2.6 Impairments Caused by DCR 23
LPF
LPF
24 OFDM Systems with Direct Conversion Receiver
LNA
Downconversion
LPF
LO
Leak signal
Fig. 2.11. Model of DC offset.
Frequency 0
DCO
Fig. 2.12. Spectrum of DC offset.
2.7 C ONCLUSIONS
In this chapter, we introduced OFDM basis and the analog impairments caused by direct conversion receiver, such as CFO, I/Q imbalance and DCO. Moreover, we analyzed this impairments in detail and expressed by matrix form. In addition, we discussed the ISI problem in OFDM systems without GI, by using matrix form. This will help us to better understand the background knowledge discussed on this thesis.
C H A P T E R 3
Robust CFO Estimation in the Presence of TV-DCO
O
FDM is known for its sensitivity to CFO, which is mainly caused by frequency mismatch between the transmitter and receiver local oscillators. Since the CFO destroys the orthogonality among subcarriers, the resulting inter-carrier interference (ICI) leads to severe performance degradation [41].On the other hand, DCR has attracted a lot of attention in recent years, for its smaller size and lower cost over the traditional superheterodyne receivers. However, the price is the additional disturbances, such as DCO, I/Q imbalance, even-order distortion and flicker noise [4]. Among these, DCO, induced by the self-mixing associated with the unperfect isolation, is known as the most serious problem [2]. Obviously, the coexistence of the CFO and DCO is a critical problem in an OFDM DCR.
In the absence of DCO, the CFO can be estimated easily from the autocorrelation of periodic pilot (PP) [14, 26, 30, 40]. Considering DCO, several joint compensation schemes have been proposed in [7, 12, 15, 37], where DCO is assumed time-invariant.
In practice, automatic gain control (AGC) is usually used to keep the received signal amplitude proper fixed level. Also, a high pass filter (HPF) is often employed in the DCR to reduce DCO [47]. Therefore, the gain shift in the low noise amplifier (LNA) will cause a TV-DCO [38], whose high frequency components may pass through the HPF. As a result, the ordinary CFO estimation in [7, 12, 15, 37] will be corrupted by the residual TV-DCO. Until now, there is only one CFO estimator taking the residual TV-DCO into account [25]. In this scheme, a differential filter is used to eliminate the residual TV-DCO, and then the CFO is estimated by the conventional
autocorrelation-25
26 Robust CFO Estimation in the Presence of TV-DCO
t10
t9
t8 t7 t6 t5 t4
t3
t2
t1 GI2 T1 T2
10×0.8µs=8µs 2×0.8µs+2×3.2µs=8µs
STS LTS
Fig. 3.1. IEEE 802.11a preamble.
based method. Since the differential filter increases the noise variance and the residual TV-DCO cannot be eliminated completely, this method will cause performance loss.
The purpose of this chapter is to develop a novel CFO estimation method in the presence of TV-DCO. As indicated in [25], the residual DCO at the output of the HPF has a linear property. Therefore, it can be approximated by a linear function. On the other hand, it has been shown in [15] that the CFO can be estimated independent of the time-invariant DCO, by exploring the periodicity of the pilot. Motivated by [15] and based on the linear property of the TV-DCO, we propose a PP-aided CFO estimator, which is able to remove the effect of TV-DCO during the CFO estimation process. After the CFO estimation, the residual TV-DCO can be obtained accordingly. Simulations confirm the effectiveness and superiority of the proposed estimator.
3.1 P ROBLEM F ORMULATION
The pilot used for the joint estimation is the preamble of the IEEE 802.11a [1] shown in Fig.3.1. This preamble has a short training sequence (STS) and a long training sequence (LTS). The STS consists of ten identical K-samples repeated symbols t1,· · · ,t10, and LTS consist of two identical N-samples repeated symbols T1 and T2, where N = 64 andK =16. Every four repeated symbols of the STS can be treated as an N-subcarriers OFDM symbol, which has only12equally-spaced subcarriers loaded, see the generation of the STS in [1]. When the channel is time-invariant during the preamble and the channel length is less than K, the received STS keeps its periodicity after discarding the first symbol. This can be understood that the pattern of the loaded subcarrier in the received STS still is equally-spaced. We can partition IFFT matrixFHinto two matrices WandV, corresponding to the loaded and unloaded subcarriers, respectively.
Figure 3.2 shows the mathematical model of the system, wheres(n)denotes thenth
3.2 Conventional Method 27
(DCO) (AWGN)
(CFO) channel
x ( n )
z ( n ) d ( n )
e
j2πεN ns ( n )
h ( n )
r ( n )
Fig. 3.2. Mathematical model.
sample of the preamble after passing through the channel, ε is the CFO normalized to subcarrier spacing, d(n) and z(n) are the possible DCO and additive white Gaussian noise (AWGN), respectively.
In the absence of DCO, the received samplers(n)corresponding to the STS can be written as
rs(n) =ss(n)ej2πεN n+z(n). (3.1) As mentioned above, we havess(n) = ss(n+K)forK≤n <9K. Then the normalized CFO can be estimated by taking the autocorrelation of the STS as
εˆ= 4
2πarg{
∑
8m=2 K−1 k
∑
=0rs∗(k+mK)rs(k+ (m+1)K)}, (3.2) wheremis the index of the repeated symbols in the STS.
In the presence of a time-invariant DCOd, the received signal becomes
rs(n) = ss(n)ej2πεN n+d+z(n). (3.3) In [7], it is known that the CFO estimation in Eq.(3.2) is biased. Hence, several PP-based joint estimation scheme have been proposed in [7, 12, 15, 37].
3.2 C ONVENTIONAL M ETHOD
In practice, in order to keep the proper fixed level of the received signal, AGC circuits are usually employed [38]. In 802.11a WLAN systems, at the middle of STS, the AGC starts to adjust the gain of the received signal, which also changes the gain of DCO.
28 Robust CFO Estimation in the Presence of TV-DCO
The output of HPF
DCO
Time
t10
t9
t8
t7
t6
t5
t4
t3
t2
t1 GI2 T1 T2 GI Signal
r
4r
3STS LTS
⇒The gain of LNA changes from t5
r
2r
1Fig. 3.3. Model of the residual TV-DCO.
On the other hand, a HPF is often used to eliminate DCO [47]. When DCO is time-invariant, a HPF is sufficient to remove the effect of DCO, then enables the CFO estimation in Eq.(3.2). However, the higher frequency components of the AGC-induced TV-DCO may pass through the HPF. In consequence, the CFO estimation will suffer from the residual TV-DCO. In [25], it has been shown that the residual DCO varies in a linear fashion, see the example given in Fig.3.3, where the LNA switches at the beginning of t5. Then, the authors propose to use a differential filter to counteract the impact of the residual TV-DCO.
At the coarse CFO estimation stage, the STS is used, whose nth sample can be written as
rs(n) = ss(n)ej2πnεN +d(n) +z(n), (3.4) whered(n)denotes the residual TV-DCO. The received STS is passed through a differ-ential filter, and thenth sample of the differential filter’s output is given by:
rsd(n) = rs(n)−rs(n−1)
= ssd(n)ej2πnεN +dsd(n) +zsd(n), (3.5) where
ssd(n) = ss(n)−e−j2πεN ss(n−1), dsd(n) = d(n)−d(n−1),
zsd(n) = z(n)−z(n−1).
3.3 Proposed Method 29 Asss(n)is a PP with a period ofK, so doesssd(n). Then the normalized CFO valueεˆ1 is estimated by replacingrs(n)withrsd(n)in Eq.(3.2).
At the fine CFO estimation stage, the obtainedεˆ1is used to compensate the CFO in the LTS, which is also passed through the differential filter. Let rld(n) denote the nth sample of the LTS after the differential filter and coarse CFO compensation. Since the LTS is a PP with a period ofN, the residual normalized CFO is calculated by
εˆ2= 1
2πarg{N
∑
−1n=0
r∗ld(n)rld(n+N)}. (3.6) Finally, the CFO estimate is given byεˆ=εˆ1+εˆ2.
3.3 P ROPOSED M ETHOD
Compared Eq.(3.4) with Eq.(3.5), it is clear that the above mentioned CFO estimation still is biased by dsd(n). Moreover, the noise variance is doubled, which also leads to performance loss. Here, by extending the method in [15], we will derive a novel CFO estimator, which can remove the effect of the TV-DCO completely. Similar to [25], we assume the LNA is switched at the beginning of t5.
At the coarse estimation stage, without loss of generality, we can form two N×1 vectorsr1 andr2 from the received STS symbolst6,t7,· · · ,t10, which are at a distance ofKshown in Fig.3.3. Note thatr1andr2are the symbols corresponding to the sampling range of t6−t9 and t7−t10, respectively. Since the TV-DCO is approximated by a linear function ax+b, a denotes the ratio of variation, whileb stands for the constant component. Then,r1andr2can be expressed as
r1 = Γ(ε)Ss+ax1+b1N +z1, (3.7) r2 = ej2πKεN Γ(ε)Ss+ax2+b1N +z2, (3.8)
x1 = [0, 1, · · · , N−1]T, (3.9)
x2 = [K,K+1,· · · ,K+N−1]T, (3.10) where
Γ(ε) = diag(1,ej2πεN , . . . ,ej2πε(N−1)N ), (3.11) Ss = [sT,sT,sT,sT]T, (3.12) ands= [ss(0),ss(1),· · · ,ss(K−1)]Tis a distortion free STS symbol,znis an AWGN vector. Meanwhile, the initial phase offset of the CFO is incorporated into the channel.
30 Robust CFO Estimation in the Presence of TV-DCO Then, in the absence of AWGN, we should have
ej2πKεN (r1−ax1−b1N) = r2−ax2−b1N. (3.13) Thus, the CFO and DCO can be estimated as
(ε, ˆˆ a, ˆb) = argmin
ε, ˜˜a,˜b
∥ r2−ej2πK˜Nεr1−a˜(x2−ej2πK˜Nεx1)−b˜(1−ej2πK˜Nε)1N ∥2 .(3.14) This equation indicates a least squares problem for three unknown scalar. The optimal solution is given by
ej2πKˆNε aˆ(1−ej2πKˆNε) bˆ(1−ej2πKˆNε) +aKˆ
= [r1x11N]†r2=R. (3.15)
The first element ofRis given as
R(1) = 1
GrH1 Dr2, (3.16)
D={x1Tx11TN1N−xT11N1TNx1}IN +1N1TNx1xT1 −x11TN1Nx1T
+x1x1T1N1TN −1NxT1x11TN, (3.17) whereIN is the N×Nidentity matrix, and
G = rH1 r1x1Tx11TN1N +rH1 x1x1T1N1TNr1 + xT1r1rH1 1N1TNx1−rH1 1N1TNr1x1Tx1
− rH1 r1x1T1N1TNx1−rH1 x1x1Tr11TN1N
= rH1 Dr1. (3.18)
Then we have
r1HDr2 =ejϕ˙SHs D ˙Ss+a˙SHs Dx2+b˙SHs D1N +aejϕxT1D ˙Ss+a2xT1Dx2+abxT1D1N
+bejϕ1TND ˙Ss+ab1TNDx2+b21TND1N+Rz, (3.19) where
Rz =ejϕzH1 D ˙Ss+azH1 Dx2+bzH1 D1N
+zH1 Dz2+ ˙SHs Dz2+axT1Dz2+b1TNDz2, (3.20)
3.3 Proposed Method 31 and ˙Ss =Γ(ε)Ss,ϕ= 2πKεN .
It can be proved that the matrix D is symmetric, D1N = 0N, Dx1 = 0N, and Dx2=0N (see Appendix B for the proof), where0N is anN×1all0vector. Then we have
R(1) = 1
GrH1 Dr2 = 1
G(ejϕ˙SHs D ˙Ss+R˙z), (3.21) where
R˙z =ejϕzH1 D ˙Ss+zH1 Dz2+ ˙SHs Dz2. (3.22) Obviously, inR(1), all terms containing DCO parameters aandb are removed. Conse-quently, we obtain a CFO estimate independent of TV-DCO as
εˆ1= 4
2πarg{R(1)}. (3.23)
At the fine estimation stage,εˆ1 is used to compensate the CFO in the received LTS symbols. Let the vectorsr3andr4express the received two LTS symbols, which can be written as
r3 = Γ(ε)Sl+ax3+b1N +z3, (3.24) r4 = ej2πεΓ(ε)Sl+ax4+b1N +z4, (3.25)
x3 = x1+Q1N, (3.26)
x4 = x3+N1N, (3.27)
where Sl is a distortion-free LTS symbol, and Q is the distance between r1 and r3. Similarly, the initial phase offset of the CFO is absorbed into the channel. After the CFO compensation, we have
˜r3 = Γ(εˆ1)H{Γ(ε)Sl+ax3+b1N +z3}, (3.28)
˜r4 = e−j2πεˆ1Γ(εˆ1)H{ej2πεΓ(ε)Sl+ax4+b1N +z4}. (3.29) Using the same approach at coarse estimation stage, we have
ej2π(εˆ−εˆ1) ae−j2πεˆ1(1−ej2πεˆ) e−j2πˆε1{b(1−ej2πˆε) +aN}
= [˜r3 ˜x3 ˜1N]†˜r4,
where ˜x3 =Γ(εˆ1)Hx3and ˜1N =Γ(εˆ1)H1N. Henceεˆ2=εˆ−εˆ1can be obtained by εˆ2= 1
2πarg{R˜(1)}, (3.30)
32 Robust CFO Estimation in the Presence of TV-DCO
Table 3.1. Computational complexity Proposed Differential
Subtraction - N+K−1
Addition N×(N+1) N Multiplication N×(N+1) N
whereR˜(1)indicates the first element of the3×1vector[˜r3 ˜x3 ˜1N]†˜r4. Finally, the fine CFO estimation is given by
εˆ=εˆ1+εˆ2. (3.31)
On the other hand, the residual TV-DCO can be estimated by usingεˆ. Forεˆ̸=0, the parameteraof the residual TV-DCO can be calculated from Eq.(3.15) as
ˆ
a= R(2) (1−ej2πKˆNε).
(3.32) Then, usingεˆanda, we haveˆ
bˆ = R(3)−aKˆ
(1−ej2πKˆNε). (3.33)
However, this method cannot work when theεˆis close to zero. As mentioned in Section 3.1, since the matrixVconsists of the columns of the IFFT matrix that correspond to the unloaded subcarriers, we knowVHSs =0. In the absence of noise, we have
VHΓ(εˆ)Hr1 =aVHΓ(εˆ)Hx1+bVHΓ(εˆ)H1N. (3.34) Therefore, using least square method, we can obtainaˆandbˆ as
aˆ bˆ
= [A B]†VHΓ(εˆ)Hr1, (3.35)
whereA =VHΓ(εˆ)Hx1andB=VHΓ(εˆ)H1N.
The computational complexity of the coarse estimation is shown in Table.3.1. The similar result can be found for the fine estimation. From the comparison result, we can see that the proposed method needs more calculation than the differential filter method.
3.4 Simulation Results 33
Table 3.2. Simulation setup
Trial number 10000 times
Modulation scheme QPSK+OFDM
Number of subcarrier 64
Number of loaded subcarrier 12(STS), 52(LTS)
Channel Rayleigh fading
HPF 1st order Butterworth
LNA gain 35/15[dB]
Normalized frequency offset -0.5∼0.5
3.4 S IMULATION R ESULTS
We performed simulations to demonstrate the validity of the proposed method. The simulation setup is shown in Table.3.2. The channel is 6-path Rayleigh fading channels.
Since the cyclic prefix (CP) exceeds the maximum channel delay spread, there is no inter symbol interference (ISI). To investigate the effect of Doppler shift fd, fd = 200Hz which corresponds to about 42km/h mobile speed is also considered. The HPF is the first order Butterworth filter, and the cutoff frequencies fc =1, 10, 50, and 100kHz are selected. The normalized CFO value ε is set to be a random value in the range of
−0.5∼0.5. The TV-DCO is generated using the same model of the LNA and the mixer as [25], where the gain of LNA changes between 15 and 35dB, the isolation between LO and LNA input is assumed to be−60dB, and the received signal power is set to be
−53dBm.
The CFO and DCO estimation results are evaluated in terms of mean square error (MSE). The acronyms DMC, DMF, LMC and LMF denote the differential method coarse and fine estimation, and least square method coarse and fine estimation, respectively.
Figures 3.4(a) and (b) show the CFO estimation results, where the cutoff frequency of HPF is fc =10kHz. From these results, it can be observed that whatever the Doppler shift exists or not, not only the coarse estimation but also the fine estimation of proposed method are better than the existing differential filter method. By comparing (a) and (b) in Fig.3.4, we observed the CFO estimation results have no obvious difference. Figure 3.5 shows TV-DCO MSE versus SNR result for fc =10kHz and fd = 200Hz, which
34 Robust CFO Estimation in the Presence of TV-DCO is obtained using the observed TV-DCO and the approximated TV-DCO based on the estimate ofaandb. From Fig. 3.5, it can be seen the estimation is performed accurately.
Figures 3.6(a), (b) and (c) show the simulation results for different cutoff frequencies fc =1, 50, and 100kHz respectively. From these figures, we can find that the proposed estimator performs quite accurately in all of these cases. Then, the similar result can be observed for various cutoff frequencies. Figure 3.7 shows CFO MSE versus variant ε, we can see the proposed method is almost independent of frequency offset. Finally, the BER curve is plotted in Fig.3.8 to show the validity of the proposed method.
3.5 C ONCLUSIONS
Orthogonality among subcarriers is the fundamental of OFDM systems. While CFO occurs, the spectrum of the received signal will be shifted. This will destroy the re-quired orthogonality and result in severe performance degradation. On the other hand, although DCR is very attractive by its small-size, low-cost, and low-power consump-tion, it introduces additional analog impairment DCO. More seriously, in practice, au-tomatic gain control (AGC) results in TV-DCO. Therefore, in OFDM-DCR, the estima-tion/compensation of CFO in the presence of TV-DCO is very crucial.
In this chapter, a novel CFO estimation scheme in the presence of TV-DCO have been proposed. It is shown the residual DCO after high-pass filtering varies in a linear fashion. Based on this observation, we have modeled the residual DCO using a linear function. Then, from the periodicity of the training sequence, we derived a CFO es-timator independent of TV-DCO. Also, the residual TV-DCO was estimated using the obtained CFO. Simulations has been performed to show the effectiveness of the pro-posed method. In the simulation, we have confirmed whatever the Doppler shift exists or not, not only the coarse estimation but also the fine estimation of proposed method are better than the existing differential filter method. Meantime, it has been shown that the proposed CFO estimator is almost independent of frequency offset. Then, TV-DCO estimation result have shown the estimation performed quite accurately. Finally, after compensating CFO and TV-DCO, BER curve has been plotted, where it has been shown the proposed method performs better than conventional method.
3.5 Conclusions 35
5 10 15 20 25 30
10−7 10−6 10−5 10−4 10−3
SNR [dB]
CFO MSE
DMC LMC DMF LMF
(a) fd =0Hz
5 10 15 20 25 30
10−7 10−6 10−5 10−4 10−3
SNR [dB]
CFO MSE
DMC LMC DMF LMF
(b) fd=200Hz
Fig. 3.4. CFO MSE to various fd, fc =10kHz
36 Robust CFO Estimation in the Presence of TV-DCO
5 10 15 20 25 30
10
−910
−810
−710
−610
−5SNR [dB]
TV−DCO MSE
Fig. 3.5. TV-DCO MSE versus SNR, fc =10kHz, fd =200Hz
3.5 Conclusions 37
5 10 15 20 25 30
10−7 10−6 10−5 10−4 10−3
SNR [dB]
CFO MSE
DMF LMF
(a) fc =1kHz
5 10 15 20 25 30
10−7 10−6 10−5 10−4 10−3
SNR [dB]
CFO MSE
DMF LMF
(b) fc =50kHz
5 10 15 20 25 30
10−7 10−6 10−5 10−4 10−3
SNR [dB]
CFO MSE
DMF LMF
(c) fc =100kHz
Fig. 3.6. CFO MSE to various fc, fd =200Hz
38 Robust CFO Estimation in the Presence of TV-DCO
−0.5 0 0.5
10
−610
−5CFO
CFO MSE