Error Probability Analysis Of Multi-carrier Systems.

Error Probability Analysis of Multi-Carrier Systems Impaired by Receiver I/Q Imbalance Marcus Windisch, Gerhard Fettweis Dresden University of Technology, Vodafone Chair Mobile Communications Systems, D-01062 Dresden, Germany Email: [email protected]

Abstract— I/Q imbalance has been identified as one of the most serious concerns in the practical implementation of the direct conversion receiver architecture. In particular, at the reception of multi-carrier signals the achievable error rate is strictly limited by the I/Q imbalance. Knowledge about the quantitative link between the hardware parameters and the resulting error rate is essential for a reasonable design of the receiver front-end. In this paper a framework for the analytical computation of the symbol error probability in multi-carrier systems is presented. We consider an arbitrary M-ary QAM modulated multi-carrier signal, which is corrupted by both a noisy frequency-selective Rayleigh fading channel and receiver I/Q imbalance. The theoretical results are validated exemplarily for the IEEE 802.11a WLAN standard. Index Terms— Error probability, theoretical analysis, I/Q imbalance, fading channel, multi-carrier systems, OFDM, IEEE 802.11a WLAN

I. I NTRODUCTION The growing number of wireless communications standards demands for highly flexible and low-cost terminals. I/Q processing architectures, such as the direct conversion receiver, are very attractive, because no costly analog image rejection filter is required [1]. Instead, a theoretically infinite image rejection is provided by the I/Q signal processing. However, the limited accuracy of the analog hardware causes mismatches between the components in the I- and the Qbranch of the receiver, known as I/Q imbalance. In the past decades, multi-carrier systems, such as OFDM, have gained a lot of acceptance for the design of high data rate communications systems. For example, the IEEE 802.11a WLAN standard [2] is an OFDM system, which is widely used in practice. While being able to easily cope with the frequency-selective nature of a wireless communication channel, multi-carrier systems are very sensitive to I/Q imbalance. In order to cope with these impairments, different approaches for a digital compensation of the I/Q imbalance have been proposed in the literature, see for example [3] and the references herein. The goal of the compensation is to provide an improved image rejection, which by nature depends on the accuracy of the digital estimation and compensation approach. Knowledge about the quantitative relationship between transceiver parameters (such as the image rejection ratio - with or without digital compensation) on the one hand and system parameters (such as the probability of a bit or symbol error) on the other hand is essential for the design and the dimensioning of communications systems. Given a targeted error probability, the hardware designer needs to know the image rejection ratio required for reaching that goal. Conventionally, this knowledge is gained for a specific system by using hardware measurements or computer simulations [3], [4]. However, a comprehensive theoretical analysis is still missing. A novel framework for a theoretical calculation of the symbol error probability in multi-carrier systems has been presented in

[5], considering signal distortions due to both a noisy frequencyselective fading channel and receiver I/Q imbalance. This approach assumes a negligible correlation between symmetric pairs of channel coefficients in the frequency domain. However, depending on the channel conditions a residual correlation may exist, leading to a small mismatch between the theoretical error probabilities and simulation results. The goal of this paper is to extend the concept of [5] towards potentially correlated channel coefficients, thus improving the accuracy of the theoretical analysis. The outline of this paper is as follows. Both the system model and the general analysis framework from previous work in [5] will be summarized in sections II and III, respectively. An extended derivation of the error vector distribution function considering channel correlation will be presented in section IV, followed by a simulative verification of the predicted theoretical error probabilities in section V. Section VI concludes the paper. II. S YSTEM M ODEL In our analysis we consider multi-carrier systems, such as OFDM. Following the approach in [5], distortions due to the fading channel and due to the I/Q imbalance are modeled in the equivalent baseband domain. Let Sk denote the transmitted frequency-domain symbol at subcarrier index k. The impact of the frequency-selective fading channel can be modeled by Y k = H k Sk + W k

(1)

for each subcarrier index k. Hk denotes the associated complexvalued channel coefficient, and Wk denotes the additive white Gaussian noise (AWGN). It has been shown in the literature that the I/Q imbalance in direct conversion receivers translates to a mutual interference between each pair of subcarriers located symmetrically with respect to the DC carrier. Thus, after being corrupted by the fading channel, the received signal Yk at subcarrier k is interfered by the received signal Y−k at subcarrier −k, and vice versa. The undesirable leakage due to the I/Q imbalance can be modeled by [3], [5] ∗ , Zk = K1,k Yk + K2,k Y−k

(2)

∗

where (·) denotes complex conjugation. The complex-valued weighting factors K1,k and K2,k are determined by the I/Q imbalance parameters, such as gain imbalance and phase imbalance [3]. Thus, the image rejection capabilities of the receiver can be quantified in terms of an image rejection ratio IRRk =

K1,k K2,k

2

(3)

or, equivalently, in terms of an image leakage ratio This work was partly supported by the German Ministry of Education and Research (BMBF) within the project Wireless Gigabit with Advanced Multimedia Support (WIGWAM) under grant 01BU370

ILRk =

1 K2,k = IRRk K1,k

2

.

(4)

Ideally, the IRR approaches infinity, corresponding to zero ILR. With today’s technologies an IRR of 30 dB up to 40 dB is achievable [1], corresponding to -30 dB to -40 dB ILR. In summary, each sent symbol Sk is rotated and scaled by both the channel coefficient Hk and by the I/Q imbalance parameter K1,k . These distortions can be removed by a proper equalization. Assuming a perfect zero-forcing equalizer, the following operation will be applied: 1 Zk . (5) S˜k = K1,k Hk The remaining error vector Δk = S˜k − Sk is a result of both the image signal and the additive channel noise: Δk =

1 Hk





K2,k ∗ K2,k ∗ ∗ H−k S−k + Wk + W−k . K1,k K1,k

(6)

In order to analyze the impact of the error vector on the system performance, two subtasks need to be solved: (i) the characterization of the error vector as a function of various distortion parameters (noise variance, image leakage, etc.), and (ii) the derivation of system performance parameters (error probabilities, etc.). Both subtasks can be efficiently approached with theory of random variables [6]. III. S TATISTICAL SIGNAL ANALYSIS It has been proposed in [5] to approach the error probability analysis by considering the statistical signal model ∗ ∗ + wk + Kk w−k Kk h∗−k S−k , (7) hk which is derived from the deterministic model in (7) with the following constraints: • The channel coefficients hk , h−k and the additive noise terms wk , w−k are modeled as complex-valued random variables (RV’s), which are written in bold style. Thus, the error vector Δk is considered as a RV which is a function of multiple RV’s. • Initially, the analysis will be done by considering a fixed instantaneous symbol at the interfering image subcarrier S−k . A generalization towards a random interfering symbol will be done thereafter. • The I/Q imbalance parameters K1,k , K2,k are considered as arbitrary fixed parameters. In order to keep the notation compact, we substitute the quotient by Kk = K2,k /K1,k . Given the distribution function of the error vector Δk , the probability of an erroneous detection can be calculated based on the decision regions of an arbitrarily shaped symbol alphabet. In the following, we focus on M -ary quadrature amplitude modulated (QAM) signals, which are most frequently used in practical multi-carrier systems. A rectangular QAM constellation is shown exemplarily for the modulation order M =16 in Fig. 1 a), where the parameter d denotes the distance between the amplitude levels of adjacent symbols in both the I and the Q dimension. With the decision threshold of the detector placed at the the midpoint of adjacent amplitude levels, 3 different areas for the location of the error vector can be distinguished, as shown in Fig. 1 b). With P (Δk ∈ Ai ) denoting the probability of the event ”Δk is located within area Ai ” for i = 1, 2 the probability of a symbol error is determined by [5] √ M− M M −1 Ps (S−k ) = P (Δk ∈ A1 ) + P (Δk ∈ A2 ). (8) M M The argument in Ps (S−k ) is to emphasize that a fixed instantaneous image signal S−k is considered initially. Thus, the total error probability for a random image signal can be obtained by integrating

Δk =

Q

Decision boundaries

Q

Symbols A1

d + d2 I

a)

d

A0 Δk

− d2

b)

A2

− d2

I

+ d2

Fig. 1. a) QAM constellation, b) distinguished areas for the occurrence of symbol errors

over the distribution function of the image signal. Moreover, in the special case of S−k being drawn from a discrete grid of equiprobable symbols, the total error probability yields Ps =

1 M

M 

Ps (S−k = sm ),

(9)

m=1

where sm denotes the mth out of M symbols of the symbol alphabet. Likewise, since (9) denotes the the total error probability at an arbitrary, but fixed subcarrier index k, the total error probability of the overall multi-carrier system can be obtained by averaging over all used subcarriers. Thus, the key challenge is to determine the desired probabilities P (Δk ∈ A1 ) and P (Δk ∈ A2 ) of (8) based on the distribution function of the error vector Δk . Due to the rectangular shape of the corresponding regions, it is reasonable to consider the cartesian coordinates of the error vector, i.e. Δk = x + jy. Furthermore, if Δk is a circular symmetric RV, both A1 and A2 are composed of 4 equiprobable partitions, respectively. In this case the desired probabilities are determined by P (Δk ∈ A1 ) = 4 P (Δk ∈ A2 ) = 4

 − ds  + ds 2 2 −∞

d − 2s

 − ds  − ds 2 2 −∞

−∞

fxy (x, y) dx dy,

(10)

fxy (x, y) dx dy,

(11)

where fxy (x, y) denotes the joint probability density function (pdf) of the cartesian coordinates x, y of the error vector. Closed form solutions of both the joint pdf of the error vector fxy (x, y) and the resulting symbol error probabilities Ps (S−k ) and Ps have been derived in [5] under the assumption of independent channel coefficients hk and h−k . A generalized derivation of the joint pdf fxy (x, y) considering a residual correlation between the channel coefficients will be presented in the next section. IV. G ENERALIZED DISTRIBUTION FUNCTION The goal of this section is to derive the joint pdf fxy (x, y) of the real/imaginary part of the error vector Δk . According to (7), Δk is a nonlinear function of multiple RV’s. In order to approach the complexity of the problem, it will be split into several subtasks which are as follows. Initially, we will determine the statistical properties of the source RV’s wk , w−k , hk and h−k in terms of a joint pdf. In order to handle the nonlinearity of the transformation, it is reasonable to split the transformation into a linear part and a nonlinear part. Let z1 denote the numerator and let z2 denote the denominator of (7), i.e. ∗ ∗ + wk + Kk w−k , z1 = Kk h∗−k S−k

(12)

z2 = hk .

(13)

Following this concept, we will first derive the joint pdf of z1 and z2 , followed by the derivation of the desired joint pdf of the error vector z1 . (14) Δk = z2

So far, we have characterized the noise terms and the channel coefficients in (7) separately. In order to derive the statistical properties of the error vector Δk , we define the 8 × 1 vector

The presented derivation rests on the known theory of random variables [6], in particular on the transformation of joint distribution functions. Due to the limited space, not all aspects of the derivation can be discussed in full detail. Therefore, we will primarily present the key ideas of the derivation.

which contains the real and imaginary parts of all right hand side RV’s of (7). With the assumption of channel coefficients and AWGN being independent, the covariance matrix of vector Xk results in

A. Characterization of the source RV’s

B. Numerator/Denominator joint pdf

Let Wk denote the 4 × 1 vector of the real and imaginary parts of the additive noise terms wk and w−k , respectively:

It has been shown that the real and imaginary parts of all right hand side RV’s of (7) are jointly gaussian distributed with mean zero and covariance matrix Cx . Since any linear transformation of jointly gaussian distributed RV’s is also jointly gaussian distributed [6], we can easily derive the joint pdf of the numerator and denominator RV’s z1 and z2 , as defined in (12) and (13), respectively. By defining the 4 × 1 vector of the corresponding real and imaginary parts





Re {wk }  Im {w }

k

 . Wk =  Re {w−k }  Im {w−k }

(15)





Cx = E Xk XTk

Cw = E Wk WkT

=

 2 σw  0   0

0

0 2 σw 0 0

0 0 2 σw 0

(16)

Hk =



Re {hk }  Im {hk }



.  Re {h−k } 

(17)

In the case of Rayleigh fading, the real and imaginary part of the channel coefficients are gaussian distributed with zero mean and equal variance σh2 . For simplicity, the average power of the channel coefficients is assumed to be frequency flat, i.e. E |hk |2 = 2σh2 for each k. It is important to notice that the channel coefficients may, in general, be correlated. The correlation between the pair of channel coefficients hk and h−k is denoted by the complex-valued parameter 2ρhk = E {hk h∗−k } .

(18)

Furthermore, it is expedient to define the complex-valued correlation coefficient 

E {hk h∗−k }

E {|hk |2 } E {|h−k |2 }

=

ρhk . σh2

(19)

The case hk = 0 corresponds to uncorrelated channel coefficients, while the case hk = 1 corresponds to fully correlated channel . Withthese definitions, the channel coefficients, i.e. hk = h−k  covariance matrix Ch = E Hk HTk yields 

σh2  0 Ch =   Re {ρh } k −Im {ρhk }

0 σh2 Im {ρhk } Re {ρhk }



(22)



(23)



Cz = E ZZT

=

 2 σ1  0   ρr

ρi

0 σ12 ρi −ρr

ρr ρi σ22 0



ρi −ρr

, 0  2 σ2

(24)

where

Im {h−k }

hk =



0 . Cw

it can be shown that Z is jointly gaussian distributed with mean zero and covariance matrix

where the operator E {·} denotes expectation. Note that the variance  2 for of the complex-valued noise sample Wk is E |wk |2 = 2 σw each k. Furthermore, let Hk denote the 4 × 1 vector of the real and imaginary parts of the channel coefficients hk and h−k , respectively: 



Ch 0

=

Re {z1 } x1 Im {z1 }

y1

= 

, Z=  Re {z2 }  x2  y2 Im {z2 }



0 0

, 0 2 σw

(21)





Assuming uncorrelated zero-mean gaussian distributed channel noise with equal power in each subcarrier, the covariance matrix of Wk is



Hk , Xk = Wk

Re {ρhk } Im {ρhk } σh2 0



−Im {ρhk } Re {ρhk }

.  0 2 σh (20)

2 , σ12 = |Kk |2 |S−k |2 σh2 + (1 + |Kk |2 )σw

(25)

σ22 = σh2 ,

(26)

ρr = ρi =

∗ Re {Kk S−k ρhk } , ∗ Im {Kk S−k ρhk } .

(27) (28)

In other words, the joint pdf of the real parts x1 , x2 and the imaginary parts y1 , y2 of numerator and denominator is determined by f1 (x1 , y1 , x2 , y2 ) =

(2π)2

  1 √ exp − 12 ZT C−1 z Z . det Cz

(29)

C. Error vector joint pdf The complexity of the problem prohibits a direct transformation of the joint pdf of the RV’s x1 , y1 , x2 , y2 to the desired joint pdf of the RV’s x, y. A promising approach is to split the direct transformation into several small transformations and to consider the polar coordinates representations of z1 , z2 , and Δk : z1 = r1 ejθ 1 , z2 = r2 e

jθ 2

,

Δk = r ejθ .

(30) (31) (32)

Furthermore, although not required for the final result, the product of z1 , z2 will be used for an intermediate representation: z1 z2 = zp = rp ejθ p .

(33)

Based on these definitions, the desired joint pdf fxy (x, y) can be derived in a four-step procedure, as will be shown in the following.

0.8 σ

1) Cartesian-to-polar transformation: Let us initially consider the joint pdf’s of the RV’s z1 and z2 . In order to derive the polar coordinates joint pdf f2 (r1 , θ1 , r2 , θ2 ) from the cartesian coordinates joint pdf f1 (x1 , y1 , x2 , y2 ), it is advisable to consider the inverse functions of the corresponding RV’s: y1 = r1 sin θ1 ,

(34)

x2 = r2 cos θ2

y2 = r2 sin θ2 .

(35)

With the Jacobian [6] of the inverse functions det Jinv = r1 r2

pdf fr (r)

x1 = r1 cos θ1

||2 = 0.95 0.6 σ

||2 = 0.9 ||2 = 0.8 ||2 = 0.6

0.4 σ

||2 = 0.4 ||2 = 0.2

0.2 σ

(36)

||2 = 0

we obtain the polar coordinates joint pdf by f2 (r1 , θ1 , r2 , θ2 ) =| det Jinv | f1 (x1 , y1 , x2 , y2 ),

(37)

0 0

which after some algebraic manipulations yields   2 2 r1 r2 1 √ f2 (r1 , θ1 , r2 , θ2 ) = exp − 2√det σ2 r1 + σ12 r22 Cz 2 (2π) det Cz

1 2

Fig. 2.



σ

σ Magnitude r = |Δk |

3 2

2σ

σ

pdf of the magnitude of the error vector Δk

− 2r1 r2 (ρr cos(θ1 + θ2 ) + ρi sin(θ1 + θ2 )) . (38) 2) Quotient/product transformation: Let us next consider the transformation of the RV’s z1 and z2 into the product zp and the quotient Δk , as defined in (33) and (14), respectively. Their corresponding polar coordinates are related by rp = r1 r2 r = r1 /r2

θp = [θ1 + θ2 ] mod 2π ,

(39)

θ = [θ1 − θ2 ] mod 2π ,

(40)

where the subscript ”mod 2π” is to emphasize that the phases θp and θ must be re-mapped into the range (−π, π] during the transformation. With the Jacobian of the forward transformation r1 (41) det Jf wd = 4 = 4r r2 the polar coordinates joint pdf of the product and the quotient is obtained by1 f3 (rp , θp , r, θ) =

2 f2 (r1 , θ1 , r2 , θ2 ), | det Jf wd |

 2 1 rp 1 √ exp − 2√det σ2 r p r Cz 2r (2π)2 det Cz (43)   rp + σ12 − 2rp (ρr cos θp + ρi sin θ) . r 3) Quotient polar coordinates joint pdf: After having derived the joint pdf of the polar coordinates of both the product and the quotient, the desired joint pdf of the quotient only can be obtained by integrating over the no longer required polar coordinates associated with the product zp by

f3 (rp , θp , r, θ) =

 π  ∞

f3 (rp , θp , r, θ) drp dθp .

(44)

0

The double integral can be solved in closed form, yielding √  −3/2 2r 4r 2 (ρ2r + ρ2i ) det Cz 1 − . frθ (r, θ) = 2π (σ22 r 2 + σ12 )2 (σ22 r 2 + σ12 )2 (45) Equation (45) is parameterized by a set of absolute parameters, namely the variances σ12 and σ22 , and the squared magnitude of the 1 The

σ12 σ2 = |Kk |2 |S−k |2 + (1 + |Kk |2 ) w2 2 σ2 σh

(46)

and the squared magnitude of the correlation coefficient of numerator and denominator ||2 =

|ρ|2 |Kk |2 |S−k |2 2 = | | . h k σ2 σ12 σ22 |Kk |2 |S−k |2 + (1 + |Kk |2 ) w2

(47)

σh

With

√

det Cz = σ12 σ22 − (ρ2r + ρ2i ) = σ12 σ22 − |ρ|2

(48)

the joint pdf in (45) yields



−π

σ2 =

(42)

which finally yields

frθ (r, θ) =

complex-valued cross-correlation |ρ|2 = |ρr + jρi |2 = ρ2r + ρ2i , as defined in (25)-(28). For the interpretation and characterization of the joint pdf it is more instructive to consider relative parameters rather than absolute parameters. Let us define the ratio of the numerator and denominator variances

factor 2 is introduced by ambiguities in the inverse functions of the phases due to the ”mod 2π” operator.

1 − ||2 2r frθ (r, θ) = 2π σ 2 ( r22 + 1)2 σ



−3/2

2

1−

4||2 σr 2 2

( σr 2 + 1)2

.

(49)

It is important to note that the joint pdf frθ (r, θ) does not depend on the phase θ. Hence, Δk is a circular symmetric RV with the polar coordinates joint pdf being separable into frθ (r, θ) = fr (r)fθ (θ),

(50)

where the pdf of the magnitude is defined as 1 − ||2 2r fr (r) = σ 2 ( r22 + 1)2 σ



2

1−

4||2 σr 2 2

( σr 2 + 1)2

−3/2

(51)

and the pdf of the phase is uniformly distributed fθ (θ) =

1 . 2π

(52)

4) Polar-to-cartesian transformation: Finally, given the joint pdf of the polar coordinates frθ (r, θ), the joint pdf of the cartesian coordinates fxy (x, y) can be easily derived considering the forward transformation functions of the corresponding RV’s: x = r cos θ,

(53)

y = r sin θ.

(54)

With the Jacobian of the forward transformation (55)

we obtain the joint pdf of the cartesian coordinates 1 fxy (x, y) = frθ (r, θ) | det Jf wd | 1 − ||2 1 2 π σ 2 ( x2 +y + 1)2 σ2



1−

2

4||2 x σ+y 2

(

x2 +y 2 σ2

+

2

1)2

−3/2

1

0.8 0.6

. (57)

0.4 0.2

D. Summary Unaffected by the correlation of the channel coefficients, the error vector Δk is still a circular symmetric RV. Its distribution function is parameterized by two parameters, namely the ratio of the numerator and denominator variances σ2 and the correlation coefficient ||2 . Similar to equation (13) in [5], σ2 is determined by both the image leakage ratio ILRk = |Kk |2 and by the variance of the channel 2 /σh2 . noise relative to the channel gain σw Furthermore, correlation between the channel coefficients of the desired subcarrier hk and the image subcarrier h−k causes correlation between the numerator z1 and the denominator z2 , as shown in (12) and (13), respectively. In other words, a non-zero channel correlation |hk |2 translates to a non-zero correlation of the quotient ||2 , as shown in (47). It is important to note that ||2 has a significant impact on the distribution function of the error vector (see Fig. 2). With rising correlation ||2 , the magnitude of the error vector becomes more concentrated around σ. In turn, the occurrence of large magnitudes becomes less likely.

0

−30

−20

−10

0

10

20

30

Subcarrier index k −1

10

Hiperlan/2 channel B Hiperlan/2 channel A Flat fading channel

Symbol Error Probability PS

=

(56)

Hiperlan/2 channel B Hiperlan/2 channel A Flat fading channel

1.2

Channel Correlation Coeff. |hk |2

det Jf wd = r

−2

10

−3

V. S YMBOL ERROR PROBABILITY The reduced likeliness of large error vector magnitudes due to correlation translates to a reduced error probability. This effect shall be illustrated exemplarily for a multi-carrier signal according to the widely used IEEE 802.11a WLAN standard [2], where the theoretical error probabilities are obtained by a numerical integration over the error vector pdf, according to (10) and (11), respectively. The upper plot of Fig. 3 shows the correlation coefficient |hk |2 as a function of the subcarrier index k for different channel models. Two frequency-selective channel models according to [7] are compared with the flat Rayleigh fading channel. Channel model A (office scenario, 50 ns RMS delay spread) exhibits a slightly higher coherence bandwidth than channel model B (large office, 100 ns RMS delay spread), resulting in a weaker decay of the correlation around the DC carrier. Note the significant reduction of the error probability due to the channel correlation, as shown in the lower plot of Fig. 3. Computer simulations have been performed in order to validate the error probabilities predicted by the theoretical analysis. The results of these simulations are denoted by blue markers with square, diamond, or circle style, respectively. As shown in Fig. 3, an exact match with the theoretically predicted error probabilities can be ascertained. VI. C ONCLUSIONS A novel framework for a theoretical computation of the symbol error probability in multi-carrier systems has been presented in this paper. We considered a M -ary QAM modulated multi-carrier signal, which is corrupted by both a noisy Rayleigh fading channel and receiver I/Q imbalance. The theoretical results have been confirmed by computer simulations. Both the theoretical results and the computer simulations show that the performance degradation due to

10 −30

−20

−10

0

10

20

30

Subcarrier index k Fig. 3. Channel correlation and resulting symbol error probabilities at individual subcarriers of an IEEE 802.11a WLAN system (64-QAM, 40 dB SNR, frequency-flat I/Q imbalance with -30 dB ILR)

receiver I/Q imbalance in terms of increased error probability is strongly affected by the channel characteristics. Correlation between the channel coefficients (near the DC carrier, or due to frequency-flat fading) attenuates the impact of the I/Q imbalance, while uncorrelated fading (as analyzed in [5]) represents the worst case. R EFERENCES [1] B. Razavi, “Design Considerations for Direct-Conversion Receivers,” IEEE Transactions on Circuits and Systems—Part II: Analog and Digital Signal Processing, vol. 44, pp. 428–435, June 1997. [2] IEEE, “Part11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications,” IEEE Std 802.11a-1999, 1999. [3] M. Windisch and G. Fettweis, “Standard-Independent I/Q Imbalance Compensation in OFDM Direct-Conversion Receivers,” in Proc. 9th Intl. OFDM Workshop (InOWo), (Dresden, Germany), pp. 57–61, 15-16 Sept. 2004. [4] C.-L. Liu, “Impacts of I/Q imbalance on QPSK-OFDM-QAM detection,” IEEE Transactions on Consumer Electronics, vol. 44, pp. 984–989, Aug. 1998. [5] M. Windisch and G. Fettweis, “Performance Degradation due to I/Q Imbalance in Multi-Carrier Direct Conversion Receivers: A Theoretical Analysis,” in Proc. IEEE Intl. Conference on Communications (ICC), (Istanbul, Turkey), 11-15 June 2006. [6] A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill, Inc., 4th ed., 2002. [7] ETSI EP BRAN, “Channel models for HIPERLAN/2 in different indoor scenarios,” Mar. 1998.