Blind Signal Separation-Based Frequency-dependent I/Q Imbalance Compensation for Direct-conversion Receivers

Research Paper / Jan 2009

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Blind Signal Separation-Based Frequency-dependent I/Q Imbalance Compensation for Direct-conversion Receivers

Research Paper / Jan 2009

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Blind Signal Separation-Based Frequency-dependent I/Q Imbalance Compensation for Direct-conversion Receivers �

Chia-Pang Yen, Yingming Tsai, and Robert Olesen

InterDigital Communications LLC

2 Huntington Quadrangle

Melville, NY 11747, USA

Chia-Pang.Yen@, Yingming.Tsai@, Robert.Olesen@interdigital.com

�

Abstract— Frequency-dependent I/Q imbalance is one of the concerns in the design of a high data rate wireless wideband direct-conversion receiver (DCR). To address this challenge, a blind signal separation method, based on a conjugate signal model is proposed. Necessary and sufficient conditions for signal separation are derived, and a gradient-based method is used to solve this problem. The performance of our approach is evaluated for orthogonal frequency division multiplexing (OFDM) systems.

Keywords: Radio frequency (RF), direct-conversion receiver (DCR), I/Q imbalance, blind signal separation, orthogonal frequency division multiplexing (OFDM) signals.

Introduction

The evolution of wireless communication systems has driven the design and implementation of the radio transceivers. Next generation of wireless communication networks (for example, 3GPP LTE system) will support a high data rate, which requires efficient wideband radio design in the terminals. Driven by those demands, I/Q down-conversion have become a key design issue in addressing the need for cost- and size-efficient transceiver implementation.

In commonly used direct-conversion front-end receivers, the received signal is I/Q down-converted from RF directly to a baseband signal. Due to an imperfect oscillator, and low pass filters in the RF front-end receiver, the In-phase and Q-phase signal paths will inevitably have mismatches of amplitude and phase. This causes interference upon down conversion to baseband. Direct-conversion receivers are vulnerable to I/Q imbalance since I/Q separation is performed early in the RF/analog portion. The impact of I/Q imbalance is more severe to a system which uses high order modulations, and a high coding rate. Therefore, I/Q imbalance correction is essential for the design of higher data rate systems.

Many I/Q imbalance estimation and compensation methods that require specific training signals patterns have been proposed, see for example [5], [6]. However, the requirement for specific training patterns may not be a practical assumption. Thus, there is a need for blind signal processing methods. To the best of our knowledge, there are not many blind frequency-dependent I/Q imbalance compensation methods for DCR published. Authors in [3] and [4] propose methods using the statistical properties of the source signal. The contributions of this paper are two-fold. First, a necessary and sufficient condition for separation of the convolutive mixtures of the desired signal and its image using blind estimation is proved. Second, based on this condition, a compensation method is proposed and demonstrated.

The rest of the paper is organized as follows. In Section II, the frequency-dependent I/Q imbalance model are described. In Section III, The conditions for signal separation are derived and a gradient based method is used to solve for a separation filter. In Section IV., the simulation results are obtained for an OFDM system. Finally, conclusions are drawn in Section V.

Notation: Bold face upper case symbols, e.g.,

denotes matrices, and bold face lower case symbols, e.g.,

, denotes column vectors. The

the element of matrix

is denoted as

.

is a composite vector obtained by stacking column vectors together. The asterisk * denotes complex conjugate.

is the expected value.

is the Frobenius norm of

.

denotes conjugate transpose of a matrix.

denotes the trace of a matrix.

is used to denote the linear convolution.

I/Q imbalance Signal model and second-order statistic property

System Model

A typical block diagram of the RF front-end for DCR is presented in Fig. 1(a), and its equivalent mathematical model is given in Fig. 1(b). The received RF signal

is expressed as,

(1)

where

is the received baseband signal

, and

, and

is the transmitted signal. The received signals

are direct downconverted by a local oscillator signal

where

and

are the the mismatched gain and phase.

The downconverted signal

is expressed as

(2)

where

and

denote low pass filters for the

and

branch shown in Fig. 1(b) and

. After carrying out the low pass filtering and transforming equation (2) to the frequency domain, the I and Q branch signals are

(3)

and

. (4)

Then the received frequency domain baseband signals

can be expressed as

where

(5)

and

. (6)

or equivalantly in the time-domain,

. (7)

Hence, we can see that due to the I/Q imbalance, the received down-conversion signal

is distorted by its image signal

.

To evaluate the I/Q imbalance distortion, the analog front-end image-reject ratio (IRR) [1], in decibels (dB), is defined as

(8)

After sampling the continuous time signal

with sampling interval

, the discrete time signal is

. (9)

Second-order statistics

The discrete-time autocorrelation function (ACF) of a complex random signal

is defined as

. Another second-order statistics, the complementary autocorrelation function (CACF), is defined as

. If a complex random signal

is proper, then its CACF is equal to zero for all lag

[7], i.e.,

. In this paper, we assume the transmitted complex signal

is a zero-mean white process, i.e.,

and is proper, i.e.,

(10)

where

denote the Dirac delta function. The above assumption is a reasonable assumption because most communication systems, specifically those which have encoders or interleavers to achieve this condition.

In this paper, we further assume that the channels are wide-sense stationary uncorrelated scattering (WSSUS). It can be shown that a white and proper process, after a WSSUS channel, remains white and proper. The signal after a time-varying channel can be expressed as,

. The autocorrelation of a WSSUS channel impulse response is given by [14],

�� EMBED Equation.3

where

is a function that depends on

,

is the zero-order Bessel function of first kind, and

represents the maximum Doppler shift. Using this autocorrelation, the ACF of received signal

can be computed as,

, thus it is still a white process. Similarly, its CACF can be evaluated as,

.

The properness of

follows from the fact that

is proper.

Blind signal separation for I/Q imbalance using conjugate signal model

If the filters

and

are modeled by

tap length FIR filters [8], the time-domain frequency-selective I/Q imbalance in (9) can be rewritten in the following conjugate model:

(11)

where

,

and

where

.

The objective of signal separation is to make the filter output a scaled and delayed version of the input [13]. Mathematically, the goal is to find a separation filter

with length

such that the signals and its image,

and

, are separated at the filter output, i.e.,

(12)

where

is a

permutation matrix, either

or

,

is a

diagonal scaling matrix and

is a delay.

In order to achieve the separation, prior information about the source signals, i.e., conjugation between these two signals needs to be imposed. This motivates us to come up with a cost function that incorporates this conjugate constraint. With this constraint, and the goal to restore the property of signal

, a necessary and sufficient condition for signal separation can be proved as in Lemma 1.

Remark 1: After the separation filter, the output is a delayed and filtered version of

.Therefore, the delay and the effective channel should be estimated then equalized prior to decoding. This is not a big issue since although a known training signal does not exist but most systems do have pilot symbols. For example, in LTE, the pilot symbols are inserted every six subcarriers in the frequency domain. These pilot symbols can be used for estimating the delay and equalization. Since these are not the scope of this paper, they are assumed to be perfectly known in the simulations. The permutation matrix

can also be identified in this manner.

Remark 2: Since

is white and proper, it can be shown that the autocorrelation matrix of

equal to a diagonal matrix,

(13)

Conditions for signal separation

In order to satisfy the conjugate condition for the elements of the filter output, i.e.

, we need to impose the following constraints on the elements of the filter matrices, i.e.

and

.

Define the composite vector

. By equation (12), this composite vector

can be written as

(14)

where

is a Toeplitz matrix composed of

and

.

We are now ready to derive the conditions which lead to signal separation.

Lemma 1: Under the constraints

and

,

are separator matrices, that is

, if and only if

Proof: The sufficient condition can be shown by plugging equation (12) into (14) then using equation (13). That is

, when

.

Now we proceed to prove the necessary condition. By looking at the matrix equation

element by element, i.e.

,

we can obtain the following conditions for

.

(15)

and

(16)

From equation (12), the filter output

can be written as

(17)

where

and

are filters depend on

and

. Then from (15)

(18)

where

is signal power and

. One can conclude that equation (18) can be satisfied when

or

is an anti-symmetric thus must be a non-causal filter. The latter case can be excluded and we can conclude that

. Without loss of generality, let us assume

. Then

in (17) becomes

(19)

Plug equation (18) into equation (15), and we have

, (20)

which implies

is an all-pass filter. It can be shown that for FIR all-pass filters, all the zeros are at the origin of the

-plane, thus, must have a linear phase. Therefore,

for delay

. One can see that the output of the filter is just a scaling and delayed version of

, i.e.

. Similarly, if

, then

which leads to

. Recall that, by construction,

, therefore

.

Implementation

In Lemma

, we prove that the autocorrelation matrix of filter output signal

is the necessary and sufficient condition for signal separation. Therefore, we define the following cost function

. (21)

Then the separation matrices

can be obtained by minimizing the cost function

. (22)

Here we proposed a gradient-based method [11] to iteratively solve the problem. At each step, the gradient of the cost function, i.e.

, is computed and the argument

is updated by stepping along the gradient direction. Note that in complex domain, the gradient direction is the derivative with respect to

instead of

.

Based on the fact that

equation (21) can be expressed as

(23)

where matrix

is defined as the

submatrix of

, where

is composed of the

submatrices

.

The term

can be expanded as

��EMBED Equation.3

(24)

Similarly, the term

is expressed as

(25)

Observing equations (24) and (25), we find out that the derivatives needed are of the form

and

, where

and

are

submatrices from

. Based on [12], the results of these derivatives are given as follows, without lengthy step by step details. Note that the gradient is the derivative w.r.t.

instead of

[11].

(26)

where

.

Similarly,

(27)

The cost function is minimized iteratively by the learning rule

(28)

where

is the matrices at

the step,

is the learning rate and

.

At each step,

is also updated by

(29)

Performance Analysis and Numerical Results

In this section, simulation results are presented to demonstrate the performance of the proposed blind compensation scheme for OFDM systems. The simulated OFDM system is with symbol length

subcarriers, guard symbol

, and subcarrier spacing

kHz [9]. Hence - the total symbol length is equal to

samples per OFDM symbol. We consider two different cases. Case 1_3-tap LPF: The I/Q imbalance parameters are; gain mismatch

, phase mismatch

and I and Q branch LPF

,

respectively [8] where the analog IRR is in the range of 18-25 dB [Fig. 2(a)]. Case 2_ 2-tap LPF: The frequency-dependent I/Q imbalance parameters are

,

,

and

[8] where the analog front end IRR is around 26-35 dB [Fig. 3(a)]. Both simulations use the learning rate

and with initial values

,

and

. This is justified by the fact that, in practical, the LPFs on both I and Q branch are close to an ideal, thus its impulse response is close to a delta function. In the simulations, 50,000 samples are used for estimating the autocorrelation function, 64-QAM modulation is used and the received SNR is 20dB. The convergence rate is shown in Fig. 2(b) and 3(b). The proposed method converges in approximately

and

iterations for case 1 and 2 respectively. In case 2 the LPFs are more ideal than in case 1, therefore it is not surprising that the simulation converges faster in case 2. In Fig.2 and 3, the performance of the proposed method is shown in terms of IRR. Figure 2(a) and 3(a) show that the IRR is improved by around 20 dB after I/Q imbalance compensation. Fig. 4 and 5 show the Symbol Error Rate (SER) performance before and after compensation. The results show that there is a need to compensate for the I/Q imbalance in both cases, since without compensation, the performance is unacceptable. Fig. 4 and 5 also show that the proposed compensation scheme provides an effective removal of the imbalance, and also improves the BER performance.

Conclusion

In this paper, a blind signal separation approach is proposed to compensate for a frequency-dependent I/Q imbalance in a wideband direct-conversion receiver. Necessary and sufficient conditions for signal separation are derived. Based on the conditions a gradient-based method is used to solve for the separation filter. Simulation results show the proposed approach effectively removes distortion caused by the I/Q imbalance.

References

P. Kiss and V. Prodanov, “One-tap wideband I/Q compensation for zero-IF filters,” IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications, vol. 51, pp. 1062–1074, June 2004.

A. A. Abidi, “Direct-conversion radio transceivers for digital communications,” IEEE Journal of Solid-State Circuits, vol. 30, pp. 1399–1410, Dec. 1995.

M. Valkama, M. Renfores, and V. K.oivunen, “Blind I/Q signal separation based solutions for receiver signal processing,” in EURASIP Journal on Applied Signal Processing - Special Issue on DSP Enabled Radios, vol. 2005, no. 16, pp. 2708-2718, Sept. 2005.

L. Anttila, M. Valkama, and M. Renfors, “Circularity-based I/Q imbalance compensation in wideband direct-conversion receivers ” Proc. IEEE,Trans. Vehicular Technology, vol 99, issue 99 pp 1-15, Sep 2007.

A. Tarighat, R. Bagheri, and A. H. Sayed, “Compensation schemes and performance analysis of I/Q imbalances in OFDM receivers,” IEEE Trans. Signal Processing, vol. 53, no. 8, pp. 3257–3268, Aug. 2005.

A. Schuchert, R. Hasholzner, and P. Antoine, “A novel IQ imbalance compensation scheme for the reception of OFDM signals,” IEEE Transactions on Consumer Electronics, vol. 47, no. 3, pp. 313–318, Aug. 2001.

F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans.

Inform. Theory, vol. 39, no. 4, pp. 1293–1302, Nov. 1993.

M. Valkama, M. Renfores, and V. K.oivunen, “Compensation of frequency-selective I/Q imbalances in wideband receivers: Models and algorithms,” in Proc. IEEE Int. Workshop on Signal Processing Advances for Wireless Commun. (SPAWC’01), 2001, pp. 42–45.

Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation, 3GPP Std. 36.211, V8.2.0, Mar. 2008. [Online]. Available: http://www.3gpp.org

J. G. Proakis, Digital Signal Processing - Principles, Algorithms, and Applications. Prentice Hall, 1996.

S. Haykin, Adaptive Filter Theory. Prentice Hall, 1996.

A. HØjrungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal

Processing, vol. 55, no. 6, pp. 2740–2746, June 2007.

A. Cichocki and S. Amari, Adaptive blind signal and image processing, Wiley, 2005.

P. Hoeher, “A statistical discrete-time model for the WSSUS multipath channel,” IEEE Trans. Vehicular Technology., vol. 41, no. 4, pp461-468, Nov. 1992.

Figure 1: (a) Frequency-dependent I/Q imbalance of direct-conversion receiver (b) Mathematical model of a DRR with I/Q imbalance (c) Equivalent mathematical model of (b).

Figure 2: 3-tap LPF case

(a) IRR in before and after compensation (b) Convergence of the cost function.

Figure 3: 2-tap LPF case

a) IRR before and after compensation (b) Convergence of the cost function.

Figure 4: Symbol Error Rate for 3-tap LPF case

Figure 5: Symbol Error Rate for 2-tap LPF case

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�PAGE �

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(overflow,LoP)

1

I

LNA

LPF

LPF

A/D

A/D

Q

I/Q LO (gd, Φ)

I/Q Imbalance compensator

Analog Front End

Digital Baseband

y(t)

r(t)

AGC

(a)

I

GI(f)

GQ(f)

I/Q LO

r(t)

Q

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Chia-Pang Yen, Yingming Tsai, and Robert Olesen

InterDigital Communications LLC

2 Huntington Quadrangle

Melville, NY 11747, USA

Chia-Pang.Yen@, Yingming.Tsai@, Robert.Olesen@interdigital.com

�

Abstract— Frequency-dependent I/Q imbalance is one of the concerns in the design of a high data rate wireless wideband direct-conversion receiver (DCR). To address this challenge, a blind signal separation method, based on a conjugate signal model is proposed. Necessary and sufficient conditions for signal separation are derived, and a gradient-based method is used to solve this problem. The performance of our approach is evaluated for orthogonal frequency division multiplexing (OFDM) systems.

Keywords: Radio frequency (RF), direct-conversion receiver (DCR), I/Q imbalance, blind signal separation, orthogonal frequency division multiplexing (OFDM) signals.

Introduction

The evolution of wireless communication systems has driven the design and implementation of the radio transceivers. Next generation of wireless communication networks (for example, 3GPP LTE system) will support a high data rate, which requires efficient wideband radio design in the terminals. Driven by those demands, I/Q down-conversion have become a key design issue in addressing the need for cost- and size-efficient transceiver implementation.

In commonly used direct-conversion front-end receivers, the received signal is I/Q down-converted from RF directly to a baseband signal. Due to an imperfect oscillator, and low pass filters in the RF front-end receiver, the In-phase and Q-phase signal paths will inevitably have mismatches of amplitude and phase. This causes interference upon down conversion to baseband. Direct-conversion receivers are vulnerable to I/Q imbalance since I/Q separation is performed early in the RF/analog portion. The impact of I/Q imbalance is more severe to a system which uses high order modulations, and a high coding rate. Therefore, I/Q imbalance correction is essential for the design of higher data rate systems.

Many I/Q imbalance estimation and compensation methods that require specific training signals patterns have been proposed, see for example [5], [6]. However, the requirement for specific training patterns may not be a practical assumption. Thus, there is a need for blind signal processing methods. To the best of our knowledge, there are not many blind frequency-dependent I/Q imbalance compensation methods for DCR published. Authors in [3] and [4] propose methods using the statistical properties of the source signal. The contributions of this paper are two-fold. First, a necessary and sufficient condition for separation of the convolutive mixtures of the desired signal and its image using blind estimation is proved. Second, based on this condition, a compensation method is proposed and demonstrated.

The rest of the paper is organized as follows. In Section II, the frequency-dependent I/Q imbalance model are described. In Section III, The conditions for signal separation are derived and a gradient based method is used to solve for a separation filter. In Section IV., the simulation results are obtained for an OFDM system. Finally, conclusions are drawn in Section V.

Notation: Bold face upper case symbols, e.g.,

denotes matrices, and bold face lower case symbols, e.g.,

, denotes column vectors. The

the element of matrix

is denoted as

.

is a composite vector obtained by stacking column vectors together. The asterisk * denotes complex conjugate.

is the expected value.

is the Frobenius norm of

.

denotes conjugate transpose of a matrix.

denotes the trace of a matrix.

is used to denote the linear convolution.

I/Q imbalance Signal model and second-order statistic property

System Model

A typical block diagram of the RF front-end for DCR is presented in Fig. 1(a), and its equivalent mathematical model is given in Fig. 1(b). The received RF signal

is expressed as,

(1)

where

is the received baseband signal

, and

, and

is the transmitted signal. The received signals

are direct downconverted by a local oscillator signal

where

and

are the the mismatched gain and phase.

The downconverted signal

is expressed as

(2)

where

and

denote low pass filters for the

and

branch shown in Fig. 1(b) and

. After carrying out the low pass filtering and transforming equation (2) to the frequency domain, the I and Q branch signals are

(3)

and

. (4)

Then the received frequency domain baseband signals

can be expressed as

where

(5)

and

. (6)

or equivalantly in the time-domain,

. (7)

Hence, we can see that due to the I/Q imbalance, the received down-conversion signal

is distorted by its image signal

.

To evaluate the I/Q imbalance distortion, the analog front-end image-reject ratio (IRR) [1], in decibels (dB), is defined as

(8)

After sampling the continuous time signal

with sampling interval

, the discrete time signal is

. (9)

Second-order statistics

The discrete-time autocorrelation function (ACF) of a complex random signal

is defined as

. Another second-order statistics, the complementary autocorrelation function (CACF), is defined as

. If a complex random signal

is proper, then its CACF is equal to zero for all lag

[7], i.e.,

. In this paper, we assume the transmitted complex signal

is a zero-mean white process, i.e.,

and is proper, i.e.,

(10)

where

denote the Dirac delta function. The above assumption is a reasonable assumption because most communication systems, specifically those which have encoders or interleavers to achieve this condition.

In this paper, we further assume that the channels are wide-sense stationary uncorrelated scattering (WSSUS). It can be shown that a white and proper process, after a WSSUS channel, remains white and proper. The signal after a time-varying channel can be expressed as,

. The autocorrelation of a WSSUS channel impulse response is given by [14],

�� EMBED Equation.3

where

is a function that depends on

,

is the zero-order Bessel function of first kind, and

represents the maximum Doppler shift. Using this autocorrelation, the ACF of received signal

can be computed as,

, thus it is still a white process. Similarly, its CACF can be evaluated as,

.

The properness of

follows from the fact that

is proper.

Blind signal separation for I/Q imbalance using conjugate signal model

If the filters

and

are modeled by

tap length FIR filters [8], the time-domain frequency-selective I/Q imbalance in (9) can be rewritten in the following conjugate model:

(11)

where

,

and

where

.

The objective of signal separation is to make the filter output a scaled and delayed version of the input [13]. Mathematically, the goal is to find a separation filter

with length

such that the signals and its image,

and

, are separated at the filter output, i.e.,

(12)

where

is a

permutation matrix, either

or

,

is a

diagonal scaling matrix and

is a delay.

In order to achieve the separation, prior information about the source signals, i.e., conjugation between these two signals needs to be imposed. This motivates us to come up with a cost function that incorporates this conjugate constraint. With this constraint, and the goal to restore the property of signal

, a necessary and sufficient condition for signal separation can be proved as in Lemma 1.

Remark 1: After the separation filter, the output is a delayed and filtered version of

.Therefore, the delay and the effective channel should be estimated then equalized prior to decoding. This is not a big issue since although a known training signal does not exist but most systems do have pilot symbols. For example, in LTE, the pilot symbols are inserted every six subcarriers in the frequency domain. These pilot symbols can be used for estimating the delay and equalization. Since these are not the scope of this paper, they are assumed to be perfectly known in the simulations. The permutation matrix

can also be identified in this manner.

Remark 2: Since

is white and proper, it can be shown that the autocorrelation matrix of

equal to a diagonal matrix,

(13)

Conditions for signal separation

In order to satisfy the conjugate condition for the elements of the filter output, i.e.

, we need to impose the following constraints on the elements of the filter matrices, i.e.

and

.

Define the composite vector

. By equation (12), this composite vector

can be written as

(14)

where

is a Toeplitz matrix composed of

and

.

We are now ready to derive the conditions which lead to signal separation.

Lemma 1: Under the constraints

and

,

are separator matrices, that is

, if and only if

Proof: The sufficient condition can be shown by plugging equation (12) into (14) then using equation (13). That is

, when

.

Now we proceed to prove the necessary condition. By looking at the matrix equation

element by element, i.e.

,

we can obtain the following conditions for

.

(15)

and

(16)

From equation (12), the filter output

can be written as

(17)

where

and

are filters depend on

and

. Then from (15)

(18)

where

is signal power and

. One can conclude that equation (18) can be satisfied when

or

is an anti-symmetric thus must be a non-causal filter. The latter case can be excluded and we can conclude that

. Without loss of generality, let us assume

. Then

in (17) becomes

(19)

Plug equation (18) into equation (15), and we have

, (20)

which implies

is an all-pass filter. It can be shown that for FIR all-pass filters, all the zeros are at the origin of the

-plane, thus, must have a linear phase. Therefore,

for delay

. One can see that the output of the filter is just a scaling and delayed version of

, i.e.

. Similarly, if

, then

which leads to

. Recall that, by construction,

, therefore

.

Implementation

In Lemma

, we prove that the autocorrelation matrix of filter output signal

is the necessary and sufficient condition for signal separation. Therefore, we define the following cost function

. (21)

Then the separation matrices

can be obtained by minimizing the cost function

. (22)

Here we proposed a gradient-based method [11] to iteratively solve the problem. At each step, the gradient of the cost function, i.e.

, is computed and the argument

is updated by stepping along the gradient direction. Note that in complex domain, the gradient direction is the derivative with respect to

instead of

.

Based on the fact that

equation (21) can be expressed as

(23)

where matrix

is defined as the

submatrix of

, where

is composed of the

submatrices

.

The term

can be expanded as

��EMBED Equation.3

(24)

Similarly, the term

is expressed as

(25)

Observing equations (24) and (25), we find out that the derivatives needed are of the form

and

, where

and

are

submatrices from

. Based on [12], the results of these derivatives are given as follows, without lengthy step by step details. Note that the gradient is the derivative w.r.t.

instead of

[11].

(26)

where

.

Similarly,

(27)

The cost function is minimized iteratively by the learning rule

(28)

where

is the matrices at

the step,

is the learning rate and

.

At each step,

is also updated by

(29)

Performance Analysis and Numerical Results

In this section, simulation results are presented to demonstrate the performance of the proposed blind compensation scheme for OFDM systems. The simulated OFDM system is with symbol length

subcarriers, guard symbol

, and subcarrier spacing

kHz [9]. Hence - the total symbol length is equal to

samples per OFDM symbol. We consider two different cases. Case 1_3-tap LPF: The I/Q imbalance parameters are; gain mismatch

, phase mismatch

and I and Q branch LPF

,

respectively [8] where the analog IRR is in the range of 18-25 dB [Fig. 2(a)]. Case 2_ 2-tap LPF: The frequency-dependent I/Q imbalance parameters are

,

,

and

[8] where the analog front end IRR is around 26-35 dB [Fig. 3(a)]. Both simulations use the learning rate

and with initial values

,

and

. This is justified by the fact that, in practical, the LPFs on both I and Q branch are close to an ideal, thus its impulse response is close to a delta function. In the simulations, 50,000 samples are used for estimating the autocorrelation function, 64-QAM modulation is used and the received SNR is 20dB. The convergence rate is shown in Fig. 2(b) and 3(b). The proposed method converges in approximately

and

iterations for case 1 and 2 respectively. In case 2 the LPFs are more ideal than in case 1, therefore it is not surprising that the simulation converges faster in case 2. In Fig.2 and 3, the performance of the proposed method is shown in terms of IRR. Figure 2(a) and 3(a) show that the IRR is improved by around 20 dB after I/Q imbalance compensation. Fig. 4 and 5 show the Symbol Error Rate (SER) performance before and after compensation. The results show that there is a need to compensate for the I/Q imbalance in both cases, since without compensation, the performance is unacceptable. Fig. 4 and 5 also show that the proposed compensation scheme provides an effective removal of the imbalance, and also improves the BER performance.

Conclusion

In this paper, a blind signal separation approach is proposed to compensate for a frequency-dependent I/Q imbalance in a wideband direct-conversion receiver. Necessary and sufficient conditions for signal separation are derived. Based on the conditions a gradient-based method is used to solve for the separation filter. Simulation results show the proposed approach effectively removes distortion caused by the I/Q imbalance.

References

P. Kiss and V. Prodanov, “One-tap wideband I/Q compensation for zero-IF filters,” IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications, vol. 51, pp. 1062–1074, June 2004.

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Figure 1: (a) Frequency-dependent I/Q imbalance of direct-conversion receiver (b) Mathematical model of a DRR with I/Q imbalance (c) Equivalent mathematical model of (b).

Figure 2: 3-tap LPF case

(a) IRR in before and after compensation (b) Convergence of the cost function.

Figure 3: 2-tap LPF case

a) IRR before and after compensation (b) Convergence of the cost function.

Figure 4: Symbol Error Rate for 3-tap LPF case

Figure 5: Symbol Error Rate for 2-tap LPF case

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(overflow,LoP)

1

I

LNA

LPF

LPF

A/D

A/D

Q

I/Q LO (gd, Φ)

I/Q Imbalance compensator

Analog Front End

Digital Baseband

y(t)

r(t)

AGC

(a)

I

GI(f)

GQ(f)

I/Q LO

r(t)

Q

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