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Iterative Channel Estimation Based on B-splines for Fast Flat Fading Channels Huiheng Mai, Yuriy V. Zakharov, Member, IEEE, and Alister G. Burr, Member, IEEE
Abstract— We propose novel low-complexity iterative channel estimators based on B-splines. Local splines are adopted for computational simplicity. Minimum Mean Square Error (MMSE) local splines with integral sampling are derived. The MSE of the proposed estimators depends on signal-to-noise ratio, fading rate, sampling interval, spline order and the number of weighting coefficients; these dependencies are investigated. The linear and cubic local splines with as few as seven weighting coefficients are capable of achieving MSE and BER performance comparable to those of the Wiener filter and the spheroidal basis expansion. However, a significantly lower complexity is achieved using Bsplines. Index Terms— Spline-approximation, iterative channel estimation, fading channels, Slepian basis expansion.
I. I NTRODUCTION
I
N mobile communications over time-varying fading channels, the accuracy of channel state information in the receiver is critical to the system performance. The difficulty of channel estimation increases with the normalized Doppler frequency. It becomes even more challenging in systems with powerful channel codes, e.g. turbo codes, which generally operate in such a low signal-to-noise ratio (SNR) that conventional pilot-based channel estimation often fails to deliver satisfactory results. Iterative channel estimation and decoding over flat fading channels has been proposed to overcome the weakness of the pilot-based estimation [1], [2]. For such estimation, the complexity issue is particularly important since not only does the estimator operate repeatedly, but also it utilizes all the symbols, i.e. pilots and data. There are two general approaches for tracking parameter changes: statistical filtering and the method of basis functions [3]. Wiener filter, an optimal statistical filter, has been used for iterative channel estimation of fast flat fading channels and dramatic performance gains over non-iterative schemes are achieved [1], [2]. However, a high price has to be paid in terms of complexity. For the method of basis functions, Fourier and polynomial basis functions are most commonly used for tracking time varying fading channels [4], Manuscript received July 25, 2005; revised March 28, 2006; accepted May 10, 2006. The associate editor coordinating the review of this paper and approving it for publication was M. Saquib. This work was supported in part by the EPSRC contract {GR/R56532/01}. This paper was presented in part at the IEEE International Conference on Communications (ICC’05), Seoul, Korea, and the IEEE Vehicular Technology Conference (VTC’05 Spring), Stockholm, Sweden, May 2005. H. Mai was with the Communications Research Group, University of York, Heslington, York, YO10 5DD, UK. He is now with IPWireless Inc., Unit 7, Bellinger Close, Chippenham, Wiltshire, SN15 1BN, UK (e-mail:
[email protected]). Y. V. Zakharov and A. G. Burr are with the Communications Research Group, University of York, Heslington, York, YO10 5DD, UK (e-mail: {yz1, alister}@ohm.york.ac.uk). Digital Object Identifier 10.1109/TWC.2007.05572.
[5], [6]. Recently, B-spline [7], [8], [9] and spheroidal [10], [11], [12] basis have also been proposed. It is shown in [10] that the spheroidal basis expansion outperforms the Fourier and polynomial basis expansion in accuracy and has a low complexity. However, a direct comparison between the Bspline and the spheroidal basis expansions is not yet available. In this paper we propose low complexity iterative channel estimators of fast flat fading channels by taking advantage of local splines [13] [14]. The work is motivated by previous success of the local spline approximation of time variation of the fading under noise-free conditions in [7], [8] and [9], which demonstrate that a good accuracy can be achieved with a low complexity. Here the noise free assumption is removed and the MMSE criterion is applied. As integral sampling is advantageous in noisy scenarios, we derive MMSE local splines based on such sampling. Due to the simplicity of the local spline approximation, the resulting estimators have low complexity while achieving performance comparable to those of the Wiener filter and the spheroidal basis expansions over a wide range of normalized Doppler frequencies. II. S YSTEM D ESCRIPTIONS The transmission system with BPSK modulation to be considered follows that in [1]. In the transmitter, information bits are firstly encoded by a turbo decoder. The output x[i] of the turbo encoder is channel-interleaved and pilot symbols are inserted periodically every (M −1) interleaved coded symbols, resulting in a frame of symbols y[k]. The received symbol r[k] at the k-th discrete time instant is given by (1) r[k] = h[k]y[k] + n [k], where h[k] is a complex channel coefficient and n [k] is additive white Gaussian noise with variance σn2 /2 in both real and imaginary components. Assuming Jakes’ model [15], the fading process h[k] has the autocorrelation function ρh [τ ] = E{h[k]h∗ [k + τ ]} = σh2 J0 (2πfd Ts τ ), where E{·} denotes statistical expectation, J0 (·) is the zero order Bessel function of the first kind, fd is the normalized Doppler frequency, Ts is the symbol period and σh2 is the fading variance. The receiver performs several iterations, in which channel estimation and decoding are refined once per iteration. In ˆ the first iteration, the channel estimates h[k] are obtained from the pilot symbols by conventional Wiener filtering based Pilot Symbol Assisted Modulation (PSAM) [16]. The received symbols are then multiplied by the complex conjugate of the ˆ ∗ [k], where (·)∗ denotes the channel estimates: f [k] = r[k]h complex conjugate. The data symbols of the sequence f [k] are channel deinterleaved and then passed to a turbo decoder.
c 2007 IEEE 1536-1276/07$25.00
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007
At the q-th iteration, the turbo decoder outputs the aposteriori log-likelihood ratio (LLR) λq [i] of the coded symbols x[i]. Soft x ˆq [i] are obtained from LLR as q decisions λ [i] q ˆq [i] are reinterleaved and the pilot x ˆ [i] =tanh 2 . Then x symbols are reinserted into x ˆq [i] to form a sequence yˆq [k], which is treated as a sequence of pilot symbols. This allows more accurate channel estimation at the next iteration. The improved channel estimates are used to re-compensate the received symbols and this leads to an improved input to the turbo decoder. For channel estimation, the effect of data modulation on the received symbols is removed by ¯ = r[k] · (ˆ h[k] y q [k])∗ .
(2)
¯ ¯ = With perfect data decisions, h[k] can be modelled as h[k] h[k] + n[k], where n[k] is additive noise that has the same statistics as those of n[k] .
where g(t) is a weighting function. Substituting (6) into (5), we have ∞ L2 ¯ am h(t)g(t − iT + mT )dt. (7) ci = In order to handle noisy situations, we seek a local spline that minimizes the integral MSE T2 1 2 ˆ 2 }dt. ε = 2 E{|h(t) − h(t)| (8) σh T − T2 Substituting (4) and (7) in (8), the integral MSE is expressed as
ˆ = h(t)
k=−L1
where 1 γη (k) = T
ci bη (t − iT ),
(4)
where ci are spline coefficients, T is the sampling interval and bη (t) is the η order B-spline basis function, which is a (η + 1) fold convolution of the zero order basis function b0 (t) = 1 for |t| ≤ T2 and b0 (t) = 0 otherwise [17]. For a local spline, the spline coefficients can be calculated by L2
am ξi−m ,
(5)
m=−L1
where am are weighting coefficients, L1 +L2 +1 is the number of weighting coefficients, and ξi are the integral samples ∞ ¯ ξi = h(t)g(t − iT )dt, (6) −∞
Akl =
L2
ak γη (k) +
L2
Akl ak al ,
(9)
k=−L1 l=−L1
(η+2)T /2
−(η+2)T /2
η
bgη (τ )ρh (τ + kT )dτ,
βη (p)Gρ (pT − kT + lT ),
(10)
(11)
p=−η
βη (p) =
1 T
(η+1)T /2
bgη (τ )
=
bη (τ )bη (τ + pT )dτ,
(12)
bη (t)g(t + τ )dt,
(13)
G(τ )ρh¯ (τ − kT )dt
(14)
−(η+1)T /2 (η+1)T /2
−(η+1)T /2
G (kT ) =
T
ρ
and
−T
G(τ ) =
T /2
−T /2
g(t)g(t − τ )dt.
(15)
Equation (9) holds for arbitrary weighting coefficients ak . To , we find the optimal MMSE weighting coefficients aMMSE k differentiate the right hand side in (9) with respect to ak and make it equal to zero. This results in a system of equations L2
aMMSE Akl = γη (l), l = −L1 , . . . , L2 . k
(16)
k=−L1
i=−∞
ci =
L2
ε2 = 1 − 2
For the ease of derivation of MMSE local splines, we start from the continuous-time domain and then proceed to the discrete-time domain. We adopt local splines for simplicity and we extend the previous works [7], [8], [9] from noisefree to noisy conditions. We assume the availability of a noisy version of h(t): ¯ = h(t) + n(t), h(t) (3)
∞
−∞
m=−L1
III. B-S PLINE BASED C HANNEL E STIMATOR
where n(t) is the additive white noise. The fading process h(t) is estimated by building a local spline ˆ h(t) from the observa¯ tion h(t). As the fading and noise are uncorrelated, the auto¯ normalized by the fading variance correlation function of h(t) 2 ¯ ¯ σh is ρh¯ (τ ) = E{h(t)h∗ (t + τ )}/σh2 = ρh (τ ) + σn2 δ(τ )/σh2 , where ρh (τ ) = E{h(t)h∗ (t + τ )}/σh2 = J0 (2πfd τ ) is the normalized autocorrelation function of h(t), and δ(τ ) is the Dirac delta function. Since h(t) and n(t) are complex random processes with independent real and imaginary components and the spline basis functions are real-valued, the real and imaginary components of h(t) can be estimated separately. To avoid extra notations and without loss of generality, in ¯ derivation of the local splines of this section, h(t), h(t) and n(t) are considered to be real-valued. A spline of order η can be represented as
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Substituting (16) in (9), we obtain the minimum MSE for the MMSE local splines as ε2MMSE = 1 −
L2
aMMSE γη (k). k
(17)
k=−L1
Depending on g(t), a local spline can be based on either discrete or integral sampling. Discrete sampling, where g(t) = δ(t), is suitable for sparse pilot symbols. Integral sampling, where g(t) = T1 for |t| ≤ T2 and g(t) = 0 otherwise, can efficiently exploit the extended set of observations of the iterative channel estimation in noisy situations. For integral sampling, (6) can be rewritten as 1 T /2 ¯ h(t − iT )dt. (18) ξi = T −T /2
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 4, APRIL 2007
SNR=0dB
−10
−10 SNR=0dB
−15
−20
b2η+1 (pT )G (kT − pT + lT ) ρ
−20 SNR=15dB SNR=15dB
−25
1 T2
−25
T
−T
b1 (τ )ρh¯ (τ − kT )dτ.
(21)
P −1 1 ¯ h[iP + p]. ξi = P p=0
−30
ci bη [k − iP ].
10 20 30 Sampling interval (P)
(22)
(23)
i=−∞
IV. N UMERICAL R ESULTS A. MSE Performance In this section, we compare the MSE performance of the MMSE and LS local splines with different number of weighting coefficients, the Wiener filter and the spheroidal basis expansions by assuming that all data symbols are perfectly known to the estimators. The SNR is defined as SNR= 1/σn2 . Aiming at a low complexity, we consider linear and cubic splines and limit L1 = L2 = L ≤ 3. For the LS splines, we further restrict L ≤ 1 and L ≤ 2 for linear and cubic splines, respectively. The reason is that in noise-free situations, the LS linear and cubic splines with such values of L already achieve the optimal approximation error [7]. Therefore in
40
0
10 20 30 Sampling interval (P)
40
Fig. 1. MSE performance as a function of the sampling interval P for MMSE linear and cubic splines with 7 weighting coefficients (L = 3); SN R = 0 and 15dB. Linear splines
Cubic splines
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L=0, LS L=1, LS L=0, MMSE L=1, MMSE L=2, MMSE L=3, MMSE
−15
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−12
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−14
3) Calculate the spline coefficients ci from (16) where corresponding weighting coefficients are obtained by numerical integration in (19), (20) and (21). ˆ 4) Build the spline h[k] by ∞
−30 0
The Least Squares (LS) weighting coefficients aLS are k the corresponding MMSE weighting coefficients aMMSE for k σn2 = 0. The corresponding MSE is obtained by substituting aLS k in (9). The advantage of the LS splines over the MMSE splines is that the weighting coefficients do not depend on the noise variance. The derived algorithm can be used with an arbitrary sampling interval T . However, for channel estimation, we shall restrict T to be an integer number P of the symbol period Ts , i.e. T = P Ts . Returning to the symbol-spaced discretetime domain, the B-spline channel estimator is summarized as follows: ¯ in discrete time by removing the modulation 1) Obtain h(t) from received signals as in (2). 2) Divide the whole frame into sub-frames with P symbols according to the optimal sampling interval and approximate the integral sample by
ˆ h[k] =
−15
(20)
MSE (dB)
Gρ (kT ) =
d s
Linear, Analytical Linear, Simulation Cubic, Analytical −5 Cubic, Simulation
−5
p=−η
and
f T =0.03
d s
MSE (dB)
Akl =
η
f T =0.01
MSE (dB)
Taking both the noise and approximation error into account, there is an optimal sampling interval that provides a minimum MSE for a specific spline degree η, number of weighting coefficients (L1 + L2 + 1) and channel condition. As g(t) = b0 (t)/T , we obtain bgη (τ ) = bη+1 (τ )/T , G(τ ) = b1 (τ )/T 2 , and βη (p) = b2η+1 (pT ). Therefore for such a weighting function g(t) the variables in (9) can be expressed as (η+2)T /2 1 bη+1 (τ )ρh (τ + kT )dτ, (19) γη (k) = 2 T −(η+2)T /2
MSE (dB)
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0.02
0.03 fdTs
0.04
L=0, LS L=2, LS L=0, MMSE L=1, MMSE L=2, MMSE L=3, MMSE
−15
0.05
−16 0.01
0.02
0.03 fdTs
0.04
0.05
Fig. 2. MSE Performance of LS and MMSE linear and cubic splines with different L with respect to the normalized Doppler frequency; SNR=0dB.
noisy situations, a further increase in L will not improve the approximation accuracy, but increase the MSE due to noise. Note that this is not the case for the MMSE local splines, as the approximation error and noise are minimized jointly. Fig. 1 shows the dependence of the MSE on the sampling interval for linear and cubic splines. The analytical results are calculated by using (17) and the numerical results are obtained using 500 simulation trials. It is observed that the simulation and analytical results are in an excellent agreement. As expected, there is an optimal sampling interval for a specific spline, normalized Doppler frequency fd Ts , and SNR. In the remainder of this section, we show the analytical MSE of the local splines only with the optimal sampling interval. Fig. 2 compares the MSE of the MMSE and LS linear and cubic splines with different L with respect to the normalized Doppler frequency at SNR=0 dB. For a fixed spline order,
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B. BER Performance We evaluate the BER performance of the iterative receiver with the channel estimators based on the MMSE linear and cubic local splines with L = 3 and optimal sampling interval, and the Wiener filter with Ntap = 61. The spheroidal basis expansion is omitted since for most situations its MSE performance is between the other two techniques. In the BER simulation, a rate Rc = 1/2, 1024 data bits, 4 state turbo code with generator polynomials 5 and 7 in octal is employed. To achieve a coding rate 1/2, puncturing has been used to delete the even index parity bits from the upper encoder and odd indexed parity bits from the lower encoder. The turbo encoder and channel interleavers are S-random with S-parameter 20 and random interleavers [20], respectively. The log-map algorithm is used for turbo decoding. There are 8 iterations between channel estimation and turbo decoding. Normalized Doppler frequencies fd Ts = 0.01, 0.03 and 0.05 with pilot spacing M = 14, 9 and 6 are considered. Note that
SNR=0dB
SNR=15dB
0
0
−2 −5 −4 −10
−6
−8
MSE (dB)
MSE (dB)
provided that L for the MMSE splines is not smaller than that of the LS splines, the MMSE splines are superior because they take into account the noise. For a fixed L, the cubic splines have better performance than that of the linear splines. The performance of the MMSE splines also improves with increasing L. The corresponding results at SNR= 15 dB leads to the same conclusion and thus are not presented. Fig. 3 compares the MSE performance of the Wiener filter, spheroidal basis expansion and cubic and linear MMSE splines with L = 3. The MSE of the Wiener filter is obtained analytically [18]. The length Ntap = 61 is chosen to be the same as in [1] while the length Ntap = 4 results in the same complexity as that of the MMSE linear splines with L = 3. = The spheroidal basis functions ψd [ψd [1], ψd [2], · · · , ψd [NF ]]T , d = 1, · · · , D, where NF is the frame length, D is the number of basis functions, are designed specifically for each normalized Doppler frequency [19]. Since the basis functions are orthonormal, the ˆ = [θˆ1 , θˆ2 , · · · , θˆD ]T are obtained as expansion coefficients θ H −1 H ¯ ˆ ¯ where ψ = [ψ , ψ , · · · , ψ ] θ = (ψ ψ) ψ h = ψ H h, 1 2 D ¯ ¯ ¯ ¯ F ]]T . Then the channel and h = [h[1], h[2], · · · , h[N ˆ = [h[1], ˆ ˆh[2], · · · , h[N ˆ F ]]T are estimated by coefficients h ˆ ˆ h = ψ θ. The performance of the spheroidal basis expansion depends strongly on the number of basis functions. Using the frame length NF = 256 as in [10], we can find the optimal number of basis functions D that provides the minimum MSE. The optimal D for fd Ts = [0.01, 0.03, 0.05] at SNR=0dB and SNR=15dB are [7, 18, 28] and [9, 19, 30], respectively. Since a fixed small number (D = 5) of basis functions has been adopted for relatively slow fading (fd Ts ≤ 0.003) in [10], [11], we will also use it to see whether it is sufficient for faster fading. It can be seen in Fig. 3 that, apart from the Wiener filter with Ntap = 4 and the spheroidal basis expansion with D = 5, which all have poor performance, the other estimators have performances which differ from one another at most 1 dB and 2 dB for SNR= 0 dB and 15 dB, respectively. It is clear that the spheroidal basis expansion with D = 5 basis functions is insufficient for fast fading.
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Linear, MMSE, L=3 −25 Cubic, MMSE, L=3 Wiener 61 taps Wiener 4 taps Spheroidal, optimal D Spheroidal, D=5 −30
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−18 0.01
0.02
0.03 fdTs
0.04
0.05
0.01
0.02
0.03 fdTs
0.04
0.05
Fig. 3. Performance of the Wiener filter, the spheroidal basis expansions and the MMSE cubic and linear splines with L = 3 as a function of the normalized Doppler frequency.
the pilot spacings have been optimized with respect to BER performance for each normalized Doppler frequency. At the first iteration, the filter length of the PSAM is 11. Eb /N0 is defined as Eb /N0 = 1/(Rc σn2 ) · M/(M − 1). Fig. 4(a) shows the BER performance when the normalized Doppler frequency and Eb /N0 are known to the channel estimators. It can be observed for all normalized Doppler frequencies that the iterative receivers with the B-spline channel estimators have performance close to that of the Wiener filter. The largest difference in the performance is at fd Ts = 0.05; however, it is only 0.15 dB and 0.3 dB at BER = 10−5 for cubic and linear splines, respectively. To see how robust the channel estimators are, Fig. 4(b) shows BER results for scenarios when actual channel parameters do not match those assumed by the estimators at fd Ts = 0.03. The results for the linear spline are omitted as its robustness is similar to that of the cubic spline. Three mismatch scenarios are considered. In the mismatched Doppler scenario, the actual fd Ts is 0.03 but the estimators are optimized for fd Ts = 0.04; the cubic splines outperform the Wiener filter by 0.25 dB at BER=10−5 . In the mismatched fading model scenario, the power spectrum of the actual channel is of rectangular shape but the estimators assume a U-shape according to Jake’s model; both estimators perform closely and the performance degradation due to imperfect channel estimates are approximately the same as for the Ushape fading model. In the mismatched SNR scenario, the estimators are designed for a fixed Eb /N0 of 11.51 dB while the actual Eb /N0 is as shown in the figure; it is seen that both estimators perform closely with a performance degradation under 0.1dB. C. Complexity The computational complexity is defined as the number of multiplications required per symbol per iteration. The extra complexity of the removal of the modulation (2) is
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Perfect Estimates, fdTs=0.05 Perfect Estimates, fdTs=0.03 Perfect Estimates, fdTs=0.01 Cubic, MMSE, L=3 Linear, MMSE, L=3 Wiener Filter, 61 taps
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fdTs=0.01 fdTs=0.03
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BER
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Eb/No (b)
Eb/No (a)
Fig. 4. BER versus Eb /N0 for different estimation schemes: (a) at fd Ts = 0.01, 0.03 and 0.05 when both fd Ts and Eb /N0 are known to the estimators, (b) at fd Ts = 0.03 when there are mismatches in fd Ts , fading model and Eb /N0 .
Linear splines
Cubic splines
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Multiplications/Symbol/Iteration
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SNR=15dB
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SNR=0dB
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60
40
20
0.02
0.03 fdTs
0.04
0.05
0 0.01
0.02
0.03 fdTs
0.04
0.05
Fig. 5. Complexity of MMSE and LS linear and cubic splines with different L; SNR=0dB.
Fig. 6. Complexity of the Wiener filter, the spheroidal basis expansions and MMSE cubic and linear splines with L = 3.
not included, being the same for all algorithms. We assume, for all estimators, that the corresponding filter and weighting coefficients, the basis functions and the optimal parameters have been obtained in advance. Fig. 5 compares the complexity of the MMSE and LS linear and cubic splines with different L at SNR=0 dB. The estimator is calculated by complexity of a B-spline + (η + 1) , where the three terms in the 2 P1 + L1+1+L2 P square bracket are due to (22), (5) and (23), respectively. As the optimal sampling interval P varies, the MMSE and LS splines with same η and L may have different complexity. Thus a MMSE spline estimator with a smaller L could have a higher complexity than that of the estimator with a larger L. It can be seen that for all normalized Doppler frequencies all spline estimators have low complexity, which is under 11 multiplications. The corresponding results at SNR=15 dB are similar and thus are not presented.
Fig. 6 compares the complexity of the Wiener filter, the spheroidal basis expansion, and the MMSE linear and cubic splines with L = 3. The complexity of the Wiener filter is 2Ntap , while for the spheroidal basis expansion it is 4D. It can be seen that the B-spline estimators have much lower complexity than the others. In contrast to the B-splines, the complexity of the spheroidal basis expansion with the optimal number of basis functions increases significantly with the normalized Doppler frequency. V. C ONCLUSION We have proposed novel iterative channel estimators based on B-splines for fast flat fading channels. MMSE local splines with integral sampling are derived to handle noisy situations. We have investigated linear and cubic MMSE and LS splines. Compared to the Wiener filter and the spheroidal basis expansion with an optimal number of basis functions for a wide
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range of normalized Doppler frequencies, the proposed spline estimators provide comparable MSE and BER performance at a significantly lower complexity. We have also shown that the Wiener filter at a similar complexity and the spheroidal basis expansion with 5 basis functions are insufficient to approximate fast fading channels with high accuracy. Due to an excellent tradeoff between performance and complexity, the iterative channel estimators based on B-splines are attractive for practical applications. R EFERENCES [1] M. C. Valenti and B. D. Woerner, “Iterative channel estimation and decoding of pilot symbol assisted turbo codes over flat-fading channels,” IEEE J. Sel. Areas Commun., vol. 19, pp. 1697–1705, Sept. 2001. [2] H.-J. Su and E. Geraniotis, “Low-complexity joint channel estimation and decoding for pilot symbol-assisted modulation and multiple differential detection systems with correlated Rayleigh fading,” IEEE Trans. Commun., vol. 50, pp. 249–261, Feb. 2002. [3] M. Niediecki, Identification of Time-Varying Processes. John Wiley and Sons, 2000. [4] A. Sayeed and B. Aazhang, “Joint multipath-doppler diversity in mobile wireless communications,” IEEE Trans. Commun., vol. 47, pp. 123–132, Jan. 1999. [5] D. Borah and B. Hart, “Frequency-selective fading channel estimation with a polynomial time-varying channel model,” IEEE Trans. Commun., vol. 47, pp. 862–873, June 1999. [6] G. Yue, X. Zhou, and X. Wang, “Performance comparisons of channel estimation techniques in multipath fading CDMA,” IEEE Trans. Wireless Commun., vol. 3, pp. 716–724, May 2004. [7] Y. V. Zakharov, T. C. Tozer, and J. F. Adlard, “Polynomial splineapproximation of Clarke’s model,” IEEE Trans. Signal Processing, vol. 52, pp. 1198–1208, May 2004.
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