ML Estimation of Channel

Due to the smoothing criterion indicated in section 3.1, Φ is replaced with

ϕϕ

The I/Q imbalance coefficients ϕ are fixed first in (3.6) and the estimated time-domain response channel response h^ˆ_ML can be treated as maximum of Equation (3.6) in least square method. It can be expressed as

( ) ( ( ) )

^†

^{( )} (

^*

)

( )

4. However, this approximation is not enough, approximate again

( )

ϕ H⁻¹_I

W D

1 Phase Imbalance θ ^(degree)

Averge Magnitude ofφ per subcarrier

Fig.3-1 The average magnitude of Φ per subcarrier

Replace ^W

( )

^{ϕ with} H¹_I

D− , hence, the likelihood function is finally approximated as

(

¹

)

¹

⁽

^*

⁾

²

The frequency domain of I/Q imbalance coefficients can be obtained through FFT matrix.

ˆ_{ML ϕ}ϕˆ_ML Φ = F

3.4 Algorithm with Frequency Offset Compensation

So far, we haven’t considered the carrier frequency offset (CFO) effect at the receiver. In this section, we propose the CFO compensation method combined with our algorithm. The received signal path considering CFO effect can be derived as following

will be removed by the following LPFs

( ) ^{j ft} ( ) ^{j ft} ( ) ^{j fc f t} ( ) ^{j fc f t}

The finally resulting baseband signal model is given by

( ) ( )

^*

2 2

0 0

[ ] ^{j n} [ ] _CH[ ] [ ] [ ] ^{j n} [ ] _CH[ ] [ ] [ ]

r n =e ^πΔ s n ⊗h n +n n ⊗h n₊ +e^{− πΔ} s n ⊗h n +n n ⊗h n₋ After conjugate cancellation and CFO compensation in time domain, the modified algorithm is shown in equation (3.11).

For frequency domain compensation, we must modify the equation as following : ( Note that ϕ[ ]n is not [ ]ϕ n at all )

(

^{e−j ωnr n[ ]}

)

^{−ϕ[ ]n ⊗}

(

^{e−j ωnr n*[ ]}

)

^{=s n[ ]⊗h n[ ]+n n[ ]} (3.11)

Take FFT of (3.11)

(3.12)

Im Im_{age Comp}

Comp age Comp Comp R

R − Φ R ₋ =R −D ₋ Φ =SH +N

( )

^Im

(

^*

)

where R_Comp =FFT e^{−j ωn}r n[ ] , R _{age Comp−} =FFT e^{−j ωn}r n[ ]

Equation (3.12) similar to equation (2.14) illustrates how to apply proposed algorithm in frequency domain with CFO compensation. Note that we assume CFO has been estimated prior to our joint estimation. Therefore, we can get and

to pre-compensate CFO effect in (3.12) and then do joint estimation based on the signal model the same with (2.14) derived previously.

RComp R_{Image Comp−}

Chapter 4 Performance Analysis

In this chapter, the mean and mean square error (MSE) of the ML estimators are examined under the assumption of a high signal-to-noise ratio (SNR). Section 4.1 provides the analysis of the mean and MSE of our proposed estimator of I/Q imbalance , and the mean and MSE of our proposed estimator of channel H are analyzed in section 4.2.

Φ

The relation between the true I/Q imbalance Φ and its estimates _N×1 Φ^ˆ_ML can be expressed as (4.2) derived from above equation.

(

^{1 *}

) (

¹

)

^{1 * 1}

⁽

^{1 *}

⁾ (

¹

)

¹ Therefore, the estimators are approximately unbiased when . Specifically, the bias of

The MSE of Φ^ˆ_ML can be approximated as following

The mean and MSE of H defined in (3.8) can be derived in the way similar to the above procedure. Start from (3.8) :

(4.3) Substitute (4.1) into (4.3)

( ) ( ) ( ) ( ) ( )

as (4.3) derived from above equation.

( )

^†

( )

^*

( )

Substitute (4.4) and (3.3) into (4.3) Therefore,

( ) ( ) ( )

Chapter 5 Derivation of Cramer-Rao Lower Bounds

The approximated conditional probability density function is shown below:

{

^{R* ,}

}

ⁿ_det¹

_{( )}

^exp

{ ⁽

^{R* h}

⁾

^{H 1}

⁽

^{* h}

⁾ }

A. We express f( )θ in scalar form :

where denotes elementwise product

1 1

C. DefineR as the deterministic part of_a :

to calculate E ^{f( )}ω_* ^{f( )}ω_* Note that for complex value

0 Then, use the above scalar form to calculate their self and cross expectation as following :

b.

{

^{0 0* 0 0*}

} ^{ _{

^{0 4}

_}

⁰ 4

^}

⁰

{ }

* *

( )

G. Then, define ^{( )}_* ^{( )}_*

CRLB( _k) _h ^H which can be nurmerically computed through computer simulation

ϕ ϕh kk

ω ⁻

∴ = F I F

( )

where _kk is its k'th diagnoal value

Chapter 6 Simulation Results

Computer simulations were conducted to evaluate the performance of proposed scheme. In the first part of simulation, the analytical results and CRLBs were confirmed and compared to the simulation results. Then, we examined performances and sensitivities of the proposed compensation scheme in the second and third part of simulation.

An OFDM system designed for IEEE 802.11a WLAN standards is considered in our simulation. The OFDM symbol is based on total 64 carriers (48 for data, 4 for pilot and others left open) uniformly distributed in 20 MHz channel bandwidth in RF band. The modulations on each carrier range from BPSK, QPSK, 16-QAM to 64-QAM.. Cyclic prefix is copy of the last quarter of each OFDM symbol. The 802.11a standards also specify one short preamble for synchronization and one long preamble for channel estimation. Our estimation is using the long preamble as prior information to do joint estimation. Following model described in Equation 1, a multi-path channel effect is constructed as a three-taps complex-valued FIR whose phases are uniformly random distributed and magnitudes are Rayleigh distributed with averaged power decaying exponentially.

For frequency-independent IQ imbalance, amplitude g and phase imbalance θ are set to be 1.08 and . As to frequency-dependent IQ imbalance, it is modeled in terms of impulse responses of baseband IQ branches and . Due to variation of analog components, these two filters are modeled based on order-3 of low-pass Butterworth filters with different cutoff frequencies at 8.5 MHz and 8.2 MHz. The sampling rate is set to be 20 MHz.

5

] [n

h_I h_Q[n]

6.1 MSE of ML Estimates

The MSE analytical expressions derived in chapter 4 were checked using fixed channel given by

h=[0.7047 + 0.7047i , 0.0578 + 0.0578i , 0.0047 + 0.0047i]

(L=3)

Also, the CRLBs derived in chapter 5 were compared to our estimators. The MSE of I/Q imbalance Φ is calculated as its average MSE per subcarrier, and channel H is also calculated as its average MSE per subcarrier.

Figure 6-1 and Figure 6-2 show means-squared-error (MSE) of IQ imbalance and delay-spreading channel estimation. The estimation error is computed averagely in frequency domain per subcarrier. It was observed that the computer simulation results of proposed estimators coincide with the MSE analyses in high SNR, and there still exists slight mismatch in the low SNR level due to some approximations that can not be made in low SNR. Furthermore, the MSE were almost identical to the CRLBs.

Figure 6-2 also compare the performance of channel estimator with and without smoothing. The “Smoothing” refers to our proposed estimator which estimates channel impulse response in time domain, while the “No Smoothing” refers to estimating the channel independently between subcarriers in frequency domain. As expected, the smoothing property indeed obtains a large performance gain over channel estimation.

0 5 10 15 20 25 30 35 40

MSE of φ (per subcarrier), Simulation V.S Analysis CRLB

SNR(dB)

MSE

Simulation MSE of φ Analysis MSE of φ CRLB of φ

Fig.6-1 MSE of IQ imbalance estimation

0 5 10 15 20 25 30 35 40

MSE of H (per subcarrier), Simulation V.S Analysis CRLB

SNR(dB)

MSE

Simulation MSE of H (No Smoothing) Simulation MSE of H (Smoothing) Analysis MSE of H

CRLB of H

Fig.6-2 MSE of delay-spreading channel estimation

6.2 Sensitivities of Proposed Estimators

In practical situation, the effective length of channel and I/Q imbalance ϕ need to be estimated, thus could not be optimal and suffers from some modeling error in a real front-end filter equivalent to an infinite impulse response (IIR) filter.

It can be observed in Fig.6-3 and 6-4 that there indeed exists an optimal length for our given I/Q imbalance and channel case. In addition, the performance will be saturate when is chosen shorter than its effective optimal length, however, it suffers from just a little acceptable performance loss when is chosen larger than its equivalent optimal length. Hence, this result suggests us to choose larger channel length to maintain a good performance level.

h 11

Fig.6-3 Channel estimation with variousL_h

0 5 10 15 20 25 30 35 40 Fig.6-4 I/Q imbalance

Φ

estimation with variousL_h

0 5 10 15 20 25 30 35 40

Fig.6-5 I/Q imbalance

Φ

estimation with various L_ϕ

0 5 10 15 20 25 30 35 40 10^-4

10^-3 10^-2 10^-1 10⁰

SNR(dB)

MSE

MSE of φ (No CFO) MSE of φ (CFO = 0.005 ) MSE of φ (CFO = 0.01 ) MSE of φ (CFO = 0.02 ) MSE of φ (CFO = 0.04 )

MSE of φ (CFO = 0.04 with Compensation)

Fig.6-6 I/Q imbalance

Φ

estimation with various CFO effect

In Fig.6-5, different length L_ϕ is checked. When SNR is smaller, AWGN noise

dominates interferences, and it suggests us to chose smaller length L_ϕ. On the contrast, when SNR is larger, I/Q imbalance may dominate interferences, then the performance will be better in choosing longer length L_ϕ. Hence, the length L_ϕ depends on operating SNR.

Finally, the severity of CFO effect on MSE and effectiveness of our proposed CFO compensation scheme in section 3.4 are examined in fig.6-6. The CFO is normalized to subcarrier spacing. It is shown that our estimation scheme can suffer CFO effect lower than 0.01. Thus, the residue of estimated CFO should be smaller

than 0.01 to make our algorithm work. “ CFO =0.04 with Compensation” legend means that applying proposed CFO pre-compensation at receiver with CFO being 0.04. It can be seen that our proposed CFO compensation is effective to combat CFO effect.

6.3 Uncoded BERs Performance

The performances of the estimators were also examined in terms of the uncoded bit error rate (BER). “IQ imbalance/No Comp” refers to a receiver with I/Q imbalance but no compensation. “IQ imbalance/Comp with No smoothing” refers to ordinary OFDM channel estimation in frequency domain; whereas, “IQ imbalance/Comp with smoothing” refers to our proposed estimators in time domain. “Ideal Receiver” refers to a receiver with no I/Q imbalance. As expected, the BER curve becomes saturated in the presence of I/Q imbalance effect; on the other hand, the performances of our proposed algorithms is close to ideal receivers and outperforms the traditional OFDM channel estimation done in frequency domain.

10 15 20 25 30 35 40

10^-4 10^-3 10^-2 10^-1 10⁰

SNR(dB)

Uncoded BER

IQ Imbalance/No Comp

IQ Imbalance/Comp with No Smoothing IQ Imbalance/Comp with Smoothing Ideal Receiver(No IQ imbalance)

Fig.6-7 Uncoded BERs with 64 QAM

0 5 10 15 20 25 30 35 40 10^-4

10^-3 10^-2 10^-1 10⁰

64 QAM Uncoded BER with CFO effect

SNR(dB)

Uncoded BER

Uncoded BER (Ideal,No IQ Imbalance/CFO) Uncoded BER (CFO = 0.005 )

Uncoded BER (CFO = 0.01 ) Uncoded BER (CFO = 0.02 ) Uncoded BER (CFO = 0.04 )

Uncoded BER (CFO = 0.04) with Comp

Fig.6-8 CFO sensitivities on 64 QAM Uncoded BERs

The last Fig.6-2 shows the sensitivity of CFO effect on uncoded BER performance and effectiveness of our proposed CFO compensation scheme in section 3.4. The same results with MSE performance can be seen in terms of uncoded BERs.

The BERs get saturate again when estimated CFO residue error is larger than 0.01. In addition, our proposed CFO compensation scheme combined with our proposed algorithm also makes it work to achieve performance target assuming CFO has been estimated previously within certain level.

Chapter 7 Conclusions

An algorithm for joint estimation of channel and IQ imbalance effects (both of frequency-independent and frequency-dependent) for OFDM system was developed.

To make use of characteristics of OFDM symbol and channel smoothing property, we construct our likelihood function using frequency data to estimate time-domain channel and IQ imbalance effects. Also, we develop carrier frequency offset compensation method combined with our algorithm assuming that carrier frequency offset has been estimated prior to our estimation. The estimation requires only one OFDM symbol as prior information to reaches performance target. No special structure for such OFDM symbol is assumed. The performance was investigated analytically and by computer simulation, which shows that the proposed algorithm reaches CRLBs as the received SNR above certain level. We also observed that our performance depends on prior information about channel length of delay-spreading channel effect and degree of difference of IQ imbalance. Some future topics about this work can be extended and researched.

(1) Apply to MIMO-OFDM systems that may take advantage of spatial diversity gains to improve performances.

(2) Take into account transmitter side I/Q imbalance and DC offset effect in MIMO technique.

(3) Develop CFO estimation method in the presence of I/Q imbalance effect.

References

[1] R. V. Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House Publisher, 2000.

[2] M. Engels, Wireless OFDM Systems: How to make them work, Kluwer Academic Publisher, 2002.

[3] IEEE, “Part 11: wireless LAN medium access control (MAC) and physical layer (PHY) specifications: High speed physical layer in the 5-GHz band,” IEEE Std 802.11a-1999, Dec. 1999.

[4] Myoung-Soo Kang and Woo-Jin Song, "A robust channel equalizer for OFDM TV receivers," IEEE Transactions on Consumer Electronics, Volume 44, Issue 3, Page(s): 1129 - 1133, Aug. 1998 OFDM one tap equalizer

[5] Whang, A.J., Yi-Uan Chen and Rui-Xin Whang, "Performance evaluation of OFDM system for wideband wireless LAN with one-tap equalizer", IEEE 2002 International Conference on Communications, Circuits and Systems and West Sino Expositions, Volume 1, Page(s): 521 - 524 vol.1, 29 June-1 July 2002 [6] Sun-Wook Kim and Kyun Hyon Tchah, "Performance analysis of adaptive

equalizer design for OFDM wireless LAN", IEEE Transactions on Consumer Electronics, Volume 50, Issue 2, Page(s): 512 - 516, May 2004

[7] A.A. Abidi; “Direct-conversion radio transceivers for digital communications”, IEEE Journal Solid-State Circuits, Volume 30, Issue 12, Page(s): 1399–1410, Dec. 1995.

[8] B. Razavi, “Design considerations for direct-conversion receivers”, IEEE Transactions Circuits and Systems II: Analog and Digital Signal Processing, Volume 44, Issue 6, Page(s): 428–435, June 1997

[9] Baier, A., "Quadrature mixer imbalances in digital TDMA mobile radio receivers", International Zurich Seminar on Digital Communications, 'Electronic

Circuits and Systems for Communications'. Proceedings, Page(s): 147 - 162 5-8 March 1990

[10] Chia-Ling Liu, "Impacts of I/Q imbalance on QPSK-OFDM-QAM detection", IEEE Transactions on Consumer Electronics, Volume 44, Issue 3, Page(s): 984 - 989, Aug. 1998

[11] Ykaczewski, P.; Brakensiek, J.; Jondral, F.K., "Towards an analytical model of I/Q imbalance in OFDM based direct conversion receivers", Vehicular Technology Conference, Volume 4, Page(s): 1831 - 1835 Vol.4, 17-19 May 2004 [12] M. Valkama, M. Renfors, V. Koivunen, “Compensation of frequency-selective I/Q imbalances in wideband receivers: models and algorithms”, IEEE 3rd Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Page(s): 42-45, March 2001.

[13] M. Valkama, M. Renfors, V. Koivunen, “Advanced methods for I/Q imbalance compensation in communication receivers”, IEEE Transactions Signal Processing, Volume 49, Issue 10, Page(s): 2335–2344, Oct. 2001.

[14] T.M. Ylamurto, “Frequency domain IQ imbalance correction scheme for orthogonal frequency division multiplexing (OFDM) systems”, IEEE Wireless Communications and Networking, Volume 1, Page(s): 20–25, March 2003.

[15] J. Tubbax, B. Come, L. VanderPerre, S. Donnay, M. Engels, H. DeMan, M.

Moonen, “Compensation of IQ Imbalance and Phase Noise in OFDM Systems”, IEEE Transactions Wireless Communications, Volume 4, Issue 3, Page(s):

872–877, May 2005.

[16] G. Gye-Tae, S. Il-Hyun, P. Jin-Kyu, Y.H. Lee, “Joint ML estimation of carrier frequency, channel, I/Q mismatch, and DC offset in communication receivers”, IEEE Transactions Vehicular Technology, Volume 54, Issue 1, Page(s): 338–349, Jan. 2005.

[17] A. Tarighat, R. Bagheri, and A. H. Sayed, “Compensation schemes and performance analysis of IQ imbalances in OFDM receivers”, IEEE Trans. Signal Process., vol. 53, no. 8, pp. 3257–3268, Aug. 2005

[18] X. Guanbin, S. Manyuan, L. Hui, “Frequency offset and I/Q imbalance compensation for direct-conversion receivers”, IEEE Transactions Wireless Communications, Volume 4, Issue 2, Page(s): 673–680, March 2005.

[19] A. can den Bos, ”A Cramer-Rao bound for complex parameters,” IEEE Trans.

Signal Processing, vol. 42, p. 2859, Oct. 1994

[20] Deneire, L.; Vandenameele, P.; van der Perre, L.; Gyselinckx, B.; Engels, M.; “A low complexity ML channel estimator for OFDM,” Communications, 2001. ICC 2001. IEEE International Conference on Volume 5, 11-14 June 2001 Page(s):1461 - 1465 vol.5

在文檔中 正交分頻多工系統之通道估計與I/Q失衡補償 (頁 20-0)