S.-G. Chen, “An efficient fast-fading channel estimation
and equalization method with self ICI cancellation,” in Eur. Signal Process. Conf., Sep. 2004, pp. 449–452.
[11] H.-w. Wang, D. W. Lin, and T.-H. Sang, “OFDM signal detection in doubly selective channels with whitening of residual intercarrier interference and noise,” in IEEE Veh. Technol. Conf., May 2010, pp. 1–5.
[12] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974.
[13] Y. Li and L. J. Cimini, Jr., “Bounds on the interchannel interference of OFDM in time-varying impairments,” IEEE Trans. Commun., vol. 49, no. 3, pp. 401–404, Mar. 2001.
[14] G. Huang, A. Nix, and S. Armour, “DFT-based channel estimation and noise variance estimation techniques for single-carrier FDMA,” in IEEE Veh. Technol. Conf. Fall, Sep. 2010, pp. 1–5.
[15] G. L. St¨uber, Principles of Mobile Communication, 2nd ed. Boston, MA: Kluwer Academic, 2001.
[16] X. Cai and G. B. Giannakis, “Bounding performance and suppressing intercarrier interference in wireless mobile OFDM,” IEEE Trans. Com-mun., vol. 51, no. 12, pp. 2047–2056, Dec. 2003.
[17] D. Huang, K. B. Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol. 53, no. 7, pp. 1191–1199, July 2005. [18] H.-D. Lin, T.-H. Sang, and D. W. Lin, “BICM-OFDM for cooperative
communications with multiple synchronization errors,” in Proc. Int. Wirel. Commun. Mobile Comput. Conf., July 2010, pp. 1055–1059. [19] K.-C. Hung and D. W. Lin, “Pilot-aided multicarrier channel estimation
via MMSE linear phase-shifted polynomial interpolation,” IEEE Trans. Wirel. Commun., vol. 9, no. 8, pp. 2539–2549, Aug. 2010.
[20] H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing, 3rd ed. Upper Saddle River, New Jersey: Prentice-Hall, 2002.
Hai-wei Wang received the B.S. degree in
con-trol engineering and the M.S. degree in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1995 and 1999, re-spectively. She was with Silicon Integrated Systems Corp., Hsinchu, during 1999–2002 and with Realtek Semiconductor Corp., Hsinchu, during 2002–2004.
She is currently pursuing the Ph.D. degree in electronics engineering from the National Chiao Tung University. Her research interests are in the areas of digital communications and communication theory.
David W. Lin (S’79-M’81-SM’88) received the
B.S. degree in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1975 and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern Cali-fornia, Los Angeles, CA, U.S.A., in 1979 and 1981, respectively.
He was with Bell Laboratories during 1981–1983 and with Bellcore during 1984–1990 and again dur-ing 1993–1994. Since 1990, he has been a Professor with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, except for a leave in 1993–1994. He has conducted research in digital adaptive filtering and telephone echo cancellation, digital subcriber line and coaxial network trans-mission, speech and video coding, and wireless communication. His research interests include various topics in signal processing and communication engineering.
Tzu-Hsien Sang (S’96-M’00) received the B.S.E.E.
degree in 1990 from National Taiwan University and Ph.D. degree in 1999 from the University of Michigan at Ann Arbor. He is currently with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Tai-wan. Prior to joining NCTU in 2003, he had worked in Excess Bandwidth, a start-up company at Sunny-vale, California, working on physical layer design for broadband technologies. His research interests include signal processing for communications, time-frequency analysis for biomedicl signals, and RF circuit noise modeling.
