quadrature-phase imbalances for orthogonal frequency division multiplexing systems

Computers and Electrical Engineering 48 (2015) 1–11

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Computers and Electrical Engineering journal homepage: www.elsevier.com/locate/compeleceng

Joint estimation of carrier frequency offset and in-phase/quadrature-phase imbalances for orthogonal frequency division multiplexing systems ✩ Yanyan Zhuang, Yi Wan∗ School of Information Science and Engineering, Lanzhou University, China

a r t i c l e

i n f o

Article history: Received 12 March 2014 Revised 13 August 2015 Accepted 13 August 2015

Keywords: Orthogonal frequency division multiplexing Carrier frequency offset In-phase/quadrature-phase imbalance Training sequence Maximum likelihood estimation

a b s t r a c t Orthogonal frequency division multiplexing (OFDM) has been adopted in various high-speed wireless communication systems. OFDM systems using direct-conversion transceivers are gaining much interest because of low cost and power. However, such systems can suffer from carrier frequency offset (CFO) and transmitter (TX)/receiver (RX) in-phase/quadrature-phase (I/Q) imbalances. These impairments may have a tremendous impact on the performance and cannot be efficiently eliminated in the analog domain. In this paper, we propose an efficient joint estimator for CFO and TX/RX I/Q imbalances in the digital baseband domain. The proposed method designs a training sequence at the transmitter that isolates the estimation of TX I/Q imbalance from the other parameters. At the receiver, we form a sinusoidal sequence from the received signal, which is used to jointly estimate the CFO and RX I/Q imbalance iteratively through the maximum likelihood estimation (MLE). Then we estimate the TX I/Q imbalance using the ratio of the estimated sequences just before CFO and RX I/Q imbalance with previously estimated parameters. Simulation results confirm the advantages of the proposed method over existing algorithms in terms of efficiency and estimation accuracy. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Orthogonal frequency division multiplexing (OFDM) [1] has become a key technology of the 4G mobile communication to increase the robustness against frequency selective fading channels and fully use the available frequency spectrum of the highspeed wireless communication systems. It has been accepted for several wireless standards (e.g. DVB-T, IEEE 802.11a/g, IEEE 802.16a, IEEE 802.20, 3GPP LTE) [2–8]. In an OFDM system, the entire channel is divided into many narrow-band orthogonal subcarriers, which are transmitted in parallel to maintain high-data-rate transmission and to increase the symbol duration to resist inter symbol interference (ISI).The most prominent characteristic of OFDM is the orthogonality among subcarriers. However, carrier frequency offset (CFO) can cause symbols or subcarriers to shift, which will destroy the orthogonality of subcarriers. Consequently, OFDM is very sensitive to CFO [9]. An OFDM system with direct-conversion transceiver instead of heterodyne transceiver has the advantage of low cost, low power consumption, small size and easy integration [10]. However, a key problem with direct-conversion transceiver when compared to heterodyne transceiver is that the baseband signals are more severely distorted by CFO and transmitter (TX)/receiver ✩ ∗

Reviews processed and recommended for publication to the Editor-in-Chief by Associate Editor Dr. M. Malek Corresponding author. Tel.: +86 186 9369 6296. E-mail address: [email protected], [email protected] (Y. Wan).

http://dx.doi.org/10.1016/j.compeleceng.2015.08.008 0045-7906/© 2015 Elsevier Ltd. All rights reserved.

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Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

(RX) in-phase/quadrature (I/Q) imbalances. Therefore, a key limitation in implementing wireless systems is the impairments associated with analog processing due to the analog component imperfections and different fabrication processes, which are generally marked by unpredictability and uncontrollability [11]. The impairments include CFO and TX/RX I/Q imbalances. CFO is mainly caused by the mismatch of the local oscillators at the transmitter and receiver. TX/RX I/Q imbalances are mainly caused by the analog processing of radio frequency (RF) signals at the direct-conversion transceiver, which can easily lead to the phase and amplitude mismatch between the In-phase (I) and Quadrature-phase (Q) paths or, equivalently, the real and imaginary parts of the complex signal [12]. These impairments can cause intercarrier interference (ICI), which will decrease the frequency spectral efficiency of the OFDM system and degrades the system performance severely. Furthermore, most of the impairments can be neither efficiently nor entirely eliminated in the analog domain due to power-area-cost tradeoff. Therefore, an efficient estimation and compensation scheme in the digital baseband domain would be highly desirable for the OFDM systems. Recently, the joint estimation of CFO and I/Q imbalance parameters based on the training sequence in OFDM systems has been an active area of research. In [13–15], only CFO and RX I/Q imbalance are taken into account. In [13], the author studied the joint estimation and compensation of CFO and RX I/Q imbalance in the frequency domain and time domain, which assumes that the channel frequency response is known and smooth. The estimation of the RX I/Q imbalance is based on the information of the estimated channel response and the true one. Its major drawback is the high computational complexity. In [14], a low-cost nonlinear least squares (NLS) CFO estimator with training sequence was proposed. The training sequence contains M identical symbols with all the even symbols rotated by π /4. The NLS method defined a maximum likelihood estimator under white Gaussian noise and the method needs a numerical search to find an estimate of CFO. An iterative estimator was proposed in [15] to solve the EM-based optimization problem for the joint estimation of CFO and RX I/Q imbalance. However, the EM estimate of the RX I/Q imbalance is sensitive to different CFO and SNR values. The accuracy of the RX I/Q imbalance estimation is especially reduced at low and high SNR values, because the approximation of the maximum likelihood function is not correct anymore; and at CFO values near zero, because the initial model lacks of information in this case. With the I/Q imbalance at both the transmitter and receiver considered, a joint CFO and TX/RX I/Q parameter estimation approach was proposed for OFDM systems in [16,17]. The CFO and TX/RX I/Q imbalances were jointly estimated by minimizing a new quantity called channel residual energy (CRE). Although the idea is novel with easy optimization formulation, the method in [16,17] incurs a heavy computational cost when minimizing the CRE, and its estimation performance may degrade if high I/Q imbalance is encountered due to its simplified calculation with higher-order term dropped. So far, there is little study on the joint estimation of CFO and TX/RX I/Q imbalances with robust performance. In this paper, we propose a new joint estimation of CFO and TX/RX I/Q imbalances with training sequence under unknown channels. We first carefully design training sequence at the transmitter in order to eliminate the influence of unknown channel and separate the estimation of the TX I/Q imbalance from the rest. At the receiver, we form a sinusoidal sequence from the received signal. We first estimate CFO and RX I/Q imbalance in an iterative manner through the maximum likelihood estimation (MLE) using the sinusoidal sequence. Then, we estimate TX I/Q imbalance using the ratio of the sequence segments obtained from the previous estimation. Simulation results show that the proposed method is able to produce high estimation accuracy and the overall performance is superior to the existing algorithms. The rest of the paper is organized as follows. Section 2 describes a typical OFDM system with direct-conversion transceiver used in the current study. The proposed method for joint estimation of CFO and TX/RX I/Q imbalances is derived in Section 3. In Section 4, computer simulation results are presented to demonstrate the performance of the proposed method. The conclusion is drawn in Section 5. Notation: The superscripts AT , AH and A∗ denote respectively the transpose, the Hermitian, and the complex conjugate of A. The symbols { · } and { · } denote the real part and imaginary part of a complex number. The operator ⊗ denotes the circular convolution. IN is an N × N identity matrix. 2. System model A typical baseband OFDM system with direct-conversion transceiver is shown in Fig. 1. In such a system, the total system bandwidth B is divided into N subcarriers. At the transmitter, assume that the input signal X consisting of QPSK or QAM modulated symbols is an N × 1 vector. We first take the N-point inverse discrete Fourier transform (IDFT) to obtain a discrete baseband OFDM signal S = [s(0), s(1), . . . , s(N − 1)]T , N−1 1  s(n) = √ x(m) exp N m=0





j2π mn , n = 0, 1, . . . , N − 1 N

Fig. 1. A typical baseband OFDM system with direct-conversion transceiver.

(1)

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

3 

Since the TX I/Q imbalance introduces an unwanted image interference, the distorted signal S in the time domain can be modeled as [18,19]

s (n) = μt s(n) + νt s∗ (n), n = 0, 1, . . . , N − 1

(2)

where μt and ν t are TX I/Q imbalance parameters. The TX and RX I/Q imbalance parameters can be formulated as

μi =

1 + i cos φi 1 + i exp (− jφi ) = − 2 2

νi =

1 − i cos φi 1 − i exp ( jφi ) = − 2 2

i sin φi 2

i sin φi

j

(3)

j

2

(4)

where i ∈ {t, r}, t and r stand for transmitter and receiver, respectively. The parameters  i and φ i are, respectively, the amplitude and phase mismatch between the I and Q branches. If no I/Q imbalance is present, i = 1 and φi = 0, which also means μi = 1 and νi = 0. After TX I/Q imbalance results in an image interference, a cyclic prefix (CP) of length Ncp is prepended to prevent intersymbol interference (ISI) before transmitting S through the channel. A finite impulse response (FIR) model with L taps h = [h(0), h(1), . . . , h(L − 1)] with its order L ≤ Ncp is usually assumed for the baseband channel. At the receiver, the received baseband signal R after removing the CP can be written as

r(n) = s (n) ⊗ h(n) + w(n), n = 0, 1, . . . , N − 1

(5)

where w(n) is the zero-mean additive white Gaussian noise (AWGN) with variance Due to CFO, the signal R can be expressed as



r (n) = exp

2π (n + Ncp )θ j N

σ 2.



r(n), n = 0, 1, . . . , N − 1

(6)

where θ is the normalized CFO parameter. Then, the received signal R being distorted by the RX I/Q imbalance is ∗

r (n) = μr r (n) + νr [r (n)] , n = 0, 1, . . . , N − 1

(7)

The matrix form of (7) is given by

R = μr R + νr [R ]∗ = μr ER + νr [ER]∗ where E is an N × N diagonal matrix and given by



E = exp

2π Ncp θ j N







diag 1, exp

2π θ j N

(8)





, . . . , exp

2π (N − 1)θ j N

 (9)

The problem is how to estimate the parameters from the received signal R , even without the knowledge of the channel response h(n). 3. The proposed joint estimation method At the transmitter, we assume the complex signal s(n) = an + jbn , where an and bn are real numbers. Because of the TX I/Q imbalance, the distorted signal s (n) can be written as

s (n) = μt (an + bn j) + νt (an − bn j), n = 0, 1, . . . , N − 1

(10)

Substituting μt and ν t from (3) and (4) into the above equation, we have



s (n) =

1 + t cos φt − 2

t sin φt 2



j



(an + bn j) +

1 − t cos φt − 2

t sin φt 2

 j

(an − bn j)

(11)

From the previous discussion, we hope to design a training sequence that will separate the estimation of TX I/Q imbalance from the rest without knowing the channel information. As a result we suppose that two OFDM blocks S1 = [1, 1, . . . , 1]TN×1 and S2 = [ j, j, . . . , j]TN×1 are available for training sequence. At the receiver, because L ≤ Ncp , we have that {Ri | ri (n) = ri (m), m, n ∈ {0, 1, . . . , N − 1}}, i ∈ {1, 2} in the absence of noise. So

r1 (n) = (μt + νt ) ⊗ h(n) = (1 − t sin φt j) ⊗ h(n) = (1 − t sin φt j)

L−1 

(12)

h(l )

l=0

r2 (n) = (μt j − νt j) ⊗ h(n) = =

(t cos φt j) ⊗ h(n) (t cos φt j)

L−1 

(13)

h(l )

l=0

Then we see that the ratio of r1 and r2 will eliminate the influence of the unknown channel response and allow us to estimate the TX I/Q parameters, if the sequence Ri can be estimated. In the following we show the process of estimating Ri , which also estimates the CFO and RX I/Q imbalance parameters.

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Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

3.1. Estimation of Ri with known θ i In the proposed method, the key component is how to estimate the sequence Ri . We first transform the received sequence Ri to a more suitable form for estimation. From (3) and (4), we have

μr + (νr )∗ = 1

(14)

Using the above equation, we can eliminate RX I/Q imbalance parameters μr and ν r from Ri . We construct the real sequence Zi as





zi (n) = ri (n) + [ri (n)]∗ = 2 exp

 = 2γi cos

2π (n + Ncp )θi + αi N

2π (n + Ncp )θi j N



 ri (n)



(15)

where ri = γi e jαi . The modulus and phase of ri and CFO parameter θ i are the sinusoidal parameters in the above equation. So we can estimate the sequence Ri (as well as θ i ) using the maximum likelihood estimation (MLE) of the amplitude, frequency and phase of the sinusoidal sequence Zi [20]. However, because the range of θ i is θ i ∈ [0, 0.5], which, when divided by N in (15), makes the frequency of sinusoidal sequence close to zero. This will render the inverse of the matrix HT H in (21) inaccurate. Hence, for now we assume that the CFO θ i is known. Since the noise w(n) is complex Gaussian with variance σ 2 , the probability density function (PDF) for = [γi , αi ]T is given as

pi (Zi ; ) =



1

(2π σ 2 ) 2

N

× exp



N−1 −1  zi (n) − 2γi cos 2 2σ n=0

The MLE of is obtained by minimizing N−1 

J( ) =

n=0 N−1 

=







zi (n) − 2γi cos

2π (n + Ncp )θi + αi N



 zi (n) − 2γi cos αi cos

n=0

2π (n + Ncp )θi + αi N

2 

2

2π (n + Ncp )θi N



 + 2γi sin αi sin

2π (n + Ncp )θi N

2 (17)

The matrix form of J( ) is

J( ) = (Zi − HA)T (Zi − HA) where

(18)

  A=

a1 , a1 = 2γi cos αi , a2 = −2γi sin αi a2



⎡

2π Ncp θi N



cos ⎢ ⎢   ⎢ ⎢ cos 2π (Ncp + 1)θi ⎢ N ⎢ H=⎢ . ⎢ .. ⎢ ⎢ ⎢   ⎣ 2π (Ncp + N − 1)θi cos

N

(19)

 sin

2π Ncp θi N



⎤

⎥  ⎥ ⎥ 2π (Ncp + 1)θi ⎥ sin ⎥ N ⎥ ⎥ .. ⎥ ⎥ . ⎥  ⎥ 2π (Ncp + N − 1)θi ⎦ 

sin

(20)

N

So the minimizing solution is [20]

 A = (HT H)−1 HT Zi

(21)

From the above equation, we obtain the estimate of Ri through



γi =

 a22 a21 + 

 i = arctan α i e jαi . then  ri = γ

(22)

2 − a2  a1

 ,

(23)

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

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3.2. CFO estimation Notice that each Si , i ∈ {1, 2}, has N identical symbols. We denote Si as two blocks of equal length and write Si = [STi,a , STi,b ]T , N 2

where Si, {a, b} is an is

× 1 vector and Ri,a = Ri,b , a and b stand for the block index. At the receiver, the received sequences from (8)

Ri,a = μr Ea Ri,a + νr (Ea Ri,a )∗

(24)

Ri,b = μr exp ( jπ θi )Ea Ri,b + νr ( exp ( jπ θi )Ea Ri,b )∗

(25)

then

Ri,a − βr (Ri,a )∗

μr Ea Ri,a =

(26)

1 − |βr |2

exp ( jπ θi )ur Ea Ri,b =

Ri,b − βr (Ri,b )∗

where βr = νr /μ∗r and the matrix Ea of size be combined to estimate CFO

θi =

1

arg

π



(27)

1 − |βr |2

N 2

×

N 2

is the upper left submatrix of E. Since μr Ea Ri,a = μr Ea Ri,b , (26) and (27) can

 (Ri,a − βr (Ri,a )∗ )H (Ri,b − βr (Ri,b )∗ )

(28)

In practice, the modulus of β r is often close to zero [16], so we can approximate (28) to estimate CFO as

θi ≈

1

arg

π



 (Ri,a )H (Ri,b )

(29)

From the estimated CFO θi , the optimal sequence Ri can be obtained from (22) and (23). 3.3. RX I/Q imbalance estimation Once CFO θ i and the sequence Ri have been estimated, we can obtain E in (9) and the sequence Ri . Then, we can estimate RX I/Q imbalance parameters μr and ν r by least squares (LS) estimation from (8)



    −1  T ∗ T ∗ r i μ    ∗ R    , R     = Ri , Ri Ri , Ri R i i i νr i

(30)

r and μ r By taking the average, we obtain the final estimations of μ

1 2

1 2

r = (μ r 1 + μ r 2 ), μ r = (νr 1 + νr 2 ). μ

(31)

3.4. TX I/Q imbalance estimation Assuming the channel to be time-invariant in a short period of the training sequences. The ratio of the received blocks r1 and r2 do not contain the channel information, and only contain TX I/Q imbalance parameters information. The ratio of r1 and r2 is given by

jη = j

r1 = r2

1

t cos φt

−

t sin φt j t cos φt

(32)

from which we get

t cos φt =

1 {− jη} , t sin φt = { jη} { jη}

(33)

t and νt . Substituting (33) into (3) and (4), we can obtain the estimate of TX I/Q imbalance parameters μ 3.5. TX/RX I/Q imbalances estimation when θ = 0 For the proposed method, the inverse of the matrix HT H in (21) does not exist when θi = 0. This will lead to a larger TX/RX I/Q imbalances estimation error. When there is no CFO, we estimate TX/RX I/Q imbalances using an improvement of Chung’s method in [16] as follows. The channel residual energy (CRE) is defined as

 2  R − βr (Ri )∗  −1 ∗ −1 −1 i   CRE = P(I + βt Scir Scir ) Scir 1− | βr |2 

(34)



where Scir is an N × N circular matrix with the first column Si and P = 0 makes the following three approximations on the right hand side of (34)

IN−L−1



. In order to simplify CRE, Chung’s method

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Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

(a) 1− | βr |2 ≈ 1. (b) (I + βt S−1 S∗ )−1 ≈ I − βt S−1 S∗ . cir cir cir cir (c) β t β r ≈ 0. We instead use the following two approximations (a’) 1− | βr |2 ≈ 1. S∗ )−1 ≈ (−βt S−1 )(I − βt S∗cir S−1 )S∗cir + I [21]. (b’) (I + βt S−1 cir cir cir cir Then we get

CRE ≈ ARi − βr A(Ri )∗ − βt BRi + βt2 CRi + βt βr B(Ri )∗ − βt2 βr C(Ri )∗  where A =

PS−1 , cir

B=

PS−1 S∗ S−1 , cir cir cir

 = ( ) 

H

−1

H

C=

2

PS−1 S∗ S−1 S∗ S−1 . cir cir cir cir cir

(35) We can estimate β t and β r by LS estimation from (35)

AR

(36)

i

where

= [βr , βt , βt2 , βt βr , βt2 βr ]T

(37)

 = [A(Ri )∗ , BRi , −CRi , −B(Ri )∗ , C(Ri )∗ ]

(38)

 and β  from . then we can obtain β r t 3.6. Summary of the proposed joint estimation procedure Notice that the accurate estimate of TX/RX I/Q imbalance parameters needs the accurate estimate of Ri , which depends on the accurate estimate of CFO θ , whose accurate estimate in (28) in turn depends on the RX I/Q imbalance β r . So we can generate the final estimates of all the parameters through iteration. Empirically we find that convergence is usually reached through 3 iterations, which we fix throughout the paper. Thus the proposed joint estimation of CFO and TX/RX I/Q imbalances can be summarized as the following six steps: Step 1. Obtain an initial estimate of CFO θ = 12 (θ1 + θ2 ) from (29). If |θ| < 0.01, we estimate TX/RX I/Q imbalance parameters using the improved algorithm derived in Section 3.5; otherwise go through the following steps. Step 2. Obtain the sequence Ri through (22) and (23) by substituting the initial θ into (21).  = ∗r by substituting θ and the sequence Ri into (30). Step 3. Obtain RX I/Q imbalance β νr /μ r  into (28). Step 4. Obtain a new CFO estimate θ = 12 (θ1 + θ2 ) by substituting β r  and most newly Step 5. Repeat steps 2 through 4 for another two times, resulting in the estimates θ, RX I/Q imbalance β r

estimated sequences Ri , i = 1, 2.  = t from (33) using the most newly estimated sequences Ri , . νt /μ Step 6. Obtain TX I/Q imbalance β t 3.7. Computational complexity

The main computational complexity of the proposed method comes from the calculation of (21), (28) and (30), all of which have the complexity of O(N). When θ = 0, the main computation of the proposed method is the estimation of TX/RX I/Q imbalances, and the complexity of the proposed method has the same or approximately with the Chung’s method, which has a complexity of O(Nlog N) in (35). 4. Simulation results In this section, a typical OFDM system with TX/RX I/Q imbalance and CFO values is simulated to examine the performance of the proposed estimator. The basic parameters used in the simulation are OFDM signal length N = 512. The channel is assumed to be a finite impulse response (FIR) filter with order L = Ncp = 64. The channel taps are assumed to be i.i.d. complex Gaussian  E {|h(l )|2 } = 1. As a randomly selected numerical random variables, and the variance of the channel tap is normalized by L−1 l=0 example, the setting for the TX I/Q imbalance parameters was chosen as t = 1.1, φt = −6◦ , so μt = 1.0470 + 0.0575 j, νt = −0.0470 + 0.0575 j; and the RX I/Q imbalance parameters were chosen as r = 1.2, φr = 8◦ , so μr = 1.0942 − 0.0835 j, νr = −0.0942 − 0.0835 j. The normalized CFO parameter is θ ∈ [0, 0.5]. For comparison, we also study Liang’s estimator in [22], Lanante’s estimator in [23] and Chung’s estimator in [16]. Table 1 lists the type and number of the estimated parameters of each algorithm. In an attempt to investigate the performance of the joint estimation of the CFO and TX/RX I/Q imbalances, the mean squared error (MSE) of the estimated parameters is defined as E[(κ −  κ )2 ]. Performance comparison of the four methods listed in Table 1 is given in Figs. 2–9.

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

7

Table 1 The estimated parameters types of each algorithm. Algorithm

CFO

TX I/Q imbalance

RX I/Q imbalance

Liang [22] Lanante [23] Chung [16] Proposed

   

− −  

   

−2

10

Proposed Chung Liang Lanante

−3

10

−4

CFO MSE

10

−5

10

−6

10

−7

10

−8

10

−9

10

0

5

10

15

20 SNR

25

30

35

40

Fig. 2. MSE of CFO for different SNR without TX I/Q imbalance (βt = 0).

−2

10

Proposed Chung Liang Lanante

−3

10

−4

CFO MSE

10

−5

10

−6

10

−7

10

−8

10

−9

10

0

5

10

15

20 SNR

25

Fig. 3. MSE of CFO for different SNR.

30

35

40

8

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11 −2

10

r r

−3

10

RX I/Q Imbalances MSE

r r

Proposed Chung Liang Lanante

−4

10

−5

10

−6

10

−7

10

−8

10

0

5

10

15

20 SNR

25

30

35

40

Fig. 4. MSE of RX I/Q imbalances for different SNR without TX I/Q imbalance (βt = 0). −1

10

t −2

TX/RX I/Q Imbalances MSE

10

r

Proposed Proposed

Chung t Chung r

−3

10

t

Liang Liang

−4

r

−5

Lanante t Lanante r

10

10

−6

10

−7

10

−8

10

0

5

10

15

20 SNR

25

30

35

40

Fig. 5. MSE of TX/RX I/Q imbalances for different SNR.

Notice that because the estimation of TX I/Q imbalance parameters is separate from the estimation of the rest of the parameters in the proposed method, its performance on these parameters is not affected by the presence or the absence of the TX I/Q imbalance. This kind of separation also exists in Chung’s method, only from a different approach. Both estimators from [22] and [23] do not consider TX I/Q imbalance. Thus Figs. 2 and3 are almost the same and so are Figs. 6 and7. Simulation 1: MSE versus SNR The MSE of CFO estimator versus the signal to noise (SNR) curves for the four methods were simulated and shown in Figs. 2 and 3. It can be seen that the performance of the proposed method outperforms the existing methods in [16,22,23], no matter if there exits TX I/Q imbalance. The MSE of TX/RX I/Q imbalances β t and β r versus SNR is shown in Figs. 4 and 5. From Figs. 4 and 5, we see that the proposed joint estimator of TX/RX I/Q imbalances provides a good performance in the SNR range of interest. It can be seen that the TX/RX I/Q imbalances estimation in Chung’s method [16] suffers an error floor under high SNR, which is due to the elimination of higher-order term in calculating the channel estimates. By

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

9

−2

10

Proposed Chung Liang Lanante

−3

10

CFO MSE

−4

10

−5

10

−6

10

−7

10

0

0.1

0.2 0.3 Normalized CFO

0.4

0.5

Fig. 6. MSE of CFO for CFO value without TX I/Q imbalance (βt = 0).

−2

10

Proposed Chung Liang Lanante

−3

10

CFO MSE

−4

10

−5

10

−6

10

−7

10

0

0.1

0.2 0.3 Normalized CFO

0.4

0.5

Fig. 7. MSE of CFO for CFO value.

contrast, the proposed method is very effective, as it produces the estimation which is nearly as accurate as the same estimation in the absence of TX I/Q imbalance, as indicated by Figs. 2–5. The results demonstrate the robustness of the proposed method under the condition of severe I/Q imbalances. Simulation 2: MSE versus the normalized CFO θ The MSE of the CFO estimator is plotted as a function of the normalized CFO θ in Figs. 6 and 7 when SNR = 20 dB. From Figs. 6 and 7, we see that our method can provide a good performance and outperform the Liang’s estimator, Lanante’s estimator and Chung’s estimator in all range of θ , whether TX I/Q imbalance is present. The estimation of CFO and I/Q imbalance from the proposed analysis can be refined by iteration, and the iterative process is one that makes estimation progress through successive refinement. The proposed iterative method can achieve convergence by less iteration numbers. Since RX I/Q imbalance results in image interference, CFO θ as the frequency of a sinusoidal signal is

10

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11 −1

10

r r

−2

10

RX I/Q Imbalances MSE

r r

Proposed Chung Liang Lanante

−3

10

−4

10

−5

10

−6

10

−7

10

0

0.1

0.2 0.3 Normalized CFO

0.4

0.5

Fig. 8. MSE of RX I/Q imbalances for CFO value without TX I/Q imbalance (βt = 0).

−1

10

t r

−2

TX/RX I/Q Imbalances MSE

10

Proposed Proposed

Chung t Chung r

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t r

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t

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r

Liang Liang Lanante Lanante

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0.2 0.3 Normalized CFO

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Fig. 9. MSE of TX/RX I/Q imbalances for CFO value.

estimated. So the Liang’s method and the Lanante’s method are more sensitive to noise which results in poor accuracy when θ → 0. Compared with the Chung’s estimator, the proposed method can increase the estimation accuracy of CFO by almost one order of magnitude, and the proposed method has a lower computing complexity. The MSEs of TX and RX I/Q imbalances β t and β r estimators are plotted as a function of the normalized CFO θ in Figs. 8 and 9 when SNR = 20 dB. Since the TX/RX I/Q imbalances estimation performance depends on the accuracy of CFO estimation for all four algorithms, the proposed method gives a much better estimation of TX/RX I/Q imbalances. Due to the error introduced by matrix inversion when the CFO θ is close to zero, the estimation performance of TX/RX I/Q imbalances tends to fall at a slower rate, which needs for improvement later. But, in general the estimation performance was robust against the normalized CFO values.

Y. Zhuang, Y. Wan / Computers and Electrical Engineering 48 (2015) 1–11

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5. Conclusion In this paper, we propose a novel joint estimator for CFO and TX/RX I/Q imbalances for OFDM systems in the absence of channel knowledge. In the proposed method, two carefully designed OFDM training sequences are first proposed in order to separate the estimation of TX I/Q imbalance parameters from that of the rest. Without the presence of the TX I/Q imbalance parameters, we transform the received signal into a sinusoidal one, whose amplitude, frequency and phase parameters can be estimated through the maximum likelihood estimation (MLE) method. These parameters can then be used to compute the CFO and RX I/Q imbalance parameters. Degenerate cases are also dealt with at this stage. Finally, the ratio of the estimated sequences just before CFO and RX I/Q imbalance is used to estimate TX I/Q imbalance. Simulation results show that the proposed method of joint estimation of CFO and RX I/Q generally is able to produce high estimation accuracy after just 3 iterations. 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In: Proceedings of the 69th IEEE vehicular technology conference (VTC). Barcelona, Spain; 2009. p. 1–5. Yanyan Zhuang received the B.S. degree in Communication Engineering from Lanzhou Jiaotong University, China, in 2011. She received the M.S degree in Signals and Information Processing from Lanzhou University, China, in 2014. Currently, she works in the Shaanxi branch of China Mobile Communication Group Design Institute Co., Ltd. Yi Wan received the B.S. degree in Electrical Engineering from Xian Jiaotong University, China, the M.S. degrees in Electrical Engineering and Mathematics from Michigan State University, East Lansing, in 1997, and the Ph.D. degree in Electrical and Computer Engineering from Rice University, Houston, TX, in 2003. He was with Bell Labs, Murray Hill, NJ, and the National Center for Macromolecular Imaging, Houston. Currently, he is a faculty member at the School of Information Science and Engineering and the Director of the Institute for Signals and Information Processing, Lanzhou University, China.