Low complexity frequency offset estimation in the ... - IEEE Xplore
Low Complexity Frequency Offset Estimation in the Presence of DC Offset Chin Keong Ho, Sumei Sun and Ping He Institute for Infocomm Research, 20 Science Park Road, #02-34/37 Tele Tech Park, Science Park II, Singapore 117674, e-mail: {hock,sunsm,heping}@i2r.a-star.edu.sg Abstract— This paper describes how to perform a low complexity frequency offset estimation in the presence of direct current (DC) offset based on a received periodic preamble. By designing a suitable preamble, we perform frequency and DC offsets estimation concurrently, followed by a simple compensator which reduces the frequency offset error induced by the presence of either a residual or actual DC offset. Its performance is illustrated based on the preamble of wireless LAN system defined in IEEE 802.11a, as well as another preamble design that offers additional performance gain. Compared to the case when no compensator is used, the proposed solution gives superior performance over a wide range of dc offset power in both AWGN and multipath quasi-static channels.
I. I NTRODUCTION The introduction of personal communication services prompted the use of a radio-frequency transceiver of low cost, low power dissipation and small form factor. The use of direct conversion architecture (DCA) [1], [2] allows these features to be realized. However, a DCA results in various detrimental effects, such as DC offset, I/Q mismatch, even-order distortion and flicker noise, to lead to performance degradation if they are not removed. In this paper, we only consider DC offset to be present and assume other effects are negligible. Orthogonal frequency division multiplexing (OFDM) [3] is a widely used modulation technique in multipath channels due to its high spectral efficiency and ability to combat intersymbol interference. However, it is sensitive to frequency offset which is caused by a mismatch between the carrier frequency of the transmitter and receiver. The presence of DC offset increases the difficulty of obtaining an accurate frequency offset estimation that is required for negligible performance degradation of OFDM. In this paper, we introduce the concept of concurrent frequency offset and DC offset estimation. Due to the presence of DC offset, the frequency offset estimation becomes inaccurate. By considering different preamble designs, in which the preamble specified in IEEE 802.11a for the use in wirless LANs [4] is also studied, we greatly simplified the frequency offset estimator implementation. Based on information of the DC offset power, we are able to achieve robust performance with only slight increase in complexity as compared to practical frequency offset estimators such as [5] [6] even in the presence of large DC offset. Section II provides the motivation for the work while Section III defines the DC offset model used. In Section IV, we investigate the frequency offset estimator proposed in [5] and
show that DC offset makes the estimation highly inaccurate due to the presence of two terms. The condition when one of the terms is considered negligible is given in Section V. Based on this condition, we develop several heuristics rules to make the undesirable term very small by a proper design of the preamble in Section VI. Based on a frequency estimator cum compensator that we propose in Section VII, simulation and analysis are then provided in Section VIII for AWGN and multipath channels. A conclusion is given in Section IX. II. M OTIVATION Many methods of estimating and eliminating the DC offset are proposed [1], [2]. In Fig. 1, a digital DC offset estimation method, based on information from the initial reception of the preamble, is first used to estimate the DC offset. The receiver then removes the DC offset, and estimates the frequency offset. We call this process Scheme A. Due to the limited time available to process the preamble (without incurring any delay), this scheme poses two problems. Firstly, since the time used for DC offset estimation is short, a residual DC offset may still be present during frequency offset estimation. Secondly, as some of the time is used for DC offset estimation and removal, this reduces the time available for frequency offset estimation, and hence the accuracy is further reduced. Since a residual DC offset may still exist in Scheme A, we re-estimate it concurrently when a frequency offset estimation is carried out; see Scheme B as illustrated in Fig. 2. A post processor then compensates for the effect of the residual DC
dc dc freq. offset offset offset est. est. removal
preamble reception
Fig. 1.
Scheme A: conventional method of estimating frequency offset.
freq. offset freq. offset dc dc est. comp. offset offset est. removal residual dc residual dc offset est. offset removal preamble reception
Fig. 2.
0-7803-7802-4/03/$17.00 © 2003 IEEE
2051
est. : estimation freq. : frequency comp: compensation
post processing
Scheme B: proposed sequential estimation.
offset on the estimation. In this case, the DC offset is estimated and removed twice. Alternatively, to maximize the scarce time resource, Scheme C allocates all the time available for DC and frequency offsets estimation concurrently as shown in Fig. 3. For Schemes B and C to work, we need to develop a compensator to improve the initial frequency offset estimate in the presence of possibly large DC offset. III. M ODEL For application in an OFDM system, we introduce the notation as the relative frequency offset (the ratio of the actual frequency offset fa to the inter-carrier spacing fc ), P as the total number of subcarriers and T = 1/(P fc ) as the sampling interval. The transmitted preamble is defined as x(m, n) with inter-slot index m and intra-slot index n. With each slot consisting of N samples, the sample index is thus defined as k = N m + n. We model the AWGN channel as shown in Fig. 4 with discrete time interval T . First, we introduce the related notations for the transmitted preamble, the complex channel gain A, and as a result of the direct conversion architecture, the DC offset δ, respectively, as: x(m, n) |x(m, n)| exp(jα(m, n)), A |A| exp(jθ) and δ |δ| exp(jβ). This model can be extended to the case of a multipath channel as detailed in Section VIII. The preamble is transmitted through a channel with a frequency offset fa = γ/(2πT ). Thus γ = 2πfa /(P fc ) = 2π/P . The received preamble with AWGN of variance σv2 is where
z(m, n) y(m, n)
and good performance. The estimator will be used to estimate γ or = P2πγ instead of fa . From [5], when the SNR is high and DC offset is absent, this estimator is unbiased, i.e. E[ˆ ] − = 0. After properly adjusting the form given in [5], the variance of the frequency offset estimator can be shown to be 2 P 1 E{(ˆ − E[ˆ ])2 } ≈ σ 2 (M, N ) N 2π (M − 1)2 N SNR (3) For unbiased estimators, the variance of the estimator is the mean square error (MSE) of the estimate. A. In the Presence of DC Offset, without AWGN Based on the model in Section III, we elaborate on the frequency offset estimator used in [5]. We define ryy (m, m + 1; n) as the multiplication of discrete point y(m, n) with its conjugate spaced N samples (or 1 slot) apart in time: ryy (m, m + 1; n) y(m, n)y ∗ (m + 1, n),
and Ryy (M, N ) as the time average of ryy (m, m + 1; n). To simplify the analysis, and for the substitution x(m, n) = x(0, n) to be valid, we only consider the case when N (M −1) terms are used for the averaging. It can be shown that in the absence of AWGN, and assuming that timing synchronization is achieved, we get Ryy (M, N )
= y(m, n) + v(m, n), (1) = Ax(m, n) exp(jγk) + δ (2)
The assumption of a time invariant A and δ made is valid when the coherence time is relatively large as compared to the packet length.
= where
freq. offset est.
freq. offset comp.
dc offset est.
residual dc offset removal
preamble reception
Fig. 3.
post processing
Scheme C: proposed concurrent estimation.
frequency offset complex gain dc-offset exp(jJk) A=|A|exp(jθ) G=|G|exp(jβ)
noise v (m, n)
y (m, n)
x (m, n)
Fig. 4.
Channel model.
z (m, n)
−1 M −2 N 1 ryy (m, m + 1; n) N (M − 1) m=0 n=0 exp(−jN γ) 1 + |δ|2 exp(jN γ) + Ψ(M, N, γ, ψ) (5) ,
N −1 Ψ(M, N, γ, ψ) = 2|Aδ| exp(jN γ/2) N1 n=0 |x(0, n)| M −2 · M1−1 m=0 cos(γ · (N m + n) + α(0, n) + ψ) , (6)
IV. F REQUENCY O FFSET E STIMATION We consider a preamble structure with M repeated slots, each slot consisting of N samples. Such preamble structure is proposed in [4] and has been explored in, for example, [5], [6], [7]. For illustrative purposes, we shall use [5] as the estimator of our choice due to its low complexity, flexibility
(4)
and ψ θ − β + N γ/2. For notational convenience, we shall drop the argument from Ψ unless emphasis is required to be made on the argument. Here, we also normalize the power of the received preamble when DC offset and noise is absent as 1, i.e. N −1 1 |Ax(0, n)|2 = 1. (7) N n=0
The implications are that we assume |A| is known, and more importantly, the values |δ|2 and Ψ are normalized to the received preamble power. In the presence of DC offset, the frequency offset estimate, for |N γ| π assuming that arg[1 + |δ|2 exp(jN γ) +Ψ] N γ, is 1 γ = − arg[Ryy (M, N )] N 1 arg[1 + |δ|2 exp(jN γ) + Ψ]. = γ− (8) N We observe that the last two terms of the argument in (8) cause errors in the frequency offset estimate even in the absence of noise.
2052
B. In the Presence of DC Offset and AWGN The MSE of an estimate can be related to the variance and bias of the estimator as [8] E{[ˆ − ]2 } = E{[ˆ − E(ˆ )]2 } + [E(ˆ ) − ]2 .
(9)
From (8), when DC offset is present, the estimator is biased: E[ˆ ] − = −
P arg(1 + |δ|2 exp(jN γ) + Ψ). 2πN
In order to observe the conditions when Ψ can be made small by some choice of x(0, n), we can re-write (13) as: N −1 1 Ψ (N, γ, ψ ) = exp(jψ ) x(0, n) exp(jγn) , (15) N n=0 where (.) represents the real part of the argument. Using (15), and the identity that in general, for any θ,
(10)
Thus, with the variance given in (3), and if Ψ 1, the MSE can be approximated as
2 P E[( − )2 ] ≈ σ 2 (M, N ) + arg(1 + |δ|2 exp(jN γ)) . 2πN (11) The MSE obtained from simulations in Section VIII shows good agreement with the analytical one. Due to its high sensitivity to DC offset, an alternative solution needs to be sought for the frequency offset estimator.
| [exp(jθ)(a + jb)]| = |a + jb| · | cos(θ + atan(b/a))| (16)
we can show that the bound of Ψ (N, γ, ψ ) is: N −1 1 |Ψ (N, γ, ψ )| ≤ x(0, n) exp(jγn) N n=0
max |Ψ (N, γ, ψ )|. ψ
Assuming that ψ follows a uniform distribution with prob1 for [0, 2π), the mean ability density function (pdf) given as 2π of Ψ (N, γ, ψ ) is Eψ [Ψ (ψ )] N −1 2π 1 1 = x(0, n) exp(jγn) exp(jψ )dψ N 2π 0 n=0
V. VALIDITY OF Ψ 1 We remarked in the previous section that the last two terms of (8) in the argument cause errors in the frequency offset estimate even in the absence of noise. However, assuming that |δ| is known, the frequency offset can then be easily estimated if Ψ 1 as shown in Section VII. Using the identity n−1 nx (n − 1)x x cos +a cos(kx + a) ≡ cosec sin 2 2 2 k=0
for the summation over index m, the terms within the square brackets of (6) becomes Ψ (M, N, γ)Ψ (N, γ, ψ ), where ψ ψ + γN (M − 2)/2 ∈ [0, 2π), and γN (M − 1)γN 1 cosec Ψ (M, N, γ) sin , M −1 2 2 (12) N −1 1 |x(0, n)| cos(γn + α(0, n) + ψ ). Ψ (N, γ, ψ ) N n=0 (13) Thus we have Ψ(M, N, γ, ψ) = 2|Aδ| exp(jN γ/2) · Ψ (M, N, γ) ·Ψ (N, γ, ψ ). A. Bound of Ψ For each received preamble, the set of parameters {A, δ, γ, ψ} can be considered to be fixed. However, when the next packet is received, we have a new set of parameters. Here, we calculate the bound of Ψ and Ψ for any given ψ ∈ [0, 2π) and γ ∈ [−π/N, π/N ). Substituting x = γN 2 and K = M − 1 into the relationship sin(Kx)/ sin(x) ≤ K where K is a positive integer, which can be easily verified, the bound of Ψ (M, N, γ) is obtained as |Ψ (M, N, γ)| ≤ 1.
(14)
(17)
=
0,
(18)
B. Variance of Ψ Next, assuming that ψ is a random variable, and noting that Ψ is independent of ψ , we analyze the mean and variance of Ψ (N, γ, ψ ). The variance can be obtained by using (16) as follows: 2π 1 var [Ψ (ψ )] = |Ψ (ψ )|2 dψ 2π 0 2 N −1 1 ≤ x(0, n) exp(jγn) 2 2N n=0 2
1 = (19) max |Ψ (N, γ, ψ )| , 2 ψ
thus relating the bound of |Ψ | and the variance of Ψ with respect to ψ . Hence, we conclude that if maxψ |Ψ (N, γ, ψ )| can be made close to 0, the assumption that Ψ(M, N, γ, ψ) 1 is valid both in the deterministic and statistical sense. In general, this cannot be achieved even if M, N → ∞ for a arbitrary preamble design. Therefore, we need to explicitly design the preamble such that the bound can be made small. This is done by scrutinizing the function maxψ |Ψ (N, γ, ψ )|. VI. P REAMBLE D ESIGN
We propose some strategies to design the preamble such that Ψ 1 is valid for small values of γ but for different ψ since in practice this is an unknown value. Three conditions, named as the weak, strong and total conditions, are deduced which leads to an increasing validity that Ψ 1 for each and every packet as well as in the statistical sense. In particular, we show that the preamble defined in the standards of IEEE 802.11a [4] satisfies the weak condition.
2053
A. Weak Condition When γn 1 for n = 0, · · · , N − 1, then we may write exp(jγn) ≈ 1, n = 0, · · · , N − 1. Thus, from (17), −1 N 1 (20) max|Ψ (N, γ)| ≈ x(0, n) . N n=0
If the weak condition defined as N −1
x(0, n) = 0
(represented as a row vector) is x2 = x(0, 0)[1, −1]; for N = 4, the preamble is x4 = [x2 , −x2 ]; and in general, the preamble is xN = [xN/2 , −xN/2 ]. Thus, the preamble is properly defined by specifying one of the preamble element, say x(0, 0). For reference, for N = 16, the preamble is x(0, 0) · [1, −1, −1, 1, −1, 1, 1, −1, −1, 1, 1, −1, 1, −1, −1, 1]. We set x(0, 0) = 1 in our simulation. Using Maclaurin’s expansion, we get N −1
(21)
n=0
is satisfied, and γn 1, we may assume that Ψ 1. This condition simply requires the mean of the preamble to be zero, and is satisfied by the short preamble used by [4] for frequency offset estimation. B. Strong Condition Assuming that N is even, we may provide another alternative representation of (17) as 1 (22) |Ψ (N, γ, ψ )| = max ψ N N/2−1 · exp(jγ2n)[x(0, 2n) + x(0, 2n + 1) exp(jγ)] . n=0
Assuming that γ 1 (note that this is a less strict assumption than that used in (20)), x(0, 2n) + x(0, 2n + 1) exp(jγ) ≈ x(0, 2n) + x(0, 2n + 1), which we may set to 0 by choice. Thus, we define the strong condition (which also satisfies the weak condition) as x(0, 2n) = −x(0, 2n + 1), n = 0, · · · , N/2 − 1.
(23)
Observe that if the strong condition is satisfied, (22) becomes max |Ψ (N, γ, ψ )| ψ N/2−1 1 |(1 − exp(jγ))| = exp(jγ2n)x(0, 2n) (24) N n=0
n=0
(1 − exp(jγ2n )) = −
N −1 n=0
(jγ2n − (γ2n )2 /2 + · · ·)
with the absolute value of the lowest order term given as N −1 |γ N · 2 n=0 n | = |γ · 2(N −1)/2 |N = (|γ| N/2)N . 2/N , the lowest order term Thus, noting (25), if |γ| < decreases at a rate of N and thus makes Ψ (N, γ, ψ ) very closeto 0 regardless of the value of ψ. The assumption γ < 2/N is less stringent than that required for the strong condition (which assumes γ 1) unless N is very large. This gives a very strong justification to design the preamble based on the total condition. VII. P ROPOSED C OMPENSATOR With a proper preamble design, Ψ(M, N, γ) 1.We propose one possible compensator here which has low complexity. We note that (5) can also be written as Ryy (M, N )
≈ exp(−jN γ) + |δ|2 ,
(27)
which is estimated using Rzz (m, m + 1; n). The DC offset power is estimated by calculating the power of the averaged received signal. Thus the frequency offset estimator cum compensator is proposed as 2 M −1 N −1 z(m, n) m=0 n=0 1 γ − arg Rzz − . (28) N (M N )2 VIII. P ERFORMANCE AND A NALYSIS
which is close to zero when the assumption γ 1 is made.
We investigate through simulations the effects of the proposed algorithms as modelled in Fig. 4. We define the signalN −1 C. Total Condition to-noise ratio as SNR = N1 n=0 |Ax(0, n)|2 σv2 = 1/σv2 . Repeating the procedure iteratively N = log2 N times in a To satisfy the weak condition, we use the preamble from [4] similar manner as (22), where N is a positive integer, we get which has practical relevance. The strong condition offers too N −1 many degrees of freedom in design and is not used. To satisfy 1 max |Ψ (N, γ, ψ )| = |x(0, 0)| (1 − exp(jγ2n )) , (25) the total condition, we use the sequence defined in Section ψ N n=0 VI-C. We set P = 64, N = 16, SNR=20 dB and a high obtained by subjecting the preamble to a set of conditions frequency offset of = 1. The MSE of are calculated over named collectively as the total condition (which satisfies the 1000 simulations (for different DC offset phase and noise) and plotted against different |δ|2 . strong condition for k = 1) defined as: x(0, k · 2n) = −x(0, k · (2n + 1)), N N1 ∀k = 1, 2, 4, · · · , , ∀n = 0, 1, 2 · · · , − 1.(26) 2 2 k To obtain the preamble which follows the total condition, we may observe the following method: for N = 2, the preamble
A. AWGN Channel We now discuss the results based on the 3 schemes described in Section II. As observed from Fig. 5, since using a compensator always give better MSE performance for the same M , we conclude that Scheme B is always better than
2054
Preamble based on total condition
Preamble based on weak condition (IEEE 802.11a) M=4 M=6 M=8
−10
−10
−15
M=4 M=6 M=8
−10
M=4 M=6 M=8
−20 −25
MSE (dB)
−30
−30
−40
−40
−50
−60 0
0.5
1
2
|δ|
1.5
2
0
0.5
1
1.5
2
|δ|2
2.5
3
3.5
−35 −35 −40
−40
Fig. 5. Original and proposed compensator using preamble satisfying weak and total conditions at SNR=20dB in AWGN channel.
Scheme A. The MSE improvement is substantial, with the exact amount depending on M and the residual DC offset power. The additional complexity required is a compensator and residual DC offset estimator. We also note that using the preamble based on the total condition makes the estimation more robust to DC offset. Next, we consider the case when Scheme C outperforms Scheme B. In [4], out of the 10 short preambles/slots available, the first few slots of the preamble are provided for automatic gain control, antenna selection, etc. We assume that MC slots are provided for DC offset correction and frequency offset estimation. All the slots can be used for estimation by Scheme C, hence M = MC when Scheme C is used. Using Scheme B, and assuming that an equivalent time of 2 slots are used for DC offset estimation/correction, we are left with MB = MC − 2 slots for frequency offset estimation. • Consider the preamble satisfying the weak condition. First, let MC = 6, MB = 4. Assuming that Schemes B has no residual DC offset (most optimistic case), it achieves a MSE of -45dB from Fig. 5. Scheme C, on the other hand, has MSE less than -45dB when the DC offset power is less than 0.4. Hence, the range when Scheme C outperforms Scheme B is |δ|2 Φ = 0.4, where the parameter Φ indicates the highest DC power when Scheme C is preferred over Scheme B. When MC = 8, Φ is approximately 0.1. This makes Scheme C useful for a small DC offset power range. • Consider the preamble satisfying the total condition. From Fig. 5, for MC = 6, MB = 4, Φ ≈ 3; for MC = 8, MB = 6, Φ ≈ 1.5. Hence, Scheme C can still be used even when DC offset power is relatively high. B. Multipath Channels When the maximum delay spread is less than 1 slot (N T ), we insert an additional slot into the preamble, and discard it after reception - in a similar manner as how OFDM handles inter-symbol interference. Thus, for a quasi-static multipath channel the noiseless received signal can be written as y(k) = Ax (k) exp(jγk)+δ, where k = N m+n, x (k) = h(k)∗x(k), and ∗ represents a N -point circular convolution. The AWGN model in Fig. 4 can now be used for multipath channels by replacing x(k) with x (k). To maintain the same SNR, we
−45
proposed compensator
−50 −55
4
Scheme A, M = 4,6,8
−25 −30
−50
proposed compensator −60
−20
−30
−45
proposed compensator
M=4 M=6 M=8
−15
Scheme A, M = 4,6,8
Scheme A, M = 4,6,8
−20
Scheme A, M = 4,6,8
MSE (dB)
−20
Preamble based on total condition
Preamble based on weak condition (IEEE 802.11a)
0
0
proposed compensator
−50 −55 0
0.5
1
2
|δ|
1.5
2
0
0.5
1
1.5
2
2 2.5
|δ|
3
3.5
Fig. 6. Original and proposed compensator using preamble satisfying weak and total conditions at SNR=20dB in multipath channels.
N −1 impose the condition k=0 |h(k)|2 = |A|2 for all realizations of the channel, generated as “Channel A” described in [9]. Since the preambles are not designed for multipath channels, slightly worse simulation results are obtained as shown in Fig. 6. Scheme B is always better than Scheme A, and preamble based on total condition are more robust than weak condition. However, the advantage of Scheme C compared to Scheme B is limited to situations when DC offset power is low. IX. C ONCLUSION We propose a frequency offset estimator and compensator which take into account the presence of DC offset with high relative power. We investigate and propose different preamble designs which allow the implementation of these simple yet robust frequency offset estimators. This method is also applicable to other multi-carrier and single carrier systems. For optimum performance, we propose transmitting a preamble that satisfies the total condition and using Scheme B when DC offset power is expected to be high, and Scheme C when the power is low. R EFERENCES [1] B. Razavi, “Design considerations for direct-conversion receivers”, IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol. 44, no. 6, pp. 428-435, June 1997. [2] A.A Abidi, “Direct-conversion radio transceivers for digital communications”, IEEE Journal of Solid-State Circuits, vol. 30, no. 12, pp. 1399-1410, Dec. 1995. [3] J. A. C. Bingham, “Multicarier modulation for data transmission: an idea whose time has come”, IEEE Commun. Mag., vol. 28, no. 5, pp. 5-14, May 1990. [4] “Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: High-speed physical layer in the 5 GHz band”, IEEE Std 802.11a-1999, Sept. 1999. [5] H. K. Song, Y. H. You, J. H. Paik, and Y. S. Cho, “Frequency-offset synchronization and channel estimation for OFDM-Based transmission”, IEEE Commun. Lett., vol. 4, no. 3, pp. 95-97, March 2000. [6] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction”, IEEE Trans. Commun., vol. 42, pp. 2908 -2914, Oct. 1994. [7] J. Li, G. Liu and G. B. Giannakis, “Carrier frequency offset estimation for OFDM based WLANs”, IEEE Signal Processing Lett., vol. 8, pp. 80-82, March 2001. [8] S. M. Kay Fundamentals of statistical signal processing vol 1: Estimation theory, Upper Saddle River, New Jersey : PTR PrenticeHall, 1993. [9] J. Medbo, H. Hallenberg, J.-E. Berg, “Propagation characteristics at 5 GHz in typical radio-LAN scenarios”, Proc. VTC Spring ’99, vol. 1, pp. 185 - 189, May 1999.
2055
4
I. I NTRODUCTION The introduction of personal communication services prompted the use of a radio-frequency transceiver of low cost, low power dissipation and small form factor. The use of direct conversion architecture (DCA) [1], [2] allows these features to be realized. However, a DCA results in various detrimental effects, such as DC offset, I/Q mismatch, even-order distortion and flicker noise, to lead to performance degradation if they are not removed. In this paper, we only consider DC offset to be present and assume other effects are negligible. Orthogonal frequency division multiplexing (OFDM) [3] is a widely used modulation technique in multipath channels due to its high spectral efficiency and ability to combat intersymbol interference. However, it is sensitive to frequency offset which is caused by a mismatch between the carrier frequency of the transmitter and receiver. The presence of DC offset increases the difficulty of obtaining an accurate frequency offset estimation that is required for negligible performance degradation of OFDM. In this paper, we introduce the concept of concurrent frequency offset and DC offset estimation. Due to the presence of DC offset, the frequency offset estimation becomes inaccurate. By considering different preamble designs, in which the preamble specified in IEEE 802.11a for the use in wirless LANs [4] is also studied, we greatly simplified the frequency offset estimator implementation. Based on information of the DC offset power, we are able to achieve robust performance with only slight increase in complexity as compared to practical frequency offset estimators such as [5] [6] even in the presence of large DC offset. Section II provides the motivation for the work while Section III defines the DC offset model used. In Section IV, we investigate the frequency offset estimator proposed in [5] and
show that DC offset makes the estimation highly inaccurate due to the presence of two terms. The condition when one of the terms is considered negligible is given in Section V. Based on this condition, we develop several heuristics rules to make the undesirable term very small by a proper design of the preamble in Section VI. Based on a frequency estimator cum compensator that we propose in Section VII, simulation and analysis are then provided in Section VIII for AWGN and multipath channels. A conclusion is given in Section IX. II. M OTIVATION Many methods of estimating and eliminating the DC offset are proposed [1], [2]. In Fig. 1, a digital DC offset estimation method, based on information from the initial reception of the preamble, is first used to estimate the DC offset. The receiver then removes the DC offset, and estimates the frequency offset. We call this process Scheme A. Due to the limited time available to process the preamble (without incurring any delay), this scheme poses two problems. Firstly, since the time used for DC offset estimation is short, a residual DC offset may still be present during frequency offset estimation. Secondly, as some of the time is used for DC offset estimation and removal, this reduces the time available for frequency offset estimation, and hence the accuracy is further reduced. Since a residual DC offset may still exist in Scheme A, we re-estimate it concurrently when a frequency offset estimation is carried out; see Scheme B as illustrated in Fig. 2. A post processor then compensates for the effect of the residual DC
dc dc freq. offset offset offset est. est. removal
preamble reception
Fig. 1.
Scheme A: conventional method of estimating frequency offset.
freq. offset freq. offset dc dc est. comp. offset offset est. removal residual dc residual dc offset est. offset removal preamble reception
Fig. 2.
0-7803-7802-4/03/$17.00 © 2003 IEEE
2051
est. : estimation freq. : frequency comp: compensation
post processing
Scheme B: proposed sequential estimation.
offset on the estimation. In this case, the DC offset is estimated and removed twice. Alternatively, to maximize the scarce time resource, Scheme C allocates all the time available for DC and frequency offsets estimation concurrently as shown in Fig. 3. For Schemes B and C to work, we need to develop a compensator to improve the initial frequency offset estimate in the presence of possibly large DC offset. III. M ODEL For application in an OFDM system, we introduce the notation as the relative frequency offset (the ratio of the actual frequency offset fa to the inter-carrier spacing fc ), P as the total number of subcarriers and T = 1/(P fc ) as the sampling interval. The transmitted preamble is defined as x(m, n) with inter-slot index m and intra-slot index n. With each slot consisting of N samples, the sample index is thus defined as k = N m + n. We model the AWGN channel as shown in Fig. 4 with discrete time interval T . First, we introduce the related notations for the transmitted preamble, the complex channel gain A, and as a result of the direct conversion architecture, the DC offset δ, respectively, as: x(m, n) |x(m, n)| exp(jα(m, n)), A |A| exp(jθ) and δ |δ| exp(jβ). This model can be extended to the case of a multipath channel as detailed in Section VIII. The preamble is transmitted through a channel with a frequency offset fa = γ/(2πT ). Thus γ = 2πfa /(P fc ) = 2π/P . The received preamble with AWGN of variance σv2 is where
z(m, n) y(m, n)
and good performance. The estimator will be used to estimate γ or = P2πγ instead of fa . From [5], when the SNR is high and DC offset is absent, this estimator is unbiased, i.e. E[ˆ ] − = 0. After properly adjusting the form given in [5], the variance of the frequency offset estimator can be shown to be 2 P 1 E{(ˆ − E[ˆ ])2 } ≈ σ 2 (M, N ) N 2π (M − 1)2 N SNR (3) For unbiased estimators, the variance of the estimator is the mean square error (MSE) of the estimate. A. In the Presence of DC Offset, without AWGN Based on the model in Section III, we elaborate on the frequency offset estimator used in [5]. We define ryy (m, m + 1; n) as the multiplication of discrete point y(m, n) with its conjugate spaced N samples (or 1 slot) apart in time: ryy (m, m + 1; n) y(m, n)y ∗ (m + 1, n),
and Ryy (M, N ) as the time average of ryy (m, m + 1; n). To simplify the analysis, and for the substitution x(m, n) = x(0, n) to be valid, we only consider the case when N (M −1) terms are used for the averaging. It can be shown that in the absence of AWGN, and assuming that timing synchronization is achieved, we get Ryy (M, N )
= y(m, n) + v(m, n), (1) = Ax(m, n) exp(jγk) + δ (2)
The assumption of a time invariant A and δ made is valid when the coherence time is relatively large as compared to the packet length.
= where
freq. offset est.
freq. offset comp.
dc offset est.
residual dc offset removal
preamble reception
Fig. 3.
post processing
Scheme C: proposed concurrent estimation.
frequency offset complex gain dc-offset exp(jJk) A=|A|exp(jθ) G=|G|exp(jβ)
noise v (m, n)
y (m, n)
x (m, n)
Fig. 4.
Channel model.
z (m, n)
−1 M −2 N 1 ryy (m, m + 1; n) N (M − 1) m=0 n=0 exp(−jN γ) 1 + |δ|2 exp(jN γ) + Ψ(M, N, γ, ψ) (5) ,
N −1 Ψ(M, N, γ, ψ) = 2|Aδ| exp(jN γ/2) N1 n=0 |x(0, n)| M −2 · M1−1 m=0 cos(γ · (N m + n) + α(0, n) + ψ) , (6)
IV. F REQUENCY O FFSET E STIMATION We consider a preamble structure with M repeated slots, each slot consisting of N samples. Such preamble structure is proposed in [4] and has been explored in, for example, [5], [6], [7]. For illustrative purposes, we shall use [5] as the estimator of our choice due to its low complexity, flexibility
(4)
and ψ θ − β + N γ/2. For notational convenience, we shall drop the argument from Ψ unless emphasis is required to be made on the argument. Here, we also normalize the power of the received preamble when DC offset and noise is absent as 1, i.e. N −1 1 |Ax(0, n)|2 = 1. (7) N n=0
The implications are that we assume |A| is known, and more importantly, the values |δ|2 and Ψ are normalized to the received preamble power. In the presence of DC offset, the frequency offset estimate, for |N γ| π assuming that arg[1 + |δ|2 exp(jN γ) +Ψ] N γ, is 1 γ = − arg[Ryy (M, N )] N 1 arg[1 + |δ|2 exp(jN γ) + Ψ]. = γ− (8) N We observe that the last two terms of the argument in (8) cause errors in the frequency offset estimate even in the absence of noise.
2052
B. In the Presence of DC Offset and AWGN The MSE of an estimate can be related to the variance and bias of the estimator as [8] E{[ˆ − ]2 } = E{[ˆ − E(ˆ )]2 } + [E(ˆ ) − ]2 .
(9)
From (8), when DC offset is present, the estimator is biased: E[ˆ ] − = −
P arg(1 + |δ|2 exp(jN γ) + Ψ). 2πN
In order to observe the conditions when Ψ can be made small by some choice of x(0, n), we can re-write (13) as: N −1 1 Ψ (N, γ, ψ ) = exp(jψ ) x(0, n) exp(jγn) , (15) N n=0 where (.) represents the real part of the argument. Using (15), and the identity that in general, for any θ,
(10)
Thus, with the variance given in (3), and if Ψ 1, the MSE can be approximated as
2 P E[( − )2 ] ≈ σ 2 (M, N ) + arg(1 + |δ|2 exp(jN γ)) . 2πN (11) The MSE obtained from simulations in Section VIII shows good agreement with the analytical one. Due to its high sensitivity to DC offset, an alternative solution needs to be sought for the frequency offset estimator.
| [exp(jθ)(a + jb)]| = |a + jb| · | cos(θ + atan(b/a))| (16)
we can show that the bound of Ψ (N, γ, ψ ) is: N −1 1 |Ψ (N, γ, ψ )| ≤ x(0, n) exp(jγn) N n=0
max |Ψ (N, γ, ψ )|. ψ
Assuming that ψ follows a uniform distribution with prob1 for [0, 2π), the mean ability density function (pdf) given as 2π of Ψ (N, γ, ψ ) is Eψ [Ψ (ψ )] N −1 2π 1 1 = x(0, n) exp(jγn) exp(jψ )dψ N 2π 0 n=0
V. VALIDITY OF Ψ 1 We remarked in the previous section that the last two terms of (8) in the argument cause errors in the frequency offset estimate even in the absence of noise. However, assuming that |δ| is known, the frequency offset can then be easily estimated if Ψ 1 as shown in Section VII. Using the identity n−1 nx (n − 1)x x cos +a cos(kx + a) ≡ cosec sin 2 2 2 k=0
for the summation over index m, the terms within the square brackets of (6) becomes Ψ (M, N, γ)Ψ (N, γ, ψ ), where ψ ψ + γN (M − 2)/2 ∈ [0, 2π), and γN (M − 1)γN 1 cosec Ψ (M, N, γ) sin , M −1 2 2 (12) N −1 1 |x(0, n)| cos(γn + α(0, n) + ψ ). Ψ (N, γ, ψ ) N n=0 (13) Thus we have Ψ(M, N, γ, ψ) = 2|Aδ| exp(jN γ/2) · Ψ (M, N, γ) ·Ψ (N, γ, ψ ). A. Bound of Ψ For each received preamble, the set of parameters {A, δ, γ, ψ} can be considered to be fixed. However, when the next packet is received, we have a new set of parameters. Here, we calculate the bound of Ψ and Ψ for any given ψ ∈ [0, 2π) and γ ∈ [−π/N, π/N ). Substituting x = γN 2 and K = M − 1 into the relationship sin(Kx)/ sin(x) ≤ K where K is a positive integer, which can be easily verified, the bound of Ψ (M, N, γ) is obtained as |Ψ (M, N, γ)| ≤ 1.
(14)
(17)
=
0,
(18)
B. Variance of Ψ Next, assuming that ψ is a random variable, and noting that Ψ is independent of ψ , we analyze the mean and variance of Ψ (N, γ, ψ ). The variance can be obtained by using (16) as follows: 2π 1 var [Ψ (ψ )] = |Ψ (ψ )|2 dψ 2π 0 2 N −1 1 ≤ x(0, n) exp(jγn) 2 2N n=0 2
1 = (19) max |Ψ (N, γ, ψ )| , 2 ψ
thus relating the bound of |Ψ | and the variance of Ψ with respect to ψ . Hence, we conclude that if maxψ |Ψ (N, γ, ψ )| can be made close to 0, the assumption that Ψ(M, N, γ, ψ) 1 is valid both in the deterministic and statistical sense. In general, this cannot be achieved even if M, N → ∞ for a arbitrary preamble design. Therefore, we need to explicitly design the preamble such that the bound can be made small. This is done by scrutinizing the function maxψ |Ψ (N, γ, ψ )|. VI. P REAMBLE D ESIGN
We propose some strategies to design the preamble such that Ψ 1 is valid for small values of γ but for different ψ since in practice this is an unknown value. Three conditions, named as the weak, strong and total conditions, are deduced which leads to an increasing validity that Ψ 1 for each and every packet as well as in the statistical sense. In particular, we show that the preamble defined in the standards of IEEE 802.11a [4] satisfies the weak condition.
2053
A. Weak Condition When γn 1 for n = 0, · · · , N − 1, then we may write exp(jγn) ≈ 1, n = 0, · · · , N − 1. Thus, from (17), −1 N 1 (20) max|Ψ (N, γ)| ≈ x(0, n) . N n=0
If the weak condition defined as N −1
x(0, n) = 0
(represented as a row vector) is x2 = x(0, 0)[1, −1]; for N = 4, the preamble is x4 = [x2 , −x2 ]; and in general, the preamble is xN = [xN/2 , −xN/2 ]. Thus, the preamble is properly defined by specifying one of the preamble element, say x(0, 0). For reference, for N = 16, the preamble is x(0, 0) · [1, −1, −1, 1, −1, 1, 1, −1, −1, 1, 1, −1, 1, −1, −1, 1]. We set x(0, 0) = 1 in our simulation. Using Maclaurin’s expansion, we get N −1
(21)
n=0
is satisfied, and γn 1, we may assume that Ψ 1. This condition simply requires the mean of the preamble to be zero, and is satisfied by the short preamble used by [4] for frequency offset estimation. B. Strong Condition Assuming that N is even, we may provide another alternative representation of (17) as 1 (22) |Ψ (N, γ, ψ )| = max ψ N N/2−1 · exp(jγ2n)[x(0, 2n) + x(0, 2n + 1) exp(jγ)] . n=0
Assuming that γ 1 (note that this is a less strict assumption than that used in (20)), x(0, 2n) + x(0, 2n + 1) exp(jγ) ≈ x(0, 2n) + x(0, 2n + 1), which we may set to 0 by choice. Thus, we define the strong condition (which also satisfies the weak condition) as x(0, 2n) = −x(0, 2n + 1), n = 0, · · · , N/2 − 1.
(23)
Observe that if the strong condition is satisfied, (22) becomes max |Ψ (N, γ, ψ )| ψ N/2−1 1 |(1 − exp(jγ))| = exp(jγ2n)x(0, 2n) (24) N n=0
n=0
(1 − exp(jγ2n )) = −
N −1 n=0
(jγ2n − (γ2n )2 /2 + · · ·)
with the absolute value of the lowest order term given as N −1 |γ N · 2 n=0 n | = |γ · 2(N −1)/2 |N = (|γ| N/2)N . 2/N , the lowest order term Thus, noting (25), if |γ| < decreases at a rate of N and thus makes Ψ (N, γ, ψ ) very closeto 0 regardless of the value of ψ. The assumption γ < 2/N is less stringent than that required for the strong condition (which assumes γ 1) unless N is very large. This gives a very strong justification to design the preamble based on the total condition. VII. P ROPOSED C OMPENSATOR With a proper preamble design, Ψ(M, N, γ) 1.We propose one possible compensator here which has low complexity. We note that (5) can also be written as Ryy (M, N )
≈ exp(−jN γ) + |δ|2 ,
(27)
which is estimated using Rzz (m, m + 1; n). The DC offset power is estimated by calculating the power of the averaged received signal. Thus the frequency offset estimator cum compensator is proposed as 2 M −1 N −1 z(m, n) m=0 n=0 1 γ − arg Rzz − . (28) N (M N )2 VIII. P ERFORMANCE AND A NALYSIS
which is close to zero when the assumption γ 1 is made.
We investigate through simulations the effects of the proposed algorithms as modelled in Fig. 4. We define the signalN −1 C. Total Condition to-noise ratio as SNR = N1 n=0 |Ax(0, n)|2 σv2 = 1/σv2 . Repeating the procedure iteratively N = log2 N times in a To satisfy the weak condition, we use the preamble from [4] similar manner as (22), where N is a positive integer, we get which has practical relevance. The strong condition offers too N −1 many degrees of freedom in design and is not used. To satisfy 1 max |Ψ (N, γ, ψ )| = |x(0, 0)| (1 − exp(jγ2n )) , (25) the total condition, we use the sequence defined in Section ψ N n=0 VI-C. We set P = 64, N = 16, SNR=20 dB and a high obtained by subjecting the preamble to a set of conditions frequency offset of = 1. The MSE of are calculated over named collectively as the total condition (which satisfies the 1000 simulations (for different DC offset phase and noise) and plotted against different |δ|2 . strong condition for k = 1) defined as: x(0, k · 2n) = −x(0, k · (2n + 1)), N N1 ∀k = 1, 2, 4, · · · , , ∀n = 0, 1, 2 · · · , − 1.(26) 2 2 k To obtain the preamble which follows the total condition, we may observe the following method: for N = 2, the preamble
A. AWGN Channel We now discuss the results based on the 3 schemes described in Section II. As observed from Fig. 5, since using a compensator always give better MSE performance for the same M , we conclude that Scheme B is always better than
2054
Preamble based on total condition
Preamble based on weak condition (IEEE 802.11a) M=4 M=6 M=8
−10
−10
−15
M=4 M=6 M=8
−10
M=4 M=6 M=8
−20 −25
MSE (dB)
−30
−30
−40
−40
−50
−60 0
0.5
1
2
|δ|
1.5
2
0
0.5
1
1.5
2
|δ|2
2.5
3
3.5
−35 −35 −40
−40
Fig. 5. Original and proposed compensator using preamble satisfying weak and total conditions at SNR=20dB in AWGN channel.
Scheme A. The MSE improvement is substantial, with the exact amount depending on M and the residual DC offset power. The additional complexity required is a compensator and residual DC offset estimator. We also note that using the preamble based on the total condition makes the estimation more robust to DC offset. Next, we consider the case when Scheme C outperforms Scheme B. In [4], out of the 10 short preambles/slots available, the first few slots of the preamble are provided for automatic gain control, antenna selection, etc. We assume that MC slots are provided for DC offset correction and frequency offset estimation. All the slots can be used for estimation by Scheme C, hence M = MC when Scheme C is used. Using Scheme B, and assuming that an equivalent time of 2 slots are used for DC offset estimation/correction, we are left with MB = MC − 2 slots for frequency offset estimation. • Consider the preamble satisfying the weak condition. First, let MC = 6, MB = 4. Assuming that Schemes B has no residual DC offset (most optimistic case), it achieves a MSE of -45dB from Fig. 5. Scheme C, on the other hand, has MSE less than -45dB when the DC offset power is less than 0.4. Hence, the range when Scheme C outperforms Scheme B is |δ|2 Φ = 0.4, where the parameter Φ indicates the highest DC power when Scheme C is preferred over Scheme B. When MC = 8, Φ is approximately 0.1. This makes Scheme C useful for a small DC offset power range. • Consider the preamble satisfying the total condition. From Fig. 5, for MC = 6, MB = 4, Φ ≈ 3; for MC = 8, MB = 6, Φ ≈ 1.5. Hence, Scheme C can still be used even when DC offset power is relatively high. B. Multipath Channels When the maximum delay spread is less than 1 slot (N T ), we insert an additional slot into the preamble, and discard it after reception - in a similar manner as how OFDM handles inter-symbol interference. Thus, for a quasi-static multipath channel the noiseless received signal can be written as y(k) = Ax (k) exp(jγk)+δ, where k = N m+n, x (k) = h(k)∗x(k), and ∗ represents a N -point circular convolution. The AWGN model in Fig. 4 can now be used for multipath channels by replacing x(k) with x (k). To maintain the same SNR, we
−45
proposed compensator
−50 −55
4
Scheme A, M = 4,6,8
−25 −30
−50
proposed compensator −60
−20
−30
−45
proposed compensator
M=4 M=6 M=8
−15
Scheme A, M = 4,6,8
Scheme A, M = 4,6,8
−20
Scheme A, M = 4,6,8
MSE (dB)
−20
Preamble based on total condition
Preamble based on weak condition (IEEE 802.11a)
0
0
proposed compensator
−50 −55 0
0.5
1
2
|δ|
1.5
2
0
0.5
1
1.5
2
2 2.5
|δ|
3
3.5
Fig. 6. Original and proposed compensator using preamble satisfying weak and total conditions at SNR=20dB in multipath channels.
N −1 impose the condition k=0 |h(k)|2 = |A|2 for all realizations of the channel, generated as “Channel A” described in [9]. Since the preambles are not designed for multipath channels, slightly worse simulation results are obtained as shown in Fig. 6. Scheme B is always better than Scheme A, and preamble based on total condition are more robust than weak condition. However, the advantage of Scheme C compared to Scheme B is limited to situations when DC offset power is low. IX. C ONCLUSION We propose a frequency offset estimator and compensator which take into account the presence of DC offset with high relative power. We investigate and propose different preamble designs which allow the implementation of these simple yet robust frequency offset estimators. This method is also applicable to other multi-carrier and single carrier systems. For optimum performance, we propose transmitting a preamble that satisfies the total condition and using Scheme B when DC offset power is expected to be high, and Scheme C when the power is low. R EFERENCES [1] B. Razavi, “Design considerations for direct-conversion receivers”, IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol. 44, no. 6, pp. 428-435, June 1997. [2] A.A Abidi, “Direct-conversion radio transceivers for digital communications”, IEEE Journal of Solid-State Circuits, vol. 30, no. 12, pp. 1399-1410, Dec. 1995. [3] J. A. C. Bingham, “Multicarier modulation for data transmission: an idea whose time has come”, IEEE Commun. Mag., vol. 28, no. 5, pp. 5-14, May 1990. [4] “Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: High-speed physical layer in the 5 GHz band”, IEEE Std 802.11a-1999, Sept. 1999. [5] H. K. Song, Y. H. You, J. H. Paik, and Y. S. Cho, “Frequency-offset synchronization and channel estimation for OFDM-Based transmission”, IEEE Commun. Lett., vol. 4, no. 3, pp. 95-97, March 2000. [6] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction”, IEEE Trans. Commun., vol. 42, pp. 2908 -2914, Oct. 1994. [7] J. Li, G. Liu and G. B. Giannakis, “Carrier frequency offset estimation for OFDM based WLANs”, IEEE Signal Processing Lett., vol. 8, pp. 80-82, March 2001. [8] S. M. Kay Fundamentals of statistical signal processing vol 1: Estimation theory, Upper Saddle River, New Jersey : PTR PrenticeHall, 1993. [9] J. Medbo, H. Hallenberg, J.-E. Berg, “Propagation characteristics at 5 GHz in typical radio-LAN scenarios”, Proc. VTC Spring ’99, vol. 1, pp. 185 - 189, May 1999.
2055
4
