Pilot Tone-based Channel Estimation for OFDM ... - Semantic Scholar
Pilot Tone-based Channel Estimation for OFDM Systems with Transmitter Diversity in Mobile Wireless Channels† I. Barhumi and M. Moonen (imad.barhumi, [email protected]) EE Dept., ESAT/SISTA-K.U.Lueven, Kasteelpark Arenberg 10 3001 Leuven-Heverlee, Belgium +32 16 32 1060 +32 16 321970 (fax) Abstract— This paper describes a channel estimation and tracking scheme for OFDM MIMO systems based on pilot tones. The mean square error of the proposed scheme is derived, and through this we derive optimal training sequences and placement of the pilot tones. It is shown in this paper that minimum mean square (MMSE) channel estimation is obtained with equipowered, equispaced, orthogonal and phase shift orthogonal training sequences. Through simulations it is shown that the optimal pilot-tone based training sequences as derived in this paper significantly outperform full-block training sequences in rapidly time varying channels. Additionally, we show that by satisfying all conditions of optimality, our training sequences outperform orthogonal and random sequences, which both appear to be sub-optimal for channel identification.
I. Introduction High-data rate techniques in communication systems have gained a considerable attraction in recent years. A technique of special interest is OFDM (Orthogonal Frequency Division Modulation) which is a multi-carrier modulation technique. This is due to its simple implementation, and robustness against frequency selective channels by converting the channel into flat fading sub channels. OFDM has been standardized for a variety of applications such as digital audio broadcasting (DAB), digital TV broadcast† This research work was carried out at the ESAT laboratory of the K.U.Leuven, in the frame of the Belgian State, Prime Minister’s Office - Federal Office for Scientific, Technical and Cultural Affairs - Interuniversity Poles of Attraction Programme - IUAP P4-02 (1997-2001): Modeling, Identification, Simulation and Control of Complex Systems, the Concerted Research Action GOA-MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology)of the Flemish Government, and was partially sponsored by IMEC (Flemish Interuniversity Microelectronics Center)
ing, wireless LAN, and asymmetric digital subscriber lines (ADSL). Combining OFDM with multiple antennas has been shown to provide significant increase in capacity through the use of transmission diversity. Commonly suggested schemes for doing this rely upon knowledge of the channel state information (CSI) at the receiver. Typical procedures for identifying the channel utilize several blocks that consist completely of training symbols for SISO[1] as well as for MIMO[3] systems. In such a system the CSI is first estimated prior to any transmission of data. When the CSI changes significantly a retraining sequence is transmitted. In a rapidly time varying environment, such systems must continuously re-train to re-estimate the CSI. Between re-training, these systems experience increased BER due to their outdated estimates of the channel. In this paper a channel tracking method based on pilot tones is used for estimation and tracking of the CSI. The ability of this scheme to track channel changes on a block-by-block basis means that it can more accurately track the CSI. The mean square error bound on channel estimation is derived through which we then derive the optimal training sequences, and the optimal placement of the pilot tones. This paper is organized as follows: in section II we briefly overview the basic system model; in section III we introduce channel estimation; the analysis of the channel estimator is derived in section IV, through which we derive an optimal training strategy; section V presents computer simulation results, and finally, conclusions are drawn in section VI.
270
L
YK
t
Nr
Y1
FFT
N
Serial/Parallel
Parallel/Serial
t
IFFT
Serial/Parallel
N
1
L
L L
X
FFT
1
h
1 Y1
Serial/Parallel
Parallel/Serial
1
IFFT
X
Serial/Parallel
L
Nr
YK
Hq,p cir is a circulant matrix with the first column is iT
T
q (n) andyq (n) = y0q (n) ... yK−1 T p XK−1 (n) is the transmitted vector from the pth transmit antenna at time index n. The eigenvalue decomposition of Hq,p cir = F H Dq,p F , where Dq,p is a diagonal matrix with the q,p frequency of hq,p ¡at its diagonal, := © q,p response ¢ ª D ¡ j0 ¢ q,p j2π(K−1)/K . Taking diag H e ..., H e the FFT of yq (n) at the receiver we obtain
h(q,p) (n) 01×K−L X0p (n) · · · Xp (n) =
Fig. 1. OFDM MIMO System Y q (n) =
Nt X
Dq,p Xp (n) + Ξ(n)
(4)
p=1
II. System Model The system under consideration is depicted in Figure 1. In OFDM systems with multiple antennas, different signals are transmitted from different antennas simultaneously. The received signal at any antenna is a superposition of these signals. At each antenna the data signal is first modulated by the IFFT, and a cyclic prefix of length υ is added. Here υ ≥ L − 1 , where L is the maximum length of all channels. Given Nt transmit antennas, the received signal at the q th receiving antenna on the k th tone can be expressed as
III. Channel Estimation In this section our channel estimation scheme is derived. Let Xp (n) = Sp (n) + Bp (n) , where Sp (n) is some arbitrary K × 1 data vector and Bp (n) is some arbitrary K × 1 training sequence (TS) vector. Substitute for Xp (n) in equation (4) we obtain
[2]
Now define Spd (n) = diag{Sp (n)} and Bpd (n) = diag{Bp (n)}, then equation (5) can be written as follows
ykq (n) =
Nt X
(q,p)
Hk
(n)Xkp (n) + ηk (n)
(1)
Y q (n) =
Nt X
(diag {Sp (n)} + diag {Bp (n)}) F0:L−1 hq,p (n)+Ξ(n)
p=1
(5)
p=1
Yq (n)
for q = 1 . . . Nr and k = 0 . . . K −1. Here Nr is the number of receiving antennas and K is the number of total (q,p) sub-carriers (tones) in the OFDM system. Hk (n) is the frequency response at the k th tone for the pth transmit antenna and the q th receive antenna at time n given by (q,p)
Hk
(n) =
L−1 X
(q,p)
hl
(q,p) h0 (n)
(q,p) hL−1 (n)
iT
, the impulse response from the p transmit antenna to the q th receive an2π tenna at time n. WK = e−j K and Xkp (n) is the symbol transmitted from the pth transmit antenna on the kth tone at time n. Given K sub channels the received time domain vector vector at the q th antenna after removing the cyclic prefix is given by: ...
Bpd (n)F0:L−1 hq,p (n) + Ξ(n)
(6)
p=1
Define
B1d F0:L−1
and
l=0
h
Nt X
+
(2)
Here, h(q,p) (n) is the lth channel tap of hq,p (n) = l
Spd (n)F0:L−1 hq,p (n)
p=1
A= kl (n)WK
Nt X
=
...
t BN d F0:L−1
(7)
b q (n) = A† Y q (n) h
(8)
where the pseudo-inverse matrix A = (A A) AH . Using (6) we can get the following: †
H
−1
th
yq (n) =
Nt X
H p Hq,p cir F X (n) + η(n)
(3)
b q (n) = h
q
h (n) + A
†
Nt X
Spd (n)F0:L−1 hq,p (n)
p=1 †
+A Ξ(n)
(9)
3 hq,1 (n) 7 6 .. hq = 4 5. . hq,Nt (n)
By imposing the following
2
where
condition
p=1
271
A† Spd (n) = 0
(10)
T
.
for p = 1, ..., Nt , we can get q
b (n) = hq (n) + A† Ξ(n) h
(11)
(subject to a constant modulus training sequences). Using a similar argument as in [4] the lower bound on the mean square error is as follows:
q
b (n) is a combination of Equation (11) indicates that h q the true h (n) plus a term affected only by the noise in b q (n)} = hq (n) + A† E{Ξ(n)} = hq (n). the system. Since E{h q b (n) forms an unbiased estimate of In other words h q h (n). H Satisfying (10) implies Brd Spd (n) = 0 for any r = 1 . . . Nt and any p = 1 . . . Nt . For such a scheme where both data and training are impeded in the same OFDM symbol, the only way of satisfying this is by choosing disjoint sets of tones for training and data. i.e. zeros in B(n) where S(n) contains data and vice versa. This allows us to write (9) in simplified form:
e = where A
h
e 1d F e B
M SE(n) ≥
A. Optimal Training Sequences in OFDM MIMO Systems
In this subsection we will derive the optimal training sequences for channel estimation and tracking in OFDM MIMO systems. Let P be the number of pilot tones required for training, with the©necessary condition of identifiability ª P ≥ LNt 1 . Let k0 , k1 , . . . , kP −1 be the set q † eq b e h (n) = A Y (n) (12) of P tones used for training. Assuming K sub-carriers i are used in the system. Rewriting equation (7) p N e = diag {the non entreis (pilots) e te , B of bp } ···
Bd F
d
e e , Y (n) and Ξ(n) , F are the corresponding rows of F0:L−1 , Yq (n) and Ξ(n) respectively, then the channel estimate becomes b q (n) = hq (n) + A e † Ξ(n) e h
(13)
=
2
bq
q
h (n) − h (n)
2 1
e†e
E A Ξ(n)
1 E LNt
H
o
M SE(n) =
σn2 LNt
½³ ´−1 ¾ He e tr A A
eH B e 1dH B e 1d F e F 6 .. e HA e =6 A . 4 H eH B e Nt B e 1d F e F d
Define
i
3 eH B e 1dH B fd Nt F e F 7 .. 7 . 5 H eH B e Nt B e Nt F e F
··· .. . ···
d
H
···
(17)
d
(18)
eH
e . For minimum as the (i, j) L × L sub-matrix of A A mean square error (MMSE) channel estimation we ree HA e to be diagonal i.e.: quire A th
cIL i = j OL×L i = 6 j
(19)
First, we will consider the case i = j in equation (19). H Assuming constant modulus training sequences H e id B e j = IP , we obtain from(18) B d eH F e Cii = F
(20)
The (r, s)th entry of the sub-matrix Cii (denoted by [Cii ]r,s ) can then be written as :
(15)
where tr {·} means trace. In [4], it was shown that for e HA e should be diagonal minimum mean square error A
e Nt F e B d
eH B e id B ejF e Cij = F d
e Ξ e (n) = σ 2 IP , P is the For AWGN where E Ξ(n) n number of pilot tones used in the system. The mean square error can then be written as follows:
···
e = f0 · · · fL−1 , fl = W lk0 W lk1 W lk2 where F 2π e HA e can be written as : and W = e−j K . Then, A
Cij =
LNt n n o o H 1 e † E Ξ(n) e e H (n) A e† tr A Ξ (14) LNt
n
e 1d F e B
2
In this section the mean square error of the channel estimate is derived through which an optimal training sequence and optimal placement of pilot tones are derived. From (13) the mean square error in estimating the channel parameters is given by
=
h
e = A
IV. Channel estimation analysis
=
(16)
where P is the number of pilot tones used in the e HA e is system. The equality obtained if and only if A diagonal.
eq
M SE(n)
σn2 P
[Cii ]r,s = frH fs 1
(21)
This condition can be relaxed. The corresponding analysis is however outside the scope of this paper.
272
W lkP −1
which is equal to
P
PP −1
[Cii ]r,s =
p=0
W
f or r = s f or r 6= s
(22)
kp (s−r)
From (22) the only set of tones that achieves the −1 diagonality of Cii is {kp }Pp=0 such that kp = p0 + pV , where p0 is some reference p0 ∈ 0, . . . , V − 1 with P V (s−r) 0, . . . , L − 1 ∈ Z and V (s−r) ∈ / Z, ∀s, r ∈ K K where Z is the set of integers. For minimum pilot tones and maximum spacing P V = K , or V = K . For cheap, P fast and simple implementation of the DFT, the total number of sub-carriers is chosen to be a power of 2 in practical systems. For this case the minimum number of pilot tones can be selected (keeping in mind that P ≥ LNt ) as P = 2dlog2 LNt e . Note that the set of equispaced tones is the optimal tone set that achieves the first part of equation (19) with the assumption that a constant modulus training sequences are used. We now investigate the conditions imposed by the second part of equation (19). From this point on we assume maximum spaced pilot tones which satisfy the . The first part of equation (19), that is kp = p0 + p K P (r, s)th entry of Cij can be written as
[Cij ]r,s
=
W p0 (s−r)
P −1 h X
∗
e id B ej B d
i p,p
p=0
=
W p0 (s−r)
P −1 h X
∗
e id B ej B d
i p,p
p=0
P −1 h X
∗
e id B ej B d
i
p=0
p,p
H
−L + 1,
nh 1
2π
e−j P
φ
...
2π
e−j K pV (s−r) (23)
e−j
...,
e−j
Equispaced+Equipowered[5] +Orthogonal
(Flat Fading Channel L = 1) Multiple Tx (Multipath Channel of L taps)
2π p(s−r) P
(24)
L−1
2π(P −1) φ P
io
In other words in MIMO systems the training sequences on different transmit antennas must be not
+Phase-Shift Orthogonal −L + 1, . . . , L − 1
shif ts ∈
TABLE I Different Configuration Requirements
only orthogonal but phase shift orthogonal for phase shifts in the range −L+1, . . . , L−1. This phase shift orthogonality in frequency corresponds to a constraint of shift orthogonality in time as well. This is intuitively satisfying since in a system with multipath propagation, the training sequence of one antenna must be orthogonal not only to the training sequence of other antennas but also to their delayed multipath components. For the purpose of comparison we summarize various scenarios and the constraint they impose on the TS in Table I. An optimal training sequence that achieves all of the above (orthogonality and phase shift orthogonality) can be designed as follows: ∗
e id B ej B d
where Dφ represents a phase shift φ, that is: Dφ = diag
Single Tx Multiple TX
h
Considering the case where s = r, we see that equation (24) implies the constraint of orthogonality on the training sequences. However, this is not sufficient, since we must also count for the case when s 6= r and s − r ∈ −L + 1, . . . , L − 1 . We can write equation (24) as Bi Dφ Bj = 0 ∀φ ∈
TS Requirement
W −rpV W spV
Substituting V = K/P in equation (23), we obtain [Cij ]r,s = W p0 (s−r)
Configuration
h
e id where B
nij ∈
Z.
i p,p
= e−j
i p,p
2π n p i P
= e−j
2π n p ij P
(25)
and nij = nj − ni where
Hence the second part of (19) is satis-
© ª 0, . . . , L − 1 , ∀i, j ∈ ∈ / Z ∀ s, r ∈ © ª 1, . . . , Nt with i 6= j . One possible choice is © ª ni = (i − 1)L ∀i ∈ 1 . . . Nt .
fied if
nij +s−r P
It is also worth to notice that for an arbitrary constant sequence a of length P , the sequences h i modulus 2π e id B = e−j P ni p . (a)p also result in an optimal trainp,p ing sequence. V. Simulations In the simulations a channel with 8-taps is used, the taps are simulated as independent Rayleigh time varying, correlated in time with the correlation function according to Jakes’ model r(τ ) = J0 (2πfd τ ), where J0 is the zeroth order Bessel function and fd is the Doppler frequency. The entire bandwidth of the channel is 120 KHz, divided into K = 128 sub-channels, and a cyclic prefix of 8 symbols is added to the head of
273
0
10
−1
10
−2
10 BER
each block to eliminate inter block interference (IBI). The OFDM frame duration is 1.13 ms. QPSK signaling is applied. 2 transmit and 4 receive antennas are assumed. Using the estimated channels, a linear Zero-Forcing (ZF) equalizer is used to recover the transmitted data. The performance of the system is measured in terms of the bit error rate (BER) vs. SNR and the mean square error (MSE) of the channel parameters estimation. Doppler frequencies of fd = 5, 20, 40, 100 Hz are assumed.
__ 16 pilot tones every symbol −− all tones every 8th symbol
−3
10
−4
As shown in Figures (2 & 3) the pilot tone-based scheme significantly outperform the full-block based system, for channels with Doppler frequency fd ≥ 20 Hz . While for fd = 5 Hz we see that the pilot-tone based scheme experiences only 2 dB loss in SNR at BER = 10−4 compared to the full-block based training system. Thus, the pilot-tone based training scheme is an effective scheme for tracking rapidly time varying channels.
10
fd=5 fd=20 fd=40 fd=100 −5
10
0
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 2. BER vs. SNR for full-block based scheme and pilot-tone based scheme 0
10
fd=5 fd=20 fd=40 fd=100
−1
10
−2
10
__ 16 pilot tones every symbol th
−− all tones every 8 symbol
−3
10
0
Additionally, in our simulations we evaluate a variety of training sequences. Specifically constant power, equispaced sequences; constant power, equispaced, orthogonal sequences; and constant power, equispaced, phase-shift orthogonal sequences. As shown in Figures (4 & 5) phase shift orthogonal training sequences outperform the orthogonal or random training sequences in terms of both BER and MSE channel estimation. From Figure (4), we can see a 2 dB SNR gain at BER = 10−4 for phase-shift orthogonal training sequences over orthogonal training sequences at Doppler frequency fd = 5 Hz , and 4 dB SNR gain at BER = 2 × 10−3 at fd = 100 Hz . Random training sequences completely failed.
2
MSE
The performance of our system was compared to a conventional system which uses full block training sequences. Both systems can be designed to have equal efficiency, where efficiency is defined as epilot = K−P K for our system, and ef ull = 1 − 1c for the full block system, where c is the period between retransmission of training blocks. For Nt = 2, P = 16 pilot tones were required for the pilot-based training scheme. To obtain the same efficiency in the full-block based training scheme we require c = 8.
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 3. MSE vs. SNR for full-block based scheme and pilot-tone based scheme
VI. Discussion In this paper a channel estimation and tracking method based on pilot tones was proposed for optimum based systems. This technique shows significant improvement over the full block based system in terms of BER, channel tracking and MSE of channel parameter estimation. It is found that for such a scheme to obtain the minimum error bound (16) the following constraints must
274
References
g=1
0
10
−1
10
−2
BER
10
−. Random TS −− Orthogonal TS − Phase−shift Orthogonal TS
−3
10
−4
10
fd=5 fd=20 fd=40 fd=100
−5
10
0
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 4. BER vs. SNR for various TS.
[1] L. Deneire, P. Vandenameele, L Van der Perre, B. Gyselinckx, and M. Engels “A Low Complexity ML Channel Estimator for OFDM” proc. of the IEEE Inter. Conf. on comm, Helsinki, Finland June 11-14, 2001 [2] Y. Li, N. Seshadri, and S. Ariyavisitakul “Channel Estimation for OFDM systems with Transmitter Diversity in Mobile Wireless Channels,” IEEE JSAC, vol. 17, no. 3, March 1999. [3] W. G. Jeon, K. H. Paik, and Y. S. Cho “An Efficient channel estimation technique for OFDM Systems with Transmitter Diversity,” IEEE Inter. Sym. on Personal, Indoor and Mobile, vol. 2, pp. 1246-50, 2000. [4] T. L, Tung, K. Yao, and R.E. Hudson “Channel Estimation and Adaptive Power Allocation for Performance and Capacity Improvement of Multiple -Antenna OFDM Systems,” Third IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, March, 2001. [5] R. Negi and J. Cioffi “Pilot Tone Selection for Channel Estimation in a Mobile OFDM System,”IEEE Tran. on Consumer Electronics, Vol. 44, No. 3, Aug. 1998.
g=1
0
10
fd=5 fd=20 fd=40 fd=100
−1
MSE
10
−. Random TS −− Orthogonal TS __ Phase−shift Orthogonal TS
−2
10
−3
10
0
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 5. MSE vs. SNR for various TS.
be satisfied. One, the pilot tones must be equispaced, equipowered. Two, the training sequences must be orthogonal in flat fading channels and phase-shift or© ª −L + 1, . . . , L − 1 thogonal with shif ts ∈ in multipath channels of L taps. It should be noted that in our discussion so far we have only considered AWGN. In the case of colored noise we predict that the use of pilot-tone hopping will increase system performance. However this is still the subject to further investigation.
275
I. Introduction High-data rate techniques in communication systems have gained a considerable attraction in recent years. A technique of special interest is OFDM (Orthogonal Frequency Division Modulation) which is a multi-carrier modulation technique. This is due to its simple implementation, and robustness against frequency selective channels by converting the channel into flat fading sub channels. OFDM has been standardized for a variety of applications such as digital audio broadcasting (DAB), digital TV broadcast† This research work was carried out at the ESAT laboratory of the K.U.Leuven, in the frame of the Belgian State, Prime Minister’s Office - Federal Office for Scientific, Technical and Cultural Affairs - Interuniversity Poles of Attraction Programme - IUAP P4-02 (1997-2001): Modeling, Identification, Simulation and Control of Complex Systems, the Concerted Research Action GOA-MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology)of the Flemish Government, and was partially sponsored by IMEC (Flemish Interuniversity Microelectronics Center)
ing, wireless LAN, and asymmetric digital subscriber lines (ADSL). Combining OFDM with multiple antennas has been shown to provide significant increase in capacity through the use of transmission diversity. Commonly suggested schemes for doing this rely upon knowledge of the channel state information (CSI) at the receiver. Typical procedures for identifying the channel utilize several blocks that consist completely of training symbols for SISO[1] as well as for MIMO[3] systems. In such a system the CSI is first estimated prior to any transmission of data. When the CSI changes significantly a retraining sequence is transmitted. In a rapidly time varying environment, such systems must continuously re-train to re-estimate the CSI. Between re-training, these systems experience increased BER due to their outdated estimates of the channel. In this paper a channel tracking method based on pilot tones is used for estimation and tracking of the CSI. The ability of this scheme to track channel changes on a block-by-block basis means that it can more accurately track the CSI. The mean square error bound on channel estimation is derived through which we then derive the optimal training sequences, and the optimal placement of the pilot tones. This paper is organized as follows: in section II we briefly overview the basic system model; in section III we introduce channel estimation; the analysis of the channel estimator is derived in section IV, through which we derive an optimal training strategy; section V presents computer simulation results, and finally, conclusions are drawn in section VI.
270
L
YK
t
Nr
Y1
FFT
N
Serial/Parallel
Parallel/Serial
t
IFFT
Serial/Parallel
N
1
L
L L
X
FFT
1
h
1 Y1
Serial/Parallel
Parallel/Serial
1
IFFT
X
Serial/Parallel
L
Nr
YK
Hq,p cir is a circulant matrix with the first column is iT
T
q (n) andyq (n) = y0q (n) ... yK−1 T p XK−1 (n) is the transmitted vector from the pth transmit antenna at time index n. The eigenvalue decomposition of Hq,p cir = F H Dq,p F , where Dq,p is a diagonal matrix with the q,p frequency of hq,p ¡at its diagonal, := © q,p response ¢ ª D ¡ j0 ¢ q,p j2π(K−1)/K . Taking diag H e ..., H e the FFT of yq (n) at the receiver we obtain
h(q,p) (n) 01×K−L X0p (n) · · · Xp (n) =
Fig. 1. OFDM MIMO System Y q (n) =
Nt X
Dq,p Xp (n) + Ξ(n)
(4)
p=1
II. System Model The system under consideration is depicted in Figure 1. In OFDM systems with multiple antennas, different signals are transmitted from different antennas simultaneously. The received signal at any antenna is a superposition of these signals. At each antenna the data signal is first modulated by the IFFT, and a cyclic prefix of length υ is added. Here υ ≥ L − 1 , where L is the maximum length of all channels. Given Nt transmit antennas, the received signal at the q th receiving antenna on the k th tone can be expressed as
III. Channel Estimation In this section our channel estimation scheme is derived. Let Xp (n) = Sp (n) + Bp (n) , where Sp (n) is some arbitrary K × 1 data vector and Bp (n) is some arbitrary K × 1 training sequence (TS) vector. Substitute for Xp (n) in equation (4) we obtain
[2]
Now define Spd (n) = diag{Sp (n)} and Bpd (n) = diag{Bp (n)}, then equation (5) can be written as follows
ykq (n) =
Nt X
(q,p)
Hk
(n)Xkp (n) + ηk (n)
(1)
Y q (n) =
Nt X
(diag {Sp (n)} + diag {Bp (n)}) F0:L−1 hq,p (n)+Ξ(n)
p=1
(5)
p=1
Yq (n)
for q = 1 . . . Nr and k = 0 . . . K −1. Here Nr is the number of receiving antennas and K is the number of total (q,p) sub-carriers (tones) in the OFDM system. Hk (n) is the frequency response at the k th tone for the pth transmit antenna and the q th receive antenna at time n given by (q,p)
Hk
(n) =
L−1 X
(q,p)
hl
(q,p) h0 (n)
(q,p) hL−1 (n)
iT
, the impulse response from the p transmit antenna to the q th receive an2π tenna at time n. WK = e−j K and Xkp (n) is the symbol transmitted from the pth transmit antenna on the kth tone at time n. Given K sub channels the received time domain vector vector at the q th antenna after removing the cyclic prefix is given by: ...
Bpd (n)F0:L−1 hq,p (n) + Ξ(n)
(6)
p=1
Define
B1d F0:L−1
and
l=0
h
Nt X
+
(2)
Here, h(q,p) (n) is the lth channel tap of hq,p (n) = l
Spd (n)F0:L−1 hq,p (n)
p=1
A= kl (n)WK
Nt X
=
...
t BN d F0:L−1
(7)
b q (n) = A† Y q (n) h
(8)
where the pseudo-inverse matrix A = (A A) AH . Using (6) we can get the following: †
H
−1
th
yq (n) =
Nt X
H p Hq,p cir F X (n) + η(n)
(3)
b q (n) = h
q
h (n) + A
†
Nt X
Spd (n)F0:L−1 hq,p (n)
p=1 †
+A Ξ(n)
(9)
3 hq,1 (n) 7 6 .. hq = 4 5. . hq,Nt (n)
By imposing the following
2
where
condition
p=1
271
A† Spd (n) = 0
(10)
T
.
for p = 1, ..., Nt , we can get q
b (n) = hq (n) + A† Ξ(n) h
(11)
(subject to a constant modulus training sequences). Using a similar argument as in [4] the lower bound on the mean square error is as follows:
q
b (n) is a combination of Equation (11) indicates that h q the true h (n) plus a term affected only by the noise in b q (n)} = hq (n) + A† E{Ξ(n)} = hq (n). the system. Since E{h q b (n) forms an unbiased estimate of In other words h q h (n). H Satisfying (10) implies Brd Spd (n) = 0 for any r = 1 . . . Nt and any p = 1 . . . Nt . For such a scheme where both data and training are impeded in the same OFDM symbol, the only way of satisfying this is by choosing disjoint sets of tones for training and data. i.e. zeros in B(n) where S(n) contains data and vice versa. This allows us to write (9) in simplified form:
e = where A
h
e 1d F e B
M SE(n) ≥
A. Optimal Training Sequences in OFDM MIMO Systems
In this subsection we will derive the optimal training sequences for channel estimation and tracking in OFDM MIMO systems. Let P be the number of pilot tones required for training, with the©necessary condition of identifiability ª P ≥ LNt 1 . Let k0 , k1 , . . . , kP −1 be the set q † eq b e h (n) = A Y (n) (12) of P tones used for training. Assuming K sub-carriers i are used in the system. Rewriting equation (7) p N e = diag {the non entreis (pilots) e te , B of bp } ···
Bd F
d
e e , Y (n) and Ξ(n) , F are the corresponding rows of F0:L−1 , Yq (n) and Ξ(n) respectively, then the channel estimate becomes b q (n) = hq (n) + A e † Ξ(n) e h
(13)
=
2
bq
q
h (n) − h (n)
2 1
e†e
E A Ξ(n)
1 E LNt
H
o
M SE(n) =
σn2 LNt
½³ ´−1 ¾ He e tr A A
eH B e 1dH B e 1d F e F 6 .. e HA e =6 A . 4 H eH B e Nt B e 1d F e F d
Define
i
3 eH B e 1dH B fd Nt F e F 7 .. 7 . 5 H eH B e Nt B e Nt F e F
··· .. . ···
d
H
···
(17)
d
(18)
eH
e . For minimum as the (i, j) L × L sub-matrix of A A mean square error (MMSE) channel estimation we ree HA e to be diagonal i.e.: quire A th
cIL i = j OL×L i = 6 j
(19)
First, we will consider the case i = j in equation (19). H Assuming constant modulus training sequences H e id B e j = IP , we obtain from(18) B d eH F e Cii = F
(20)
The (r, s)th entry of the sub-matrix Cii (denoted by [Cii ]r,s ) can then be written as :
(15)
where tr {·} means trace. In [4], it was shown that for e HA e should be diagonal minimum mean square error A
e Nt F e B d
eH B e id B ejF e Cij = F d
e Ξ e (n) = σ 2 IP , P is the For AWGN where E Ξ(n) n number of pilot tones used in the system. The mean square error can then be written as follows:
···
e = f0 · · · fL−1 , fl = W lk0 W lk1 W lk2 where F 2π e HA e can be written as : and W = e−j K . Then, A
Cij =
LNt n n o o H 1 e † E Ξ(n) e e H (n) A e† tr A Ξ (14) LNt
n
e 1d F e B
2
In this section the mean square error of the channel estimate is derived through which an optimal training sequence and optimal placement of pilot tones are derived. From (13) the mean square error in estimating the channel parameters is given by
=
h
e = A
IV. Channel estimation analysis
=
(16)
where P is the number of pilot tones used in the e HA e is system. The equality obtained if and only if A diagonal.
eq
M SE(n)
σn2 P
[Cii ]r,s = frH fs 1
(21)
This condition can be relaxed. The corresponding analysis is however outside the scope of this paper.
272
W lkP −1
which is equal to
P
PP −1
[Cii ]r,s =
p=0
W
f or r = s f or r 6= s
(22)
kp (s−r)
From (22) the only set of tones that achieves the −1 diagonality of Cii is {kp }Pp=0 such that kp = p0 + pV , where p0 is some reference p0 ∈ 0, . . . , V − 1 with P V (s−r) 0, . . . , L − 1 ∈ Z and V (s−r) ∈ / Z, ∀s, r ∈ K K where Z is the set of integers. For minimum pilot tones and maximum spacing P V = K , or V = K . For cheap, P fast and simple implementation of the DFT, the total number of sub-carriers is chosen to be a power of 2 in practical systems. For this case the minimum number of pilot tones can be selected (keeping in mind that P ≥ LNt ) as P = 2dlog2 LNt e . Note that the set of equispaced tones is the optimal tone set that achieves the first part of equation (19) with the assumption that a constant modulus training sequences are used. We now investigate the conditions imposed by the second part of equation (19). From this point on we assume maximum spaced pilot tones which satisfy the . The first part of equation (19), that is kp = p0 + p K P (r, s)th entry of Cij can be written as
[Cij ]r,s
=
W p0 (s−r)
P −1 h X
∗
e id B ej B d
i p,p
p=0
=
W p0 (s−r)
P −1 h X
∗
e id B ej B d
i p,p
p=0
P −1 h X
∗
e id B ej B d
i
p=0
p,p
H
−L + 1,
nh 1
2π
e−j P
φ
...
2π
e−j K pV (s−r) (23)
e−j
...,
e−j
Equispaced+Equipowered[5] +Orthogonal
(Flat Fading Channel L = 1) Multiple Tx (Multipath Channel of L taps)
2π p(s−r) P
(24)
L−1
2π(P −1) φ P
io
In other words in MIMO systems the training sequences on different transmit antennas must be not
+Phase-Shift Orthogonal −L + 1, . . . , L − 1
shif ts ∈
TABLE I Different Configuration Requirements
only orthogonal but phase shift orthogonal for phase shifts in the range −L+1, . . . , L−1. This phase shift orthogonality in frequency corresponds to a constraint of shift orthogonality in time as well. This is intuitively satisfying since in a system with multipath propagation, the training sequence of one antenna must be orthogonal not only to the training sequence of other antennas but also to their delayed multipath components. For the purpose of comparison we summarize various scenarios and the constraint they impose on the TS in Table I. An optimal training sequence that achieves all of the above (orthogonality and phase shift orthogonality) can be designed as follows: ∗
e id B ej B d
where Dφ represents a phase shift φ, that is: Dφ = diag
Single Tx Multiple TX
h
Considering the case where s = r, we see that equation (24) implies the constraint of orthogonality on the training sequences. However, this is not sufficient, since we must also count for the case when s 6= r and s − r ∈ −L + 1, . . . , L − 1 . We can write equation (24) as Bi Dφ Bj = 0 ∀φ ∈
TS Requirement
W −rpV W spV
Substituting V = K/P in equation (23), we obtain [Cij ]r,s = W p0 (s−r)
Configuration
h
e id where B
nij ∈
Z.
i p,p
= e−j
i p,p
2π n p i P
= e−j
2π n p ij P
(25)
and nij = nj − ni where
Hence the second part of (19) is satis-
© ª 0, . . . , L − 1 , ∀i, j ∈ ∈ / Z ∀ s, r ∈ © ª 1, . . . , Nt with i 6= j . One possible choice is © ª ni = (i − 1)L ∀i ∈ 1 . . . Nt .
fied if
nij +s−r P
It is also worth to notice that for an arbitrary constant sequence a of length P , the sequences h i modulus 2π e id B = e−j P ni p . (a)p also result in an optimal trainp,p ing sequence. V. Simulations In the simulations a channel with 8-taps is used, the taps are simulated as independent Rayleigh time varying, correlated in time with the correlation function according to Jakes’ model r(τ ) = J0 (2πfd τ ), where J0 is the zeroth order Bessel function and fd is the Doppler frequency. The entire bandwidth of the channel is 120 KHz, divided into K = 128 sub-channels, and a cyclic prefix of 8 symbols is added to the head of
273
0
10
−1
10
−2
10 BER
each block to eliminate inter block interference (IBI). The OFDM frame duration is 1.13 ms. QPSK signaling is applied. 2 transmit and 4 receive antennas are assumed. Using the estimated channels, a linear Zero-Forcing (ZF) equalizer is used to recover the transmitted data. The performance of the system is measured in terms of the bit error rate (BER) vs. SNR and the mean square error (MSE) of the channel parameters estimation. Doppler frequencies of fd = 5, 20, 40, 100 Hz are assumed.
__ 16 pilot tones every symbol −− all tones every 8th symbol
−3
10
−4
As shown in Figures (2 & 3) the pilot tone-based scheme significantly outperform the full-block based system, for channels with Doppler frequency fd ≥ 20 Hz . While for fd = 5 Hz we see that the pilot-tone based scheme experiences only 2 dB loss in SNR at BER = 10−4 compared to the full-block based training system. Thus, the pilot-tone based training scheme is an effective scheme for tracking rapidly time varying channels.
10
fd=5 fd=20 fd=40 fd=100 −5
10
0
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 2. BER vs. SNR for full-block based scheme and pilot-tone based scheme 0
10
fd=5 fd=20 fd=40 fd=100
−1
10
−2
10
__ 16 pilot tones every symbol th
−− all tones every 8 symbol
−3
10
0
Additionally, in our simulations we evaluate a variety of training sequences. Specifically constant power, equispaced sequences; constant power, equispaced, orthogonal sequences; and constant power, equispaced, phase-shift orthogonal sequences. As shown in Figures (4 & 5) phase shift orthogonal training sequences outperform the orthogonal or random training sequences in terms of both BER and MSE channel estimation. From Figure (4), we can see a 2 dB SNR gain at BER = 10−4 for phase-shift orthogonal training sequences over orthogonal training sequences at Doppler frequency fd = 5 Hz , and 4 dB SNR gain at BER = 2 × 10−3 at fd = 100 Hz . Random training sequences completely failed.
2
MSE
The performance of our system was compared to a conventional system which uses full block training sequences. Both systems can be designed to have equal efficiency, where efficiency is defined as epilot = K−P K for our system, and ef ull = 1 − 1c for the full block system, where c is the period between retransmission of training blocks. For Nt = 2, P = 16 pilot tones were required for the pilot-based training scheme. To obtain the same efficiency in the full-block based training scheme we require c = 8.
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 3. MSE vs. SNR for full-block based scheme and pilot-tone based scheme
VI. Discussion In this paper a channel estimation and tracking method based on pilot tones was proposed for optimum based systems. This technique shows significant improvement over the full block based system in terms of BER, channel tracking and MSE of channel parameter estimation. It is found that for such a scheme to obtain the minimum error bound (16) the following constraints must
274
References
g=1
0
10
−1
10
−2
BER
10
−. Random TS −− Orthogonal TS − Phase−shift Orthogonal TS
−3
10
−4
10
fd=5 fd=20 fd=40 fd=100
−5
10
0
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 4. BER vs. SNR for various TS.
[1] L. Deneire, P. Vandenameele, L Van der Perre, B. Gyselinckx, and M. Engels “A Low Complexity ML Channel Estimator for OFDM” proc. of the IEEE Inter. Conf. on comm, Helsinki, Finland June 11-14, 2001 [2] Y. Li, N. Seshadri, and S. Ariyavisitakul “Channel Estimation for OFDM systems with Transmitter Diversity in Mobile Wireless Channels,” IEEE JSAC, vol. 17, no. 3, March 1999. [3] W. G. Jeon, K. H. Paik, and Y. S. Cho “An Efficient channel estimation technique for OFDM Systems with Transmitter Diversity,” IEEE Inter. Sym. on Personal, Indoor and Mobile, vol. 2, pp. 1246-50, 2000. [4] T. L, Tung, K. Yao, and R.E. Hudson “Channel Estimation and Adaptive Power Allocation for Performance and Capacity Improvement of Multiple -Antenna OFDM Systems,” Third IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, March, 2001. [5] R. Negi and J. Cioffi “Pilot Tone Selection for Channel Estimation in a Mobile OFDM System,”IEEE Tran. on Consumer Electronics, Vol. 44, No. 3, Aug. 1998.
g=1
0
10
fd=5 fd=20 fd=40 fd=100
−1
MSE
10
−. Random TS −− Orthogonal TS __ Phase−shift Orthogonal TS
−2
10
−3
10
0
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 5. MSE vs. SNR for various TS.
be satisfied. One, the pilot tones must be equispaced, equipowered. Two, the training sequences must be orthogonal in flat fading channels and phase-shift or© ª −L + 1, . . . , L − 1 thogonal with shif ts ∈ in multipath channels of L taps. It should be noted that in our discussion so far we have only considered AWGN. In the case of colored noise we predict that the use of pilot-tone hopping will increase system performance. However this is still the subject to further investigation.
275
