A Simple Estimation Scheme for Joint Estimation of Carrier Frequency Offset and I/Q Imbalance Leonardo Lanante Jr., Masayuki Kurosaki, Hiroshi Ochi Graduate School of Computer Science and Systems Engineering Kyushu Institute of Technology Iizuka City, Fukuoka, Japan 820-8502 Email:{leonardo,kurosaki}@dsp.cse.kyutech.ac.jp,
[email protected]
Abstract—This paper proposes a simple algorithm for joint estimation of carrier frequency offset and I/Q imbalance. The reduced complexity algorithm is derived from a closed form solution of the carrier frequency offset and I/Q imbalance parameters given two identical training blocks. The proposed algorithm also uses the least squares algorithm to give accurate estimate in the presence of noise. It was shown that for as little as 10 samples each from two identical OFDM symbols, the proposed algorithm can produce accurate estimate of CFO and I/Q imbalance. While the target system is OFDM, the proposed algorithm applies to any system as long as two identical data blocks are received in the receiver. Simulation results show that performance of the proposed algorithm that uses only 10 samples for complexity reduction is only 1dB below a system without CFO nor I/Q imbalance degradation. Index Terms—Carrier Frequency Offset, I/Q Imbalance, Least Squares Estimate
I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) has been more and more adapted for various wireless communication systems [1]. These systems include DVB-T, IEEE 802.11a/g [2] and 802.16a [3]. The main reason is OFDM’s intrinsic ability to mitigate frequency selective fading. With the use of sufficient cyclic prefix, the frequency selective channel becomes a one-tap channel in the frequency domain which makes it easy for the receiver to compensate for the channel effects leading to a very reliable wireless communication. Another interesting technology for wireless systems is the direct conversion receiver (DCR) as shown in Fig. 1. As opposed to heterodynes, this receiver doesn’t have an IF stage. The RF signal is directly mixed to baseband and low pass filters are used to suppress high frequency interference. Due to zero intermediate frequency, image rejection filters are not needed hence neither bulky surface acoustic wave (SAW) filters nor high Q analog filters are needed making low part number receivers possible [4]. Though acknowledged as a relatively simple design in paper, DCR is known to have severe issues with front end imperfections. These issues include DC offset, I/Q imbalance, and carrier frequency offset (CFO)[1]. While these issues can This work was (partly) supported by a grant of Knowledge Cluster Initiative 2nd stage implemented by Ministry of Education, Culture, Sports, Science and Technology(MEXT).
be solved by proper analog techniques or calibration, cost and size limits make these implementations difficult [4]. Recently, digital I/Q imbalance and CFO compensation in OFDM systems have been an active area of research [5]-[9]. Besides its direct application to DCRs, another motivation is the fact that OFDM is very sensitive to front end effects [1]. Both I/Q imbalance and CFO cause intercarrier interference (ICI) which decreases the spectral efficiency of the OFDM system. In [5], a simple data aided approach to I/Q imbalance compensation was proposed. This estimation technique uses the smoothness of the estimated channel vector to estimate the I/Q imbalance parameters. In [8], the same authors used the maximum likelihood estimation scheme to cancel the effect of CFO [1] when it is present along with I/Q imbalance. This same approach was used in [6] but to further improve performance, another compensation block was added. The added block was a decision feedback equalizer block that uses demodulated data symbols as reference for adaptive removal of the residual I/Q imbalance from [5]. The authors claimed that additional 3dB gain of performance can be obtained by this revised method. The problem with [5] and [6] is that both disregard I/Q imbalance when estimating CFO. This causes errors in both CFO and I/Q imbalance estimation and limits the bit error rate (BER) performance of the estimator. To improve the accuracy of CFO and I/Q imbalance parameter estimation, [10] proposed a joint estimation scheme. The authors claimed that this algorithm aside from providing better accuracy, it is also computationally simpler compared to the earlier proposed algorithm in [9]. The contribution of this paper is the same as [10] but as will be shown in this paper, the proposed algorithm is significantly more accurate, more stable, and more scalable in terms of complexity. The proposed algorithm is actually a closed form solution in obtaining CFO and I/Q imbalance parameters. In the absence of noise, the proposed algorithm only need two samples each from two received identical data blocks. Any excess samples will help when estimating in the presence of uncorrelated input noise using the least squares algorithm. This paper is organized as follows. Section II presents the I/Q imbalance and CFO Model. In Section III, the proposed estimation algorithm is derived. Simulation results showing the performance of the proposed algorithm is presented on Section
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Fig. 1.
Fig. 2.
hence knowledge of the transmitted frequency domain symbols is not needed. Also, the algorithm assumes that sufficient guard interval is present such that in the absence of CFO, the two received training symbols are identical. The notations R and I are the real and imaginary components of the complex number while the (·) operation is the element by element vector multiplication or simply scalar multiplication. Many variables in the following derivation are used only for enhancing readability unless otherwise stated.
Direct Conversion Receiver
Direct Conversion Receiver with I/Q imbalance and CFO
IV. Finally, conclusions are given in Section V. II. F RONT E ND M ODEL Here, we briefly discuss the CFO and I/Q imbalance models used in this paper. For a more thorough discussion on these CFO and I/Q imbalance models, the reader is referred to [1], [4], and [8] and the references therein. I/Q imbalance exists whenever there is a mismatch between the In-phase (I) and Quadrature-phase (Q) branches of the receiver. Without loss of generality, let the magnitude imbalance 1+ε and phase difference be 2θ. In of the I and Q branch be 1−ε the absence of CFO, I/Q imbalance causes mirror subcarriers to interfere with each other and hence limits the performance of the system. CFO on the other hand is the result of frequency mismatch between the receiver local oscillator and the received signal carrier frequency and is unavoidable. In the absence of I/Q imbalance effect, CFO manifests as a simple cumulative phase rotation in the time domain. Fig. 3 shows the model of a system with both CFO and I/Q imbalance degradation. Normal methods that specialize in either I/Q imbalance or CFO estimation will usually not work due to the significant change in the output characteristics. In equation form the output baseband signal of Fig. 2 is xCIQ (t) = αx(t)ej2πΔf t + βx∗ (t)e−j2πΔf t . α = cos(θ) + jεsin(θ) β = εcos(θ) − jsin(θ)
(1) (2) (3)
where xCIQ (t) and x(t) are the received baseband signal with and without CFO and I/Q imbalance degradation respectively. The superscript (*) denotes complex conjugation while the variable Δf is the carrier frequency offset parameter. The new parameters for I/Q imbalance α and β were chosen since these variables produce easier to manage equations. III. P ROPOSED E STIMATION S CHEME The proposed algorithm needs two identical OFDM training symbols to estimate the CFO parameter Δf , and the two I/Q imbalance parameters α and β. It is a time domain algorithm
A. Derivation Let xCIQ,1 and xCIQ,2 be two received OFDM training symbol vectors from a receiver with both I/Q imbalance and CFO degradation, we can solve x1 and x2 from (1). Note that both lengths are equal to the FFT size N after the removal of the guard interval samples. We get, α∗ xCIQ,1 − βx∗CIQ,1 −j2πΔf t e (4) x1 = |α|2 − |β|2 α∗ xCIQ,2 − βx∗CIQ,2 −j2πΔf (t+T ) x2 = e (5) |α|2 − |β|2 where T is the period between the two symbols. Since, x1 = x2 , (4) and (5) can be combined to xCIQ,1 − Ψx∗CIQ,1 = Φ(xCIQ,2 − Ψx∗CIQ,2 ). β Ψ= ∗ α Φ = e−j2πΔf T
(6) (7) (8)
Notice that (6) can be thought of as a nonlinear system of equations consisting of N complex equations. These complex equations only have one complex unknown variable Ψ and one real unknown variable Δf . Hence, this is solvable by using only two of the N complex equations. Without loss of generality, we choose the first two equations to solve all the needed parameters. xCIQ,1,1 − Ψx∗CIQ,1,1 = Φ(xCIQ,2,1 − Ψx∗CIQ,2,1 ) xCIQ,1,2 −
Ψx∗CIQ,1,2
= Φ(xCIQ,2,2 −
Ψx∗CIQ,2,2 )
(9) (10)
The variable Ψ in (9) and (10) can then be eliminated algebraically as shown in (11). xCIQ,1,1 − ΦxCIQ,2,1 xCIQ,1,2 − ΦxCIQ,2,2 Ψ= ∗ = ∗ (11) xCIQ,1,1 − Φx∗CIQ,2,1 xCIQ,1,2 − Φx∗CIQ,2,2 This results in a quadratic equation in Φ with complex coefficients but can easily be simplified to a quadratic equation with real coefficients. m1 Φ2 + m2 Φ + m3 = 0 m1 = I{xCIQ,2,1 ·x∗CIQ,2,2 }
(12) (13)
m3 = I{xCIQ,1,2 ·x∗CIQ,1,1 }
(15)
m2 = I{xCIQ,1,2 ·x∗CIQ,2,1 } + I{xCIQ,2,2 ·x∗CIQ,1,1 } (14)
From (8) and (12), it is straightforward to show that m2 R{Φ} = − 2m1 Φ = R{Φ} + j 1 − R{Φ}2 .
(16) (17)
Then from (8) and (16), the estimated CFO is −1 ˜ = cos (R{Φ}) . Δf 2πT
estimate. To obtain the Least Squares estimate in the presence of noise, we generalize (16) for all samples as (18)
After computing Φ, the value of Ψ is solved directly from (11). This value will then be used to decouple the parameters α and β from (7). To do this, we need another equation in (19) relating α and β. R{β}I{β} α = 1 − I{β}2 − j 1 − I{β}2
The derivation of the solution will not be shown to save space. But it can be easily verified that the following is a solution to (20-23) [11]. χ = −I{Ψ}2 + R{Ψ}2 − 1 χ − χ2 + 4R{Ψ}I{Ψ} κ= 2R{Ψ}I{Ψ} 1 ϕ= 2 2 2 κ I {Ψ} + κ − 2κR{Ψ}I{Ψ} + R2 {Ψ} β˜ = κϕR{Ψ} + ϕI{Ψ} + j(κϕI{Ψ} − ϕR{Ψ})
(24)
α ˜ = κϕ + jϕ
(28)
(25) (26) (27)
To simplify this solution, it is desirable to remove the two square root operations using approximation methods. Again, it can be verified that using typical values of I/Q imbalance the following set of equations is a good approximation for (24-28).
α ˜ = κϕ + jϕ
(29) (30) (31) (32)
B. Least Squares Solution In (16), an expression for obtaining R{Φ} that leads to the CFO variable Δf was derived. With very high signal to noise ratio (SNR), (16) can give proportionally accurate estimate. In the typical scenario of low to moderate SNR, the rest of the samples of xCIQ,1 and xCIQ,2 is used to get a noise robust
(33)
bi =
−I{xCIQ,1,i+1 ·x∗CIQ,2,i }
(34) +
I{xCIQ,2,i+1 ·x∗CIQ,1,i } (35)
It then follows that the least squares estimate of R{Φ} is R{Φ}LS = (aT a)−1 aT b.
(19)
This equation can be verified from the definitions of α and β in (2) and (3). To finally compute for α and β, we need to solve a nonlinear system of equations that we will set up from (7) and (19). (20) R{α} = 1 − I{β}2 R{β}I{β} (21) I{α} = R{α} R{β} = R{α}R{Ψ} + I{α}I{Ψ} (22) I{β} = −I{α}R{Ψ} + R{α}R{Ψ} (23)
−I{Ψ}2 + R{Ψ}2 − 1 κ= R{Ψ}I{Ψ} 2 ϕ= 2 κ(I {Ψ} + 2) ˜ β = κϕR{Ψ} + ϕI{Ψ} + j(κϕI{Ψ} − ϕR{Ψ})
aR{Φ} = b. ai = 2I{xCIQ,2,i ·x∗CIQ,2,i+1 }
(36)
In a similar way, the least squares estimate for Ψ can be computed as ΨLS = (cH c)−1 cH d.
(37)
Φx∗CIQ,2,i
(38)
di = xCIQ,1,i − ΦxCIQ,2,i
(39)
ci =
x∗CIQ,1,i
−
C. Summary of Proposed Algorithm As a summary, from the two received OFDM training symbols xCIQ,1 and xCIQ,2 , the proposed algorithm to obtain I/Q imbalance and CFO parameters is as follows. ai = 2I{xCIQ,2,i ·x∗CIQ,2,i+1 }
bi = −I{xCIQ,1,i+1 ·x∗CIQ,2,i } + I{xCIQ,2,i+1 ·x∗CIQ,1,i }
R{Φ}LS = (aT a)−1 aT b Φ = R{Φ} + j 1 − R{Φ}2 ci = x∗CIQ,1,i − Φx∗CIQ,2,i di = xCIQ,1,i − ΦxCIQ,2,i ΨLS = (cH c)−1 cH d −I{Ψ}2 + R{Ψ}2 − 1 κ= R{Ψ}I{Ψ} 2 ϕ= κ(I2 {Ψ} + 2) −1 ˜ = cos (R{Φ}) Δf 2πT β˜ = κϕR{Ψ} + ϕI{Ψ} + j(κϕI{Ψ} − ϕR{Ψ}) α ˜ = κϕ + jϕ IV. S IMULATION R ESULTS The following simulations were performced to test the estimation accuracy as well as BER performance of the proposed algorithm compared to algorithms [5] and [10]. In regard to [5], since it’s only an I/Q imbalance algorithm, we used the CFO algorithm of the same authors in [8]. In all simulations, a system model compliant with 802.11a was used. The proposed algorithm similar to [5] uses the two long training symbols with 64 samples each defined in the 802.11a standard while [10] uses 9 of the 10 short training symbols with 16 samples each. Other common parameters used are 64QAM modulation, coding rate of 3/4 for convolutional encoder, and N=64 FFT size. For the channel, a pseudo stationary frequency selective channel based on TGN channel model B with additive white Gaussian noise (AWGN) was used [12]. The compensation
Fig. 3. Estimation Error of CFO vs. SNR. I/Q = [ε=0.1 and θ = 100 ], CFO=0.3
Fig. 4. Estimation Error of CFO vs. CFO value. I/Q = [ε=0.1 and θ = 100 ], SNR=30dB
method used after I/Q imbalance and CFO parameter estimation is exactly the same as in [5] and [11]. A. CFO Estimation Accuracy For CFO estimation accuracy, the following parameters were used, ε=0.1 and θ = 100 while the fractional CFO was 0.3 multiplied by the subcarrier spacing. All CFO values mentioned in the paper are all normalized by the subcarrier spacing unless otherwise stated. The choice for I/Q imbalance being higher than practical is mainly to get worst case condition performance for all the algorithms. As seen in Fig. 3, the proposed algorithm that completely used the two 64 sample LTS symbols has the best performance. The reduced complexity version that used only two samples has the worst performance in higher noise regions but surpasses [5] at about 35dB SNR. As seen the proposed algorithm needs only about 10 samples labeled as Proposed 10 to be robust from noise. However, Proposed 64 is still about 8dB better compared to Proposed 10 while Proposed 20 is seen to be 3dB better than Proposed 10. In Fig. 4, the CFO estimation error vs. CFO value is shown. The I/Q imbalance parameters used is the same as in Fig. 4 but this time the SNR is fixed to 30dB. This figure shows that the accuracy of CFO estimation varies greatly with the value of CFO being estimated. While the accuracy in [10] is better in higher CFO values, the accuracy becomes highly unstable at lower CFO values. The proposed algorithm doesn’t show this instability as seen in the discontinuity at CFO=0. A ceiling function must be applied however on (16) to prevent the noise from pushing its value outside the range of the cosine function. For CFO estimation, the proposed algorithm only consists of 8(Ns-1)+1 real multiplications, 5(Ns-1) additions and 1 division to obtain R{Φ} where Ns is the number of samples used . It follows that the complexity of Proposed 10 is only 73 multiplications, 45 additions and 1 division. B. I/Q Imbalance Estimation Accuracy For I/Q imbalance estimation accuracy, the same CFO and I/Q parameters as in Fig. 3 were used. In both Fig. 5 and Fig. 6,
Fig. 5. Estimation Error of α vs SNR. I/Q = [ε=0.1 and θ = 100 ], CFO=0.3
the full length proposed algorithm shows the best performance. The trend for all algorithms is almost the same as in Fig. 3. Again, it is observed that 10-20 samples are enough to be comparable with the performance of previous algorithms using significantly more samples. In Fig. 5, Proposed 10 is about 10dB below Proposed 64 and about 4dB below [10] in higher SNR regions while in Fig. 6, it is about 9dB below Proposed 64 and 5dB below [10]. It should be noted that I/Q imbalance parameter estimation performance depends on the accuracy of CFO estimation for all algorithms. Hence, in Fig. 5, the choice of CFO matters in the relative performance of the algorithms. The CFO of 0.3 was chosen to avoid unfair advantage of the proposed algorithm or disadvantage to either [5] or [10]. C. Bit Error Rate Performance Fig. 7 shows the BER performance of the different algorithms. The I/Q imbalance parameters used are ε = 0.1 and θ = 100 while the CFO is 0.1. For [5], I/Q imbalance estimation low accuracy creates a very high BER floor at our chosen system. On the other hand, the BER performance of the reduced sample version of the proposed algorithm is actually
Fig. 6. Estimation Error of β vs SNR. I/Q = [ε=0.1 and θ = 100 ], CFO=0.3
The algorithm shows excellent accuracy and stability in the estimation of CFO and I/Q imbalance parameters. While the derivation and simulation assumed an OFDM system, estimation can be done whenever two identical data blocks is received.In terms of complexity, the proposed method is O(N) meaning that the complexity is only linearly increasing with the symbol block size. The scalability of the proposed algorithm allows one to use only 10 samples each from two OFDM symbols to estimate CFO and I/Q imbalance with good accuracy. The same 10 sample algorithm results in BER performance of only about 1dB below a system without CFO nor I/Q imbalance issues. These results will benefit DCR-OFDM systems by reducing the effect of front end issues that limit theoretical performance of these systems. R EFERENCES
Fig. 7. BER vs SNR Performance of the Proposed Algorithm. I/Q = [ε=0.1 and θ = 100 ], CFO=0.1
very good despite the limited noise reduction gain compared to the full 64 sample estimate. Proposed 10 is seen to be only 1dB below a system without CFO nor I/Q imbalance issues. On the other hand, Proposed 16, 32 and 64 are all squeezed in between Proposed 10 and the case without front end degradation. V. C ONCLUSION A simple preamble based algorithm for CFO and I/Q imbalance compensation was proposed.
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