Robust MMSE channel estimation in OFDM systems with ... - CiteSeerX

Robust MMSE Channel Estimation in OFDM Systems with Practical Timing Synchronization Vineet Srivastava, Chin Keong Ho, Patrick Ho Wang Fung and Sumei Sun Institute for Infocomm Research 21, Heng Mui Keng Terrace Singapore 119613 Email: {vineet, hock, fungpatrick, sunsm} @i2r.a-star.edu.sg Abstract— Robust minimum mean-square error (MMSE) channel estimation for orthogonal frequency-division multiplexing (OFDM) systems with practical timing synchronization is considered. The correlation between the channel coefficients at the different subcarriers depends upon the channel power delay profile (PDP) and the probability density function (pdf) of the timing synchronization offset. Since both of these are not known at design time, there is a need for a robust estimator that results in mean-square error performance independent of the actual channel power delay profile (PDP) and the actual timing synchronization offset pdf. We develop the notion of effective PDP, given by the convolution of the channel PDP and the pdf of the timing synchronization offset. We show that designing with a uniform PDP of the same length as the effective PDP leads to a robust solution. This follows from the robustness of the uniform PDP for MMSE estimation, a known result for which we provide an alternative derivation. We illustrate the performance of the robust estimator using a numerical example.

I. I NTRODUCTION In orthogonal frequency division multiplexing (OFDM) systems, the frequency domain channel coefficients at the different subcarriers are correlated. If this correlation between the channel coefficients at the different subcarriers is known, minimum mean-square error (MMSE) channel estimation can be performed in the frequency domain [1], [2]. In practical systems (e.g. IEEE 802.11a [3]) however, such knowledge is not available at design time. There are two reasons for this. Firstly, the correlation between the channel coefficients at the different subcarriers in the frequency domain depends upon the power delay profile (PDP) of the time domain channel, which is a function of the wireless environment in which the device is operated. Secondly, with practical timing (or frame) synchronization, the position of the timing synchronization inevitably fluctuates from packet to packet. This is seen from the non-zero timing estimation variance at practical signal-to-noise ratios for the various timing synchronization methods proposed in the literature which include employing a preamble with repeated parts [4], fine-tuning the coarse timing estimation by estimating the delay of the first channel tap [5] or with iterative processing ([6], [7]), and cyclic-prefix based methods [8], just to name a few. A timing synchronization offset causes a phase rotation of the channel coefficients in the frequency domain [9]. As such, with fluctuating timing synchronization offset, the correlation between the channel coefficients at the different subcarriers is also a function of WCNC 2004 / IEEE Communications Society

the probability density function (pdf) of the timing synchronization offset. However, determining the pdf for the timing synchronization offset with the methods listed above is nontrivial (see e.g. [10], [11]). Thus, to sum up, the correlation between the channel coeffients at the subcarriers depends upon the channel PDP and the timing synchronization offset pdf, both of which are not known at design time. In light of the aforementioned observations, there is a need for designing a robust MMSE estimator [1] which guarantees a certain mean-square error (MSE) performance regardless of the actual channel PDP, as well as the actual timing synchronization offset pdf. An estimator that results in MSE performance that is independent of the channel PDP is presented in [1]. In this paper, we extend the robust estimator result from [1] to include the case of fluctuating timing synchronization resulting from practical timing synchronization. To do so, we come up with the notion of effective PDP, given by the convolution of the channel’s actual PDP and the pdf of the timing synchronization offset. We show that the design performed using a uniform PDP of the same length as the effective PDP is robust. That is, the MSE performance is independent of the actual channel PDP and the actual pdf of the timing synchronization, as long as the actual effective PDP is of the same length as the robust PDP employed for design. This follows in a rather straightforward manner from the result in [1] which states that designing with a uniform power delay profile (PDP) leads to a robust MMSE estimator. Thus, by knowing the maximum spread of the timing synchronization offset (which can be determined empirically) and the worstcase length of the channel PDP, robust MMSE estimator can be designed. We also show that the MSE performance deteriorates if the actual effective PDP is longer than the robust PDP employed for design, but it remains unchanged if the actual effective PDP is shorter than the PDP employed for design. In the rest of this paper, we describe the system model in Section II and present the equivalent channel with fluctuations in timing synchronization offset in Section III. We then present the mathematical expressions for the MMSE filter in Section IV. Next, in Section V, we derive the result that designing with a uniform PDP leads to a robust solution and extend this result to the case of fluctuating timing synchronization offset in Section VI. Subsequently, we present a numerical example in Section VII and summarize our conclusions in Section VIII.

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II. S YSTEM M ODEL

Ng No

A. Transmitted Baseband Signal We consider an N -subcarrier OFDM system. Let Xk be the frequency domain symbol modulated onto the kth subcarrier. We assume that there are no null subcarriers. The complex baseband time domain data xn for n = 0, 1, . . . , N −1 is given by the Inverse Discrete Fourier Transform (IDFT) of the N -pt sequence X = [X0 X1 . . . XN −1 ] [9]. Prior to transmission, a cyclic prefix of Ng complex symbols is put at the beginning of the time domain sequence x = [x0 x1 . . . xN −1 ]. We denote this N + Ng term complex sequence by xCP and the nth CP is term of the sequence xCP by xCP n . The sequence x transmitted through the multipath channel, which is described next.

N -pt Sequence yr N + Ng + L -1 pt Received Sequence

Fig. 1. Notation for the received signal. Frequency domain data is obtained by the N -pt DFT of the sequence yr . Timing Offset No = 0 corresponds to the perfect timing synchronization position.

III. E QUIVALENT C HANNEL WITH N ON -Z ERO T IMING O FFSET

B. Multipath Channel Description

A. Fixed Non-Zero Timing Offset

We consider a sample-spaced [1] multipath channel described by an L-tap complex random vector h, such that

In order to prevent inter-symbol interference (ISI) from the previous OFDM symbol, we must have 0 ≤ No ≤ Ng − L − 1 [9]. When No = no is within this range, the equivalent channel in the frequency domain at the kth subcarrier is [9]   j2πno k ˜ Hk = Hk exp (5) N

T

h = [h0 h1 . . . hL−1 ]

(1)

where hi ’s for 0 ≤ i ≤ L − 1 are independent zero-mean complex Gaussian random variables. The channel PDP is given by an L-element vector p, whose elements pi are pi =

1
2 E |hi | 2

i = 0, 1, . . . , L − 1.

(2)

where Hk , for k = 0, 1, . . . N − 1 is obtained by the N -point DFT of the the zero-padded channel coefficients vector h [9]. The equivalent channel in time-domain is   ˜ n = 0 0 . . . 0 h0 h1 . . . hL−1  . h o  

We L−1assume that the channel is normalized such that i=0 pi = 1.

(6)

no zeros

C. Received Signal

Thus, a non-zero timing offset No = no has the effect of delaying the channel PDP by no samples.

The time-domain channel output when the sequence xCP is transmitted through the channel h in (1) is

B. Fluctuating Timing Offset

ym =

L−1 

hi xCP m−i + wm

m = 0, 1, . . . , N + Ng + L − 2

i=0

(3) where wm represents additive white Gaussian noise (AWGN). = 0∀i < In deriving (3), we have assumed that the xCP i 0. This is equivalent to ignoring the impact of the previous OFDM symbols. The timing synchronization mechanism extracts an N -point sequence yr from the received data. We define a discrete random variable No , which represents the timing offset in the number of samples, counted from the perfect synchronization point (No = 0), as illustrated in Fig. 1. If No = no , we have T  yr = yNg −no yNg −no +1 . . . yNg −no +N −1 .

(4)

An N -point Discrete Fourier Transform (DFT) transforms the sequence yr into the frequency domain [9]. WCNC 2004 / IEEE Communications Society

As we have mentioned, in systems with practical timing synchronization, the timing offset No is a discrete random variable. Let the pdf of No be denoted by Π, a vector whose ith element πi is given by πi = P (No = i)

i = 0, 1, . . . , Nomax

(7)

where Nomaxmax is the maximum timing synchronization offset. No Clearly, i=0 πi = 1. The channel PDP p from (2) is of length L. We have seen from (6) that a timing synchronization offset No = no samples causes a delay of no samples in the channel PDP. The effective power delay profile pE with fluctuating timing synchronization offset is thus given by the convolution of the channel PDP p and the pdf of the timing synchronization offset Π. The intuition here is that the equivalent channel PDP is delayed by i samples with probability πi , for i = 0, 1, . . . , Nomax . Thus, the effective PDP is of length L + Nomax and it represents the average channel that encompasses both the channel PDP and the timing synchronization statistics. Mathematically, pE = MT Π,

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where the M is an (Nomax +1)×(L+Nomax ) circulant matrix formed from p = {p0 p1 . . . pL−1 } as follows:

    M=  

L+Nomax columns

p0 0 0 .. .

p1 p0 0 .. .

... ... p0 .. .

pL−1 pL−2 ...

 0 pL−1 pL−2

0

0

0

p0

p1

0 0 pL−1 .. .

... ... ... .. .

0 0 0

...

...

pL−1



     

(14) and P is an N × N matrix with only L nonzero elements, which are along its principal diagonal and are equal to the L elements of the channel PDP. That is, if the PDP of the channel is p = {p0 p1 . . . pL−1 }, then the P matrix is given by diag{p0 p1 . . . pL−1 }. We note that WW† = N I. The expression for the Φ matrix in (13) is exact in case of samplespaced channels. In case of non sample-spaced channels too, this result holds, with some loss of accuracy [1]. We substitute (13) into (11) and perform simple matrix algebraic manipulations to obtain

(8)

ˆ LS , is The least-squares estimate [2] of Hk , denoted by H k (9)

where Yk is the received pilot symbol at the kth subcarrier and Nk is the additive noise in the frequency domain. We have assumed that Xk = 1 ∀ k. B. MMSE Estimation ˆ M M SE [2] is The MMSE estimate of the channel, H H

†

ˆ LS

=F H



W (k1 , k2 ) = exp

A. A One-Tap Least-Squares Estimation

ˆ M M SE

−j2πk1 k2 N



IV. C HANNEL E STIMATION

ˆ kLS = Yk = Hk + Nk = Hk + Nk H Xk Xk

where † as before refers to the Hermitian operation, W is the N × N DFT matrix defined as

−1  WPW† WPW† + σ 2 I −1  = WX−1 W† WPW† 1 WXPW† = N where X is an N × N diagonal matrix defined as F =

 X=

(10)

ˆ M M SE and H ˆ LS are N × 1 column vectors with where H elements that are the MMSE channel estimates and the leastsquares channel estimates (from (9)) respectively for the various k values, with each k representing a subcarrier. The † operation refers to the complex conjugate transpose, or the Hermitian operation. The MMSE filter matrix F, which can be obtained by applying the orthogonality principle [2], is an N × N matrix given by

k1 , k2 = 0, 1, . . . , N −1

σ2 P+ I N

(15)

−1 .

(16)

Note that the matrix X too has only L nonzero elements that are all along its principal diagonal. V. ROBUST MMSE E STIMATOR

We see from (11) and (15) that the MMSE estimator F can only be computed if the channel correlation matrix Φ, or equivalently the channel PDP p = {p0 p1 . . . pL−1 } is known. However, since the channel PDP is a function of the wireless environment in which the device is operated, it is 
−1 Φ (11) not possible to know it at the design time. Moreover, with F = Φ + σ2 I where σ 2 is the AWGN variance which we assume is known, I fluctuation in the timing synchronization offset from packet is an N ×N identity matrix and Φ is the N ×N autocorrelation to packet, the effective PDP is also a function of the pdf of the timing synchronization offset which, as we have noted matrix [12] of the channel with elements given by is difficult to obtain. Thus, there is motivation to design the MMSE estimator F in such a way that regardless of what the   −1  N −j2π (k1 − k2 ) i 1  actual PDP and the actual pdf of the timing synchronization ∗ Φ (k1 , k2 ) = E Hk1 Hk2 = pi exp . offset are, the same MSE performance is obtained. 2 N i=0 To derive the robust estimator with fluctuations in timing (12) Note that Φ (k1 , k2 ) = Φ∗ (k2 , k1 ). The kth column of the F synchronization, we start with a result similar to the one matrix, fk is the N -tap MMSE filter needed for the estimation presented in [1], where the problem of robust estimation with unknown channel PDP is considered. It is shown in [1] that of the channel at the kth subcarrier. when the F matrix is designed using a uniform PDP, it is C. An Alternative Expression for the F matrix robust. That is, for all PDPs with same maximum delay, An alternative expression for the MMSE estimator F in (11) identical MSE performance is obtained. In this section, we can be obtained by using the relation between the correlation present a new derivation of this result. Subsequently, we matrix Φ and the channel PDP seen in (12). Specifically, we extend this result to include the case of fluctuating timing have [1] synchronization offset in the next section. Our derivation makes use of the alternative expression for (13) the F matrix given in (15). Before presenting the result and Φ = WPW† WCNC 2004 / IEEE Communications Society

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its derivation, we first obtain an expression for the MSE when the actual PDP is different from the PDP that is employed for the design of the MMSE filter. This can be found by using the expression for the MSE when an arbitrary N -tap filter gk (not necessarily MMSE filter) is employed for the estimation at the kth subcarrier.

the MSE expression when the actual correlation matrix of the channel is ΦA , while the estimation filter F has been designed assuming the correlation matrix to be ΦD . Next, we look at how the MSE changes with ΦA for a given design ΦD , and that leads us to the result that designing with the uniform PDP leads to a robust MSE performance.

A. MSE with Power Delay profile Mismatch

B. Designing with Uniform Power Delay Profile is Robust.

The estimation error in the kth subcarrier is denoted by ek .  We define the MSE as k = 12 E |ek |2 , and for an arbitrary filter gk , simplify it to obtain

Theorem 1 (Robust Design with unknown PDP): If a uniform PDP of length L is employed for design, the channel estimation mean-square error performance is independent of the actual channel PDP, as long as the length of the actual channel PDP is also L. That is, for any arbitrary PDP of length L, the same mean-square error performance is obtained. Proof: We first note from (23) and (24) that for a fixed ΦD , we can study the effect of correlation mismatch by just looking at a quantity tA , defined as


 1  E |Hk |2 − Φ†k gk − gk† Φk + gk† Φ + σ 2 I gk (17) 2 where Φk is the kth column of the Φ matrix in (12). The average MSE over all subcarriers is k =

N −1
  1  1  k = Tr I − Φ† G − G† Φ + G† Φ + σ 2 I G N N k=0 (18) where Tr[•] refers to the trace of the matrix and the filter matrix G = [g0 g1 . . . gN −1 ], with gk for k = 0, 1, . . . , N −1 being the filter employed for the  kth subcarrier. In deriving (18), we have assumed that 12 E |Hk |2 = 1 for all k. Now, suppose the channel correlation matrix using which we design the MMSE filter F is ΦD , such that

  tA = Tr WPA QD W† .

=

ΦD = WPD W† D

(19)

D

where PD = diag{p }, with p being the power delay profile employed for design. The F matrix is then given by 1 F = WXD PD W† (20) N where XD is obtained from (16) by replacing P with PD . Let’s say that the actual correlation matrix ΦA is ΦA = WPA W†

(21)

where PA = diag{pA }, with pA being the actual power delay profile. We assume pD = pA , and hence PD = PA . Using the MSE expression from (18), we have
  1  A = Tr I − Φ†A F − F† ΦA + F† ΦA + σ 2 I F N We substitute (20) and (21) into (22) and obtain

    1 1 σ2 † A = Tr W I − PA QD + XD PD XD PD W N N N (23) where QD is an N × N matrix defined as QD = 2XD PD − XD PD XD PD .

Note that an increase in tA leads to a decrease in the MSE. In what follows, we prove that when ΦD corresponds to a uniform PDP, tA is independent of the actual PDP pA . This just proves that identical MSE performance is obtained for any arbitrary actual PDP of length L. We simplify tA in (25) by noting that the matrix PA QD has only L non-zero components which are along its principal D D diagonal. If pD and pA are given by pD = {pD 0 p1 . . . pL−1 } A A A A and p = {p0 p1 . . . pL−1 } respectively, the ith diagonal component of the matrix PA QD is D d i = pA i qi

(24)

(26)

values are the diagonal elements of the QD where the matrix in (24), and are given by
D D 2 D qiD = 2xD i pi − xi pi

i = 0, 1, . . . , L − 1.

(27)

The xD i values in (27) are the diagonal elements of the XD matrix in (16). From (16), we note that  =

pD i

σ2 + N

−1 i = 0, 1, . . . , L − 1.

(28)

The diagonal structure of the matrix PA QD enables us to simplify the tA expression to tA =

L−1  i=0

di =

L−1 

D pA i qi .

(29)

i=0

When pD corresponds to the uniform PDP, we have pD 0 = D D D = . . . = pD L−1 , hence x0 = x1 = . . . = xL−1 and as a D . Thus, consequence, q0D = q1D = . . . = qL−1 pD 1

The simplification of A and the derivation of the QD matrix is shown in the Appendix. In (23) and (24), we have presented WCNC 2004 / IEEE Communications Society

i = 0, 1, . . . , L − 1

qiD

xD i (22)

(25)

714

tA =

L−1  i=0

di =

L−1  i=0

D D pA i qi = qi

L−1 

D pA i = qi

(30)

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L−1 L−1 D where we have used i=0 pA i = i=0 pi = 1. From (30), we note that tA is independent of the actual PDP. This just proves that the uniform PDP is robust, because when the design is performed using a uniform PDP, all PDP’s of the same length exhibit identical MSE performance. C. Impact of Length Mismatch in the Power Delay Profile Corollary 1: When the actual PDP is longer than the PDP employed for design, the MSE performance deteriorates. On the other hand, when the actual PDP is shorter than the PDP employed for design, the MSE performance remains unchanged. Proof: Let’s say that the length of the PDP pA is L , such that L = L. In this case, we have tA

=

∗ L −1

di =

i=0

∗ L −1

D pA i qi

(31)

i=0



where L∗ = min {L, L }. When L < L (actual PDP longer), we have tA

=

L−1 

di =

i=0

qiD

L−1 

pA i


L (actual PDP shorter), we have tA

=

 L −1 

i=0

di =

 L −1 

i=0

qiD pA i

=

qiD

 −1 L 

pA i = tA

(33)

i=0

L −1 because i=0 pA i = 1. In this case thus, the MSE performance remains unchanged. VI. ROBUST E STIMATOR W ITH P RACTICAL T IMING S YNCHRONIZATION We introduced the notion of effective PDP in Section IIIB as the average channel that encompasses the effect of both the channel PDP as well as the statistics of the timing synchronization. The length of the effective PDP is L+Nomax , where L and Nomax are the length of the channel PDP and the maximum spread of the timing synchronization offset respectively. Our objective is to design the MMSE filter in such a way that regardless of the actual PDP and the actual timing synchronization pdf, identical MSE performance is obtained. This is straightforward in light of Theorem 1, and we can state the result as follows: Theorem 2 (Robust Design with unknown effective PDP): If a uniform PDP of length L + Nomax is employed for design, identical MSE performance is obtained for all channel PDP’s of length L and all timing synchronization prabability density functions with non-zero values in the range 0 to Nomax . Thus, the PDP that leads to robust performance is pR , given by WCNC 2004 / IEEE Communications Society

 pR =



1  1 1 ... 1 .   Nomax + L max No

(34)

+L terms

Proof: From Theorem 1, we see that uniform PDP leads to robust design. With timing synchronization fluctuations, we have an effective power delay profile of length Nomax + L. Hence, a robust PDP pR for design is simply a uniform PDP of length Nomax + L. A. Effect of Length Mismatch Let’s say that the length of the robust power delay profile pR is LR = Nomax + L, but the actual length of the effective power delay profile is LA , such that LA = LR . We note from Corollary 1 that when LA < LR , the performance is unchanged. However, when LA > LR , the MSE performance deteriorates. This means that the design should be performed for the worst-case, that is for the longest power delay profile as well as the largest timing synchronization spread. VII. N UMERICAL E XAMPLE We consider an N = 64 subcarrier OFDM system with Ng = 16 symbol cyclic-prefix. We consider a 16-tap exponential channel PDP, characterized by the parameter α, such that the ith multipath component has a power given by exp (−2αi) pi = 15 i=0 exp (−2αi)

i = 0, 1, . . . , 15.

(35)

We set α = 0.5 for our simulations. Due to the exponential decay of the power delay profile in (35), we note that with α = 0.5, the power of the 6th tap, p5 is about 21.7dB below the power of the first tap. Hence, we set the channel PDP length L at 6. We set Nomax = 4, which means that the effective PDP is of length 10. Hence, in accordance with (34), the robust PDP for design is a 10-tap vector given by pR = [0.1, 0.1, . . . , 0.1]. We design an MMSE estimator using the robust PDP pR = [0.1, 0.1, . . . , 0.1]. Next, we run simulation for a total of 3000 trials. In each trial, an independent 16-tap exponential channel realization and a random timing offset according to a certain timing synchronization offset pdf are generated. The average MSE over the 3000 trials is obtained and reported. We consider three different 5-point pdf’s of the timing synchronization T offset (see (7)), namely Π1 = [0.35 0.275 0.200 0.125 0.05] , T T Π2 = [0.1 0.2 0.4 0.2 0.1] and Π3 = [0.2 0.2 0.2 0.2 0.2] which represent decaying, peaky and uniform pdf’s respectively, as shown in Fig. 2. As expected, the MSE performance obtained for the three pdf’s is identical, as we note from Fig. 3. The discrepancy in the MSE performance at high signal to noise ratio is due to the fact that the channel PDP length is actually more than L = 6. For reference, we have also presented the MSE performance obtained with an exactly 6-tap channel, and as expected, all three pdf’s lead to identical MSE performance. Next, we consider the impact of having a channel with PDP length different from the designed length. We accomplish this by varying the exponential decay factor α in (35). When α is

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0

0.35

Decaying pdf

0.3

10

Peaky pdf

0.3

0.25

0.25

0.25

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0

0

α = 0.5, Decaying Timing Sync. pdf α = 0.5, Peaky Timing Sync. pdf α = 0.5, Uniform Timing Sync. pdf Exactly 6−Tap PDP α = 0.6, Uniform Timing Sync. pdf α = 0.4, Uniform Timing Sync. pdf One−Tap Channel Estimation

Uniform pdf

0.3

−1

Mean Square Error

0.35

10

−2

10

−3

10

4

2

0

Timing Offset No

4

2

0

4

4

Timing Offset N

o

2

6

8

10

12

14

16

18

20

Signal−to−Noise Ratio in dB

0

Timing Offset N

o

Fig. 2. The three pdf’s of the timing synchronization offset considered in the numerical example.

Fig. 3. Channel Estimation MSE performance for an exponential channel PDP with robust design. The MSE performance is independent of the actual timing synchronization offset pdf.

made 0.6, the channel PDP effectively becomes shorter, and we note that the MSE performance is practically unchanged. On the other hand, when α is changed to 0.4, the effective PDP becomes longer than 10, and hence the MSE performance deteriorates. Finally, we note that the MSE performance obtained with the robust MMSE estimator is significantly better than the MSE performance obtained by the one-tap estimator in (9).


 F† ΦA + σ 2 I F = †
 
1
WXD PD W† WPA W† + σ 2 I WXD PD W† 2 N   σ2 XD PD XD PD W† (37) = W PD XD PA XD PD + N

VIII. C ONCLUSIONS

Substituting these values into the A expression in (22) , we obtain (23), with QD matrix as defined in (24).

We considered the problem of robust MMSE channel estimation in packet-by-packet OFDM systems. We presented a new derivation for the robustness of the uniform power delay profile. We introduced the notion of effective power delay profile that encompasses both the channel power delay profile and the pdf of the timing synchronization. Using the robustness of the uniform power delay profile, we concluded that in OFDM systems with practical timing synchronization, a robust estimator is simply the MMSE estimator designed using a uniform power delay profile of the same length as the effective power delay profile. When the robust estimator is employed, the MSE performance is independent of the actual channel power delay profile and the pdf of the timing synchronization. We also studied the effect of length mismatch in the design and actual power delay profile, and concluded that the design should be performed using the worst case channel length and timing synchronization spread. APPENDIX

In this appendix, we derive the expressions for A in (23) and Q matrix in (24). The approach is to simplify the terms in (22). Thus, we have † 1
WXD PD W† = WPA XD PD W† Φ†A F = WPA W† N (36) Likewise, F† ΦA = WPA XD PD W† . Finally, WCNC 2004 / IEEE Communications Society

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