Pilot-Aided Multicarrier Channel Estimation via
MMSE Linear Phase-Shifted
Polynomial Interpolation
Kun-Chien Hung and David W. Lin, Senior Member, IEEE
Abstract—Current wireless multicarrier systems commonly
adopt pilot-aided channel estimation, for which a simplest and least restrictive technique is polynomial interpolation in the frequency domain. For channels with large delay spreads, however, the performance of low-order polynomial interpolation suffers from modeling error. The problem may be remedied by adding a linear phase shift to the interpolator, or equivalently, a “window shift" in the time domain. We derive a method to estimate the optimal window shift, in the minimum mean-square error (MMSE) sense, for polynomial-interpolative channel estimation of arbitrary order. As a practical application, we show how to apply the resultant technique to Mobile WiMAX downlink channel estimation. In addition, we propose a method to automatically select the interpolation order based on some estimated MSE.
Index Terms—Channel estimation, delay estimation,
orthog-onal frequency-division multiple access (OFDMA), orthogorthog-onal frequency-division multiplexing (OFDM), polynomial interpola-tion.
I. INTRODUCTION
C
OHERENT multicarrier transmission requires channelstate information. Hence wireless multicarrier systems today commonly transmit pilots to facilitate channel estima-tion. Pilots are subcarriers that carry known signals. They may be dispersed in frequency and/or in time. A typical channel estimator starts by estimating the channel responses at the pilot subcarriers. Then it “fleshes out" the estimate for other subcarriers in some way. Three primary types of pilot-aided channel estimation methods are model-based, Wiener filtering, and channel-independent interpolation.
Model-based methods attempt to identify the the multipath delays. The channel is then estimated via projection onto the so-called “delay subspace." Such methods can achieve good performance with few pilots, but two main concerns are the Manuscript received May 4, 2009; revised September 28, 2009, January 27, 2010, and April 16, 2010; accepted May 25, 2010. The associate editor coordinating the review of this paper and approving it for publication was X. Gao.
K.-C. Hung was with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu, Taiwan 30010. He is now with MediaTek Inc., 1 Dusing Rd. 1, Hsinchu Science Park, Hsinchu, Taiwan 30078 (e-mail: hkc.ee90g@nctu.edu.tw).
D. W. Lin is with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu, Taiwan 30010 (e-mail: dwlin@mail.nctu.edu.tw).
This work was supported in part by the National Science Council of the Republic of China under Grant NSC 97-2219-E-009-009. Part of this work was presented at IEEE Globecom 2008, New Orleans, LA, USA.
Digital Object Identifier 10.1109/TWC.2010.061010.090467
complexity and the robustness of delay subspace estimation [1]–[4], especially when the multipath delays are not sample-spaced.
Wiener, or linear minimum mean-square error (LMMSE), filtering, has been considered widely [5]–[9]. However, it re-quires the channel correlation function, which raises problems not only in the complexity of correlation estimation but also in the impact of estimation accuracy on channel estimation performance. A suboptimal solution is to use a predefined shape for the correlation function, such as that associated with a uniform [9] or an exponential power-delay profile (PDP) [5], [10]. Nevertheless, one still needs to estimate the parameters of the PDP.
Channel-independent interpolation methods seek to recon-struct the channel responses at data subcarriers by channel-independent interpolation of the channel estimates at pilot subcarriers. The simplest of them is low-order polynomial interpolation in the frequency domain [11]. The maximum-likelihood (ML) interpolator [7] is a more sophisticated mem-ber of them, but it requires that the nummem-ber of pilots be no smaller than the length of the channel impulse response.
In short, an ideal channel estimator should require as few pilots as possible, have good estimation perfomance and low computational complexity, and be robust to variations in channel and operating conditions. Existing techniques are open to improvement in one or another of these aspects. In this work, we consider the lowest-complexity approach, namely, polynomial interpolation, which also appears to be the most robust and most broadly applicable of all approaches. We seek to enhance its performance without significant increase in complexity. We assume that the synchronization of carrier frequency and multicarrier symbol timing is done prior to channel estimation. Further, we assume that the carrier fre-quency synchronization is rather accurate while the symbol timing synchronization may be less so. These are reasonable assumptions, for example, in downlink (DL) Mobile WiMAX transmission [12], [13].
To begin, we first note that, of all polynomial interpolation schemes, a simplest and most commonly considered is linear interpolation. For this scheme, it has been found that the channel estimation performance may suffer greatly not only in poor symbol synchronization [14] but also when the channel has a large delay spread [15]. One way to alleviate the problem is to use a higher interpolation order [11]. The reason why different interpolation orders may give different performance 1536-1276/10$25.00 c⃝ 2010 IEEE
−800 −60 −40 −20 0 20 40 60 80 0.2 0.4 0.6 0.8 1 1.2 1.4 Sample Time Amplitude Response Linear Window Quadratic Window Shifted Linear Window Channel Response
Fig. 1. Comparison of different ways of interpolation in terms of the equivalent time-windowing effects.
can be interpreted via some well-known results in signal processing theory, which not only provide a useful perspective for understanding and analysis of the phenomenon but also hint at improved interpolator design. Specifically, polynomial interpolation amounts to a sort of linear filtering [16]. Since convolution in the frequency domain corresponds to multipli-cation in the time domain, different orders of interpolation of the frequency response correspond to different kinds of win-dowing on the channel impulse response. Fig. 1 is a conceptual illustration [17]. Linear interpolation (dashed line) corresponds to a smaller window size than quadratic interpolation (dash-dot line) and hence results in greater distortion of the channel response. However, if one can shift the window corresponding to linear interpolation by some amount to better capture the time range of significant channel response samples (solid line), then one may improve its performance. The same also applies to higher-order interpolation.
Now the question is: how to determine the optimal window shift for a given interpolation order? As shifting a signal in time amounts to modulating its frequency spectrum with a single complex exponential, Hsieh and Wei [11] adopt the single frequency estimators of Kay [18] to find the desired offset. Unfortunately, for channel estimation these estimators are not optimal in the MSE sense [15]. We will derive the optimal window shift that attains MMSE in channel estimation for arbitrary polynomial order. Another interesting question is how the required window shifting (in time) or the equivalent linear phase rotation (in frequency) may be realized. Earlier studies [11], [15] have employed straightforward linear phase rotation in the frequency domain. We will see that, when the amount of shift is an integer multiple of the sample period, a much simpler alternative exists which involves mere circular shifting of signal samples.
It is instinctively expected that the MMSE in channel esti-mation should depend on the amount of channel noise. In fact, later analysis will show that, in high channel noise, higher-order interpolation may be affected so more adversely than lower-order interpolation as to yield worse channel estimates. Therefore, it would be desirable if the interpolation order can
be selected adaptively according to the channel condition to attain the best possible channel estimation performance. We will also address this issue.
In summary, the contributions of this paper are: 1) a low-complexity channel estimation technique based on linear phase-shifted polynomial interpolation, 2) specialization of the technique to DL Mobile WiMAX transmission, and 3) a technique for adaptive selection of the polynomial inter-polation order. In what follows, Section II considers the estimation of MMSE window shift under given polynomial order. Section III evaluates its performance in noisy channel under comb-type pilots. Section IV adapts the technique to suit the Mobile WiMAX DL signal, which has an interesting pilot structure that necessitates some additional design work. We also investigate the adaptive selection of interpolation order in this context. Finally, Section V concludes the paper.
II. MMSE WINDOWSHIFTING
Consider a sampled equivalent lowpass channel impulse response
ℎ(𝑛) =𝐿−1∑
𝑙=0
𝛼𝑙𝛿(𝑛 − 𝑙) (1)
where 𝑛 and 𝑙 are integers in units of the sampling period
𝑇𝑠, 𝛼𝑙 is the complex gain of path 𝑙, and (𝐿 − 1)𝑇𝑠 is the
maximum path delay. The corresponding frequency response is given by
𝐻(𝑓) =
𝐿−1∑
𝑙=0
𝛼𝑙𝑒−𝑗2𝜋𝑙𝑓/𝑁 (2)
where 𝑁 is the size of discrete Fourier transform (DFT)
used in multicarrier transformation and the division by 𝑁
in the exponent term normlizes the subcarrier spacing in the multicarrier system to unity as well as normalizes the period of𝐻(𝑓) in 𝑓 to 𝑁.
Consider comb-type pilot allocation and let the 𝑚th pilot
subcarrier be located at frequency 𝑝0 + 𝑚𝐹 where 𝑝0 is
the lowest pilot subcarrier frequency and 𝐹 is the spacing
of pilot subcarriers. (The treatment can be readily extended to noncomb-type pilots, but the equations become somewhat
cumbersome.) Let ˆ𝐻(𝑝) denote the channel estimate at any
pilot subcarrier𝑝, however it may be obtained. Then
conven-tional polynomial interpolation between two pilot locations
𝑝 = 𝑝0+ 𝑚𝐹 and 𝑝 + 𝐹 = 𝑝0+ (𝑚 + 1)𝐹 can be written as
ˆ
𝐻(𝑝 + 𝑛) =∑𝐾
𝑘=0
𝑐𝑛𝑘𝐻(𝑝 + 𝑥ˆ 𝑛𝑘) (3)
where 0 < 𝑛 < 𝐹 , 𝐾 is the interpolation order, 𝑥𝑛𝑘 is an
integer multiple of𝐹 that defines the 𝑘th pilot location used
in interpolation, and 𝑐𝑛𝑘 is the corresponding interpolation
coefficient. Usually,𝑥𝑛𝑘 should be as small in magnitude as
possible so that the interpolating pilots𝑝+𝑥𝑛𝑘are as close to
the frequency𝑝 + 𝑛 as possible, but this point does not matter
for now. The coefficients that yield error-free interpolation of
They are real and can be written in the Vandermonde form as c′ 𝑛𝐾 ≜ [𝑐𝑛0 𝑐𝑛1 ⋅ ⋅ ⋅ 𝑐𝑛𝐾] =[1 𝑛 ⋅ ⋅ ⋅ 𝑛𝐾] ≜v′ 𝑛𝐾 ⎡ ⎢ ⎢ ⎢ ⎣ 1 𝑥𝑛0 ⋅ ⋅ ⋅ 𝑥𝐾𝑛0 1 𝑥𝑛1 ⋅ ⋅ ⋅ 𝑥𝐾𝑛1 .. . ... ... ... 1 𝑥𝑛𝐾 ⋅ ⋅ ⋅ 𝑥𝐾𝑛𝐾 ⎤ ⎥ ⎥ ⎥ ⎦ ≜X𝑛𝐾 −1 , (4)
where ′ denotes transpose, or in the Lagrange form as [20,
§25.2.2] 𝑐𝑛𝑘= 𝐾 ∏ 𝑚=0,𝑚∕=𝑘 𝑛 − 𝑥𝑛𝑚 𝑥𝑛𝑘− 𝑥𝑛𝑚, 0 ≤ 𝑘 ≤ 𝐾. (5) Now consider phase-rotated interpolation corresponding to
a time window delay of𝜏 samples, where the optimal value of
𝜏 is to be found. (In principle, 𝜏 needs not be an integer.) To
see how it affects (3), consider an arbitrary frequency-domain
filter with input𝑎(𝑓), output 𝑏(𝑓), and impulse response 𝑐(𝑓).
A𝜏-sample delay of the time window corresponding to 𝑐(𝑓)
amounts to a linear phase rotation of the frequency-domain
impulse response to𝑒−𝑗2𝜋𝜏𝑓/𝑁𝑐(𝑓). That is, the input-output
relation is changed from𝑏(𝑓) =∑𝑓′𝑐(𝑓 −𝑓′)𝑎(𝑓′) to 𝑏(𝑓) =
∑ 𝑓′𝑒−𝑗2𝜋𝜏(𝑓−𝑓 ′)/𝑁 𝑐(𝑓 − 𝑓′)𝑎(𝑓′). Identifying 𝑓 with 𝑝 + 𝑛 and𝑓′ with𝑝 + 𝑥 𝑛𝑘, we get, for (3), ˆ 𝐻(𝑝 + 𝑛) = 𝐾 ∑ 𝑘=0 𝑒−𝑗2𝜋𝜏(𝑛−𝑥𝑛𝑘)/𝑁𝑐 𝑛𝑘𝐻(𝑝 + 𝑥ˆ 𝑛𝑘) = 𝑒−𝑗2𝜋𝜏(𝑝+𝑛)/𝑁∑𝐾 𝑘=0 𝑐𝑛𝑘 [ 𝑒𝑗2𝜋𝜏(𝑝+𝑥𝑛𝑘)/𝑁𝐻(𝑝 + 𝑥ˆ 𝑛𝑘) ] . (6)
By substituting 𝑝 + 𝑥𝑛𝑘 with 𝑓 in the last bracketed term,
we see that this term indicates a phase rotation of the pilot subcarrier channel estimates by an amount corresponding to a𝜏-sample time advance. The right-hand side (RHS) of (6)
can thus be interpreted as indicating a four-step procedure for
channel estimation under given𝜏 as follows: 1) obtain the pilot
subcarrier channel estimates ˆ𝐻(𝑝 + 𝑥𝑛𝑘) by some method,
2) phase-rotate the pilot subcarrier channel estimates by an
amount corresponding to a 𝜏-sample time advance (as
indi-cated by the bracketed term in (6)), 3) perform conventional polynomial interpolation to obtain the data subcarrier channel estimates (as indicated by the last summation in (6)), and 4) phase-derotate the channel estimates from the last step by an
amount corresponding to a𝜏-sample time delay (as indicated
by the premultiplication by𝑒−𝑗2𝜋𝜏(𝑝+𝑛)/𝑁 in (6)).
As mentioned previously, when 𝜏 is an integer, a simple
time-domain equivalent exists for steps 2 and 4. Specifically,
for an integer𝜏, if we shift the received signal circularly by
−𝜏 samples before taking its DFT, the channel response is
effectively advanced by𝜏 samples. Working with the circularly
shifted signal, the phase rotation and derotation can be saved. More details will be given in Section IV when we consider Mobile WiMAX DL transmission. But for convenience, the theoretical derivation below employs the formulation based on phase rotation.
Now we turn to the key issue of determining 𝜏. For this
we first find how the channel estimation MSE depends on 𝜏
under arbitrary polynomial interpolation order. Then we derive
a suboptimal solution for 𝜏 and consider its computational
complexity.
A. MSE as Function of Window Shift
Consider advancing the channel impulse response by 𝜏
samples. The resultant frequency response is given by
𝐻𝜏(𝑓) ≜ 𝑒𝑗2𝜋𝜏𝑓/𝑁𝐻(𝑓) =
𝐿−1∑
𝑙=0
𝛼𝑙𝑒−𝑗2𝜋(𝑙−𝜏)𝑓/𝑁. (7)
Laying together the first RHS in (7) and the bracketed term in (6), we see that the bracketed term may be viewed as an
estimate of𝐻𝜏(𝑝 + 𝑥𝑛𝑘). More generally, let
ˆ
𝐻𝜏(𝑓) = 𝑒𝑗2𝜋𝜏𝑓/𝑁𝐻(𝑓).ˆ (8)
Then (6) can be rewritten as ˆ 𝐻𝜏(𝑝 + 𝑛) = 𝐾 ∑ 𝑘=0 𝑐𝑛𝑘𝐻ˆ𝜏(𝑝 + 𝑥𝑛𝑘). (9)
The MSE in ˆ𝐻𝜏(𝑝 + 𝑛) (as estimate of 𝐻𝜏(𝑝 + 𝑛)) is the
same as that in ˆ𝐻(𝑝 + 𝑛) (as estimate of 𝐻(𝑝 + 𝑛)), for the
two estimates are related by mere phase rotation. Hence we derive the former. To concentrate on the modeling error of interpolation, we ignore the effect of channel noise for the
moment. That is, we assume temporarily that ˆ𝐻𝜏(𝑝 + 𝑥𝑛𝑘) =
𝐻𝜏(𝑝 + 𝑥𝑛𝑘). In fact, mathematics below shows that white
channel noise does not affect the optimal window shift. The𝐾th-order Taylor series expansion of 𝐻𝜏(𝑝 + 𝑛) about
a subcarrier𝑝 is given by 𝐻𝜏(𝑝 + 𝑛) = 𝐾 ∑ 𝑘=0 𝐻𝜏(𝑘)(𝑝)𝑛𝑘 𝑘! + 𝜌𝐾(𝑝 + 𝑛) (10)
where 𝐻𝜏(𝑘)(𝑝) denotes the 𝑘th derivative of 𝐻𝜏(𝑝) and
𝜌𝐾(𝑝 + 𝑛) is the remainder term. By the integral form of
the remainder term [20, §25.2.25],
𝜌𝐾(𝑝 + 𝑛) = ∫ 𝑛 0 𝐻𝜏(𝐾+1)(𝑝 + 𝑢) 𝐾! (𝑛 − 𝑢)𝐾𝑑𝑢 =𝐿−1∑ 𝑙=0 𝛼𝑙 [ −𝑗2𝜋(𝑙 − 𝜏) 𝑁 ]𝐾+1 𝑒−𝑗2𝜋(𝑙−𝜏)𝑝/𝑁 ⋅ ∫ 𝑛 0 𝑒−𝑗2𝜋(𝑙−𝜏)𝑢/𝑁 𝐾! (𝑛 − 𝑢)𝐾𝑑𝑢. (11)
From (9) and (4), the interpolation error for 𝐻𝜏(𝑝 + 𝑛) is
given by 𝐻𝜏(𝑝 + 𝑛) − ˆ𝐻𝜏(𝑝 + 𝑛) = 𝐻𝜏(𝑝 + 𝑛) − v𝑛𝐾′ X−1𝑛𝐾 ⋅ vec [ ˆ 𝐻𝜏(𝑝 + 𝑥𝑛0), ˆ 𝐻𝜏(𝑝 + 𝑥𝑛1), ⋅ ⋅ ⋅ , ˆ𝐻𝜏(𝑝 + 𝑥𝑛𝐾) ] (12) where vec[⋅] denotes a vector of the items given in the brackets. Since no error arises in interpolating a polynomial function of
order𝐾 or below, all errors in (12) are from the remainder
terms𝜌𝐾(𝑝 + 𝑛) and 𝜌𝐾(𝑝 + 𝑥𝑛𝑘), 0 ≤ 𝑘 ≤ 𝐾. Hence
𝐻𝜏(𝑝 + 𝑛) − ˆ𝐻𝜏(𝑝 + 𝑛)
= 𝜌𝐾(𝑝 + 𝑛) − v′𝑛𝐾X−1𝑛𝐾⋅ vec [𝜌𝐾(𝑝 + 𝑥𝑛0),
𝜌𝐾(𝑝 + 𝑥𝑛1), ⋅ ⋅ ⋅ , 𝜌𝐾(𝑝 + 𝑥𝑛𝐾)] . (13)
From (11), if subcarriers𝑝 and 𝑝 + 𝑛 are spaced closely
com-pared to the coherence bandwidth such that𝑒−𝑗2𝜋(𝑙−𝜏)𝑢/𝑁 ≈
1 for 0 ≤ 𝑢 ≤ 𝑛, then ∫ 𝑛 0 𝑒−𝑗2𝜋(𝑙−𝜏)𝑢/𝑁 𝐾! (𝑛 − 𝑢)𝐾𝑑𝑢 ≈ 𝑛𝐾+1 (𝐾 + 1)! (14) and thence 𝜌𝐾(𝑝 + 𝑛) ≈ 𝑛 𝐾+1 (𝐾 + 1)! ⋅ 𝐿−1∑ 𝑙=0 𝛼𝑙 [ −𝑗2𝜋(𝑙 − 𝜏) 𝑁 ]𝐾+1 𝑒−𝑗2𝜋(𝑙−𝜏)𝑝/𝑁. (15)
In fact, (15) gives an approximation of𝜌𝐾(𝑝+𝑛) for any value
of𝑛, not limited to 0 < 𝑛 < 𝐹 , although its accuracy should
tend to diminish as the magnitude of𝑛 increases. Using it to
approximate𝜌(𝑝 + 𝑥𝑛𝑘) (0 ≤ 𝑘 ≤ 𝐾) also and substituting
the result into (13), we get
𝐻𝜏(𝑝 + 𝑛) − ˆ𝐻𝜏(𝑝 + 𝑛) ≈ (𝐾 + 1)!1 [𝑛𝐾+1− v′ 𝑛𝐾X−1𝑛𝐾x𝑛𝐾]𝜉𝐾(𝑝, 𝜏) (16) where x𝑛𝐾 =[𝑥𝐾+1𝑛0 , 𝑥𝐾+1𝑛1 , ⋅ ⋅ ⋅ , 𝑥𝐾+1𝑛𝐾 ]′ , (17) 𝜉𝐾(𝑝, 𝜏) = 𝐿−1∑ 𝑙=0 𝛼𝑙 [ −𝑗2𝜋(𝑙 − 𝜏) 𝑁 ]𝐾+1 𝑒−𝑗2𝜋(𝑙−𝜏)𝑝/𝑁. (18)
The interpolation MSE at a subcarrier located an offset𝑛 from
a pilot can be estimated by taking the average of∣𝐻𝜏(𝑝+𝑛)−
ˆ
𝐻𝜏(𝑝 + 𝑛)∣2 over all pilots. It is shown in Appendix A that
the bracketed term in (16) is equal to∏𝐾𝑘=0(𝑛 − 𝑥𝑛𝑘). As a
result, ⟨⟨∣𝐻𝜏(𝑝 + 𝑛) − ˆ𝐻𝜏(𝑝 + 𝑛)∣2⟩⟩ ≈ [∏𝐾 𝑘=0(𝑛 − 𝑥𝑛𝑘) (𝐾 + 1)! ]2 ⟨⟨∣𝜉𝐾(𝑝, 𝜏)∣2⟩⟩ = [∏𝐾 𝑘=0(𝑛 − 𝑥𝑛𝑘) (𝐾 + 1)! ]2 ⋅ 𝐿−1∑ 𝑙=0 (2𝜋)2𝐾+2 𝑁2𝐾+2 ∣𝛼𝑙∣2(𝑙 − 𝜏)2𝐾+2 ≜𝜎2 𝜉𝐾(𝜏) (19)
where ⟨⟨⋅⟩⟩ denotes averaging over all pilot subcarriers. The
second equality in (19) indicates that the MSE is dominated
by the channel paths with larger values of∣𝛼𝑙∣2(𝑙 − 𝜏)2𝐾+2.
B. Estimation of Optimal Window Shift
From the above results, the channel estimation MSE can be
minimized by minimizing 𝜎2
𝜉𝐾(𝜏). By elementary calculus,
the approximately optimal window shift𝜏 can be obtained by
solving 𝑑𝜎2 𝜉𝐾(𝜏) 𝑑𝜏 = − (2𝜋)2𝐾+2(2𝐾 + 2) 𝑁2𝐾+2 𝐿−1∑ 𝑙=0 ∣𝛼𝑙∣2(𝑙 − 𝜏)2𝐾+1 = 0. (20)
Unfortunately, this requires knowing the multipath gains and delays, which are most likely unknown until after channel estimation. To sidestep this dilemma, note that since
𝐻(𝐾+1) 𝜏 (𝑓) = 𝐿−1∑ 𝑙=0 𝛼𝑙 [ −𝑗2𝜋(𝑙 − 𝜏) 𝑁 ]𝐾+1 𝑒−𝑗2𝜋(𝑙−𝜏)𝑓/𝑁, (21)
by Parseval’s theorem we get 𝜎2
𝜉𝐾(𝜏) = ⟨∣𝐻𝜏(𝐾+1)(𝑓)∣2⟩
where ⟨⋅⟩ denotes averaging over all frequencies. Thus
min-imizing𝜎2
𝜉𝐾(𝜏) is equivalent to minimizing ⟨∣𝐻𝜏(𝐾+1)(𝑓)∣2⟩.
While 𝐻𝜏(𝐾+1)(𝑓) is usually not available, either, it can be
approximated, for example, by the𝐾 +1st forward difference
given by 𝐷𝐾+1𝐻𝜏(𝑓) = 𝐹𝐾+11 𝐾+1∑ 𝑘=0 (−1)𝑘(𝐾 + 1 𝑘 ) 𝐻𝜏(𝑓 + (𝐾 + 1 − 𝑘)𝐹 ). (22)
The target of minimization can then be approximated by
𝜎2 𝜉𝐾(𝜏) ≈ 𝛾 ≜ 𝐹2𝐾+21 ⋅〈〈 𝐾+1∑ 𝑘=0 (−1)𝑘 ⋅ ( 𝐾 + 1 𝑘 ) ˆ 𝐻𝜏(𝑝 + (𝐾 + 1 − 𝑘)𝐹 ) 2〉〉 . (23)
Some algebraic manipulations result in
𝛾 = 𝐹2𝐾+21 [𝑃𝐾𝑅0+ 2𝛾𝐾(𝜏)] (24) where 𝑃𝐾 = 2𝐾+1(2𝐾 + 1)!!/(𝐾 + 1)!, 𝑅𝑘 = ⟨⟨ ˆ𝐻(𝑝 + 𝑘𝐹 ) ˆ𝐻∗(𝑝)⟩⟩, and 𝛾𝐾(𝜏) = ℜ {𝐾+1 ∑ 𝑚=1 (−1)𝑚𝐴 𝐾𝑚𝑒−𝑗2𝜋𝑚𝐹 𝜏/𝑁𝑅𝑚 } (25) withℜ{𝑎} denoting the real part of quantity 𝑎, 𝑚!! = 𝑚(𝑚−
2)(𝑚 − 4) ⋅ ⋅ ⋅ 𝑥 where 𝑥 = 2 for even 𝑚 and 𝑥 = 1 for odd
𝑚, and 𝐴𝐾𝑚 = 𝐾+1∑ 𝑘=𝑚 ( 𝐾 + 1 𝑘 )( 𝐾 + 1 𝑘 − 𝑚 ) = (𝐾 + 1 − 𝑚)!(𝐾 + 1 + 𝑚)!2𝐾+1(2𝐾 + 1)!!(𝐾 + 1)! . (26)
Noting that only 𝛾𝐾(𝜏) depends on 𝜏, we may estimate the
optimal window shift as
ˆ𝜏 = arg min
TABLE I
ORDER-OF-MAGNITUDECOMPARISON OFCHANNELESTIMATOR
COMPLEXITY
Method No. of Real Multiplications
Proposed 2(𝐾 + 1)(𝑁 + 𝑀 + 𝐿/𝑔)
ML with radix-2 FFT 2𝑀 log2𝑀 + 2𝑁 log2𝑁
ML with radix-4 FFT 3𝑀2 log2𝑀 +3𝑁2 log2𝑁
For linear, quadratic and cubic interpolators, we have
𝛾1(𝜏) = ℜ{𝑒−𝑗2𝜃𝑅2− 4𝑒−𝑗𝜃𝑅1}, (28)
𝛾2(𝜏) = ℜ{−𝑒−𝑗3𝜃𝑅3+ 6𝑒−𝑗2𝜃𝑅2− 15𝑒−𝑗𝜃𝑅1}, (29)
𝛾3(𝜏) = ℜ{𝑒−𝑗4𝜃𝑅4− 8𝑒−𝑗3𝜃𝑅3+ 28𝑒−𝑗2𝜃𝑅2
−56𝑒−𝑗𝜃𝑅
1}, (30)
where 𝜃 = 2𝜋𝐹 𝜏/𝑁. To compute ˆ𝜏 by (27), one way is
to solve the equation 𝑑𝛾𝐾(𝜏)/𝑑𝜏 = 0 for 𝜏 that minimizes
𝛾𝐾(𝜏). Unfortunately, algebraic solution exists only in the
case of linear interpolation, for it is the only case where the equation has order under 5 [15]. Numerical solutions have to be resorted to in other cases. This being so, one might
as well search over the range [0, 𝐿 − 1] for the value of
𝜏 that minimizes 𝛾𝐾(𝜏) rather than find it through solving
𝑑𝛾𝐾(𝜏)/𝑑𝜏 = 0. The density of searched points can be chosen
according to the desired accuracy. By experience, we find that a suitable granularity is roughly 2 to 6 sample periods.
C. Computational Complexity
In summary, the proposed procedure to computeˆ𝜏 for
𝐾th-order interpolation is as follows: 1) Given pilot subcarrier
channel response estimates ˆ𝐻(𝑝), calculate 𝑅𝑘 = ⟨⟨ ˆ𝐻(𝑝 +
𝑘𝐹 ) ˆ𝐻∗(𝑝)⟩⟩, 1 ≤ 𝑘 ≤ 𝐾+1, and 2) search over the maximum range of path delays at a chosen granularity for the delay value
𝜏 that minimizes 𝛾𝐾(𝜏) and let ˆ𝜏 be equal to it. Step 1 requires
on the order of 4(𝐾 + 1)𝑀 real multiplications (RMULTs)
where𝑀 is the number of pilot subcarriers used to compute
𝑅𝑘. For step 2, the number of 𝛾𝐾(𝜏) values to be computed
is on the order of 𝐿/𝑔 where 𝑔 is the search granularity.
The 𝑒𝑗𝜃 values corresponding to the searched delay values
can be precomputed and stored. Then step 2 requires on the
order of2(𝐾 +1)𝐿/𝑔 RMULTs. Thus the total computational
complexity to obtainˆ𝜏 is on the order of (𝐾 +1)(4𝑀 +2𝐿/𝑔)
RMULTs. In most system designs, it should be a fraction of
the order of2(𝐾 + 1)(𝑁 − 𝑀) RMULTs for the interpolation
proper.
It is of interest to compare the above with the complexity of ML interpolation. Given pilot subcarrier channel estimates
ˆ
𝐻(𝑝), ML interpolation can be achieved by taking the
𝑀-point inverse DFT (IDFT) of ˆ𝐻(𝑝), truncating the result to
have maximum delay 𝐿 − 1, and taking the 𝑁-point DFT.
Table I gives a comparison in number of RMULTs, assuming that radix-2 or radix-4 fast Fourier transform (FFT) can be used to carry out the IDFT and DFT in ML interpolation.
Note that ML interpolation requires that𝑀 > 𝐿 − 1.
In current systems, the channel path delays usually do not vary significantly over several multicarrier symbols. Nor do the path gains vary violently over consecutive multicar-rier symbols. These can be exploited to further reduce the
complexity of the proposed technique. For example, after initialization, we may perform incremental search for the
optimal ˆ𝜏 within a small range around the previous solution,
rather than conducting a full search over the entire interval. Indeed, the optimal window shift may need be estimated only once every few symbols.
III. PERFORMANCEWITHCOMB-TYPEPILOTS
Now consider the overall MSE of channel estimation by the proposed method in noisy channel for a system with comb-type pilots. Two reasons why we consider such a system are its relative ease of analysis and its ability to provide insights concerning the proposed method’s performance.
To start, note that the estimation error arises from three sources, namely, 1) error from optimal interpolation (i.e., the
minimum modeling error𝐻𝜏(𝑝 + 𝑛) − ˆ𝐻𝜏(𝑝 + 𝑛) attainable
with optimal window shift), 2) use of the suboptimal window
shiftˆ𝜏 of (27), and 3) the channel noise (which has been
omit-ted thus far). The effect of source 2 (suboptimal estimation of window shift) defies a simple and general characterization because, as can be seen from the derivation from (19) to (27), it depends on the channel property in a complicated way. For-tunately, experience shows that when the pilots are reasonably dense, this error source contributes a minor amount in the total MSE. So we omit its effect in the analysis. Actually, we do not have an exact expression for source 1 (error of optimal interpolation), either, but only an approximation as given in (19). Thus the analysis is necessarily approximate. We assume that the channel noise is additive white Gaussian, i.e., AWGN, and independent of the modeling error.
For simplicity, let the pilot spacing𝐹 be an integer fraction
of𝑁. And let 𝜎2
𝑒𝐾(𝑛) denote the RHS of (19). Then from (19),
the average MSE due to modeling error is approximately
𝜎2 𝑒𝐾≜ 𝐹 − 11 𝐹 −1∑ 𝑛=1 𝜎2 𝑒𝐾(𝑛) = 𝐹 − 11 𝐹 −1∑ 𝑛=1 [∏𝐾 𝑘=0(𝑛 − 𝑥𝑛𝑘) (𝐾 + 1)! ]2 ≜𝑞𝐾(𝐹 ) ⋅𝜎2 𝜉𝐾(𝜏𝑜) (31)
where𝜏𝑜is the approximately optimal delay defined by (20).
To be more specific of the values of𝑞𝐾(𝐹 ), we need to define
𝑥𝑛𝑘specifically. For this, consider letting𝑥𝑛𝑘lie in the range
−⌊𝐾/2⌋𝐹 to ⌊(𝐾 +1)/2⌋𝐹 so that it is as small in magnitude
as possible. And discard the nonpilot subcarriers that are too
near a bandedge that there do not exist 𝐾 + 1 pilots in the
above range for interpolation use. Then for linear, quadratic, and cubic interpolations we get
𝑞1(𝐹 ) = 1201 𝐹 (𝐹 + 1)(𝐹2+ 1), (32)
𝑞2(𝐹 ) = 75601 𝐹 (𝐹 + 1)(2𝐹 − 1)(2𝐹 + 1)(4𝐹2+ 5), (33)
𝑞3(𝐹 ) = 3628801 𝐹 (𝐹 + 1)
⋅(103𝐹6+ 103𝐹4+ 61𝐹2+ 21), (34)
respectively. But𝜎2
𝜉𝐾(𝜏𝑜) is channel-dependent and no general
For the contribution of the AWGN, consider (3). Let the pilot signals have unity magnitude and let the pilot
fre-quency responses𝐻(𝑝) be estimated by the least-squares (LS)
method, that is, by simply dividing the received signal value at
each pilot subcarrier𝑝 by the pilot signal value there. Then the
MSE in each ˆ𝐻(𝑝) is equal to the AWGN variance, which we
denote by𝜎2
𝜂. Hence, the AWGN contribution in the channel
estimation MSE at subcarrier𝑝 + 𝑛 is
𝜎2
𝑤𝐾(𝑛) ≜ 𝜎2𝜂∥c𝑛𝐾∥2= 𝜎𝜂2v′𝑛𝐾(X𝑛𝐾X′𝑛𝐾)−1v𝑛𝐾 (35)
(wherec𝑛𝐾 is as defined in (4)) and the average over all data
subcarriers is given by 𝜎2 𝑤𝐾 = ∑𝐹 −1 𝑛=1𝜎𝑤𝐾2 (𝑛)/(𝐹 − 1). In particular, 𝜎2 𝑤1= ( 0.667 −0.333 𝐹 ) 𝜎2 𝜂, (36) 𝜎2 𝑤2= ( 0.8 −0.2𝐹 +0.05𝐹2 +0.05𝐹3 ) 𝜎2 𝜂, (37) 𝜎2 𝑤3= ( 0.776 −0.224 𝐹 − 0.058 𝐹2 − 0.058 𝐹3 − 0.013 𝐹4 +0.013𝐹5 ) 𝜎2 𝜂. (38)
The overall approximate channel estimation MSE is given
by𝜎2
𝑒𝐾+ 𝜎𝑤𝐾2 .
A. Numerical Examples
As an example, consider an orthogonal frequency-division multiplexing (OFDM) system with bandwidth = 10 MHz, DFT size𝑁 = 1024, cyclic prefix (CP) length = 𝑁/8 = 128, and
pilot spacing𝐹 = 4. The window shift is obtained by search
over all even-sample delays between 0 and 127. That is, the search granularity is 2 samples. We simulate the SUI-4 and
SUI-5 PDPs [21], whose power profiles are [0, −4, −8] and
[0, −5, −10] (in dB), respectively, and whose delay profiles
are (approximately) [0, 14, 36] and [0, 45, 112] (in samples),
respectively. The path coefficients are Rayleigh and block-static (constant over one symbol period). A total of 2000 simulation runs are used to obtain the MSE statistics for each signal-to-noise (SNR) value.
Fig. 2 shows histograms of the estimated window shifts for different interpolation orders over SUI-4 and SUI-5. Most shifts are quite a distance from zero, and they span a broad range within the limits of the delay spread, especially for the lower interpolation orders. Together with the MSE results below, this confirms the importance of proper window shifting on interpolation performance.
Fig. 3 illustrates the channel estimation performance of the proposed technique and compares it with conventional poly-nomial interpolation, LMMSE (4 taps), and ML (truncating to 128 samples between IDFT and DFT), where the MSE is normalized (NMSE) relative to the channel power gain. For LMMSE channel estimation, two conditions are simulated, one with filter coefficients calculated using the exact channel correlation functions (the “exact LMMSE" curves) and the other with them calculated using the channel correlation function corresponding to a uniform PDP of length equal to the CP (the “approx. LMMSE" curves). For SUI-4, the approximate analysis for phase-shifted interpolation yields
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25
Window Shift (Samples)
Percentage (%) Linear Quad. Cubic 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12
Window Shift (Samples)
Percentage (%)
Linear Quad. Cubic
Fig. 2. Histograms of estimated window shifts for different orders of polynomial interpolations at 30 dB SNR. Top: over 4; bottom: over SUI-5.
almost indistinguishable results from simulation. That for conventional polynomial interpolation is less accurate, as can be expected. At higher SNR values, conventional polynomial interpolation suffers significant loss from the MSE floors due to modeling errors, whereas phase-shifted interpolation can
maintain a −1 slope in NMSE to much higher SNR values.
For SUI-5, which has a greater delay spread than SUI-4, the approximate analysis tends to overestimate the MSE. And the error floors begin at lower SNR values than in SUI-4. However, the MSE floors of phase-shifted interpolation are much lower than that of conventional interpolation for all interpolation orders.
Not surprisingly, the best performer at any SNR is either the ML or the exact LMMSE. For ML channel estimation, since there are 256 pilots and we truncate to 128 samples between IDFT and DFT, its NMSE is uniformly better than the SNR by 3 dB. From Table I, phase-shifted polynomial
interpolation takes on the order of 5376 (for 𝐾 = 1) to
10752 (for 𝐾 = 3) RMULTs, whereas ML interpolation on
the order of 24576 (with 2 FFT) or 18432 (with radix-4 FFT) RMULTs. Regarding LMMSE channel estimation, exact LMMSE is not realizable. The approximate LMMSE
5 10 15 20 25 30 35 40 45 50 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 SNR (dB) NMSE (dB) 5 10 15 20 25 30 35 40 45 50 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 SNR (dB) NMSE (dB) Linear (Simulation) Quad. (Simulation) Cubic (Simulation) Linear (Analysis) Quad. (Analysis) Cubic (Analysis) PS Linear (Simul.) PS Quad. (Simul.) PS Cubic (Simul.) PS Linear (Analy.) PS Quad. (Analy.) PS Cubic (Analy.) Approx. MMSE (Simul.) Exact MMSE (Simul.) ML (Simulation)
Fig. 3. NMSE of different channel estimation methods, where “PS" stands for “phase-shifted." Top: under SUI-4 PDP; bottom: under SUI-5 PDP.
5 10 15 20 25 30 35 40 45 50 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 SNR (dB) NMSE (dB) Linear (Simulation) Quad. (Simulation) Cubic (Simulation) PS Linear (Simul.) PS Quad. (Simul.) PS Cubic (Simul.) PS Linear (Analy.) PS Quad. (Analy.) PS Cubic (Analy.) Approx. LMMSE (Simul.) Exact LMMSE (Simul.) ML (Simulation)
Fig. 4. NMSE of different channel estimation methods under the SUI-4 PDP with 25 samples of delay, where “PS" stands for “phase-shifted."
solution is quite poor in SUI-4 compared to phase-shifted polynomial interpolation, but is comparable to phase-shifted cubic interpolation in SUI-5 that has a longer delay spread. In a channel with reasonably short delay spread, such as SUI-4, phase-shifted cubic interpolation can provide comparable performance to exact LMMSE in medium to high SNR.
To verify that the proposed method is able to deal with inaccurate OFDM symbol timing, we simulate SUI-4 with 25
5 10 15 20 25 30 35 40 45 50 −35 −30 −25 −20 −15 −10 −5 SNR (dB) NMSE (dB) PS Linear (Simulation) PS Quad. (Simulation) PS Cubic (Simulation) Brute−Force PS Linear (Simul.) Brute−Force PS Quad. (Simul.) Brute−Force PS Cubic (Simul.)
Fig. 5. Comparison of NMSE in channel estimation of the proposed method for finding the window shift with brute-force search at integer-sample granularity, where “PS" stands for “phase-shifted."
samples of initial delay. Fig. 4 shows the resultant performance of various channel estimators. Comparing with Fig. 3 (top plot), we see that the performance of the proposed method, as well as that of ML and exact LMMSE, stays unchanged. But conventional polynomial interpolation yields a much degraded performance due to the mechanism explained via Fig. 1. In contrast, the approximate LMMSE yields an improved performance, presumably because the 25 samples of initial delay makes the PDP closer to being uniform.
To see how much suboptimality the several approximations in Sections II-A and B may lead to, we compare the NMSE in channel estimation of the proposed method for finding the window shift to that achieved with brute-force search at integer-sample granularity. It turns out that the difference is very small. Fig. 5 shows the result for the SUI-5 channel.
IV. PERFORMANCEUNDER THEMOBILEWIMAX
DOWNLINKSIGNALSTRUCTURE
The Mobile WiMAX specifications furnish an interesting setting to test the performance of the proposed technique, especially since the number of pilots therein may be so few that ML interpolation cannot be applied to channels with moderate to long delay spreads.
The DL transmission of Mobile WiMAX is organized into “subframes." A subframe consists of an all-pilot OFDM preamble symbol followed by an even number of OFDM data symbols. The subcarriers in a data symbol are divided into “clusters" that contain 14 consecutive subcarriers each. A DL user signal consists of a number of subcarriers from a number of pseudo-randomly distributed (in frequency) clusters. The pattern of pilot subcarriers alternates in temporally successive OFDM symbols. Fig. 6 illustrates how pilots are placed in
clusters. For convenience, in this section let 𝐻(𝑠, 𝑛) denote
the channel response at the𝑛th subcarrier of some cluster in
symbol 𝑠. In a typical system with 10 MHz bandwith and
𝑁 = 1024 [12], [13], there can be as few as 24 pilots in an
OFDM symbol.
As the pilots are no longer of the comb type, the channel estimation method described earlier needs to be adjusted, and
( )
1,0
H
H
(
1,12
)
( )
4, 4
H
H
( )
4,8
Fig. 6. Cluster structure in Mobile WiMAX downlink and corresponding channel estimation method. Each circle indicates a subcarrier and dark circles indicate pilot subcarriers. Four temporally consecutive clusters are shown, from top to bottom. The 14 subcarriers in a cluster are ordered from left to right.
so is the MSE analysis. Below we tailor the proposed channel estimation technique to suit the given DL signal structure and investigate the ensuing performance. We also propose a method to adaptively select the interpolation order for the best performance.
A. Channel Estimator Design
By the pseudo-random distribution of clusters, two “neigh-boring" clusters in a DL signal may not be adjacent in frequency. Thus it is simpler to perform channel estimation on clusters at different frequency positions separately than jointly. Now, in each cluster, there are only two pilot subcarriers whose distance varies with symbol index. In a typical system
with 10 MHz bandwidth and𝑁 = 1024, two adjacent
subcar-riers are approximately 10 kHz apart. Hence in odd-numbered symbols the pilots are spaced by over 100 kHz, which is about the same order of magnitude as the coherence bandwidth of a typical urban outdoor channel [22]. Linear interpolation, even with optimal window shift, is not expected to yield good performance over this distance. In even-numbered symbols, extrapolation is needed for the many data subcarriers that lie outside the span of the pilots. Such extrapolation should be minimized because it amplifies the noise, for at least one extrapolation coefficient is greater than 1. Therefore, a good channel estimator should not limit itself to using the two only pilots in the current cluster, but should avail itself of the channel information at the two other pilot frequencies in temporally close clusters.
Concerning the window shift ˆ𝜏, since an integer value
suffices for performance and can simplify the computation,
we let ˆ𝜏 be an integer. In addition, to save computation, we
estimate ˆ𝜏 only once per DL subframe from the preamble
symbol employing the method of Section II-B. The resultant channel estimation method for the data symbols is as follows. For convenience, we describe it for the middle two (dash-boxed) symbols illustrated in Fig. 6 together.
1) Recall the value of ˆ𝜏 that has been estimated using the
preamble symbol.
2) Circularly shift the signal samples in symbols 1 to 4 by
−ˆ𝜏 samples before taking their DFT. This is equivalent
to effecting a channel with phase-rotated frequency
response 𝐻𝜏(𝑠, 𝑛) = 𝑒𝑗2𝜋ˆ𝜏𝑓(𝑛)/𝑁𝐻(𝑠, 𝑛) where 𝑓(𝑛)
denotes the normalized frequency of subcarrier𝑛. (We
have used an adapted version of the notations in (7) here.)
3) Do LS channel estimation for pilot subcarriers in sym-bols 1 to 4. This yields, for each pilot with
time-frequency index(𝑠, 𝑛), a phase-rotated channel estimate
ˆ
𝐻𝜏(𝑠, 𝑛) = 𝑒𝑗2𝜋ˆ𝜏𝑓(𝑛)/𝑁𝐻(𝑠, 𝑛). (We have used anˆ
adapted version of the notations in (8) here.)
4) Linearly interpolate in time to obtain 𝐻ˆ𝜏(2, 0),
ˆ
𝐻𝜏(2, 12), ˆ𝐻𝜏(3, 4) and ˆ𝐻𝜏(3, 8).
5) Perform conventional𝐾th-order interpolation (and
ex-trapolation) in frequency to obtain channel estimates for the remaining subcarriers in symbols 2 and 3. This corresponds to the operation described in (9), or equivalently, that in the last summation in (6).
Note that we do not have to carry out the phase-derotation
indicated by the premultiplication with 𝑒−𝑗2𝜋𝜏(𝑝+𝑛)/𝑁 in the
RHS of (6), for by the circular signal shifting in step 2, the phase-rotated channel becomes what is needed for signal detection, not the phase-derotated. Note also that step 4 (temporal interpolation) results in four reference data points per cluster. Hence in step 5 we may employ an interpolation
order up to three. To minimize the modeling error, in
𝐾th-order interpolation of a data subcarrier’s channel response the
𝐾 + 1 nearest pilots’ responses are used. The performance of
the proposed channel estimator is analyzed in Appendix B.
B. Simulation Examples
We simulate a system with carrier frequency = 2.5 GHz, bandwidth = 10 MHz, DFT size = 1024, and cyclic prefix length = 128. We let a DL subframe contain 24 OFDM symbols following the preamble. Again, we simulate the SUI-4 and SUI-5 PDPs with block-static fading at a rate corresponding to 100 km/h of mobile speed.
Fig. 7 shows some channel estimation performance results. (See curves marked by the first six legend items in each plot.) For SUI-4, the approximate analysis matches the simulation results almost exactly. For SUI-5, which has a larger delay spread than SUI-4, the approximate analysis is less accurate, but still follows the general behavior of the simulation results. The channel estimators show poorer performance under SUI-5 than under SUI-4, but the relative performance of the three interpolation orders are characteristically similar in both cases. In either channel, the MSE floors of lower-order interpolation start to manifest at lower SNR values than higher-order inter-polation. As a rule, in higher pilot SNR, higher-order interpo-lators perform better by having lower frequency interpolation errors, but in lower pilot SNR, lower-order interpolators are better due to smaller AWGN effects.
C. Adaptive Selection of Interpolation Order
That the best interpolation order depends on the channel SNR suggests adaptive selection of the interpolator order. From the analysis in Appendix B, this can be accomplished
if we can estimate𝜎2
𝜂,𝜎𝑡2, and𝜎2𝜉𝐾(𝜏𝑜) and use the results to predict the MSE via (44), (47), (48), and (49). The interpola-tion order that yields the least MSE can then be chosen.
5 10 15 20 25 30 35 40 −40 −35 −30 −25 −20 −15 −10 −5 Pilot SNR (dB) NMSE (dB) Linear (Simulation) Linear (Analysis) Quad. (Simulation) Quad. (Analysis) Cubic (Simulation) Cubic (Analysis) Adaptive (Simulation) Approx. LMMSE (Simul.) Exact LMMSE (Simul.)
5 10 15 20 25 30 35 40 −35 −30 −25 −20 −15 −10 −5 Pilot SNR (dB) NMSE (dB) Linear (Simulation) Linear (Analysis) Quad. (Simulation) Quad. (Analysis) Cubic (Simulation) Cubic (Analysis) Adaptive (Simulation) Approx. LMMSE (Simul.) Exact LMMSE (Simul.)
Fig. 7. NMSE of channel estimation in WiMAX DL transmission at 100 km/h mobile speed with fixed-order and adaptive interpolation. Top: over SUI-4 channel; bottom: over SUI-5 channel.
As the window shift ˆ𝜏 is determined once per DL subframe
(see Section IV-A), the interpolation order is also determined
once per DL subframe. As the quantity 𝜎2
𝑡 (variance of
temporal interpolation error) is a function of the time-variation of the channel, it is hard to estimate using only the preamble symbol. Thus we disregard it (and the center RHS term in (49)) in the selection of the interpolation order. For the other
two quantities, 𝜎2
𝜂 (variance of LS channel estimation error)
can be relatively easily estimated using the null subcarriers
in the preamble. As to𝜎2
𝜉𝐾(𝜏𝑜), from (23) and (24) we can
estimate it from the preamble symbol as ˆ𝜎2𝜉𝐾(𝜏𝑜) =𝐹2𝐾+21
[
𝑃𝐾𝑅ˆ0+ 2 min𝜏 𝛾𝐾(𝜏) ]
(39)
where ˆ𝑅0 = ⟨⟨∣ ˆ𝐻(𝑓)∣2⟩⟩ − ˆ𝜎𝜂2 with ˆ𝜎2𝜂 being an estimate of
𝜎2
𝜂, and 𝑃1 = 6, 𝑃2 = 20, and 𝑃3 = 70. In summary, for
each DL subframe, we partially predict the MSEs for linear, quadratic, and cubic interpolations based on (49), with its center RHS term omitted, using the received preamble symbol. The interpolation order yielding the smallest predicted partial MSE is selected.
The performance of adaptive interpolation over the SUI-4 and SUI-5 channels is also illustrated in Fig. 7. We see
that, by and large, the proposed scheme can choose the optimal interpolation order, though at times it may make an unfavorable decision. For comparison, the figure also shows the performance of 4-tap LMMSE estimation based on exact channel correlation (marked “exact LMMSE") and that based on a correlation function corresponding to uniform PDP of length equal to CP (marked “approx. LMMSE"). Again, the exact LMMSE has the best performance of all, but is unrealizable. On the other hand, the performance of the approximate LMMSE estimator is more erratic, because a uniform PDP may or may not be close to the characteristics of the instantaneous channel response at any given time.
V. CONCLUSION
We considered channel estimation by polynomial inter-polation for multicarrier transmission. We showed that, by shifting the effective time window by a proper amount, the performance could be improved significantly. We derived a method to estimate the optimal window shift for arbitrary interpolation order. The method operates on frequency-domain quantities and is thus particularly suitable for pilot-aided multicarrier systems. It can be used where ML interpolation cannot for scarcity of pilots.
As a practical example, we considered the structure of Mobile WiMAX downlink signal and specialized the proposed technique for it. We also proposed a way to carry out adaptive selection of interpolation order based on some estimated MSE values. Simulation results showed that the adaptive scheme could yield nearly optimal performance over a wide range of SNR values. It may even compare favorably to LMMSE channel estimation. APPENDIXA SIMPLIFICATION OF[𝑛𝐾+1− v′𝑛𝐾X−1𝑛𝐾x𝑛𝐾] Let z = [𝑧(𝑥𝑛0), 𝑧(𝑥𝑛1), ⋅ ⋅ ⋅ , 𝑧(𝑥𝑛𝐾)]′ where 𝑧(𝑛) ≜ 𝑛𝐾+1− v′ 𝑛𝐾X−1𝑛𝐾x𝑛𝐾. Then z = x𝑛𝐾 − [v𝑥0𝐾, v𝑥1𝐾, ⋅ ⋅ ⋅ , v𝑥𝐾𝐾]′X−1𝑛𝐾x𝑛𝐾 = x𝑛𝐾 − X𝑛𝐾X−1𝑛𝐾x𝑛𝐾 = 0. (40)
Thus𝑥𝑛𝑘 is a root of𝑧(𝑛) ∀𝑘 ∈ {0, ⋅ ⋅ ⋅ , 𝐾}. Now that 𝑧(𝑛)
is a𝐾 + 1st-order polynomial in 𝑛 by definition, we have 𝑧(𝑛) = 𝐶∏𝐾
𝑘=0
(𝑛 − 𝑥𝑛𝑘) (41)
for some𝐶. Further, since the 𝑛𝐾+1 term in𝑧(𝑛) has unity
coefficient by definition,𝐶 = 1.
APPENDIXB
PERFORMANCEANALYSIS FOR THEPROPOSEDWIMAX
CHANNELESTIMATOR
Four factors contribute to the channel estimation error. They are, in order of appearance in the algorithm steps, 1) suboptimality in the estimated window shift (introduced in step 1), 2) the AWGN (introduced in step 3), 3) modeling error due to temporal interpolation (introduced in step 4), and 4) modeling error due to frequency interpolation (introduced in step 5). Compared to the analysis in Section III, the only
additional factor is item 3. However, because of it the AWGN propagates among the subcarriers differently than in the earlier analysis. The temporal interpolation error itself propagates through step 5, too. Moreover, the pilots used for frequency interpolation in step 5 are chosen somewhat differently than in Section III and the bandedge subcarriers are no longer discarded. In short, we will need to redo some analysis. Again, item 1 is difficult to analyze but, fortunately, constitutes a minor contribution in the total MSE. Thus only the other three factors are analyzed. We assume that these three errors are uncorrelated.
First, consider the AWGN. It enters channel estimator computation through the LS channel estimation at the
pi-lot subcarriers. Again, let 𝜎2
𝜂 denote the noise variance of
LS channel estimation at a pilot. This noise propagates to other subcarriers via step 4 (temporal interpolation) and step 5 (frequency interpolation). Via step 4, it contaminates
ˆ
𝐻𝜏(𝑠, 𝑛) where (𝑠, 𝑛) ∈ {(2, 0), (2, 12), (3, 4), (3, 8)}. Since
ˆ
𝐻𝜏(𝑠, 𝑛) = [ ˆ𝐻𝜏(𝑠−1, 𝑛)+ ˆ𝐻𝜏(𝑠+1, 𝑛)]/2, the noise variance
in ˆ𝐻𝜏(𝑠, 𝑛) is given by 𝜎2 𝑤𝐾(𝑠, 𝑛) = ( 1 2 )2 𝜎2 𝜂+ ( 1 2 )2 𝜎2 𝜂 = 12𝜎𝜂2 (42)
for all𝐾. Via step 5, it results in a noise variance
𝜎2 𝑤𝐾(𝑠, 𝑛) = 𝐾 ∑ 𝑘=0 𝑐2 𝑛𝑘𝜎𝑤𝐾2 (𝑠, 𝑥𝑛𝑘) (43) where 𝑠 ∈ {2, 3}, 𝑛 ∈ {1, 2, 3, 5, 6, 7, 9, 10, 11, 13}, and
𝑥𝑛𝑘 ∈ {0, 4, 8, 12}, with 𝑐𝑛𝑘 indexed similarly to (3). Av-eraging over all data subcarriers, we obtain the average noise variance as
𝜎2
𝑤1= 0.5130𝜎𝜂2, 𝜎2𝑤2= 0.6699𝜎2𝜂, 𝜎𝑤32 = 0.8121𝜎𝜂2, (44)
where the second subscript to𝜎 gives the frequency
interpo-lation order𝐾 as before (and similarly below).
Next, consider the modeling error in temporal interpolation. Its mean-square value is
𝜎2
𝑡𝐾(𝑠, 𝑛)
= 𝐸 𝐻𝜏(𝑠, 𝑛) − 12[𝐻𝜏(𝑠 − 1, 𝑛) + 𝐻𝜏(𝑠 + 1, 𝑛)]
2
(45)
for (𝑠, 𝑛) ∈ {(2, 0), (2, 12), (3, 4), (3, 8)} for all 𝐾. With
Rayleigh faded paths [15], [22],
𝜎2 𝑡𝐾(𝑠, 𝑛) = 𝐿−1∑ 𝑙=0 𝐴2 𝑙 [ 3 2 − 2𝐽0(2𝜋𝑓𝑙) + 1 2𝐽0(4𝜋𝑓𝑙) ] ≈3𝜋24 𝐿−1∑ 𝑙=0 𝑓4 𝑙𝐴2𝑙 ≜ 𝜎2𝑡 (46) where 𝐴2
𝑙 = 𝐸∣𝛼𝑙∣2, 𝑓𝑙 is the peak Doppler shift of path 𝑙
times the OFDM symbol period,𝐽0(⋅) is the Bessel function
of the first kind of order 0, and the approximation is obtained by expanding the Bessel function into a second-order Taylor series. Assume that the channel responses at different subcar-riers are uncorrelated. Then a relation similar to (43) exists concerning the propagation of the temporal modeling error in the frequency domain via step 5. We get the average MSE as
𝜎2
𝑡1= 0.4531𝜎2𝑡, 𝜎2𝑡2= 0.5577𝜎𝑡2, 𝜎2𝑡3= 0.6525𝜎𝑡2. (47)
Finally, consider the modeling error in frequency interpo-lation. The MSE at any data subcarrier is as given in (19), but the averages are different from that given in (31) due to the difference in choice of interpolating pilots as well as the presence of extrapolation for subcarrier 13. Straightforward numerical calculation yields the following average MSE:
𝜎2 𝑒1= 2.6458 𝜎2𝜉1(𝜏𝑜), 𝜎2 𝑒2= 12.8125 𝜎𝜉22(𝜏𝑜), 𝜎2 𝑒3= 93.0820 𝜎𝜉32(𝜏𝑜). (48)
Put together, the overall average channel estimation MSE is given by
𝜎2
𝐾= 𝜎2𝑤𝐾+ 𝜎2𝑡𝐾+ 𝜎2𝑒𝐾. (49)
ACKNOWLEDGMENT
The authors would like to thank the reviewers whose comments helped improve the paper.
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Kun-Chien Hung received the B.S. and Ph.D.
degrees in electronics engineering from the National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 2001 and 2008, respectively. He is currently a senior engineer with MediaTek Inc., Hisnchu. His research interests are in digital signal processing, communi-cation receiver design, and wireless communicommuni-cation.
David W. Lin (S’79-M’81-SM’88) received the B.S.
degree in electronics engineering from the 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 subscriber 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.
