Frequency offset and I/Q imbalance compensation for Direct-Conversion receivers

Transcription

1 Frequency offset and I/Q imbalance compensation for Direct-Conversion receivers Hung Tao Hsieh Department of Communication Engineering National Chiao Tung University May 12th,

2 Reference M.Valkama, M.Renfors, and V.Koivunen, Advanced methods for I/Q imbalance compensation in communication receivers, IEEE Trans. Signal Processing, vol.49, no.10, pp , Oct G. Xing, M. Shen, and Hui Liu, Frequency offset and I/Q imbalance compensation for Direct-Conversion receivers, IEEE Trans. Wireless Commun., vol.4, No. 2, pp , Mar Hai Lin; Xu Zhu; Yamashita, K., Pilot-Aided Low-Complexity CFO and I/Q Imbalance Compensation for OFDM Systems, Communications, ICC apos;08. IEEE International Conference on, May 2008, pp Hai Lin; Xu Zhu; Yamashita, K., Subcarrier allocation based compensation for CFO and I/Q Imbalance in OFDM Systems, IEEE Trans. Wireless Commun., vol.8, No. 1, pp.18-23, Jan

3 Outline Introduction The DCR signal model CFO estimation with a modified pilot I/Q imbalance compensation LLS-based compensation scheme Complexity Comparison Simulations Conclusions SCA (subcarrier allocation) System model CFO estimation in SCA based system SCA -based I/Q compensation scheme Blind compensation scheme Simulations Conclusions 3

4 Introduction The I/Q imbalance and the imperfectly balanced LO are commonly seen in any RF front-end that exploits analog quadrature down-mixing. Due to the remarkable merits in circuit board size, cost and power consumption, direct conversion receiver (DCR) is recently considered as a replacement of traditional receivers. There re two types of imbalance having different frequency characteristics: Frequency independent : CFO and the LO imbalance from I/Q branches. Frequency selective : The LPF imbalance from I/Q branches. 4

5 The DCR signal model (1/3) Imperfect 90 phase difference and unequal amplitudes from LO The imbalance from LPF If Phase error = 0 5

6 The DCR signal model (2/3) The imbalance effects on the individual channel signals can be explicitly characterized as [1] 6

7 The DCR signal model (3/3) The received radio frequency signal [2] The down-converted baseband signal where 7

8 CFO estimation with a modified pilot (1/3) After the GI/CP removal (STO = 0), we stack the pilot sequences in a matrix as follows Rˆ For conventional pilot sequences, the n th column of, denoted as, can be expressed as * * CFO, p(k) h(k) c ( k), p (k) h (k) c2( n CFO : exp( j2 πε ), n = k+(m-1)k N 1 k ) 8

9 CFO estimation with a modified pilot (2/3) The 1st column of the CFO matrix is For conventional pilot sequences, the phase bios in e should be 0, instead of. Then E becomes ill-conditioned when the initial offset is very close to zero. In this paper, they use a modified pilot (MP) sequence as follows Only the even pilot symbols are rotated, therefore, GI is necessary to prevent the IBI. 9

10 CFO estimation with a modified pilot (3/3) The [a(k) b(k)] T can be estimated by nonlinear least squares method So, we can obtain a CFO estimator by the ML algorithm T rk ˆ( ) = E( ε )[ ak ( ) bk ( )] + noise T ML => min{ rˆ ( k) E( ε )[ a( k) b( k)] } T => max{ rk ˆ( ) E( ε )[ ak ( ) bk ( )] } 2 10

11 I/Q imbalance compensation (1/5) In practical systems, the I/Q imbalance induced by LPF should be relatively flat across frequency. As a result, it s reasonable to compensate with a FIR of L taps on either I or Q branch. If Phase error = 0 11

12 I/Q imbalance compensation (2/5) Let g(t)= F -1 {G I (f)}. The signal after the filter-based compensation becomes Then, β χ 12

13 I/Q imbalance compensation (3/5) Utilizing the special structure of the pilot symbols, two adjacent pilot symbols after compensation is related by where 2πεˆ K π +, m=odd N 4 Ω m = 2πεˆ K π, m=even N 4 13

14 I/Q imbalance compensation (4/5) Then Define 14

15 I/Q imbalance compensation (5/5) Therefore, the above equation can be further expressed as where 15

16 LLS-based compensation scheme (1/3) To shorten the pilot sequence in [2], a novel modified pilot scheme was proposed [3] With I/Q imbalance and noise are all absent, the relation between the compensated pilot symbols becomes ψ 16

17 LLS-based compensation scheme (2/3) Therefore, all the unknown parameters can estimate by where and = ri ri

18 LLS-based compensation scheme (3/3) We can calculate the above column vector cos( ψ) + βsin( ψ) u = cos( ψ) βsin( ψ) = xsin( ψ ) Let θ=pi/2 + Λ ri ψ = 2 πε K / N + θ 18

19 Complexity Comparison 19

20 Simulations (1/4) The normalized CFO is 0.288, For [2], K=16, \hat{m} = 6, L=5 For [3], K=16, M = 10, L=5 The estimated resolution for CFO in [2] is 1E-2 Case A : Case B : 20

21 Simulations(2/4) 21

22 Simulations(3/4) 22

23 Simulations(4/4) 23

24 Conclusions Some difficulties in [2] have been overcame in [3], such as The complexity of estimating CFO The solution of x and \beta cannot be calculated until the CFO estimation ends. The STO may not easily estimated before the compensation in [2]. The redundancy samples (GI) In [3], the I/Q imbalance and the CFO can obtain the closedform solution. 24

25 SCA (subcarrier allocation) System model (1/2) The system model of the SCA is the same with [3]. r' ( k) = H H * r Γ F h Γ F h where r '(0) ' = = ( ε) + ( ε)( ) r'( N 1) r' ( k) = 25

26 SCA (subcarrier allocation) System model (2/2) r '(0) ' = = ( ε) + ( ε)( ) r'( N 1) H H * r Γ F h Γ F h Usually, there are some null subcarriers in practical OFDM systems. Without loss of generality, it is assumed that there re P modulated and K=N-P null subcarriers. Therefore, we can partition the F H into W=[w 1 w P ] and V=[v 1 v K ], corresponding to two index sets {a p } p=1p and {b k } k=1k. * d r ' = Γ( ε) W Γ( ε) W + z = A( ε) d+ z d 26

27 CFO estimation in SCA based system (1/2) Recall the filtered signal, r' = A( ε ) d+ z Then the ML estimation of ε can be obtain by minimizing Using the NLS, we can derive the d Ls. Substituting that into likelihood function, we have A is a N - by - 2P matrix. In order to apply the ML equation, A should be a full-rank tall matrix. Then we have P<N/2 (for this cost function). 27

28 CFO estimation in SCA based system (2/2) Recall * d r ' = Γ( ε) W Γ( ε) W + z = A( ε) d+ z d, W is a NxP matrix. When R(W*)=R(W), N(A(ε))= N(A(-ε)), which in turn leads to the likelihood function having two equal minimums at ε and ε. Therefore, W should not be symmetric. So we can choose W, whose 28

29 SCA -based I/Q compensation scheme (1/3) Recall the I/Q-imbalance compensation scheme in [2],.. Using the NLS algorithm for r = Γ ε Wd +, we have.. (ˆ, x β) = arg min r I.. = arg min r Γ( ε ) ( ) noise [ ( ) ] 1 H W WΓ( ε ) Γ( ε ) W W [ VV ] H.. Γ( ε ) r.. r Blind estimation 29

30 SCA -based I/Q compensation scheme (2/3) Then, the cost function is Let where L = (L x -1)/2. Substituting into the cost function and differentiating with respect to x and \beta, we have 30

31 SCA -based I/Q compensation scheme (3/3) Note that U I and U Q are the real and imaginary parts of. Note that Then the I/Q compensator are 31

32 Blind compensation scheme For a trial ε, once V exists, we can obtain the corresponding.. ( ˆε ) r = Γ Wd H H J( ˆ ε) = r Γ( ε) VV Γ( ˆ ε) r= V Γ( ˆ ε) r Then ˆ ε = arg min J( ε) Once εis obtained, itself and the corresponding x( ˆ) ε and β ( ˆ) ε can be estimated. 32

33 L x = 5 Simulation For SCA-based scheme (pilot):p=12, N =64 For blind-based scheme :P=30+22, N =64, 10informationbearing OFDM symbols For conventional scheme[2] :128 samples containing 4 pilot symbol, K=16 33

34 Simulation 34

35 Conclusions They propose a matrix representation of OFDM signals with CFO and I/Q imbalances. A SCA-based approach has been proposed. 35