Estimation and Compensation of I/Q Imbalance in OFDM Systems

Transcription

1 Estimation and Compensation of I/Q Imbalance in OFDM Systems Kuang-Yu Sung Institute of Communications Engineering ational Tsing Hua University Hsinchu 30013, Taiwan, ROC Chi-chao Chao Institute of Communications Engineering ational Tsing Hua University Hsinchu 30013, Taiwan, ROC Abstract In this paper, we propose two methods to compensate the I/Q imbalance generated by the direct conversion receiver in orthogonal frequency division multiplexing (OFDM) systems In our proposed methods, the channel is considered to be an unknown multipath channel with additive white Gaussian noise Two methods for I/Q imbalance compensation are proposed: the maximum-likelihood estimation (MLE)-based method and the least-square estimation (LSE)-based method Computer simulations are conducted for comparisons of the performance Effects of timing and frequency offsets on performance are also considered We find that the MLE-based method generally performs well and is robust against timing and frequency offsets The LSE-based method performs well at high signal-to-noise ratio and is robust against the timing offset I ITRODUCTIO In orthogonal frequency division multiplexing (OFDM) systems, it is attractive to employ the direct conversion architecture in the radio frequency (RF) module of receivers for low cost and low power consumption, but one of the main problems is I/Q imbalance The imperfect RF circuits cause that the amplitudes in inphase and quadrature oscillators are not the same and the phase shift is not exactly 90 degrees The mismatches of amplitude and phase shift are called gain and phase imbalance Since I/Q imbalance degrades the system performance considerably and the analog circuits are not easily done perfectly, we should compensate the I/Q imbalance by some digital methods in baseband receivers I/Q imbalance compensation can be approached in many ways [1] [8], but most of the previous methods ignored the unknown multipath channel in communication systems The unknown channel is treated in [7] and [8], but the considered case is single carrier and the maximum-likelihood (ML) solution in [8] requires employing a numerical method We assume the multipath channel to be unknown and consider OFDM systems in this paper, and the ML estimation method results in a closed-form solution This paper is organized as follows In Section II, the I/Q imbalance model and signal analysis are presented Section III and IV introduce the two proposed compensation methods, and the effect of frequency offset is considered in Section V Simulation results are shown in Section VI, and conclusion is drawn in Section VII This work was supported by the ational Science Council of the Republic of China under Grant SC 9 13 E multipath channel z (t) RF Fig 1 y(t) o 90 + θ cos(w ct) 1+ ^ y I (t) LPF ADC LPF ^ y (t) Q ADC y^i,i,n ^ y Q,i,n I/Q Imbalance Compensation OFDM receiver architecture with I/Q imbalance ^ y i,n FFT OFDM DeMOD II I/Q IMBALACE MODEL I OFDM SYSTEMS A OFDM Receiver Architecture In the OFDM receiver, we assume that the gain imbalance is, the phase imbalance is θ, and both of them are concentrated totally on the quadrature branch The OFDM receiver architecture with I/Q imbalance is shown in Fig 1 In practical receivers, the I/Q imbalance may be frequency-dependent [] For simplicity, we assume that the I/Q imbalance is frequencyindependent, yet the compensation method may need to be re-performed periodically [3] B Signal Analysis After quadrature demodulation of the received RF signal y(t) without I/Q imbalance, the ideal low-pass filtered inphase and quadrature signals denoted by ỹ I (t) and ỹ Q (t) are given by ỹ I (t) LPF {y(t) cos(πf C t)} {05 x(t) h(t)+z(t)} ỹ Q (t) LPF {y(t) sin(πf C t)} {05 x(t) h(t)+z(t)} x(t) is the transmitted baseband signal, h(t) is the equivalent baseband impulse response of an unknown multipath channel, assumed quasi-static during an entire packet, z(t) is the equivalent baseband additive white Gaussian noise (AWG), f C is the carrier frequency, { } and { } denote the real and imaginary parts, respectively, LPF{ } represents low-pass filtering, and is the linear convolution operation If I/Q imbalance exists, quadrature demodulation and low-pass filtering of the received RF signal y(t) yield the baseband inphase and quadrature components ŷ I (t) and ŷ Q (t) given by ŷ I (t) ỹ I (t) ŷ Q (t) (1+)cosθ ỹ Q (t) (1 + )sinθ ỹ I (t) /05/$000 (c)005 IEEE

2 After analog-to-digital conversion, the ith sample of the nth received OFDM signal with I/Q imbalance denoted by ŷ i,n can be given by ŷ i,n ỹ I,i,n +j [(1+)cosθ ỹ Q,i,n (1+)sinθ ỹ I,i,n ] (1) ỹ I,i,n and ỹ Q,i,n represent the real and imaginary parts of ỹ i,n, which is the ith sample of the nth received OFDM signal without I/Q imbalance By taking the discrete Fourier transform (DFT) of (1), we can obtain γỹ k,n + λỹ k,n γ[ k,n H k +Z k,n ]+λ[ k,nh k+z k,n]() and Ỹ k,n are the DFT values of ŷ i,n and ỹ i,n, respectively, is the number of subcarriers, k,n and Z k,n are the modulation symbol and noise at the kth subcarrier of the nth symbol, respectively, H k is the channel response at the kth subcarrier, denotes complex conjugation, and γ and λ are complex values given by γ 05 {1+(1+)(cos θ j sin θ)} and λ 05 {1 (1 + )(cos θ + j sin θ)} Equation () can then be taken as our I/Q imbalance model in the frequency domain C oise Statistics The time-domain samples z i,n of the AWG for the nth OFDM symbol are independent, identically distributed complex Gaussian random variables, with mean zero and variance σz in real and imaginary parts Then the DFT values {Z k,n } and {Z k,n } are both Gaussian random variables with mean zero and variance σz/ The real parts of Z k,n at different subcarriers are independent, so are the imaginary parts And the real and imaginary parts of Z k,n at the same or different subcarriers are independent Also Z k,n in different data symbols are independent III MAIMUM-LIKELIHOOD ESTIMATIO (MLE)-BASED COMPESATIO METHOD Since the channel is assumed unknown, we need to estimate the channel response first The ML estimation is first used to get the channel estimate as a function of imbalance parameters, and we then proceed to estimate the I/Q imbalance again by ML estimation Finally, the compensation of the received signal can be performed A Channel Estimation If we transmit the same training symbol d k M times, the M received signals at subcarrier k can be expressed as γ(d k H k + Z k,n )+λ(d kh k + Z k,n), n 0,,M 1 Given H k and H k, the received signals Ŷk,n are complex Gaussian random variables with E[ ]γd k H k + λd kh k We can then find the relationships of the received signals at the same or different subcarriers by using the relationships of Z k,n in Section II If the number of subcarriers is even, the results can be summarized as follows k 1 k 0or /: Cov({Ŷ k1,n}, {}) σ z Cov({Ŷ k1,n}, {Ŷ k,n}) (1+) σ z Cov({Ŷ k1,n}, {Ŷ k,n}) (1 + )sinθ σ z k 1 k 60or /: Cov({Ŷ k1,n}, {Ŷ k,n}) 1+(1+) σ z Cov({Ŷ k1,n}, {}) 1+(1+) σ z Cov({Ŷ k1,n}, {Ŷ k,n}) 0 k 1 + k, butk 1 6 / and k 6 /: Cov({Ŷ k1,n}, {}) 1 (1+) σ z Cov({Ŷ k1,n}, {Ŷ k,n}) 1+(1+) σ z Cov({Ŷ k1,n}, {}) (1 + )sinθ σ z Otherwise: Cov({Ŷ k1,n}, {}) 0 Cov({Ŷ k1,n}, {Ŷ k,n}) 0 Cov({Ŷ k1,n}, {}) 0 Similarly, we can show that the received signals corresponding to different data symbols are uncorrelated The log-likelihood function for H k and H k can then be derived as Γ(H k,h k ) M ln(4π [det(k)] 1 ) 1 (y k,n µ k,n ) T K 1 (y k,n µ k,n ) y k,n ({ }, {Ŷ k,n }, { }, {Ŷ k,n }) T, µ k,n (E[{ }], E[{Ŷ k,n }], E[{ }], E[{Ŷ k,n }]) T, and the covariance matrix K is given by K E (y k,n µ k,n )(y k,n µ k,n ) T a b 0 c b a c 0 0 c a b c 0 b a σ z with a[1+(1+) ]/, b[1 (1+) ]/, andc (1+)sinθ The ML estimates of H k and H k are thus the solution to Γ(H k,h k ) {H k } Γ(H k,h k ) {H k } 0 and Γ(H k,h k ) {H k } 0 and Γ(H k,h k ) {H k } 0 (3) 0 (4) By solving (3) and (4), we can obtain the ML estimate of the unknown channel in functions of and θ as {Ĥk} µ1 {Ŷ k,nd k }+µ {Ŷ k,nd k } M d k + µ3 {Ŷ k,nd k }+µ {Ŷ k,nd k } (5) M d k {Ĥk} µ {d k } µ 1 {d k } M d k + µ {Ŷ k,nd k } µ 3 {Ŷ k,nd k } (6) M d k

3 µ secθ/((1 + )), µ tanθ/, andµ 3 05 sec θ/((1 + )) The channel estimates need not be found explicitly; instead, (5) and (6) are substituted into the log-likelihood function for estimation of I/Q imbalance B I/Q Imbalance Estimation and Compensation We proceed to estimate the gain and phase imbalance Assume the used subcarriers are 1 to F and F to 1 The log-likelihood function of and θ can be derived as Γ(, θ) FM ln(4π [det(k)] 1 ) 1 F (y k,n µ k,n ) T K 1 (y k,n µ k,n ) The ML estimates of and θ are then given by (ˆ, ˆθ) arg maxγ(, θ) (,θ) which are the solution to Γ(, θ) 0 and θ Γ(, θ) 0 (7) The channel estimate Ĥ k obtained in (5) and (6) is substituted for the actual channel response H k in Γ(, θ) Let x 1+ and y sinθ; (7) can then be simplified as 1 A x (1 y ) + B 1 x (1 y ) +C y x(1 y ) D (8) A (1+x )y x (1 y ) +B ( 1+x )y x (1 y ) +C 1+y x(1 y D y (9) ) F A Ŷk,n + Ŷ k,n 1 F + Ŷ k,n M B C F F Ŷk,n Ŷ k,n 1 M Ŷk,n Ŷ k,n 1 M F F Ŷ k,n Ŷ k,n D4FM σ z otice that A, B, andc are only related to the received signals Solving (8) and (9) after some complicated calculations, we can obtain the estimates of and θ as ˆ p (A B)/D (A+B D)C /(A+B) D 1 (10) ˆθ sin 1³ C p D/((A+B) (A B) (A+B D)C ) (11) Finally, we can compensate the I/Q imbalance and obtain the compensated received signal ŷi,n 0 by ŷ0 I,i,n 1 0 ŷi,i,n ŷq,i,n 0 tan ˆθ 1/((1+ˆ)cosˆθ) ŷ Q,i,n (1) ŷi,i,n 0 and ŷ0 Q,i,n are the real and imaginary parts of ŷ0 i,n while ŷ I,i,n and ŷ Q,i,n are the real and imaginary parts of ŷ i,n In the MLE-based method, we only need the received signals for calculation of A, B, andc and the SR value for D Then we can obtain the estimates of the gain and phase imbalance by (10) and (11) We do not need to know the training symbols for estimation and compensation of the I/Q imbalance IV LEAST-SQUARE ESTIMATIO (LSE)-BASED COMPESATIO METHOD In this method, the least-square (LS) estimation is used to estimate the channel and then the I/Q imbalance A Channel Estimation Given the training symbols d k,n, n 0, 1,,,, M is assumed at least, we can obtain for each subcarrier k Y 1 D H + 1 (13) Y 1 {Ŷk,0} + {Ŷ k,0} {Ŷk,0} {Ŷ k,0} {Ŷk,} + {Ŷ k,} {Ŷ k, } {Ŷ k, } {d k,0 } {d k,0 } {d k,0 } {d k,0 } {d k,0 } {d k,0 } {d k,0 } {d k,0 } D {d k, } {d k, } {d k, } {d k, } {d k, } {d k, } {d k, } {d k, } {Z k,0 } + {Z k,0 } {H k } {Z k,0 } {Z k,0 } {H H k } {H k } and 1 {H k } {Z k, } + {Z k, } {Z k, } {Z k, } As there are no and θ in (13), we can use (13) for channel estimation Since E[ 1 ]0 and Var( 1 )(σ z/) I M, I M is a M M identity matrix, the LS estimate of the channel response is then given by Ĥ (D T D) 1 D T Y 1 (14) B I/Q Imbalance Estimation and Compensation For all subcarriers, we can obtain Y M b + (15) {Ŷ1,0} {Ŷ 1,0} {Ŷ1,0} + {Ŷ 1,0} {Ŷ1,} {Ŷ 1,} {Ŷ 1, } + {Ŷ 1, } Y {Ŷ F,0 } {Ŷ F,0 } {ŶF,0} + {Ŷ F,0} {ŶF,} {Ŷ F,} {ŶF,} + {Ŷ F,}

4 with M M 1,0,1 M 1,0, M 1,0,3 M 1,0,4 M 1,,1 M 1,,3 M F,0,1 M F,0,3 M F,,1 M F,,3 M 1,, M 1,,4 M F,0, M F,0,4 M F,, M F,,4 M k,n,1 {H kd k,n} {H kd k,n} M k,n, {H k d k,n } {H k d k,n } M k,n,3 {H k d k,n } + {H k d k,n } M k,n,4 {H k d k,n } {H k d k,n } Also b is the imbalance vector defined as b (1 + )cosθ (1 + )sinθ and is a vector consisting of noise only given by (1+) cosθ {Z 1,0 Z 1,0 }+sinθ {Z 1,0 Z 1,0 } cosθ {Z 1,0 +Z 1,0 } sinθ {Z 1,0 +Z 1,0 } cosθ {Z 1, Z 1,}+sinθ {Z 1, Z 1,} cosθ {Z 1,+Z 1,} sinθ {Z 1,+Z 1,} cosθ {Z F,0 Z F,0}+sinθ {Z F,0 Z F,0} cosθ {Z F,0+Z F,0} sinθ {Z F,0+Z F,0} cosθ {Z F, Z F, }+sinθ {Z F, Z F, } cosθ {Z F, +Z F, } sinθ {Z F, +Z F, } The channel estimate Ĥk obtained in (14) is substituted for the actual channel response H k in (15), and then we proceed for I/Q imbalance estimation Since E[ ]0 and Var( ) (1 + ) (σ z/ ) I MF, the LS estimate of the imbalance vector b can be given by ˆb (M T M) 1 M T Y Finally, we can compensate the I/Q imbalance with the estimated imbalance vector ˆb and obtain the I/Q imbalance compensated received signal ŷi,n 0 by (1) V EFFECT OF FREQUECY OFFSET I MLE-BASED COMPESATIO In order to study the effect of the frequency offset, we first simplify A, B, andc obtained in Section III as follows A B C F γz k,n + λz k,n + γz k,n + λzk,n (γz k,n +λz k,n) + (γz k,n +λz k,n) (16) F (γz k,n + λz k,n)(γz k,n + λzk,n) 1 (γz k,n + λz k,n) (γz k,n + λzk,n) (17) M F (γz k,n + λz k,n)(γz k,n + λzk,n) 1 (γz k,n + λz k,n) (γz k,n + λzk,n) (18) M ote that they are functions of noise only If the frequency offset ν exists, the received signal with I/Q imbalance becomes γỹk,n 0 + λỹ k,n 0 (19) Ỹk,n 0 is given by Ỹ 0 k,n d k H k + 1 m0,m6k 1 e jπnν 1 d m H m e j πn(m k+ν) + Z k If the received signal with frequency offset in (19) is used to reduce A, B, andc, we can obtain the same results as in (16) (18) Therefore, the frequency offset does not affect the estimates of and θ in the MLE-based method VI SIMULATIO RESULTS We employ the data format of long training symbols in the IEEE 8011a standard [9] Performance evaluation is carried out by the normalized mean-squared error (MSE) of compensated signals Assume the gain imbalance is 01 and the phase imbalance is 10 Fig shows the MSE of the compensated received signals at different SR on AWG and multipath channels, the multipath channel follows the IEEE 8011 channel model in [10] We find that both methods perform well on AWG channels and multipath channels The MLE-based method performs well at all SR even when the SR is very low The LSE-based method works better as the SR increases and outperforms the MLE-based method at SR higher than Fig 3 shows that the effects of timing and frequency offsets over multipath channels The MLE-based method is robust against the timing and frequency offsets while the LSE-based method is only robust against the timing offset

5 10 0 ormalized MSE of Received Signals at Different SR AWG channel Multipath channel with rms delay spread 50 ns Multipath channel with rms delay spread 100 ns 5 (a) ormalized MSE of Received Signals with Different Gain Imbalance MLE based method LSE based method Gain imbalance (b) ormalized MSE of Received Signals with Different Phase Imbalance 5 MLE based method LSE based method SR () Phase imbalance (degrees) Fig MSE of compensated received signals at different SR ote that the solid lines are for the MLE-based method and the dashdot lines for the LSE-based method Fig 4 MSE of compensated received signals for both methods with different gain or phase imbalance over multipath channels with rms delay spread 100 ns (at SR 15 ) ormalized MSE of Received Signals with Timing and Frequency Offset o timing and frequency offset With timing offset 16 samples With frequency offset 00 khz is stable at different SR, and it is robust to the delay spread, timing offset, and frequency offset The LSE-based method performs better as the SR increases, and it outperforms the MLE-based method at higher SR The LSE-based method works well with the delay spread and timing offset but poorly with the frequency offset Both of them have high tolerance to different gain and phase imbalance SR () Fig 3 MSE of compensated received signals with timing and frequency offsets at different SR over multipath channels with rms delay spread 100 ns ote that the solid lines are for the MLE-based method and the dashdot lines for the LSE-based method In Figs and 3,we can find that the MLE-based method is insensitive to different SR We can find the expectations of A, B, andc given by E[A] 1+(1+) F (M 1)/SR E[B] 1 (1 + ) F (M 1)/SR E[C] (1 + ) F (M 1)/SR SR F/(σz ) All the expectations of A, B, C, and D are proportional to 1/SR, and the numerator and denominator of the estimates of and θ in (10) and (11) have thesameorderofa, B, C, andd So the estimates of and θ change very slightly as SR changes In Fig 4, we change the gain and phase imbalance to show the performance of these two methods with difference I/Q imbalance We can see that the two proposed methods are insensitive to different gain and phase imbalance VII COCLUSIO In this paper, we have proposed two I/Q imbalance compensation methods: the MLE-based method and the LSE-based method The compensation ability of the MLE-based method REFERECES [1] M Valkama, M Renfors, and V Koivunen, Advanced methods for I/Q imbalance compensation in communication receivers, IEEE Trans Signal Processing, vol 49, pp , Oct 001 [] M Valkama, M Renfors, and V Koivunen, Compensation of frequency -selective I/Q imbalances in wideband receivers: models and algorithms, in 001 IEEE Third Workshop on Signal Processing Advances in Wireless Communications, Taiwan, Mar 001, pp 4 45 [3] S Fouladifard and H Shafiee, On adaptive cancellation of IQ mismatch in OFDM receivers, in Proc IEEE Int Conf Acoust, Speech, and Signal Processing, Hong Kong, Apr 003, pp IV [4] H Q Mu and Y Peng, An approach to the correction of I and Q imbalance in time domain, in Proc IEEE/CIE Int Conf Signal Processing, Beijing, China, Oct 001, pp [5] J P F Glas, Digital I/Q imbalance compensation in a low-if receiver, in Proc IEEE Globecom, Sydney, Australia, ov 1998, pp [6] G ing, M Shen, and H Liu, Frequency offset and I/Q imbalance compensation for OFDM direct-conversion receivers, in Proc IEEE Int Conf Acoust, Speech, and Signal Processing, Hong Kong, Apr 003, pp IV [7] I H Sohn, E R Jeong, and Y H Lee, Data-aided approach to I/Q mismatch and DC offset compensation in communication receivers, IEEE Commun Lett, vol 6, pp , Dec 00 [8] GGil,IHSohn,YHLee,YISong,andJKPark, JointML estimation of I/Q mismatch, DC offset, carrier frequency, and channel for direct-conversion receivers, in Proc IEEE Vehicular Technology Conf, Jeju, South Korea, Apr 003, pp [9] IEEE 8011, Supplement to IEEE standard for information technology Telecommunications and information exchange between systems Local and metropolitan area networks Specific requirements Part 11: Wireless LA medium access control (MAC) and physical layer (PHY) specifications: High-speed physical layer in the 5 GHz band, Sept 1999 [10] B O Hara and A Petrick, The IEEE 8011 Handbook: A Designer s Companion ew York: IEEE Press, 1999