Performance Degradation Due To Iq Imbalance In

Performance Degradation due to I/Q Imbalance in Multi-Carrier Direct Conversion Receivers: A Theoretical Analysis 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 novel 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 Rayleigh fading channel and receiver I/Q imbalance. The theoretical results are validated exemplarily for the IEEE 802.11a WLAN standard.

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 Q-branch 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 frequencyselective nature of a wireless communication channel, multicarrier 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 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

a)

i(k)

s(k)

−fSC

+fSC

b)

iRx (k)

sRx (k)

−fSC

+fSC

c)

˜iRx (k)

s˜Rx (k)

−fSC

+fSC

Fig. 1. Equivalent baseband representation of a) transmitted RF signal, b) received RF signal (corrupted by the fading channel), c) received baseband signal after direct conversion with I/Q imbalance

with or without digital compensation) on the one hand and system parameters (such as the symbol error probability) 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. The goal of this paper is to contribute towards closing this gap. The outline of this paper is as follows: Section II defines the system model, which is the basis of our analysis. The symbol error probability of a single subcarrier will be derived in section III. Both exact solutions and reasonable approximations are presented. A generalization of the results to multi-carrier systems is done in section IV, followed by the conclusions in section V. II. S YSTEM M ODEL The degradation of the system performance due to the I/Q imbalance is a consequence of the imperfect conversion of the received radio frequency (RF) signal down to the baseband (BB). Appropriate models for the impact of the I/Q imbalance on the received signal have been derived in the literature. It has been shown, that multi-carrier signals are affected by a mutual inter-carrier interference between each pair of symmetric subcarriers, see for example [3]. Because the effect of the I/Q imbalance is the same for all subcarriers, it is sufficient to consider a representative pair of symmetric subcarriers, as depicted in Fig. 1.

Let sRx (k) denote the received symbol at an arbitrary subcarrier frequency fSC at time index k, corresponding to the received signal before down conversion (Fig. 1b). Similarly, let iRx (k) denote the received symbol at the image subcarrier frequency −fSC . The interference due to an imperfect direct conversion with I/Q imbalance can be modelled by [3] s˜Rx (k) = Ks sRx (k) + Ki i∗Rx (k).

Q

Decision boundaries

Q

Symbols A1

d + d2 I

A2

A0 ∆

− d2

I

(1)

The asterisk denotes complex conjugation. The complex valued weighting parameters Ks and Ki are determined by the image rejection capabilities of the receiver. Ideally, the image rejection ratio


Ks
2

(2) IRR =
Ki


a)

b)

d

− d2

+ d2

Fig. 2. a) QAM constellation, b) distinguished areas for the occurrence of symbol errors

III. S TATISTICAL S IGNAL A NALYSIS

approaches infinity. However, the IRR which is achievable with today’s technologies is limited to only 30-40 dB [1]. Most practical multi-carrier systems are designed such that subcarrier spacing is much smaller than the coherency bandwidth of the wireless channel. Hence, the frequency-selective fading channel is split into frequency-flat subchannels in each subcarrier. Considering the representative pair of subcarriers at fSC and −fSC , the following channel models are used:

The goal of this section is to derive the probability of a symbol error due to the I/Q imbalance and the channel noise. Therefore, we model the samples of the channel coefficients and the noise as complex valued random variables (RV’s), which are written in bold style:

sRx (k) = hs (k)s(k) + ns (k),

(3)

iRx (k) = hi (k)i(k) + ni (k).

(4)

The resulting displacement ∆ is also a RV, whose properties we will analyze in this section. The I/Q imbalance parameters Ki and Ks are arbitrary fixed parameters. Furthermore, the actual value of the image signal i is considered as an instantaneous fixed parameter. A generalization to a non-constant i will be done at the end of this section. The resulting error probability due to the error vector ∆ depends on both the structure (alphabet) of the desired signal and the distribution of the error vector. In the following subsections we will first consider the symbol alphabet. Then we will analyze the distribution of the error vector.

s(k), i(k) denote the transmitted symbols. hs (k), hi (k) denote the corresponding time-variant channel coefficients. ns (k), ni (k) denote the additive channel noise in each subcarrier. From (1) and (3) one can see, that the desired signal s(k) is rotated and scaled by both the channel coefficient hs (k) and the I/Q imbalance parameter Ks . These effects can be removed by a proper equalization. Assuming a perfect zeroforcing equalizer, the following operation will be applied: 1 s˜Rx (k). (5) sˆ(k) = Ks hs (k) The remaining error vector ∆(k) = sˆ(k) − s(k) is a result of both the image signal and the additive channel noise: ∆(k) =

1 Ki 1 Ki h∗i (k) ∗ i (k)+ ns (k)+ n∗ (k). (6) Ks hs (k) hs (k) Ks hs (k) i

The severeness of the additional impairment due to the I/Q imbalance strongly depends on the properties of the communication channel. While the performance degradation is only moderate for AWGN channels, a significant performance degradation can be observed for frequency-selective fading channels [3]. This result stems from the fact, that the quotient h∗i (k)/hs (k) in (6) can take arbitrary large values in the case of a frequency-selective channel. For fading channels, the occurrence of large values is likely, if the individual fading processes hi (k) and hs (k) are independent. Independency holds, if the distance 2fSC between the considered pair of symmetric subcarriers is larger than the coherency bandwidth of the frequency-selective fading channel. Because this scenario is most critical from the I/Q imbalance point of view, it will be considered in our analysis.

∆=

1 Ki 1 ∗ Ki hi ∗ ∗ i + ns + ni . Ks hs hs Ks hs

(7)

A. Effect of the Error Vector on the Signal Constellation In our analysis we focus on the reception of 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. 2a. The parameter d denotes the distance between the amplitude levels of adjacent symbols in both the I and the Q dimension. Assuming equally probable symbols, the average power of a QAM signal is [5] 1 (M − 1)d2 . (8) 6 In order to distinguish between the parameters of the desired signal and the image signal, we will use the subscripts s and i, respectively. Both the desired signal s(k) and the image signal i(k) are assumed to be rectangular QAM signals of the order Ms and Mi , respectively. Nevertheless, the analysis framework presented in this paper can be easily adapted to alternative constellations. 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 (see Fig. 2b): σ2 =

If the error vector is located within area A0 , no symbol error occurs. • A location within area A1 (consisting of 4 partitions, which are shaded in light gray) may result in a symbol error. No symbol error occurs, if the desired symbol is located at the corresponding boarder of the constellation √ ( M out of M symbols). • Similarly, a location within area A2 (consisting of 4 partitions, which are shaded in dark gray) may result in a symbol error. No symbol error occurs, if the desired symbol is located at the corresponding corner of the constellation (1 out of M symbols). Hence the symbol error probability of a QAM signal corrupted by an additive random error vector ∆ is determined by: √ Ms − M s Ms − 1 P (∆ ∈ A1 )+ P (∆ ∈ A2 ), (9) Ps (i) = Ms Ms •

where P (∆ ∈ Ai ) denotes the probability, that ∆ is located within area Ai . The argument in Ps (i) stresses the fact, that the error probability is based on an instantaneous value of the image signal i. In order to calculate the probabilities of ∆ being located in a certain area, the probability density function (pdf) of the error vector ∆ will be derived in the next section. B. Distribution of the Error Vector In our analysis we consider a Rayleigh fading channel [5], i.e. the RV’s hs and hi representing the channel coefficients are complex Gaussian distributed with zero mean and variances σh2 s and σh2 i , respectively. Furthermore, we assume an independent additive Gaussian noise. Therefore, the RV’s ns and ni are complex Gaussian distributed with zero mean and variances σn2 s and σn2 i , respectively. Based on (7), the resulting RV ∆ can be rewritten as a quotient of two RV’s: ∆=

∆N = ∆D

∗ ∗ Ki Ks hi i

+ ns + hs

Ki ∗ Ks ni

.

(10)

The numerator ∆N is a linear combination of Gaussian distributed RV’s. Hence ∆N is also Gaussian distributed, if the channel coefficient hs and the noise terms ns , ni are mutually independent.With this assumption, ∆N has zero mean and a variance of

2

2



2 2
Ki
2 2 2
Ki
|i| + σns + σni
. (11) σN = σhi
Ks
Ks
Similarly, the denominator ∆D is found to have zero mean 2 = σh2 s . and a variance of σD Hence ∆ is the quotient of two zero-mean Gaussian distributed RV’s. While the derivation of its distribution function is quite extensive in general, the derivation becomes much simpler in the case of ∆N and ∆D being independent. This assumption holds, if hs is independent from both the noise terms ns , ni and the channel coefficient hi . In order to calculate the distribution function of the displacement vector, we represent ∆ = rejϕ in polar coordinates. The real RV’s r and ϕ denote the magnitude and the phase of the the complex RV ∆, respectively. Similarly, the complex

Gaussian distributed RV’s of the numerator and the denominator can be represented as ∆N = rN ejϕN and ∆D = rD ejϕD . The magnitudes rN and rD are Rayleigh distributed with 2 2 and σD , respectively. It can be shown (see the variances σN Appendix), that the pdf of the magnitude r = rN /rD yields  2 r a2 r≥0 (r 2 +a2 )2 fr (r) = (12) 0 r < 0. The distribution function is parameterized by the parameter a2 , which is defined as



σh2 i
Ki
2 2 σn2 s σn2 i

Ki

2 σ2

|i| + a2 = N = + . (13) 2 σD σh2 s
Ks
σh2 s σh2 s
Ks
The phases ϕN and ϕD are uniformly distributed. Hence the phase ϕ = ϕN − ϕD is also uniformly distributed, i.e. its pdf is 1 . (14) fϕ (ϕ) = 2π It has been shown, that the distribution of the complex RV ∆ can be easily described in polar coordinates using the pdf’s of the magnitude r and the phase ϕ. However, for the following calculations it is reasonable to consider cartesian coordinates ∆ = x + jy instead. In general, the joint pdf’s of the polar coordinates r, ϕ and the cartesian coordinates x, y are linked by [6] 1 (15) fxy (x, y) = frϕ (r, ϕ). r In our case, the joint pdf of the polar coordinates is separable, i.e. frϕ (r, ϕ) = fr (r)fϕ (ϕ). Merging (12), (14), (15) and using the dependency r2 = x2 + y 2 , the joint pdf of the cartesian coordinates results in: fxy (x, y) =

a2 1 . π (x2 + y 2 + a2 )2

(16)

C. Symbol Error Probability Given the joint pdf in cartesian coordinates, the desired probabilities P (∆ ∈ A1 ) and P (∆ ∈ A2 ) of equation (9) can be calculated by a two-dimensional integration within the appropriate integration boundaries:  − d2s  + d2s fxy (x, y) dx dy (17) P (∆ ∈ A1 ) = 4  P (∆ ∈ A2 ) = 4

−∞

− d2s −∞

− d2s



− d2s

−∞

fxy (x, y) dx dy

(18)

Here we used the circular symmetry of the joint pdf fxy (x, y), resulting in 4 equiprobable partitions of the areas A1 and A2 , respectively. The equations (17) and (18) are solvable in closed form. By merging the results into (9), the symbol error probability can be represented as √ Ms − 1 2 Ms − 1  Ps (i) = (19) − Ms Ms 1 + γ 2 (i) √ 1 1 ( Ms − 1)2 4  arctan  , − Ms π 1 + γ 2 (i) 1 + γ 2 (i)

where γ 2 (i) =

σh2 i σh2 s




Ki
2 2 σn2 s σn2 i


Ks
|i| + σ 2 + σ 2 hs hs




Ki
2 4


Ks
d2 . (20) s

Recall, that (19) represents the symbol error probability for an instantaneous value of the complex-valued image signal i. In order to calculate the total symbol error probability, the distribution function of the image signal must be considered. With the realistic assumption, that the samples of the image signal are generated based on an alphabet with discrete equally probable symbols, the total symbol error probability yields Ps =

M 1 i Ps (im ), Mi m=1

(21)

where Mi denotes the modulation order of the image signal and im denotes the mth complex-valued symbol of the modulation alphabet. D. Second order approximation Based on the set of equations (19)-(21) the exact symbol error probability can be calculated for arbitrary settings of the individual variances. However, due to the high complexity of the formulas, their usage might be impractical. Therefore we aim for a simplification of the formulas, while preserving the accuracy of the predicted error probability in regions of interest. Our approach is to approximate the instantaneous symbol error probability in (19) by a second order Taylor series, which results in: √  1 Ms − 1 1 ( Ms − 1)2 2 [2] + (22) γ (i). Ps (i) = 2 Ms π Ms By merging (20),(21) and (22), the second order approximation of the total symbol error probability yields: M 1 i [2] P (im ) Mi m=1 s 

Mi σh2 i
Ki
2 1  σn2 s σn2 i 2

= 2
|i | + + m σhs Ks
Mi m=1 σh2 s σh2 s √  1 ( Ms − 1)2 4 1 Ms − 1 + · 2 . d s 2 Ms π Ms

Ps[2] =

(23)


Ki
2


Ks
(24)

Interestingly, the specific structure of the image signal (modulation order Mi , symbol alphabet) is irrelevant in the second order approximation of the symbol error probability. The only relevant parameter is the average power of the image signal σi2 =

M 1 i |im |2 . Mi m=1

(25)

By using (8), the symbol distance ds can be replaced by the average power of the desired signal σs2 : 4 1 2 = 2 (Ms − 1). 2 ds σs 3

(26)

Correction factor alpha

0.6



0.5 0.4

0.3

0.2

4

16

64 256 Modulation Order M

1024

4096

s

Fig. 3.

Shape of the correction function α(Ms )

In order to derive a concise representation of (24), the impact of the modulation order Ms is separated from the remaining parameters. We introduce the correction parameter √  2 Ms − 1 1 Ms − 1 1 ( Ms − 1)2 + , (27) α(Ms ) = 3 Ms 2 Ms π Ms which is approximated by α(Ms ) ≈ 1/2 for higher order modulations (see Fig. 3). Using this correction factor, (24) can be rewritten as 



 2 2
Ki
2 σh2 i σi2
Ki
2 σ σ n n [2] s i
+
Ms α(Ms ). Ps = 2 2

+ 2 2

σhs σs Ks
σh2 s σs2 σhs σs Ks
(28) The 3 terms of the sum inside the brackets of (28) deserve a more detailed discussion. σi2 is the average power of the image subcarrier at the transmitter side, while σh2 i is the variance of its associated channel coefficient. Hence σh2 i σi2 is the average power of the image subcarrier measured at the receiver side. Similarly, the term σh2 s σs2 represents the average power of the desired subcarrier measured at the receiver side. Therefore, the term σ 2 σs2 (29) SIRSC = h2s 2 σhi σi denotes the signal-to-image power ratio of the considered pair of subcarriers, measured at the receiver. Similarly, the term SN RSC =

σh2 s σs2 σn2 s

(30)

denotes the subcarrier-based signal-to-noise power ratio, measured at the receiver. Finally, the third term describes the residual impact of the additive noise in the image signal. This term is much smaller than the second term for similar noise powers σn2 s ≈ σn2 i , because IRR  1 for practical I/Q imbalance parameters. Hence, the third term of the sum will be neglected. With these definitions, (28) simplifies to an easy-to-use approximation of the symbol error probability for a single subcarrier:  1 1 1 [2] + (31) Ms α(Ms ). Ps ≈ SIRSC IRR SN RSC Equation (31) indicates an interchangeability between distortions due to the channel noise (SN RSC ) on the one hand and distortions due to the I/Q imbalance (SIRSC IRR) on

0

10

−1

Symbol Error Probability

10

−2

10

M = 256 −3

M = 64

10

M = 16 −4

10

M=4 −5

10

Theory: Approximation Theory: Exact solution Simulation

−6

10

0

10

20 30 40 Image Rejection Ratio [dB]

50

60

Single subcarrier symbol error probability: SIR = 1, no channel

Fig. 4. noise

60

0.0

01

50 45 40

0.1

Signal to Noise Ratio [dB]

0.001

0.01

0.01

0.1

55

0.01

35

0.1

30

0.1 25 20 20

0.1

25

30

35 40 45 50 Image Rejection Ratio [dB]

55

60

Fig. 5. Contour plot of the single subcarrier symbol error probability: SIR = 1, modulation orders: Ms = 64 (solid lines), Ms = 256 (dashed lines)

the other hand. In the presence of a dominating channel noise (SN RSC  SIRSC IRR), the set of equations (19)(21) simplify to the special case of an M -ary QAM signal transmitted over a noisy Rayleigh fading channel, which has been analyzed in various textbooks [5], [7]. Equivalently, for SN RSC  SIRSC IRR, the system performance is mainly determined by the I/Q imbalance. Figure 4 shows the symbol error probability as a function of the image rejection ratio for different modulation orders Ms = Mi = M . The theoretical results are confirmed by computer simulations. Approximation (31) matches sufficiently with the exact solution (19)-(21) for symbol error probabilities of Ps ≤ 0.1. Finally, Fig. 5 addresses the performance degradation due to both channel noise and I/Q imbalance for two exemplary modulation orders.

IV. M ULTI -C ARRIER SYSTEMS The derived equations for the symbol error probability of a single representative subcarrier constitute a basis for the calculation of the total symbol error probability in arbitrarily designed multi-carrier systems. The total error probability can be obtained by merging the individual error probabilities of all data subcarriers. Note, that this approach is capable to deal with frequency-selective I/Q imbalance as well. In this case, the parameter IRR becomes a function of the subcarrier frequency. For example, we consider a simple multi-carrier system with the following properties: • all data subcarriers are transmitted with the same power, i.e. σs2 = σi2 = σ 2 for all SC, • the average power of the channel coefficients is frequency-flat, i.e. σh2 s = σh2 i = σh2 for all SC, 2 2 2 • the additive channel noise is white, i.e. σns = σni = σn for all SC, • the I/Q imbalance is frequency-flat, i.e. the IRR is constant for all SC. In this case, the subcarrier-based symbol error probability Ps will be identical for all data subcarriers. Consequently, the total symbol error probability of the multi-carrier system P¯s equals Ps as well. When considering the total system performance, it is more meaningful to determine P¯s as a function of the signal-to-noise power ratio SN RM C , which is defined based on the entire multi-carrier signal. The scaling factor between SN RM C and SN RSC depends on the actual structure of the multi-carrier signal. Let Ntotal denote the total number of subcarriers. Assuming Nused used subcarriers with equal transmit power, we get: Nused · σh2 σ 2 Nused = SN RSC . (32) Ntotal · σn2 Ntotal The validity of the derived theoretical results will be demonstrated exemplarily for the IEEE 802.11a WLAN standard [2]. Figure 6 shows the simulation results for the ETSI Hiperlan/2 channel models A and B [8]. For high SN Rs, the error floor due the I/Q imbalance becomes obvious. A careful comparison between the theoretical and the simulative results reveals a small mismatch, if the I/Q imbalance is the dominating impairment. This effect can be understood by considering the individual symbol error rates in each subcarrier (see Fig. 7). In the theoretical analysis, the channel coefficients of the desired subcarrier and the image subcarrier are assumed to be independent. However, in practical systems a residual dependency between the channel coefficients may exist, in particular around the DC subcarrier. Obviously, the degree of residual dependency is strongly related to the coherency bandwidth of the channel. An increased coherency bandwidth corresponds to a reduced symbol error probability (here: channel ETSI-A). The gap between the theoretical and the simulative results vanishes as the coherency bandwidth gets smaller. Therefore, the theoretical results presented in this paper can be considered as a worst case analysis. SN RM C =

A PPENDIX Given two independent Rayleigh distributed RV’s rN , rD 2 2 with the variances σN and σD , respectively. The pdf of these RV’s is  2  2rx − σrxr2 x rx ≥ 0 , frx (rx ) = σr2x e (33)  0 rx < 0

0

10

IRR = 20dB −1

Symbol Error Probability

10

IRR = 30dB −2

10

IRR = 40dB −3

10

where the index x is replaced by N and D, respectively. We want to calculate the pdf fr (r) of the random variable r = rN /rD , which is a function of the random variables rN and rD . According to [6], the pdf of the quotient of two RV’s is  ∞ |rD |f (r rD , rD ) drD , (34) fr (r) =

IRR = 50dB

−4

10

Theory: Approximation Theory: Exact solution Simulation: ETSI−A channel Simulation: ETSI−B channel 0

10

IRR → ∞

−∞

20 30 40 Signal to Noise Ratio SNR [dB]

50

60

MC

Fig. 6. Performance degradation in a 64-QAM mode IEEE 802.11a WLAN system due to I/Q imbalance for different image rejection ratios

−∞

In order to calculate fr (r), the cases r ≥ 0 and r < 0 must be distinguished. First we consider the case r < 0. According to (33), frN (r rD ) = 0 for rD ≥ 0. On the other hand, frD (rD ) = 0 for rD < 0. Hence (35) becomes  ∞ fr (r) = |rD | frN (r rD ) frD (rD ) drD = 0 (36)

  −∞

−1

10

Symbol Error Probability

where f (rN , rD ) denotes the joint pdf of rN and rD . Because rN and rD are independent, the joint pdf can be separated as  ∞ |rD |frN (r rD ) frD (rD ) drD . (35) fr (r) =

0

−2

10

−3

10 −30

Theory Simulation: ETSI−A channel Simulation: ETSI−B channel −20

−10

0 Subcarrier index

10

20

30

for r < 0. Next we consider the case r ≥ 0. According to (33), frD (rD ) = 0 for rD < 0. Hence the integral in (35) becomes  ∞ r2 r2 r2 2 r rD − σ2 D 2 rD − σD 2 N D dr rD (37) fr (r) = D 2 e 2 e σ σ 0 N D   ∞ 2 2 − r2 + 12 rD 4r 3 rD e σN σD drD (38) = 2 2 σN σD 0

 

Fig. 7. Comparison between theoretical and measured symbol error probability of individual subcarriers: 64-QAM, SN RM C =40dB, IRR=30dB

= 2r for r ≥ 0.

V. C ONCLUSION AND FUTURE WORK A novel framework for the theoretical computation of the symbol error probability in multi-carrier systems has been derived 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. In addition to analyzing the symbol error probability, the presented framework can be used as a basis for calculating the bit error probability of the multi-carrier system. In many practical applications, the bit error probability is even more expressive than the symbol error probability alone. Furthermore, an extension of the system model towards correlated channel coefficients is desirable. A solution of these issues is under active research.

2 σN 2 σD



1 2

r2 +



r2 σ2 N

2 σN 2 σD

+

1 σ2 D

 −2

−2

.

(39)

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] J. G. Proakis, Digital Communications. McGraw-Hill, Inc., 3rd ed., 1995. [6] A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill, Inc., 3rd ed., 1991. [7] K. D. Kammeyer and V. K¨uhn, MATLAB in der Nachrichtentechnik. J. Schlembach Fachverlag, 2001. [8] ETSI EP BRAN, “Channel models for HIPERLAN/2 in different indoor scenarios,” Mar. 1998.