# Joint Least Squares Estimation of Frequency, DC Offset, I-Q Imbalance, and Channel in MIMO Receivers

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(3) HSU et al.: JOINT LEAST SQUARES ESTIMATION OF FREQUENCY, DC OFFSET, I-Q IMBALANCE AND CHANNEL. Taking into account the frequency-independent I-Q imbalance at the RF front end, the complex sinusoidal signal coming into the mixer of the jth branch is given by. 2203. 1, θj = 0, and hIj (n) = hQ j (n) = hj (n), we have the no I-Q imbalance case, i.e.,   rj (n) = hj (n) ⊗ yj (n)ej2πνn + w0,j (n) + d0,j .. cj (t) = 2 cos(2πfc t) − j2αj sin(2πfc t + θj ) III. R ECEIVER A RCHITECTURE. = γj e−j2πfc t + ϕj ej2πfc t where αj and θj are the gain and the phase imbalance, respectively, γj = (1 + αj e−jθj ), and ϕj = (1 − αj ejθj ). Note that αj and θj are different from one branch to another. After down conversion, hIj (t) and hQ j (t) are the baseband filters that are used to remove out-of-band noise and high-frequency components. If hIj (t) = hQ j (t), we say that there exists frequencydependent I-Q imbalance. The frequency-dependent I-Q imbalance is mostly encountered in a wide-band RF receiver because it is generally difficult to maintain baseband filters to have the same response over a wide frequency range , . After a simple manipulation, baseband signal rj (t) is given by . rj (t) = rjI (t) + jrjQ (t)   = h+,j (t) ⊗ yj (t)ej2πΔf t + w0,j (t)  ∗ + h−,j (t) ⊗ yj (t)ej2πΔf t + w0,j (t) + d0,j. (1). −jθj ], h−,j (t) = 1/2 · where h+,j (t) = 1/2 · [hIj (t) + hQ j (t)αj e Q Q I jθj I [hj (t) − hj (t)αj e ], d0,j = d0,j + jd0,j is the dc offset, and [x]∗ denotes the complex conjugate of x. After sampling, the end-to-end equivalent discrete system can be modeled as (up to a constant for the case of no aliasing).   rj (n) = h+,j (n) ⊗ yj (n)ej2πνn + w0,j (n)  ∗ + h−,j (n) ⊗ yj (n)ej2πνn + w0,j (n) + d0,j. (2). −jθj ], h−,j (n) = where h+,j (n) = 1/2 · [hIj (n) + hQ j (n)αj e Q I jθj 1/2 · [hj (n) − hj (n)αj e ], ν = Δf TS is the normalized t si (n) ⊗ hj,i (n), and w0,j (n) frequency offset, yj (n) = ni=1 2 . = is the zero-mean additive white Gaussian noise with σw E[|w0,j (n)|2 ], j = 1, . . . , nr . For use in Section V, rj (n) is rewritten as. rj (n) = rj,d (n) + rj,r (n). (3). where rj,d (n) = h+,j (n) ⊗ yj (n)ej2πνn + h−,j (n) ⊗ yj∗ (n)e−j2πνn + d0,j is the deterministic part, and rj,r (n) = ∗ (n) is the random part h+,j (n) ⊗ w0,j (n) + h−,j (n) ⊗ w0,j due to noise. {rj,r (n)} is generally the colored Gaussian noise. Equation (2) says that the effect of the I-Q imbalance introduces self-image interference in the received signal. With hIj (n) = hQ j (n) = hj (n), (2) degenerates to the case of no frequency-dependent I-Q imbalance, with h+,j (n) = 1/2 · γj hj (n) and h−,j (n) = 1/2 · ϕj hj (n). In addition, if αj =. Motivated by (2), one way to process the received signal is to cancel out first the self-image interference due to the I-Q imbalance. By introducing filter ρj (n), we have rj (n)−ρj (n)⊗rj∗ (n)  . = h+,j (n)−ρj (n)⊗h∗−,j (n) ⊗ yj (n)ej2πνn +w0,j (n).  + h−,j (n)−ρj (n)⊗h∗+,j (n).

(4). =0. . j2πνn. ⊗ yj (n)e. ∗  +w0,j (n) + d0,j −ρj (n)⊗d∗0,j . (4). To completely cancel out the self-image interference, ρj (n) = (h∗+,j (n))−1 ⊗ h−,j (n), where (h∗+,j (n))−1 is the inverse filter of h∗+,j (n). [For the case of no frequency-dependent I-Q imbalance, ρj (n) = ϕj /γj∗ as is given in .] Thus, (4) becomes rj (n) − ρj (n) ⊗ rj∗ (n) . = h+,j (n) − ρj (n) ⊗ h∗−,j (n)

(5). =gj (n).  ⊗ yj (n)ej2πνn + w0,j (n) + d0,j − ρj (n) ⊗ d∗0,j  n t  = ej2πνn si (n) ⊗ gj,i (n) + dj + wj (n) (5) . i=1. . . where gj (n) = h+,j (n) − ρj (n) ⊗ h∗−,j (n), gj,i (n) = hj,i (n) ⊗ (gj (n)e−j2πνn ), dj = d0,j − ρj (n) ⊗ d∗0,j , and wj (n) = gj (n) ⊗ w0,j (n). gj,i (n) is the overall impulse response from the ith transmit to the jth receive antenna after canceling out the self-image interference, dj is the equivalent dc offset, and wj (n) is the additive Gaussian noise, but generally not white. In (5), ρj (n), gj,i (n), ν, and dj are the deterministic unknown parameters to be estimated. In what follows, ρj (n) and gj,i (n) will be approximated as FIR filters with large enough taps, although they are generally infinite impulse response ones, as can be seen in (4) and (5). Following (5), we propose the receiver architecture as in Fig. 2; after canceling the self-image interference, the dc offset is compensated next, and then, the compensation for the frequency offset follows. Parameters ρj (n), gj,i (n), ν, and dj will be jointly estimated in the LS sense, which will be discussed in Section IV, and the MIMO detection is done with the MMSE detector based on estimated channel responses {ˆ gj,i (n)} along with the compensated received signals from all branches. Some other types of MIMO detectors can be used as well . IV. J OINT L EAST S QUARES E STIMATION Here, the set of parameters ρj (n), gj,i (n), ν, and dj are jointly estimated in the sense of LS. To do that, ρj (n).

(6) 2204. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009. Fig. 2. MIMO receiver with joint LS estimation/compensation of RF parameters.. and gj,i (n) are approximated by FIR filters ρj = [ρj (0), ρj (1), . . . , ρj (Lρj − 1)]T and gj,i = [gj,i (0), gj,i (1), . . . , gj,i (Lgj,i − 1)]T , with length Lρj and Lgj,i , respectively.1 In the following, Lρj = Lρ , ∀j, and Lgj,i = Lg , ∀i, j. In addition, Lρ and Lg are assumed to be large enough (to contain 99% of the energy) here and in the next sections, and, thus, the approximation error is negligible. The impact of Lρ and Lg will be investigated by computer simulations in Section VI. A. LS Estimators Consider the case of Lg ≤ Ng ; thus, there will be no interblock interference. Let rj (k) = [rj,k (0), rj,k (1), . . . , rj,k (N − 1)]T be the useful part of the kth received block, . where rj,k (n) = rj (k(Ng + N ) + n); let Rj (k) be the N × Lρ received signal matrix with the (m, l)th entry [Rj (k)]m,l = rj,k (m − l) for 0 ≤ m ≤ N − 1 and 0 ≤ l ≤ Lρ − 1; and let Si (k) be the N × Lg signal matrix with [Si (k)]m,l = si,k (m − l) for 0 ≤ m ≤ N − 1 and 0 ≤ l ≤ Lg − 1. From (5), the useful part of the jth received signal can be written as the following vector form: n  t  ∗ rj (k) − Rj (k)ρj = Γk (ν) Si (k)gj,i + dj 1N + wj (k) i=1. (6) j2πk(Ng +N )ν. · for k = 0, . . . , P − 1, where Γk (ν) = e diag{1, ej2πν , . . . , ej2πν(N −1) } is a diagonal matrix, 1N is the vector of ones with dimension N , and wj (k) = [wj,k (0), wj,k (1), . . . , wj,k (N − 1)]T , with wj,k (n) = wj (k(Ng + N ) + n). Furthermore, let rj = [rTj (0), rTj (1), . . . , rTj (P −1)]T , Rj = [RTj (0), RTj (1), . . . , RTj (P − 1)]T , S(k) = [S1 (k), S2 (k), T T T T . . . , Snt (k)], gj = [gj,1 , gj,2 , . . . , gj,n ] , and wj = [wjT (0), t 1 Throughout this paper, bold uppercase letters denote matrices, and bold lowercase letters denote vectors. (·)T and (·)H represent the operations of conjugate transpose and transpose of a matrix or a vector, respectively. In addition, I and 1 denote IN ·P and 1N ·P , respectively.. wjT (1), . . . , wjT (P − 1)]T . The total useful received signal for training is rj − R∗j ρj = Γ(ν)Sgj + dj 1 + wj. (7). where S = [ST (0), ST (1), . . . , ST (P − 1)]T , and Γ(ν) = diag{Γ0 (ν), Γ1 (ν), . . . , ΓP −1 (ν)}. From (7), the joint LS estimates of all parameters are obtained by minimizing the cost function2 ˜ j , d˜j , g ˜j , j = 1, . . . , nr ) = Λ(˜ ν, ρ. nr . ˜ j , d˜j , g ˜j ) (8) Λj (˜ ν, ρ. j=1. with  2   ˜ j − d˜j 1 − Γ(˜ ˜ j , d˜j , g ˜j ) = rj − R∗j ρ ν, ρ ν )S˜ gj  . (9) Λj (˜ Recall that each branch of the receiver has its own I-Q imbalance, dc offset, and channel response; however, the frequency offset is the same for all branches. Therefore, given fixed trial frequency offset ν˜, ρj , dj , and gj can be estimated ˜ j , d˜j , g ˜j ). In other by simply minimizing cost function Λj (˜ ν, ρ words, ρj , dj , and gj can be independently estimated from one branch to another. On the other hand, (8) can be used to jointly estimate the frequency offset to increase the performance by exploiting the diversity and the power gain that are inherent in MIMO systems. The LS solution of (8) or (9) can be successively obtained as ˜ j , d˜j ), the LS estimate follows . First, under fixed trial (˜ ν, ρ ˆj is given by  of channel g   ˜ j , d˜j ) = (Γ(˜ ˜ j − d˜j 1 ˆj (˜ ν, ρ ν )S)† rj − R∗j ρ g. (10). 2 The estimators that jointly maximize (8) are the ML estimators, provided that wj is white Gaussian noise, which is a case that is not true here. The joint ML estimation of ν, ρj , dj , and gj in this case is very complicated if not impossible..

(7) HSU et al.: JOINT LEAST SQUARES ESTIMATION OF FREQUENCY, DC OFFSET, I-Q IMBALANCE AND CHANNEL. 2205. where (X)† denotes the pseudoinverse of X. Using singular value decomposition   Σ 0 X=U VH 0 0. ν ))H (I − C(˜ ν ))(I − 1f H (˜ ν )). Coinwhere Q(˜ ν ) = (I − 1f H (˜ cidentally, in (19), the relevant elements to be calculated are in ν )u, where v = [ v0 v1 · · · vN ·P −1 ]T the form of vH Q(˜ and u = [ u0 u1 · · · uN ·P −1 ]T are N · P × 1 vectors. In addition, from Appendix A. we have. vH Q(˜ ν )u = vH u − vH C(˜ ν )u  H . v 1 − vH C(˜ ν )1 1H u − 1H C(˜ ν )u . − 1H (I − C(˜ ν )) 1. (X)† = V. . Σ−1 0. 0 0.  UH. where Σ = diag(λ1 , λ2 , . . . , λr ), r and {λk } are the rank and singular values of X, respectively, and U and V are some unitary matrices. If X has a full column rank, then (X)† = (XH X)−1 XH . ˆj in (9), we have Second, substituting g  2   ˜ j − d˜j 1) ˜ j , d˜j ) = (I − C(˜ ν )) (rj − R∗j ρ Λj (˜ ν, ρ. Define vT = [v1T , v2T , . . . , vPT ], uT = [uT1 , uT2 , . . . , uTP ], and ⎡. (15). (16). ν ) = [(I − C(˜ ν ))(I − 1f H (˜ ν ))R∗j ]† . Finally, substiwhere Pj (˜ ˆ j in (15) and using (8), we have tuting ρ Λj (˜ ν). BP 2. ···. ⎤ B1P B2P ⎥ .. ⎥ . ⎦ BP P. vH C(˜ ν )u = . P  P . e−j2π(s−r)(N+Ng )˜ν. r=1 s=1 N −1 . × −ηr,s (0)+. . −j2πm˜ ν. ηr,s (m)e. j2πm˜ ν. +ςr,s (m)e.  (21). where ηr,s (m) =. N −1 . ∗ [Brs ]k−m,k vr,k−m us,k. k=m N −1 . ∗ [Brs ]k,k−m νr,k us,k−m .. k=m. ˆ j is given by and the LS estimate ρ. nr . BP 1. ςr,s (m) =. Third, substituting dˆj in (11), one obtains. Λ(˜ ν) =. ··· ···. m=0. . f H (˜ ν) =  ((I − C(˜ ν )) 1)† 1H (I−C(˜ ν )) , if (I − C(˜ ν )) 1|2 = 0 (13) = (I−C(˜ν ))1 2 0T , otherwise.  ˜j . (14) dˆj = f H (˜ ν ) rj − R∗j ρ. ˆ j = Pj (˜ ν )(I − 1f H (˜ ν )) rj ρ. B12 B22 .. .. where vr and us are N × 1 vectors, and Brs is an N × N matrix, with 1 ≤ r and s ≤ P . It can be shown in the following that. In addition, it is easy to show that.  2 . ˜ j ) = (I − C(˜ ˜ j ) ν )) I − 1f H (˜ Λj (˜ ν, ρ ν ) (rj − R∗ ρ. B11 ⎢ B21 B=⎢ ⎣ .... (11). where C(˜ ν ) = Γ(˜ ν )BΓH (˜ ν ), with B = S(SH S)−1 SH . Note that C(˜ ν ) and B are projection matrices. By minimizing (11) ˜ j ), the LS estimate for dj is ν, ρ with respect to d˜j , given fixed (˜ given by.  ˜j . (12) ν )) 1)† (I − C(˜ ν )) rj − R∗j ρ dˆj = ((I − C(˜. (20). (17). xr,k = [xr ]k and x = v, u. Thus, the terms vH C(˜ ν )u, ν )u, and vH C(˜ ν )1 in (20) are in the same form of (21), 1H C(˜ which can be efficiently calculated by using FFT in searching for νˆ. Given νˆ in (18), the other estimates are obtained as follows: . ˆ j = Pj (ˆ ν ) I − 1f H (ˆ ν ) rj (22) ρ.  ˆj (23) ν ) rj − R∗j ρ dˆj = f H (ˆ   ˆj = (Γ(ˆ ˆ j − dˆj 1 . ν )S)† rj − R∗j ρ (24) g. j=1. ˆ j (˜ with Λj (˜ ν ) = (I − C(˜ ν ))(I − 1f H (˜ ν ))(rj − R∗j ρ ν )) 2 , and the LS estimate νˆ is given by νˆ = arg min {Λ(˜ ν )} . ν ˜. (18). Generally, no closed form is available for νˆ; an exhaustive search has to be performed for the solution. From Appendix A, it is shown that Λj (˜ ν ) = rH ν )rj j Q(˜ H  T −1 T  T Rj Q(˜ − Rj Q(˜ ν )rj ν )R∗j Rj Q(˜ ν )rj. (19). B. Low-Complexity Implementation From (14), (16), and (21), it is observed that the calculation of projection matrix B = S(SH S)−1 SH in C(˜ ν) = ν ) plays a key role in determining the complexity Γ(˜ ν )BΓH (˜ ˆ j , and νˆ. Motivated by the design of the estimators of dˆj , ρ for SISO systems in , we design training format S as in (25), shown at the bottom of the next page, where A is an nt Lg × nt Lg full-rank matrix, N = K · nt Lg with K ≥ 1, −1 and {φk }P k=0 are the parameters for optimizing the estimation performance. Example {φk } for K = P = 2 will be given in Section VI; nevertheless, the issue of their optimum design will.

(8) 2206. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009. not be pursued any further in this paper. This way, projection matrix B becomes ⎡ ⎤ F ej(φ0 −φ1 ) F · · · ej(φ0 −φP −1 ) F j(φ −φ ) j(φ −φ ) F · · · e 1 P −1 F⎥ e 1 0 F 1 ⎢ ⎢ ⎥ .. .. .. B= ⎣ ⎦ P ·K . . . j(φP −1 −φ0 ) j(φP −1 −φ1 ) Fe F ··· F e (26) with. ⎡. Int Lg ⎢ Int Lg ⎢ F=⎢ . ⎣ .. Int Lg. Int Lg Int Lg .. . Int Lg. ··· ··· ···. ⎤ Int Lg Int Lg ⎥ ⎥ .. ⎥ . ⎦ Int Lg N ×N. which contains the K 2 matrices of Int Lg . Recall that A is a full-rank square matrix, and, therefore, its projection matrix A(AH A)−1 AH is equal to Int Lg . Since B is a sparse matrix ν ) can be calculated much easily. In now, C(˜ ν ) = Γ(˜ ν )BΓH (˜ ν )u in (21) becomes particular, using (26), vH C(˜ P P  . vHC(˜ ν )u =.  . −j2πmnt Lg ν ˜ j2πmnt Lg ν ˜ ηr,s (m)e +ςr,s (m)e. m=0. (27) where N −1 . ηr,s (m) =. =. ej(φr−1 −φs−1 ) ∗ [F]k−mnt Lg ,k vr,k−mn u t Lg s,k P ·K. k=mnt Lg. =. ej(φr−1 −φs−1 ) P ·K j(φr−1 −φs−1 ). ςr,s (m) =. e. P ·K. N −1 . ∗ vr,k−mn u t Lg s,k. −. \$2 \$ T \$r Q(˜ ν )rj \$ j. rTj Q(˜ ν )r∗j. (30). V. P ERFORMANCE A NALYSIS In this section, the mean and the variance of estimators νˆ, ˆj are analyzed under the conditions of SNR 1 ˆ j , dˆj , and g ρ . 2 . We start with νˆ. From  and N 1, where SNR = σs2 /σw and , using the fact that νˆ is close to true carrier frequency offset ν for SNR 1 and N 1, one has ˙ E{Λ(ν)} ¨ E{Λ(ν)} %& '2 ( ˙ E Λ(ν). E{ˆ ν} ≈ ν − ∗ vr,k us,k−mnt Lg .. k=mnt Lg. Hence, the complexity of the carrier-frequency offset estimation is reduced by about nt Lg times. However, the frequency range that can be estimated is also reduced by the same factor with this design. For SISO systems (nt = nr = 1) with P = 1, (25) degenerates to the one in . Furthermore, for the unde-. (29). νs ) = rTj Q(ˆ νs )rj /rTj Q(ˆ νs )r∗j . It will be shown in with ρˆj (ˆ Section VI that there is no degradation on the BER performance when using the simplified estimators in (28) and (30).. k=mnt Lg N −1 . rH ν )rj j Q(˜. where no matrix inversion is needed, and . dˆj,s (ˆ νs ) = f H (ˆ νs ) rj − r∗j ρˆj (ˆ νs ). ∗ [Brs ]k−mnt Lg ,k vr,k−mn u t Lg s,k. k=mnt Lg N −1 . with. e−j2π(s−r)(N +Ng )˜ν. K−1 . × −ηr,s (0)+. j=1. Λj,s (˜ ν) =. r=1 s=1. . ν ))1 2 |ν˜=0 = sirable case of φ0 = φ1 = · · · = φP −1 , (I − C(˜ 0 in (13). In other words, with this design of the training sequence, it is not able to estimate the dc offset when the frequency offset is zero because dj 1 is now located in the space that is spanned by the column vectors of S, and its effect is ˆj . During the real data transmission, included in the estimate g however, the receiver needs the estimate dˆj for the dc offset compensation. In addition, from the simulation results in Section VI, the frequency-dependent I-Q imbalance has little effect on the estimation of ν and dj , and can be neglected with almost no loss in performance. With this observation, we propose the simplified estimators for the frequency and dc offset by replacing Rj as rj in (14) and (19) as follows, i.e., the case of Lρ = 1 ⎫ ⎧ nr ⎬ ⎨  νˆs = arg min Λs (˜ ν) = Λj,s (˜ ν) (28) ν ˜ ⎩ ⎭. E{ˆ ν − ν)2 } ≈ &  '2 ¨ E Λ(ν). (31). (32). ˙ ¨ where Λ(ν) = ∂Λ(ν)/∂ν, and Λ(ν) = ∂ 2 Λ(ν)/∂ 2 ν. Unfortunately, it is quite cumbersome to evaluate (31) and (32) with. ⎤H. ⎡. ⎥ ⎢ ⎢  jφ H  jφ H  jφ H  jφ H  jφ H ⎥ ⎥ ⎢ jφ0 H e 1 A · · · e 1 A · · · e P −1 A · · · e P −1 A ⎥ S = ⎢ e A · · · e 0A ⎢.

(9).

(10) .

(11) ⎥ ⎦ ⎣ K K K.

(12). K·P. (25).

(13) HSU et al.: JOINT LEAST SQUARES ESTIMATION OF FREQUENCY, DC OFFSET, I-Q IMBALANCE AND CHANNEL. ˆ j , dˆj , and Λ(ν) given in (17), where ν is jointly estimated with ρ ˆj . To simplify the analysis, the effects of the I-Q imbalance g and the dc offset on the frequency estimation will be neglected ν ) = 0 and f (˜ ν) = here, i.e., ρj = 0 and dj = 0. By setting Pj (˜ 0, (17) becomes Λ(˜ ν) =. nr . (I − C(˜ ν )) rj 2. j=1. =. nr  . rj 2 − rH ν )rj j C(˜. . j=1. =. nr  .  ν )BΓH (˜ ν )rj . rj 2 − rH j Γ(˜. It will be shown in Section VI by computer simulations that the simplified analysis is very accurate for the ranges of ρj and dj of practical interest. From Appendix B, it is shown that E{(ˆ ν − ν)} ≈ 0 (an unbiased estimator), and 1 2. nr  j=1. (34). zH j (I − B)zj.   ˆ j (ν) = Pj (ν) I − 1f H (ν) R∗j ρj +Γ(ν)Sgj + dj 1 + wj . ρ (35) In addition, by using (45) and the identities . Pj (ν) I − 1f H (ν) R∗j ρj = ρj . Pj (ν) I − 1f H (ν) 1 dj = 0

(14). Pj (ν) I − 1f H (ν) Γ(ν)Sgj −1 T  = RTj Q(˜ ν )R∗j Rj Q(˜ ν )Γ(ν)S gj = 0.

(15) .  ˆj ρj , ν) = f H (ν) rj − R∗j ρ dˆj (ˆ ˆj ) = f H (ν)1 dj + f H (ν)R∗j (ρj − ρ.

(16) + f H (ν)Γ(ν)S gj + f H (ν)wj.

(17). =0. ˆ j ) + f H (ν)wj . (39) = dj + f (ν)R∗j (ρj − ρ H. Similar to (37), we have E{dˆj − dj |ν} ≈ 0, and under SNR 1, the MSE is given by   E |dˆj − dj |2 |ν . ˆ j ) + f H (ν)wj ≈ E f H (ν)R∗j,d (ρj − ρ H   ˆ j ) + f H (ν)wj × f H (ν)R∗j,d (ρj − ρ ) * ˆ j )(ρj − ρ ˆ j )H |ν RTj,d f (ν) = f H (ν)R∗j,d E (ρj − ρ. where we have used the approximation ) * * ) . ˆ j )wjH |ν = E Pj (ν) I − 1f H (ν) wj wjH E (ρj − ρ . ≈ Pj,d (ν) I − 1f H (ν) Kwj .. =0. one has (36). Following the notation in (3), Rj can be rewritten as Rj = Rj,d + Rj,r , where Rj,d and Rj,r are the deterministic and random parts of Rj , respectively. Taking the expectation on both sides of (36), we have . . where tr{X} denotes the trace of matrix X, and Kwj = E{wj wjH } is the correlation matrix of wj . The explicit expression of Kwj can be found in Appendix C. For the estimator dˆj , from (14). + f H (ν)Kwj f (ν) ) * * ) ˆ j )wjH |ν f (ν) (40) + 2Re f H (ν)R∗j,d E (ρj − ρ. =0. . ˆ j = ρj + Pj (ν) I − 1f H (ν) wj . ρ. (38). =1. where zj = j2πΦSgj , Φ = diag{κ(1), κ(2), . . . , κ(P )}, and κ(p) = [(Ng +N )p+1, (Ng +N )p+2, . . . , (Ng +N )p+N ]. ˆ j , dˆj , and g ˆj with no influence Next, we analyze estimators ρ of the frequency offset; that is, νˆ = ν is assumed in the analysis. It will be shown in Section VI by computer simulations that the analysis predicts the MSE performance very well even when νˆ = ν. For estimator ρj , from (7) and (16), it is easy to show that. . where Pj,d (ν) is the deterministic part of Pj (ν). The approximation is justifiable with SNR 1. Furthermore, the MSE is derived as ) * E |ˆ ρj − ρj |2 |ν ) ) ** ˆ j )(ρj − ρ ˆ j )H |ν = tr E (ρj − ρ   . = tr E Pj (ν) I − 1f H (ν)   H × wj wjH I−1f H (ν) PH j (ν)    . H (ν) ≈ tr Pj,d (ν) I−1f H (ν) Kwj I − 1f H (ν) PH j,d. (33). j=1. * ) 2 E (ˆ ν − ν)2 = σw. 2207. E{ˆ ρj |ν} ≈ ρj + Pj,d (ν) I − 1f H (ν) E{wj } = ρj (37). ˆj Finally, for the estimator g   ˆ j , ν) = (Γ(ν)S)† rj − R∗j ρ ˆj (dˆj , ρ ˆ j − dˆj 1 g  = (Γ(ν)S)† R∗j ρj + Γ(ν)Sgj.  ˆ j − dˆj 1 + dj 1 + wj − R∗j ρ. ˆj ) = gj + S† ΓH (ν)R∗j (ρj − ρ † H ˆ + S Γ (ν)1(dj − dj ) + S† ΓH (ν)wj .. (41).

(18) 2208. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009. Similarly, E{ˆ gj − gj |ν} ≈ 0, and the MSE is given by * ) E ˆ gj − gj 2 |ν * ) ≈ tr S† ΓH (ν)Kwj Γ(ν)(S† )H    ˆ j )(ρj − ρ ˆ j )H |ν + tr S† ΓH (ν)R∗j,d E (ρj − ρ  × RTj,d Γ(ν)(S† )H     H + tr S† ΓH (ν)1E |dˆj − dj |2 |ν 1H Γ(ν)(S† )     ˆ j )(dj − dˆj )H |ν + 2Re tr S† ΓH (ν)R∗j,d E (ρj − ρ  × 1H Γ(ν)(S† )H   ) * ˆ j )wjH |ν + 2Re tr S† ΓH (ν)R∗j,d E (ρj − ρ  × Γ(ν)(S† )H     + 2Re tr S† ΓH (ν)1E (dj − dˆj )wjH |ν  (42) × Γ(ν)(S† )H .. TABLE I SYSTEM PARAMETERS. In deriving (42), we have used the following approximations:   ˆ j )(dj − dˆj )H |ν E (ρj − ρ  H   ˆ j ) f H (ν)R∗j,d (ρj − ρ ˆ j ) + f H (ν)wj ≈ E (ρj − ρ ) * ˆ j )(ρj − ρ ˆ j )H |ν RTj,d f (ν) = E (ρj − ρ * ) ˆ j )wjH |ν f (ν) + E (ρj − ρ   E (dj − dˆj )wjH |ν ) * ˆ j )wjH |ν + f H (ν)Kwj . ≈ f H (ν)R∗j,d E (ρj − ρ. VI. N UMERICAL R ESULTS The performance of the proposed estimators is evaluated for an uncoded MIMO OFDM system. Table I gives the system parameters. The transmission is done on a packet-by-packet basis, with the training portion consisting of two OFDM symbols at the beginning of each packet. A wide-sense stationary uncorrelated scattering discrete  channel is considered, with impulse response hj,i (τ ) = L l=0 hj,i (l)δ(τ − lTs ), where L + 1 is the length of the channel, and {hj,i (l)} are tap gains that are mutually independent complex Gaussian random variables with zero mean and variance σl2 . Exponential multipath intensity profile is employed with σl2 = σ02 · exp(−lTs /TRMS ), where TRMS is the root-mean-square delay spread and, to maintain the unit power gain, σ02 = 1 − exp(−Ts /TRMS ). The channel remains unchanged during a packet. The parameters are set as TRMS = 50 ns, L = 10, and Lg = 16. In Figs. 3–7, the training sequence is the one that is given in  for the case of nt = 2. In Fig. 8, square matrix A of the low-complexity training sequence is designed as 5230F 641H given in  for. Fig. 3.. MSE performance in a static channel.. the first transmit antenna and its circular shift by Lg for the second transmit antenna. In addition, K = P = 2, φ0 = 0, and φ1 = π/2. In Fig. 3, simulations are given to verify the MSE analysis with + the system nt = nr = 2 under the static channel hj,i (l) = (1 − exp(−1))(1 − exp(−l)), ∀j, i, 0 ≤ l ≤ 10. Only the results of the first receive antenna are shown; similarity is observed for other receive antennas. As is shown, the analysis predicts the MSE performance very well for all the estimators in the SNRs of interest. (Note that the simulations for estimators ˆj have used the real estimated frequency νˆ, which ˆ j , dˆj , and g ρ may not be equal to the true frequency ν.) In addition, the variance of the estimators approaches the respective CRLB. One observation that is worthy of noting is that the analysis of the frequency estimation is done under the perfect condition of no.

(19) HSU et al.: JOINT LEAST SQUARES ESTIMATION OF FREQUENCY, DC OFFSET, I-Q IMBALANCE AND CHANNEL. Fig. 4.. 2209. MSE performance of the joint estimators in Rayleigh fading channels.. Fig. 6. Effects of Lρ on the MSE in Rayleigh fading channels.. Fig. 5. MSE performance of the frequency estimator in Rayleigh fading channels.. dc offset and I-Q imbalance. This explains why the analytical MSE of the frequency estimator is slightly smaller than its corresponding CRLB in Fig. 3. In Fig. 4, the MSE performance is evaluated for the Rayleigh fading channel. In this case, the frequency offset is set to uniformly vary between −0.5 and 0.5 of subcarrier spacing. Again, the estimators performed very ˆ tends to closely to the CRLBs. In Fig. 4, the variance of ρ have a floor at the high SNR region. This may be attributed to having a modeling error by using Lρ = 5 in this case. The error, however, does not cause too much loss in the BER performance, as shown in Fig. 7. The performance of the frequency-offset estimation is shown in Fig. 5 with a different number of received antennas. It is clearly shown that more than one receive antenna branch can be used in the estimation to improve the performance by exploiting the power and the diversity gain that are offered by multiple receive antennas.. Fig. 7. Effects of Lρ on the BER performance in Rayleigh fading channels.. Fig. 6 investigates the effect of Lρ on the MSE performance of the joint estimators by computer simulations. As can be seen, Lρ has a significant impact on the channel and I-Q imbalance estimation, particularly at the high SNR region, and that causes an error floor in the BER performance, as can be seen in Fig. 7. Nevertheless, it affects the estimation of the dc offset and the frequency offset in a very insignificant way; this motivates us to use the simplified estimators proposed in (28) and (30)..

(20) 2210. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 5, JUNE 2009. A PPENDIX A D ERIVATION OF (19) From (16) and a simple manipulation   † . ˆ j (˜ ρ ν ) = (I − C(˜ ν )) I − 1f H (˜ ν ) R∗j I − 1f H (˜ ν ) rj −1 T  = RTj Q(˜ ν )R∗j Rj Q(˜ ν )rj (43) where  H . ν ) = I − 1f H (˜ ν ) (I − C(˜ ν )) I − 1f H (˜ ν) . Q(˜ ν ) = QH (˜ Using (43) in (15), we have    2 ˆ j (˜ ν )) I − 1f H (˜ ν ) = (I − C(˜ ν ) rj − R∗j ρ ν)  Λj (˜   . ˆ j (˜ ˆH ν ) rj − R∗j ρ = rH ν )RTj Q(˜ ν) j (˜ j −ρ Fig. 8. BER performance with low-complexity and/or simplified estimators in Rayleigh fading channels.. Fig. 7 shows the impact of Lρ on the BER performance in the Rayleigh fading channel. The modulation scheme is the 64 quadratic-amplitude modulation. The receiver is the one given in Fig. 2; after compensating the I-Q imbalance and the dc and frequency offset, MMSE MIMO detection is performed based on the channel estimation coming out of the joint LS estimators. Clearly, the modeling error due to the use of a small Lρ incurs an error floor in the BER performance, as predicted in Fig. 6, where a small Lρ results in a large MSE for estimating the channel and the I-Q imbalance. On the other hand, an Lρ that is too large, e.g., Lρ = 15, slightly degrades the BER performance, as can be seen in the figure, due to the extra noise that is induced by using a large filter length. In the figure, an ideal receiver is the one with the perfect RF compensation. Finally, in Fig. 8, we show the BER performance by using low-complexity training sequence and/or simplified frequency and dc offset estimators. The low-complexity training works very well, and almost no performance loss is observed with the low-complexity implementations. In fact, the low-complexity training performs a little better than the training that is used in  for Lρ = 3.. H  = rH ν )rj − RTj Q(˜ ν )rj j Q(˜ −1 T  × RTj Q(˜ ν )R∗j Rj Q(˜ ν )rj . Furthermore, from (13), for f (˜ ν ) = 0. .  ν )) I − 1f H (˜ ν) Q(˜ ν ) = I − f (˜ ν )1H (I − C(˜   (I − C(˜ ν )) 11H (I − C(˜ ν )) = I− (I − C(˜ ν )) 1 2   ν )) 11H (I − C(˜ × I− (I − C(˜ ν )) 1 2 = (I − C(˜ ν )) −. ν )) (I − C(˜ ν )) 11H (I − C(˜ . (45) 2 (I − C(˜ ν )) 1. A PPENDIX B ν − ν)} AND E{(ˆ ν − ν)2 } D ERIVATION OF E{(ˆ From (33), it can be shown that ˙ Λ(ν) = j2π ·. nr . H rH j Γ(ν)DΓ (ν)rj. (46). H rH j Γ(ν)EΓ (ν)rj. (47). j=1. ¨ Λ(ν) = 4π 2 ·. VII. C ONCLUSION The theory of the joint LS estimation of the frequency, the I-Q imbalance, the dc offset, and the channel has been developed for MIMO receivers with direct-conversion RF architecture. Frequency-independent and frequency-dependent I-Q imbalances have been included. Previously, RF parameters were estimated separately, and that leads to inferior performance. The estimators have been shown through analysis to be unbiased and approach the CRLB for the SNRs of interest. Special attention has been paid to the implementation complexity issue; several measures have been proposed, including a special training-sequence design and low-complexity estimators for the frequency and dc offset. Simulation results have shown that the performance degradation is negligible when using the lowcomplexity designs.. (44). nr  j=1. with D = ΦB − BΦ and E = 2ΦBΦ − BΦ2 − Φ2 B, where Φ = diag{κ(1), κ(2), . . . , κ(P )}, and κ(p) = [(Ng + N )p + 1, (Ng + N )p + 2, . . . , (Ng + N )p + N ]. Recall that B = S(SH S)−1 SH . Using rj = Γ(ν)Sgj + wj in (46) and (47), one obtains ˙ Λ(ν) = j2π ·. nr . ˜j + w ˜ jH DSgj + w ˜ jH Dw ˜ j (48) gjH SH Dw. j=1. ¨ Λ(ν) = 4π 2 ·. nr . ˜j + w ˜ jH ESgj gjH SH ESgj + gjH SH Ew. j=1. +. ˜ jH Ew ˜j w. (49).

(21) HSU et al.: JOINT LEAST SQUARES ESTIMATION OF FREQUENCY, DC OFFSET, I-Q IMBALANCE AND CHANNEL. 2 ˜ jH } ≈ σw ˜ j = ΓH (ν)wj with E{w ˜ jw where w I. In addition ⎧ ⎫ nr ⎨ ⎬   ˙ ˜j ˜ jH Dw E Λ(ν) = j2π · E w ⎩ ⎭ j=1. = j2π ·. nr . ** ) ) ˜ jH ˜ jw tr D · E w. j=1 2 · ≈ j2πσw. nr . tr{D}. j=1. =0. (50). nr    * ) H ¨ ˜ j Ew ˜j = 4π 2 · E Λ(ν) gjH SH ESgj + E w. A PPENDIX C D ERIVATION OF THE CRLB ˆ j , dˆj , and g ˆj are unbiased In Section V, it is shown that νˆ, ρ estimators provided that SNR 1 and N 1. Here, we derive the CRLB for those estimators under the same conditions. The derivation follows that given in  and . Define ∗ H ω T = (ν, λT1 , . . . , λnTr ), with λTj = (ρTj , dj , gjT , ρH j , dj , gj ) with length Lλ . From (7), since wj for j = 1, . . . , nr are independent of each other, the probability density function of observation rj , given ω, is shown as follows: . f (ω) = f (r1 , . . . , rnr |ω) nr = p(rj − R∗j ρj |gj , dj , ρj , ν). j=1 2. ≈ 4π ·. nr . gjH SH ESgj. +. =. 2 σw tr{E}. j=1. = 4π 2 ·. nr . 2211. gjH SH ESgj. j=1 nr -. 1 det(Kwj ) j=1   H × exp − rj − R∗j ρj − dj 1 − Γ(ν)Sgj   ∗ × K−1 wj rj − Rj ρj − dj 1 − Γ(ν)Sgj πN. j=1. =2 ·. nr . (54). zH j (B − I)zj. (51). j=1. where zj = j2πΦSgj . On the other hand % E. 2 ( ˙Λ(ν)  2. = −4π E. nr . 2 , ˜j gjH SH Dw. +. ˜ jH DSgj w. +. ˜ jH Dw ˜j w. j=1. ≈ −4π 2 ·. nr . * ) ˜ jH DSgj ˜ jw 2gjH SH DE w. P. Let M = E{(∂ ln f (ω)/∂ω T )H (∂ ln f (ω)/∂ω T )} be the Fisher information matrix. From , the CRLB for each respective parameter is given by. j=1 2 = 2σw ·. nr . . where Kwj = E{wj wjH } is the correlation matrix of wj . Recall that wj = [ wjT (0) wjT (1) · · · wjT (P − 1) ]T with wj (n) = gj (n) ⊗ w0,j (n). We have wj (k) = Φgj w0,j (k) and w0,j (k) = [w0,j,k (−Lgj +1), . . . , w0,j,k (0), w0,j,k (1), . . . , . w0,j,k (N − 1)]T with w0,j,k (n) = w0,j (k(Ng + N ) + n) and Φgj , shown at the bottom of the page, where Lgj is ˜ j w0,j with the length of gj (n). Therefore, wj = Φ T T T ˜j = w0,j = [w0,j (0) w0,j (1) · · · w0,j (P − 1)]T and Φ 2 ˜ ˜H Φj Φj . diag{Φgj , . . . , Φgj }. Finally, we can obtain Kwj = σw.

(22). H zH j (I − B)zj .. (52). var(ˆ ν ) ≥ [M−1 ]11. j=1. The approximation in (52) is justifiable for SNR 1, and the last equality is obtained by using SH (ΦB − BΦ) = SH (ΦB − Φ), (ΦB − BΦ)S = (Φ − BΦ)S, and the fact that (I − B) is a projection matrix. Therefore, from (31), (32), and (50)–(52), we have E{(ˆ ν − ν)} ≈ 0 and * ) 2 E (ˆ ν − ν)2 ≈ σw. 1 2. nr  j=1. zH j (I. ⎡. Φg j. gj (Lgj − 1) ⎢ 0 ⎢ =⎢ ⎣ 0. . − B)zj. ··· gj (Lgj − 1) ... (53). . ···. 1+(j−1)Lλ +Lρ. . * ) E ˆ ρj − ρj 2 ≥. [M−1 ]kk. k=1+(j−1)Lλ +1. var(dˆj ) ≥ [M−1 ]kk|k=1+(j−1)Lλ +Lρ +1 ). 2. E ˆ gj − gj . *. 1+(j−1)Lλ +Lρ +1+Lg. . ≥. [M−1 ]kk .. k=1+(j−1)Lλ +Lρ +2. gj (0) ··· .. . 0. 0 gj (0) gj (Lgj − 1). ··· 0 .. . ···. ···. 0 0 gj (0). ⎤ ⎥ ⎥ ⎥ ⎦ N ×(N +Lgj −1).

(24) HSU et al.: JOINT LEAST SQUARES ESTIMATION OF FREQUENCY, DC OFFSET, I-Q IMBALANCE AND CHANNEL. Racy Cheng received the B.S. degree in mathematics from the National Central University, Taoyuan, Taiwan, in 1988, the M.S. degree in electrophysics from Polytechnic University, Brooklyn, NY, in 1992, and the Ph.D. degree in electrical engineering from the State University of New York at Stony Brook in 1997. From 1997 to 2000, he was with the Computer and Communications Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, working on Digital Enhanced Cordless Telecommunications and Global System for Mobile Communications projects. From 2000 to 2005, he was with InProComm, Hsinchu, where he led a team for the 802.11g Baseband/Medium Access Control Application Specific Integrated Circuit design. He is currently with Minghsin University of Science and Technology, Hsinchu. In addition to teaching, he is also a Technical Consultant for research institutes and industrial companies. The projects he is working on range from 4G cellular systems to proprietary wireless communication devices. His research interest includes digital signal processing with applications to wireless communication and multimedia.. 2213. Wern-Ho Sheen (M’91) received the B.S. degree from the National Taiwan University of Science and Technology, Taipei, Taiwan, in 1982, the M.S. degree from the National Chiao Tung University, Hsinchu, Taiwan, in 1984, and the Ph.D. degree from Georgia Institute of Technology, Atlanta, in 1991. From 1993 to 2001, he was with the National Chung Cheng University, Chiayi, Taiwan, where he held the positions of Professor with the Department of Electrical Engineering and Managing Director with the Center for Telecommunication Research. Since 2001, he has been a Professor with the Department of Communication Engineering, National Chiao Tung University. He has been extensively consulting for the industry and research institutes in Taiwan. His research interests include the general areas of communication theory, cellular mobile and personal radio systems, adaptive signal processing for wireless communications, spreadspectrum communications, and very-large-scale integration design for wireless communications systems..

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