Blind Estimation Algorithms for I/Q Imbalance in Direct Down-conversion Receivers
BBlind Estimation Algorithms for I/Q Imbalancein Direct Down-conversion Receivers
Peiyang Song ∗† , Nan Zhang ∗ , Hang Zhang † and Fengkui Gong ∗∗ State Key Laboratory of ISN, Xidian University, Xi’an 710071, China † Science and Technology on Communication Networks Laboratory, Shijiazhuang, ChinaEmail: [email protected], [email protected], [email protected], [email protected]
Abstract —As known, receivers with in-phase and quadrature-phase (I/Q) down conversion, especially direct-conversion archi-tectures, always suffer from I/Q imbalance. I/Q imbalance iscaused by amplitude and phase mismatch between I/Q paths. Theperformance degradation resulting from I/Q imbalance can notbe mitigated with simply higher signal to noise ratio (SNR). Thus,I/Q imbalance compensation in digital domain is critical. Thereare two main contributions in this paper. Firstly, we proposed ablind estimation algorithm for I/Q imbalance parameters basedon joint first and second order statistics (FSS) which has a lowercomplexity than conventional Gaussian maximum likelihoodestimation (GMLE). This can be used for further precessingsuch as equalization in presence of receiver IQ imbalance. Inaddition, we find out the reason of error floor in conventional I/Qimbalance compensation method based on conjugate signal model(CSM). The proposed joint first order statistics and conjugatesignal model (FSCSM) compensation algorithm can reach theideal bit error rate (BER) performance.
Index Terms —I/Q imbalance, blind estimation algorithm, firstand second order statistics, conjugate signal model
I. I
NTRODUCTION
With increasingly demanding requirements for low-costand low-power wireless receivers, I/Q imbalance problemhas attracted more attention in both industrial and academiccommunities. I/Q imbalance is caused by amplitude and phasemismatch between in-phase and quadrature-phase branches inanalog front-end, which is unavoidable in practical imple-mentation. It always leads to serious performance degradationin receivers, especially those with low-cost radio frequencyfront-end (e.g. direct conversion architecture). This distortion,unfortunately, can not be mitigated by simply improving SNR.Hence, dealing with I/Q imbalance in digital domain becomescritical, especially for communications with large modulationorder.Most literatures so far focus on data-aided estimation al-gorithms, such as least mean squares (LMS) [1], decision-directed (DD) [2], the expectation maximization (EM) [3],maximum likelihood (ML) [4], minimum mean squared error(MMSE) [5], etc. More recently, [6] describes an iterative I/Qcompensation algorithm using both the training symbols anddata symbols. Estimation by known symbols usually achievesa rapid convergence. However, the pilot symbols can alsoresult in low spectral efficiency. This problem has becomeparticularly severe as the spectrum resources are increasinglyvaluable. A few effective blind algorithms can also be found for I/Qimbalance estimation. [7], for the first time, introduces theblind source separation (BSS) into I/Q imbalance estimation.The advanced BSS technology, like joint approximative diago-nalization of eigenmatrix (JADE) algorithm [8], is an efficientapproach to correct I/Q imbalance, whereas it always suffersfrom significantly higher complexity and severe performancedegradation when interfered by frequency offset. [9] treatsthe I/Q imbalance problem as a conjugate signal model(CSM), where the observed signal is a linear combination ofthe desired signal and its complex conjugate. However, thealgorithm always suffers from error floor which is especiallysevere in high order modulation. In addition, [10] and [11]derive estimated values of amplitude and phase mismatchby approximating quadrature amplitude modulation (QAM)symbols as two-dimensional Gaussian variables. The estimatedparameters are used for equalization in presence of receiverI/Q imbalance. Especially, [12] Addresses the compensationof transmitter I/Q imbalances and carrier frequency offset(CFO) for uplink single-carrier interleaved frequency-divisionmultiple-access (SC-IFDMA) systems, which is of greaterinterest for the 5G networks.Throughout the letter, we define the notation as follows. Weuse bold-face upper case letters like X to denote matrices,bold-face lower case letters like x to denote column vectors,and light-face italic letters like x to denote scalers. x i is the i th element of vector x . x i is the i th column vector of matrix X . The complex conjugate of a complex x is representedas x ∗ . I is the identity matrix. (cid:60){·} and (cid:61){·} denote thereal and imaginary part of complex numbers, respectively. E [ · ] represents the expectation. [ · ] T and [ · ] H denote the transposeand conjugate transpose operations, respectively. ˆ x (or ˆ X ) isthe estimated value of x (or X ).II. I/Q I MBALANCE S YSTEM M ODEL
Mathematical model of I/Q imbalance has been establishedand widely used [1]–[14]. In this section, we consider thetransmission of baseband signals over a flat-frequency noisychannel and I/Q imbalance in direct conversion receivers.Bandpass signal is given by x RF ( t ) = I ( t ) cos (2 πf c t ) − Q ( t ) sin (2 πf c t ) , (1) a r X i v : . [ ee ss . SP ] A p r here f c is the carrier frequency, I ( t ) is the in-phase compo-nent of baseband signal and Q ( t ) is the quadrature-phase one.The received signal can be written as r ( t ) = x RF ( t ) + ω RF ( t )= r c ( t ) cos (2 πf c t ) − r s ( t ) sin (2 πf c t ) , (2)where ω RF ( t ) is real-valued additive white Gaussian noise(AWGN). r c ( t ) and r s ( t ) are equivalent baseband signalswhich contain transmitted signals and the noise. In directconversion architecture, down conversion is realized by multi-plying received signal r RF ( t ) by local carrier and then passingthe result through a low-pass filter. As described before, the in-phase local oscillator (LO) signal z LO,c ( t ) and the quadrature-phase one z LO,s ( t ) always exhibit both amplitude and phasemismatch, i.e. z LO,c ( t ) = 2(1 + α ) cos (2 πf c t + θ ) , (3a) z LO,s ( t ) = − − α ) sin (2 πf c t − θ ) , (3b)where α and θ are amplitude and phase imbalance parameters,respectively. It is worth noting that amplitude mismatches indifferent branches do not need to be exactly equal. The symbol α is used to make the formula more symmetrical. Thus, thefinal output baseband signal influenced by I/Q imbalance inreceiver can be written as y c ( t ) = LP F { z LO,c ( t ) r ( t ) } = (1 + α ) cos ( θ ) r c ( t ) (cid:124) (cid:123)(cid:122) (cid:125) signal + (1 + α ) sin ( θ ) r s ( t ) (cid:124) (cid:123)(cid:122) (cid:125) interference , (4a) y s ( t ) = LP F { z LO,s ( t ) r ( t ) } = (1 − α ) sin ( θ ) r c ( t ) (cid:124) (cid:123)(cid:122) (cid:125) interference + (1 − α ) cos ( θ ) r s ( t ) (cid:124) (cid:123)(cid:122) (cid:125) signal , (4b)where LP F {·} accounts for low-pass filter. y c ( t ) and y s ( t ) arebaseband signals of I/Q paths. As can be seen from (4a) and(4b), when ideal LO is taken into consideration (i.e. α = 0 and θ = 0 ), receivers can successfully recover the originalbaseband signals as y c ( t ) = r c ( t ) and y s ( t ) = r s ( t ) . But inpractice, when amplitude and phase imbalance occur, each ofthese two branches will be interfered by the other. (4a) and(4b) can be expressed by y ( t ) = [ y c ( t ) y s ( t )] T as y ( t ) = (cid:20) (1 + α ) cos ( θ ) (1 + α ) sin ( θ )(1 − α ) sin ( θ ) (1 − α ) cos ( θ ) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Γ (cid:20) r c ( t ) r s ( t ) (cid:21) . (5)Fig. 1 is the block diagram of this system model. LO LPF
LPF x RF (t) r(t) z LO,c (t) z LO,s (t) y c (t) y s (t) AWGN w RF (t) Fig. 1. I/Q imbalance system model with real signals
The model can be derived by the complex envelope [15] ofreal signal as well. Received signal r ( t ) can be given as r ( t ) = (cid:60) (cid:8)(cid:101) r ( t ) e j πf c t (cid:9) = 12 (cid:0)(cid:101) r ( t ) e j πf c t + (cid:101) r ∗ ( t ) e − j πf c t (cid:1) , (6)where (cid:101) r ( t ) = r c ( t ) + jr s ( t ) is the complex envelope ofreceived signals r ( t ) . Here, local carrier generated by LO canbe written as (cid:101) z LO ( t ) = 2 (cid:0) K e − j πf c t + K e j πf c t (cid:1) , (7)where K = [(1 − α ) e jθ + (1 + α ) e − jθ ] / , (8a) K = [(1 + α ) e jθ − (1 − α ) e − jθ ] / . (8b)Equivalent complex signal after down conversion can begiven as (cid:101) y ( t ) = LP F { r ( t ) (cid:101) z LO ( t ) } = K (cid:101) r ( t ) (cid:124) (cid:123)(cid:122) (cid:125) signal + K (cid:101) r ∗ ( t ) (cid:124) (cid:123)(cid:122) (cid:125) interference , (9)which is shown in Fig. 2. ( • ) * r(t)~ y(t) AWGN w RF (t)~ K K x RF (t)~ ~ LPF
Fig. 2. I/Q imbalance system model with equivalent complex enve-lope, where ( · ) ∗ denotes conjugate value. As can be seen from (9), when ideal LO is implemented, K = 1 and K = 0 . Baseband signal can be accuratelyrecovered without interference. If α and θ can not be ignored,the entire complex baseband signal will be interfered by theconjugate value of itself.Obviously, (9) is equivalent to (4a) and (4b), i.e. y c ( t ) = (cid:60){ (cid:101) y ( t ) } and y s ( t ) = (cid:61){ (cid:101) y ( t ) } . (10)Then, we consider the QAM modulation and assume thein-phase and quadrature-phase components of the transmittedsignal as I ( t ) = (cid:88) (cid:96) a (cid:96) ψ ( t − (cid:96)T ) , (11a) Q ( t ) = (cid:88) (cid:96) b (cid:96) ψ ( t − (cid:96)T ) , (11b)where, a (cid:96) and b (cid:96) belong to the QAM alphabet. ψ ( t ) is asquare-root raised-cosine pulse, and T is the symbol period.The discrete-time in-phase and quadrature-phase componentsafter down-conversion and matched filtering can be written as y c,k = y c ( kT )= (1 + α )[( a k + n c,k ) cos ( θ ) + ( b k + n s,k ) sin ( θ )] , (12a) y s,k = y s ( kT )= (1 − α )[( a k + n c,k ) sin ( θ ) + ( b k + n s,k ) cos ( θ )] , (12b)where n c,k and n s,k , which uncorrelate with each other, are thesamples of zero-mean real-valued Gaussian random variableswith variance σ n / .II. P ROPOSED JOINT FIRST AND SECOND ORDERSTATISTICS (FSS)
ALGORITHM
Firstly, α and θ are set as the trial values of I/Q imbalanceparameters. [10] proposed an estimation algorithm for α and θ by Gaussian maximum likelihood. The estimated parametersare used for equalization in presence of I/Q imbalance. Here,we propose a simpler algorithm for estimation of α and θ . Itis easy to break (12a) and (12b) into Y = (cid:20) y c y s (cid:21) = (cid:20) α
00 1 − α (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Ψ (cid:20) cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) (cid:21) (cid:20) r c r s (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Ω , (13)where r c,k = a k + n c,k and r s,k = b k + n s,k are samples of r c ( t ) and r s ( t ) which contain both QAM symbols and noise.(13) can be written as Y = Γ R , where R = [ r c , r s ] T . When α and θ have been estimated, original signal can be recoveredby R = Γ − Y .We use C R , C Ω and C Y represent covariance matrix of R , Ω and Y respectively. With E [ Ω k ] = [0 , T , C Ω can begiven as (14) (see the bottom of this page).Then, we assume that the received signals before downconversion are circular symmetry, i.e. E [( r c,k + jr s,k ) ] = 0 , which can be further written as E [ r c,k ] = E [ r s,k ] and E [ r c,k r s,k ] = 0 .We set P r = 2 E [ r c,k ] = 2 E [ r s,k ] as the average power ofreceived symbols. C R and C Ω can be written as C R = 12 P r I , (15a) C Ω = 12 P r (cid:20) sin (2 θ ) sin (2 θ ) 1 (cid:21) , (15b)which reveals that phase mismatch θ will not change theaverage power (i.e. the variance) of received symbols of eachbranch. Similar to C Ω , C Y can be written as C Y = E [ ΨΩΩ T Ψ T ] = E [ Ψ C Ω Ψ T ]= 12 P r (cid:20) (1 + α ) (1 − α ) sin (2 θ )(1 − α ) sin (2 θ ) (1 − α ) (cid:21) . (16)To avoid power measurement, we give a calculation of P r bybaseband signal y s and y c . Resulting from the i.i.d . propertyof a k and b k (also n c,k and n s,k ), it is obvious that η = E [ | ( a k + n c,k ) cos ( θ ) + ( b k + n s,k ) sin ( θ ) | ]= E [ | ( a k + n c,k ) sin ( θ ) + ( b k + n s,k ) cos ( θ ) | ] . (17)Thus, relationship between α and received symbols can bederived by (12a) and (12b) as E [ | y c,k | ] = (1 + α ) η and E [ | y s,k | ] = (1 − α ) η. (18) Then, we replace the statistical average by actual receivedsymbols. And (18) can be written as α − α = (cid:80) Nk =1 | y c,k | (cid:80) Nk =1 | y s,k | , (19)where α can be easily calculated by received symbols. Also,we give the expression of θ from (16) as
12 (1 − α ) P r sin (2 θ ) = E [ y c,k y s,k ] = E [ y s,k y c,k ] . (20)Then, we replace the statistical average by actual receivedsymbols. And (20) can be written as
12 (1 − α ) P r sin (2 θ ) = 1 N (cid:88) Nk =1 ( y c,k y s,k ) , (21)where P r can be obtained from (16) by received symbols andcalculated α as P r = 1 N (cid:34) (cid:80) Nk =1 y c,k (1 + α ) + (cid:80) Nk =1 y s,k (1 − α ) (cid:35) . (22)At last, α and θ can be written as α = (cid:80) Nk =1 | y c,k | − (cid:80) Nk =1 | y s,k | (cid:80) Nk =1 | y c,k | + (cid:80) Nk =1 | y s,k | , (23a) θ = 12 arcsin (cid:20) − α ) ρ cs (1 − α ) ρ c + (1 + α ) ρ s (cid:21) , (23b)where ρ cs = (cid:80) Nk =1 ( y c,k y s,k ) , ρ c = (cid:80) Nk =1 y c,k and ρ s = (cid:80) Nk =1 y s,k .As known, frequency offset does not change the circularsymmetry property of received signal. Hence, the proposedalgorithm can achieve a strong robustness to frequency offset.IV. P ROPOSED JOINT FIRST ORDER STATISTICS ANDCONJUGATE SIGNAL MODEL (FSCSM)
ALGORITHM
Firstly, we give a brief introduction of conventional CSMalgorithm. The blind I/Q imbalance estimation is addressed in[9] with conjugate signal model, i.e. (cid:101) Y = (cid:20) (cid:101) y (cid:101) y ∗ (cid:21) = (cid:20) K K K ∗ K ∗ (cid:21) (cid:20) (cid:101) r (cid:101) r ∗ (cid:21) = K (cid:101) R . (24)Assuming that the target signal (cid:101) r is circular or proper, thetarget here is to find a matrix W to whiten or decorrelate thecomponents of (cid:101) r as (cid:101) z = W (cid:101) Y = W K (cid:101) R = T (cid:101) R , (25)where T is the equivalent system matrix. When perfect estima-tion is taken into consideration, T ≈ I . W can be calculatedas W = U Λ − / U H , (26) C Ω = E (cid:2) ΩΩ T (cid:3) = (cid:20) cos ( θ ) E [ r c,k ] + sin ( θ ) E [ r s,k ] sin ( θ ) cos ( θ )( E [ r c,k ] + E [ r s,k ]) sin ( θ ) cos ( θ )( E [ r c,k ] + E [ r s,k ]) sin ( θ ) E [ r c,k ] + cos ( θ ) E [ r s,k ] (cid:21) (14)here U and Λ are calculated by eigenvalue decompositionof C (cid:101) Y , i.e. C (cid:101) Y = E ( (cid:101) Y (cid:101) Y H ) = U Λ U H . (27)Compared to earlier BSS technology, CSM algorithm hasa much lower complexity. However, its performance seemsnot to be reliable enough. In practical, we find that CSMalgorithm always suffers from a certain error floor, which isespecially severe in high order modulations. A large numberof simulations revealed that the error floor usually mitigatesas the amplitude mismatch α reduces. When α is not takeninto consideration, BER performance of CSM algorithm canbe improved to ideal bound.(13) has proved that the influence of α and θ on impairedsignals is independent. Thus, one possible solution to eliminatethe error floor is compensating amplitude mismatch in ad-vance. Our proposed FSCSM algorithm add a pre-processingmodule before conventional CSM method. The pre-processingcan be given as Y (cid:48) = (cid:20) y (cid:48) c y (cid:48) s (cid:21) = ˆ Ψ − Y = ˆ Ψ − ΨΩ = (cid:20) α − ˆ α (cid:21) (cid:20) y c y s (cid:21) = (cid:20) α α − α − ˆ α (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Ψ (cid:48) (cid:20) cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) (cid:21) (cid:20) r c r s (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Ω , (28)where ˆ α is the estimated value of amplitude mismatch cal-culated by (23a). Ψ (cid:48) = ˆ Ψ − Ψ is the remaining amplitudemismatch matrix. When accurate estimation of α is performed, Ψ (cid:48) ≈ I and Y (cid:48) ≈ Ω . Then, the equivalent complex basebandsignal for CSM is Y (cid:48) = [ (cid:101) y (cid:48) , (cid:101) y (cid:48) ∗ ] T , where (cid:101) y (cid:48) = y (cid:48) c + j y (cid:48) s . −2 −1 0 1 2−2−1.5−1−0.500.511.52 (a)In−phase Q uad r a t u r e − pha s e −1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 In−phase Q uad r a t u r e − pha s e −2 −1 0 1 2−2−1.5−1−0.500.511.52 (b)In−phase Q uad r a t u r e − pha s e −1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 In−phase Q uad r a t u r e − pha s e Fig. 3. Constellation comparison pre- and post- FSCSM algorithm.(a) signal is impaired by I/Q imbalance. (b) signal is impaired byboth I/Q imbalance and frequency offset.
The effect of proposed FSCSM algorithm can be shown inFig. 3. Signals impaired by I/Q imbalance can be perfectly compensated. Frequency offset will be completely reservedand wait for further processing. Here we consume that thefrequency offset is known and has been fully correct after ourFSCSM module. Further performance analysis will be givenin Section V. V. N
UMERICAL R ESULTS
In this section, we assess the performance and robustnessof our proposed algorithms. Table I shows the operationsrequired for GMLE and proposed FSS. As can be seen,compared to GMLE, our proposed FSS estimation is free ofcomplicated square root operations. Also, less additions andmultiplications are required. Additional bit shift and absolutevalue operations introduced by proposed FSS are relativelyeasy for implementation. Fig. 4 reports the mean square errorof estimated α and θ achieved by GMLE and FSS versusthe number of received 16-QAM symbols N , for SNR=18dB.Estimation results of both algorithms are quite close whichmeans the accuracy degradation of proposed FSS caused bysimplification is slight enough.
100 200 300 400 50000.20.40.60.811.2 x 10 −3 N M SE o f α
100 200 300 400 50000.511.522.5 x 10 −3 N M SE o f θ GMLE [11]proposed FSS GMLE [11]proposed FSS
Fig. 4. Mean square error (MSE) of estimated α and θ achieved byGMLE and proposed FSS Fig. 5 reports the convergence of different algorithms in16-QAM modulation with SNR=18dB. I/Q imbalance is setas α = 0 . and θ = 10 ◦ . ∆ f is the normalized CFO. Toignore the effect of frequency offset compensation algorithmson the BER performance, we assume that ∆ f is known atreceiver and is perfectly compensated after I/Q imbalancecompensation.Simulation shows that our proposed FSCSM algorithmperforms a fast BER performance convergence. The BER canbe eventually upgraded to AWGN bound (i.e. BER of idealreceiver without I/Q imbalance). Moreover, when frequencyoffset is taken into consideration, BER performance of boththe proposed algorithms does not degrade.Fig. 6 compares the performance of conventional CSMwith proposed FSCSM blind algorithm in 16-QAM, 64-QAMand 256-QAM, respectively. Parameters are set as α = 0 . , θ = 10 ◦ and ∆ f = 0 . And N is large enough to guaranteethat the BER performance has converged. As can be seen, forconventional CSM algorithm, the degradation of performancecan not be mitigated by simply increase N . Our proposedFSCSM algorithm solves this problem without knowledgeof any other additional information and improve the BERperformance to AWGN bound. ABLE IO
PERATION NUMBER REQUIRED FOR
GMLE
AND PROPOSED
FSS
ALGORITHMS
Algorithm Square Root Multiplication Addition Absolute Value of Real Number Bit Shift Division ArcsinGMLE [11] N + 2 3 N − proposed FSS N + 4 2 N
500 1000 150010 −4 −3 −2 −1 N(a) BE R −4 −3 −2 −1 N(b) BE R JADE [7] [8]CSM [9]proposed FSCSMAWGN bound JADE [7] [8]CSM [9]proposed FSCSMAWGN bound
Fig. 5. Convergence of different blind algorithms in 16-QAM, forSNR=18dB. (a) ∆ f = 0 ; (b) ∆ f = 0 .
15 20 25 30 3510 −4 −3 −2 −1 SNR (dB) BE R AWGN boundCSMproposed FSCSM
Fig. 6. Performance of conventional CSM and proposed FSCSMalgorithm in different QAMs
VI. C
ONCLUSION
This letter has addressed the I/Q imbalance problem insingle-carrier direct conversion receivers. Two blind estima-tion and compensation algorithms have been put forward.The proposed FSS algorithm achieves less complexity thanconventional GMLE estimation with slight enough perfor-mance degradation. In addition, the reason of error floorin conventional CSM algorithm is found out. Our proposedFSCSM algorithm can eliminate the error floor of CSM andachieve ideal BER performance. It is worth noting that theproposed two algorithms both perform a strong robustness to frequency offset, which makes it work well before frequencyoffset estimation algorithms. Furthermore, both the proposedalgorithms are not sensitive to modulation order. Our resultsshow that they also perform well with 4096-QAM and 256-APSK. A
CKNOWLEDGEMENT
This work is supported in part by joint fund of ministryof education of China (6141A02022338) and the openingproject of science and technology on communication networkslaboratory (KX162600027).R
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