Performance Analysis of Coherent and Noncoherent Modulation under I/Q Imbalance
Bassant Selim, Sami Muhaidat, Paschalis C. Sofotasios, Bayan S. Sharif, Thanos Stouraitis, George K. Karagiannidis, Naofal Al-Dhahir
aa r X i v : . [ ee ss . SP ] A ug Performance Analysis of Coherent andNoncoherent Modulation under I/Q Imbalance
Bassant Selim,
Student Member, IEEE , Sami Muhaidat,
Senior Member, IEEE ,Paschalis C. Sofotasios,
Senior Member, IEEE , Bayan S. Sharif,
Senior Member,IEEE , Thanos Stouraitis,
Fellow, IEEE , George K. Karagiannidis,
Fellow, IEEE and Naofal Al-Dhahir,
Fellow, IEEE
Abstract
In-phase/quadrature-phase Imbalance (IQI) is considered a major performance-limiting impairmentin direct-conversion transceivers. Its effects become even more pronounced at higher carrier frequenciessuch as the millimeter-wave frequency bands being considered for 5G systems. In this paper, we quantifythe effects of IQI on the performance of different modulation schemes under multipath fading channels.This is realized by developing a general framework for the symbol error rate (SER) analysis of coherentphase shift keying, noncoherent differential phase shift keying and noncoherent frequency shift keyingunder IQI effects. In this context, the moment generating function of the signal-to-interference-plus-noise-ratio is first derived for both single-carrier and multi-carrier systems suffering from transmitter(TX) IQI only, receiver (RX) IQI only and joint TX/RX IQI. Capitalizing on this, we derive analyticexpressions for the SER of the different modulation schemes. These expressions are corroborated by
B. Selim, B. Sharif and T. Stouraitis are with the Department of Electrical and Computer Engineering, Khalifa University of Sci-ence and Technology, PO Box 127788, Abu Dhabi, UAE (e-mail: { bassant.selim; bayan.sharif; thanos.stouraitis } @kustar.ac.ae).S. Muhaidat is with the Department of Electrical and Computer Engineering, Khalifa University of Science and Technology,PO Box 127788, Abu Dhabi, UAE and with the Institute for Communication Systems, University of Surrey, GU2 7XH, Guildford,UK (email: muhaidat@ieee . org ).P. C. Sofotasios is with the Department of Electrical and Computer Engineering, Khalifa University of Science and Technology,PO Box 127788, Abu Dhabi, UAE, and with the Department of Electronics and Communications Engineering, Tampere Universityof Technology, 33101 Tampere, Finland (e-mail: p . sofotasios@ieee . org ).G. K. Karagiannidis is with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki,54124 Thessaloniki, Greece (e-mail: [email protected]).N. Al-Dhahir is with the Department of Electrical Engineering, University of Texas at Dallas, TX 75080 Dallas, USA (e-mail:[email protected]). comparisons with corresponding results from computer simulations and they provide insights into thedependence of IQI on the system parameters. We demonstrate that the effects of IQI differ considerablydepending on the considered system as some cases of single-carrier transmission appear robust to IQI,whereas multi-carrier systems experiencing IQI at the RX require compensation in order to achieve areliable communication link. Index Terms
Hardware Impairments, I/Q imbalance, coherent detection, non coherent detection, differential PSK,FSK, performance analysis, symbol error rate.
I. I
NTRODUCTION
The emergence of the Internet of Things (IoT) along with the ever-increasing demands of themobile Internet, impose high spectral efficiency, low latency and massive connectivity require-ments on fifth generation (5G) wireless networks and beyond. Accordingly, next-generationwireless communication systems are expected to support heterogeneous devices for variousstandards and services with particularly high throughput and low latency requirements. Thisapplies to both large scale and small scale network set ups, which calls for flexible and softwarereconfigurable transceivers that are capable of supporting the desired quality of service demands.To this end, direct conversion transceivers have attracted considerable attention owing to theirsuitability for higher levels of integration and their reduced cost and power consumption sincethey require neither external intermediate frequency filters nor image rejection filters. However,in practical communication scenarios, direct-conversion transceiver architectures inevitably sufferfrom radio-frequency (RF) front-end related impairments, including in-phase/quadrature-phaseimbalances (IQI), which limit the overall system performance. In this context, IQI, which refersto the amplitude and phase mismatch between the I and Q branches of a transceiver, leads toimperfect image rejection, which results in performance degradation of both conventional andemerging communication systems [1], [2]. In ideal scenarios, the I and Q branches of a mixerhave equal amplitude and a phase shift of °, providing an infinite attenuation of the imageband; however, in practice, direct-conversion transceivers are sensitive to certain analog front-end related impairments that introduce errors in the phase shift as well as mismatches betweenthe amplitudes of the I and Q branches which corrupt the down-converted signal constellation,thereby increasing the corresponding error rate [1]. It is recalled that depending on the receiver’s (RX) ability to exploit knowledge of the carrier’sphase to detect the signals, the detection can be classified into coherent and noncoherent [2].In the former, exact knowledge of the carrier phase as well as the channel state information(CSI) is required at the receiver, which is a challenging task in certain practical applications.On the contrary, this information is not required in noncoherent detection, which ultimatelyreduces the corresponding receiver complexity at the expense of a decreased spectral efficiencyor a performance penalty. Therefore, the associated complexity-performance tradeoff must bethoroughly quantified in order to optimize the overall system efficiency and performance.I/Q signal processing is widely utilized in today’s communication transceivers which gives riseto the problem of matching the amplitudes and phases of the branches, resulting in an interferencefrom the image signal. Motivated by this practical concern, several recent works have proposedto model, mitigate or even exploit IQI, see [3]–[5] and the references therein. Specifically, theauthors in [6] derive the signal-to-interference-plus-noise-ratio (SINR), taking into account thechannel correlation between the subcarriers, in the context of orthogonal frequency divisionmultiplexing (OFDM) systems. Assuming IQI at the receiver only, the SINR probability dis-tribution function (PDF) of generalized frequency division multiplexing under Weibull fadingchannels was derived and the average symbol error rate (SER) of M -ary quadrature ampli-tude modulation ( M − QAM) was formulated in [7]. For Rayleigh fading channels, the ergodiccapacity of OFDM systems with receiver IQI and single-carrier frequency-division-multiple-access (SC-FDMA) systems with joint transmitter (TX)/receiver IQI was investigated in [8] and[9], respectively. Likewise, the bit error rate (BER) of differential quadrature phase shift keying(DQPSK) was recently derived in [10] for single-carrier and multi-carrier systems in the presenceof IQI. Moreover, the authors in [11] derived the SER of OFDM with M − QAM constellation,over frequency selective channels with RX IQI, whereas the authors in [12] quantified theeffects of IQI on the outage probability of both single-carrier and multi-carrier systems over N *Nakagami- m fading conditions. Likewise, the error rate of free-space optical systems usingsubcarrier intensity modulated QPSK over Gamma-Gamma fading channels with receiver IQI wasinvestigated in [13], while Chen et al. recently analyzed the impact of IQI on differential spacetime block coding (STBC)-based OFDM systems by deriving an error floor and approximationsfor the corresponding BER [14], [15]. Finally, IQI has also been studied in half-duplex (HD) andfull duplex (FD) amplify and forward (AF) and decode and forward (DF) coopeartive systems[16]–[19], as well as two-way relay systems and multi-antenna systems [20]–[22]. A. Motivation
It is well known that coherent information detection requires full knowledge of the CSI atthe receiver, which is typically a challenging task as sophisticated and often complex channelestimation algorithms are required. In this context, noncoherent detection has been proposedas an efficient technique particularly for low-power wireless systems such as wireless sensornetworks and relay networks [23]. The main advantage of this scheme stems from the fact thatit simplifies the detection since it eliminates the need for channel estimation and tracking, whichreduces the cost and complexity of the receiver [24], [25]. However, this comes at a cost ofhigher error rate or lower spectral efficiency; as a result, selecting the most suitable modulationscheme depends on the considered application and both noncoherent and coherent detectionare efficiently implemented in practical systems. Moreover, it is recalled that the detrimentaleffects of RF front-end impairments on the system performance are often neglected. This alsoconcerns the effects of IQI on M- ary phase shift keying ( M- PSK), M- ary differential phase shiftkeying ( M- DPSK) and M- ary frequency-shift keying ( M -FSK), which, to the best of the authors’knowledge, have not yet been addressed in the open technical literature. To this end, this articleis devoted to the quantification and analysis of these effects in wireless communications overmultipath fading channels. B. Contribution
The main objective of this paper is to develop a general framework for the comprehensiveanalysis of coherent and noncoherent modulation schemes under different IQI scenarios. To thisend, we consider both single-carrier and multi-carrier systems and we quantify the effects ofTX IQI, RX IQI and joint TX/RX IQI for M -PSK, M -DPSK and M -FSK constellations overRayleigh fading channels. In more details, the main contributions of this work are summarizedas follows: • We derive novel analytic expressions for the SINR PDF and the cumulative distributionfunction (CDF) for single-carrier systems over Rayleigh fading channels with TX and/orRX IQI along with a novel generalized closed form expression for the corresponding SINRMGF. • We derive novel closed form expressions for the SINR PDF, CDF and MGF for the caseof multi-carrier systems over Rayleigh fading channels with TX and/or RX IQI. • Using the derived MGFs, we derive the corresponding SER expressions for the cases of M -PSK, M -DPSK and M -FSK constellations. • We derive simple and fairly tight upper bounds for the SER of the different investigatedmodulation schemes with TX and/or RX IQI, which provide insights into the effect of eachparameter on the system performance.
C. Organization and Notations
The remainder of the paper is organized as follows: Section II provides a brief overview ofthe considered modulation schemes. In Section III, the SINR PDF, CDF and MGF is derivedfor single-carrier and multi-carrier systems with IQI, while Section IV presents the SER of M -PSK, M -DPSK and M -FSK with IQI. Upper bounds on the SER of the considered scenarios arederived in Section V whereas the corresponding numerical results and discussions are providedin Section VI. Finally, closing remarks are given in Section VII. Notations:
Unless otherwise stated, ( · ) ∗ denotes conjugation and j = √− . The operators E [ · ] and |·| denote statistical expectation and absolute value operations, respectively. Also, f X ( x ) and F X ( x ) denote the PDF and CDF of X , respectively while M X ( s ) is the MGF associated with X . Finally, the subscripts t/r denote the up/down-conversion process at the TX/RX, respectively.II. S YSTEM MODEL
We assume that a signal, s , is transmitted over a flat fading wireless channel, h , which followsa Rayleigh distribution and is subject to additive white Gaussian noise, n . Assuming that theTX/RX are equipped with a single antenna, we first revisit briefly the signal model for theconsidered M -ary PSK, DPSK and FSK modulation schemes. A. Coherent Detection of M-PSK Symbols
Assuming M -PSK modulation, it is recalled that θ m = (2 m − πM , m = 1 , , . . . , M (1)Hence, the complex baseband signal at the transmitter in the l th symbol interval is given by s [ l ] = A c e jθ [ l ] (2) where θ [ l ] is the information phase in the l th symbol. Assuming that the receiver has perfectknowledge of the CSI as well as carrier phase and frequency, the complex baseband signal atthe receiver is represented as x [ l ] = A c e jθ [ l ] + n [ l ] . (3) B. Noncoherent Detection of M-DPSK Symbols
Assuming M -ary DPSK modulation, the information phase in (1) is modulated on the carrieras the difference between two adjacent transmitted phases. Considering that the channel is slowlyvarying and remains constant over two consecutive symbols, the receiver takes the differenceof two adjacent phases to reach a decision on the information phase without knowledge ofthe carrier phase and channel state [26]. In this context, the information phases ∆ θ [ l ] are firstdifferentially encoded to a set of phases as follows θ [ l ] = ( θ [ l −
1] + ∆ θ [ l ]) mod π (4)where ∆ θ m = (2 m − π/M, m = 1 , ..., M and ∆ θ [ l ] is the information phase in the l th symbolinterval. The modulated symbol s [ l ] is then obtained by applying a phase offset to the previoussymbol s [ l − , namely s [ l ] = s [ l − e jθ [ l ] (5)where s [1] = 1 . Similarly, the decision variable is obtained from the phase difference betweentwo consecutive received symbols as follows b s [ l ] = r ∗ [ l − r [ l ] . (6) C. Noncoherent Detection of M-FSK Symbols
Assuming M -FSK modulation, the M information frequencies are given by f m = (2 m − − M ) ∆ f , m = 1 , , . . . , M (7)and thus the l th complex baseband symbol at the transmitter is given by s [ l ] = A c e j πf [ l ] . (8)The decision variable at the receiver is then obtained by multiplying the received signal by theset of complex sinusoids e j πf m , m = 1 , , ..., M and passing them through M matched filters.For orthogonal signals, the frequency spacing is chosen as ∆ f = N/T s , where T s is the symbolperiod and N is an integer. III. MGF
OF THE R ECEIVED
SINR
WITH
IQIAt the receiver RF front end, the received RF signal undergoes various processing stagesincluding filtering, amplification, and analog I/Q demodulation (down-conversion) to basebandand sampling. Assuming an ideal RF front end, the baseband equivalent received signal isrepresented as r id = hs + n (9)where h denotes the channel coefficient and n is the circularly symmetric complex additive whiteGaussian noise (AWGN) signal. The instantaneous signal to noise ratio (SNR) per symbol atthe receiver input is given by γ id = E s N | h | (10)where E s is the energy per transmitted symbol and N denotes the single-sided AWGN powerspectral density.Likewise, in the case of multicarrier systems, the corresponding baseband equivalent receivedsignal at the k th carrier is represented as r id ( k ) = h ( k ) s ( k ) + n ( k ) (11)where s ( k ) is the transmitted signal at the k th carrier, whereas h ( k ) and n ( k ) denote thecorresponding channel coefficient and the circular symmetric complex AWGN, respectively.Hence, the corresponding instantaneous SNR can be represented as γ id ( k ) = E s N o | h ( k ) | . (12)It is assumed that the RF carriers are up/down converted to the baseband by direct conver-sion architectures, while we assume frequency independent IQI caused by the gain and phasemismatches of the I and Q mixers. In this context, the time-domain baseband representation ofthe IQI impaired signal is given by [27] g IQI = µ t/r g id + ν t/r g ∗ id (13)where g id is the baseband IQI-free signal and g ∗ id is due to IQI. In addition, the correspondingIQI coefficients µ t/r and ν t/r are given by µ t ν t = 1 {±} ǫ t e {±} jφ t (14) and µ r ν r = 1 {±} ǫ r e {∓} jφ r (15)where ǫ t/r and φ t/r denote the TX/RX amplitude and phase mismatch levels, respectively. It isnoted that for ideal RF front-ends, φ t/r = 0 ° and ǫ t/r = 1 , which implies that µ t/r = 1 and ν t/r = 0 . Moreover, the TX/RX image rejection ratio (IRR) is given by IRR t/r = (cid:12)(cid:12) µ t/r (cid:12)(cid:12) (cid:12)(cid:12) ν t/r (cid:12)(cid:12) . (16)It is recalled that in single-carrier systems, IQI causes distortion to the signal from its owncomplex conjugate while in multi-carrier systems, IQI causes distortion to the transmitted signalat carrier k from its image signal at carrier − k . In the following, assuming that both thetransmitter and receiver are equipped with a single antenna, we revisit the signal model ofboth single-carrier and multi-carrier systems in the presence of IQI at the transmitter and/orreceiver. Then, we derive novel analytic expressions for the SINR PDF, CDF and MGF in eachscenario. A. Single-Carrier Systems
Single-carrier modulation is receiving increasing attention due to its robustness towards RFimpairments compared to multi-carrier modulation; see [28] and the references therein. Hence,it is considered more suitable for low complexity and low power applications. In what follows,we derive unified closed form expressions for the SINR PDF, CDF and MGF of single-carriersystems in the presence of IQI.
1) Signal Model: • TX IQI and ideal RX:
This case assumes that the RX RF front-end is ideal, while the TXexperiences IQI. Based on this, the baseband equivalent transmitted signal is expressed as s IQI = µ t s + ν t s ∗ (17)whereas the baseband equivalent received signal is given by hs IQI + n = µ t hs + ν t hs ∗ + n. (18)Hence, the instantaneous SINR per symbol at the input of the receiver is given by γ IQI = | µ t | | ν t | + γ id . (19) • RX IQI and ideal TX:
This case assumes that the TX RF front-end is ideal, while the RXis subject to IQI. Hence, the baseband equivalent received signal is given by r IQI = µ r hs + ν r h ∗ s ∗ + µ r n + ν r n ∗ . (20)Therefore, at the RX input, the instantaneous SINR per symbol is expressed as γ IQI = | µ r | | ν r | + | µ r | + | ν r | γ id . (21) • Joint TX/RX IQI:
This case assumes that both TX and RX are impaired by IQI and thebaseband equivalent received signal is given by r IQI = ( ξ h + ξ h ∗ ) s + ( ξ h + ξ h ∗ ) s ∗ + µ r n + ν r n ∗ . (22)Based on this, the instantaneous SINR per symbol at the RX input is given by γ IQI = | ξ | + | ξ | | ξ | + | ξ | + | µ r | + | ν r | γ id (23)where ξ = µ r µ t , ξ = ν r ν ∗ t , ξ = µ r ν t , and ξ = ν r µ ∗ t .
2) SINR Distribution:
From (19), (21) and (23), the SINR of single-carrier systems in thepresence of IQI can be expressed as γ IQI = αβ + Aγ id (24)where the parameters α , β , and A are given in Table I.TABLE I: Single-carrier systems impaired by IQI parameters α β A TX IQI | µ t | | ν t | RX IQI | µ r | | ν r | | µ r | + | ν r | Joint TX/RX IQI | ξ | + | ξ | | ξ | + | ξ | | µ r | + | ν r | Hence, the CDF of γ IQI is obtained as F γ IQI ( x ) = F γ id (cid:18) A αx − β (cid:19) (25)where γ id is the IQI free SNR, which follows an exponential distribution with CDF and PDFgiven by F γ id ( x ) = 1 − exp (cid:18) − xγ (cid:19) (26) and f γ id ( x ) = exp (cid:16) − xγ (cid:17) γ (27)respectively, where γ = E s /N is the average SNR. Hence, assuming TX and/or RX IQI, thecorresponding SINR CDF is given by F γ IQI ( x ) = 1 − e − Aγ ( αx − β ) , ≤ x ≤ αβ (28)Given that f γ IQI ( x ) = dd x F γ IQI ( x ) , the SINR PDF, in the presence of IQI, is given by f γ IQI ( x ) = αAe − Aγ ( αx − β ) γ ( α − xβ ) (29)which is valid for ≤ x ≤ αβ .
3) Moment Generating Function (MGF):
The MGF is an important statistical metric andconstitutes a convenient tool in digital communication systems over fading channels [26]. In whatfollows, we derive a generalized closed form expression for the SINR MGF of single-carriersystems in the presence of IQI, which will be particularly useful in the subsequent analysis.
Proposition 1.
For single-carrier systems impaired by IQI, the MGF of the instantaneous fadingSINR is given by M γ IQI ( s ) = e αβ s + Aβγ Γ (cid:18) , Aγβ ; sαAβ γ (cid:19) (30) where Γ ( α, x ; b ) = R ∞ x t α − e − t − bt d t is the extended upper incomplete Gamma function [29].Proof. By recalling that [26] M γ IQI ( s ) = Z ∞ e sx f γ IQI ( x ) d x (31)and substituting (29) into (31) yields M γ IQI ( s ) = Z αβ e sx αAe − Aγ ( αx − β ) γ ( α − xβ ) d x. (32)By also considering the change of variable y = α − γβ and after some mathematical manipula-tions, one obtains M γ IQI ( s ) = αAγβ e αβ s + Aβγ Z α e − syβ − αAβγy d y. (33)Based on this and by taking z = αAβγy , equation (30) is deduced, which completes the proof. B. Multi-carrier systems
It is recalled that multi-carrier systems divide the signal bandwidth among K carriers, whichprovides several advantages including enhanced robustness against multipath fading. Based onthis, Long-Term Evolution (LTE) employs orthogonal frequency division multiplexing (OFDM)in the downlink. In this subsection, we derive the SINR PDF, CDF and MGF of multi-carriersystems in the presence of IQI, which creates detrimental performance effects. To this end, weassume that the RF carriers are down converted to the baseband by wideband direct conversion.We also denote the set of signals as S = {− K , . . . , − , , . . . , K } and assume that there is adata signal present at the image subcarrier and that the channel responses at the k th carrier andits image are uncorrelated.
1) Joint TX/RX impaired by IQI:
Here, we consider the general scenario where both the TXand RX suffer from IQI. The baseband equivalent received signal in this case is given by r IQI = ( ξ h ( k ) + ξ h ∗ ( − k )) s ( k ) + ( ξ h ( k ) + ξ h ∗ ( − k )) s ∗ ( − k ) + µ r n ( k ) + ν r n ∗ ( − k ) (34)where the carrier − k is the image of the carrier k . To this effect, the instantaneous SINR persymbol at the input of the RX is given by γ = | ξ | + | ξ | γ id ( − k ) γ id ( k ) | ξ | + | ξ | γ id ( − k ) γ id ( k ) + | µ r | + | ν r | γ id ( k ) (35)where γ id ( − k ) = E s N | h ( − k ) | . (36)Therefore, for the case of given γ id ( − k ) and with the aid of (35) and (26), the conditional SINRCDF can be expressed as F γ IQI ( x | γ id ( − k )) = 1 − exp − x (cid:0) | ξ | γ id ( − k ) + | µ r | + | ν r | (cid:1) − | ξ | γ id ( − k ) γ (cid:0) | ξ | − x | ξ | (cid:1) ! . (37)Based on this, the unconditional CDF is obtained by integrating (37) over (27), yielding F γ IQI ( x ) = 1 − exp (cid:18) − x ( | µ r | + | ν r | ) γ ( | ξ | − x | ξ | ) (cid:19) x | ξ | −| ξ | ( | ξ | − x | ξ | ) , ≤ x ≤ | ξ | | ξ | (38)whereas the SINR PDF is obtained as f γ IQI ( x ) = exp (cid:18) − x ( | µ r | + | ν r | ) γ ( | ξ | − x | ξ | ) (cid:19) | ξ | ( | µ R | + | ν R | ) γ + | ξ | | ξ | −| ξ | | ξ | x | ξ | −| ξ | | ξ | − x | ξ | ! ( | ξ | − x | ξ | ) ( | ξ | − | ξ | + x ( | ξ | − | ξ | )) , (39) which is valid for ≤ x ≤ | ξ | / | ξ | . Proposition 2.
The MGF of multi-carrier systems impaired by joint TX/RX IQI is given by M γ IQI ( s ) = C + | ξ | s ( | ξ | − | ξ | ) e s | ξ | | ξ | + | µr | | νr | | ξ | γ γ (cid:18) , s | ξ | | ξ | ; s | ξ | ( | µ r | + | ν r | ) | ξ | γ (cid:19) (40) for | ξ | = | ξ | , M γ IQI ( s ) = | ξ | | ξ | − | ξ | + ∞ X k =0 ( − k s k d k e | µR | | νR | | ξ | γ + s | ξ | | ξ | ( | ξ | − | ξ | ) k +1 | ξ | k − × γ (cid:18) − k, s | ξ | | ξ | ; s | ξ | ( | µ r | + | ν r | ) | ξ | γ (cid:19) (41) for (cid:12)(cid:12)(cid:12) | ξ | | ξ | −| ξ | | ξ | | ξ | −| ξ | (cid:12)(cid:12)(cid:12) < | ξ | , and M γ IQI ( s ) = C + e s | ξ | | ξ | + | µr | | νr | | ξ | γ ∞ X k =0 ( − k ( | ξ | − | ξ | ) k | ξ | k +4 d k +1 s k +1 × γ (cid:18) k + 2 , s | ξ | | ξ | ; s | ξ | ( | µ r | + | ν r | ) | ξ | γ (cid:19) (42) for (cid:12)(cid:12)(cid:12) | ξ | | ξ | −| ξ | | ξ | | ξ | −| ξ | (cid:12)(cid:12)(cid:12) > | ξ | , where γ ( α, x ; b ) = R x t α − e − t − bt d t is the extended lowerincomplete Gamma function [29], while C = | ξ | | ξ | − | ξ | (43) and d = | ξ | | ξ | − | ξ | | ξ | . (44) Proof.
The proof is provided in Appendix A.
2) TX Impaired by IQI:
Assuming that the RX RF front-end is ideal, while the TX experiencesIQI, the baseband equivalent received signal is s IQI = µ t s ( k ) h ( k ) + ν t s ∗ ( − k ) h ( k ) + n ( k ) (45)and the instantaneous SINR per symbol at the input of the RX is given by γ IQI = | µ t | | ν t | + γ id ( k ) . (46)Hence, by setting µ r = 1 and ν r = 0 in (38), it follows that F γ IQI ( x ) = 1 − e − γ (cid:18) | µt | x −| νt | (cid:19) , ≤ x ≤ | µ t | | ν t | (47) which yields straightforwardly the corresponding SINR PDF, namely f γ IQI ( x ) = | µ t | e − γ (cid:18) | µt | x −| νt | (cid:19) γ ( | µ t | − x | ν t | ) (48)which is valid for ≤ x ≤ | µ t | / | ν t | . It is noted that (48) is similar to (29) for α = | µ t | , β = | ν t | , and A = 1 . Hence, with the aid of (30), the instantaneous SINR MGF of multi-carriersystems experiencing TX IQI only is given by M γ IQI ( s ) = e | µt | | νt | s + | νt | γ Γ (cid:18) , γ | ν t | ; s | µ t | | ν t | γ (cid:19) . (49)
3) RX Impaired by IQI:
Assuming that the TX RF front-end is ideal, while the RX is impairedby IQI, the baseband equivalent received signal is represented as r IQI = µ r h ( k ) s ( k ) + ν r h ∗ ( − k ) s ∗ ( − k ) + µ r n ( k ) + ν r n ∗ ( − k ) . (50)Likewise, the instantaneous SINR per symbol at the input of the RX is expressed as γ IQI = | µ r | | ν r | γ id ( − k ) γ id ( k ) + | µ r | + | ν r | γ id ( k ) . (51)Hence, substituting µ t = 1 and ν t = 0 in (38) one obtains F γ IQI ( x ) = 1 − | µ r | | µ r | + x | ν r | e − xγ (cid:18) | νr | | µr | (cid:19) , ≤ x ≤ ∞ (52)which with the aid of (39) and after some algebraic manipulations yields the respective SINRPDF, namely f γ IQI ( x ) = exp (cid:16) − xγ (cid:16) | ν r | | µ r | + 1 (cid:17)(cid:17) γ (cid:16) x | ν r | | µ r | + 1 (cid:17) | ν r | | µ r | + γ | ν r | | µ r | (cid:16) x | ν r | | µ r | + 1 (cid:17) (53)which is valid for ≤ x ≤ ∞ .Finally, from (31) and (53), the corresponding MGF is obtained as M γ IQI ( s ) = 1 + | µ r | | ν r | γ Z ∞ e − x (cid:18) γ + | νr | γ | µr | − s (cid:19) x + | µ r | | ν r | dx + Z ∞ e − x (cid:18) γ + | νr | γ | µr | − s (cid:19) (cid:16) x + | µ r | | ν r | (cid:17) d x (54)which with the aid of [30, eq. (3.352)] and [30, eq. (3.353)], eq. (54) can be expressed by thefollowing closed-form representation M γ IQI ( s ) = 1 − s | µ r | | ν r | e γ + | µr | γ | νr | − s | µr | | νr | Ei (cid:18) − γ − | µ r | γ | ν r | + s | µ r | | ν r | (cid:19) (55)where Ei ( z ) = − R ∞− z e − t /t d t denotes the exponential integral function [30].The different MGF expressions derived are summarized in Table II, where Λ = | µ r | + | ν r | . Itis noted that with the aid of the derived MGFs, the SER of various M -ary modulation schemesunder different IQI effects as well as multi-channel reception schemes can be readily determined. TABLE II: SINR MGFs
Single-carrier systems Multi-carrier systemsTX IQI M γ IQI ( s ) = e | µt | | νt | s + | νt | γ Γ (cid:16) , γ | ν t | ; s, | µ t | | ν t | γ (cid:17) RX IQI M γ IQI ( s ) = e | µr | s | νr | + Λ | νr | γ Γ (cid:16) , Λ γ | ν r | ; s | µ r | Λ | ν r | γ (cid:17) M γ IQI ( s ) = 1 − s | µ r | | ν r | e γ + | µr | γ | νr | − s | µr | | νr | × Ei (cid:16) − γ − | µ r | γ | ν r | + s | µ r | | ν r | (cid:17) M γ IQI ( s ) = C + | ξ | s ( | ξ | −| ξ | ) e s | ξ | | ξ | + Λ | ξ | γ × γ (cid:16) , s | ξ | | ξ | ; s | ξ | Λ | ξ | γ (cid:17) , for | ξ | = | ξ | Joint IQI M γ IQI ( s ) = e | ξ | | ξ | | ξ | | ξ | s + | µr | | νr | ( | ξ | | ξ | ) γ M γ IQI ( s ) = C + P ∞ k =0 ( − s ) k d k e s | ξ | | ξ | ( | ξ | −| ξ | ) k +1 | ξ | k − × Γ (cid:18) , Λ γ ( | ξ | + | ξ | ) ; s ( | ξ | + | ξ | ) Λ ( | ξ | + | ξ | ) γ (cid:19) × e Λ | ξ | γ γ (cid:16) − k, s | ξ | | ξ | ; s | ξ | Λ | ξ | γ (cid:17) , for (cid:12)(cid:12)(cid:12) | ξ | | ξ | −| ξ | | ξ | | ξ | −| ξ | (cid:12)(cid:12)(cid:12) < | ξ | M γ IQI ( s ) = C + P ∞ k =0 | ξ | k +4 e s | ξ | | ξ | ( | ξ | −| ξ | ) − k d k +1 s k +1 × e Λ | ξ | γ γ (cid:16) k + 2 , s | ξ | | ξ | ; s | ξ | Λ | ξ | γ (cid:17) , for (cid:12)(cid:12)(cid:12) | ξ | | ξ | −| ξ | | ξ | | ξ | −| ξ | (cid:12)(cid:12)(cid:12) > | ξ | IV. S
YMBOL E RROR R ATE A NALYSIS
This section capitalizes on the derived MGF representation and evaluates the SER performanceof both single-carrier and multi-carrier systems employing different coherent and non-coherent M -ary modulation schemes in the presence of IQI and multipath fading. A. Coherent M -PSK Symbol Error Rate Analysis For coherently detected M -PSK, the SER under AWGN is given by [26, eq. (8.22)] P s, PSK = 1 π Z ( M − πM exp (cid:18) − γ g PSK sin ( θ ) (cid:19) d θ (56) where γ = E s /N and g PSK = sin (cid:0) πM (cid:1) . Under slow fading conditions, the average SER isobtained by averaging (56) over the considered channel’s SINR PDF, namely P s, PSK = 1 π Z ∞ Z ( M − πM exp (cid:18) − x g PSK sin ( θ ) (cid:19) f γ ( x ) d θ d x (57)which is equivalent to P s, PSK = 1 π Z ( M − πM M γ IQI (cid:18) − x g PSK sin ( θ ) (cid:19) d θ. (58)Therefore, by assuming PSK modulation, the average SER in the presence of IQI is obtained bysubstituting the derived MGF expressions into (58), which for single-carrier systems is given by P s, PSK = 1 π Z ( M − πM e − g PSK α sin2( θ ) β + Aβγ Γ (cid:18) , Aγβ , − g PSK αA sin ( θ ) β γ , (cid:19) d θ (59) B. Differential M -PSK Symbol Error Rate Analysis Considering differential detection of M -PSK under AWGN, the exact SER is given by [26,eq. (8.90)], namely P s, DPSK = 1 π Z ( M − πM exp (cid:18) − γ g PSK ρ cos ( θ ) (cid:19) d θ (60)where ρ = √ − g PSK . Based on this and assuming Rayleigh fading conditions, and TX and/orRX IQI, the above expression can be expressed as P s, DPSK = 1 π Z ( M − πM M γ IQI (cid:18) − x g PSK ρ cos ( θ ) (cid:19) d θ. (61)The average symbol error rate for M -DPSK over Rayleigh fading channels in the presence ofIQI is obtained by substituting the derived MGF expressions in (61), which for multi-carriersystems with TX IQI only is given by P s, DPSK = 1 π Z ( M − πM e − | µt | g PSK | νt | ρ cos( θ )) + | νt | γ Γ (cid:18) , γ | ν t | , − g PSK | µ t | (1 + ρ cos ( θ )) | ν t | γ , (cid:19) d θ. (62) C. Noncoherent M -FSK Symbol Error Rate Analysis Assuming noncoherent detection of orthogonal signals, corresponding to a minimum frequencyspacing ∆ f = 1 /T s , the SER of M -FSK under AWGN is given by [26, eq. (8.66)], namely P s, FSK = M − X k =1 ( − k +1 (cid:18) M − k (cid:19) exp (cid:0) − γ kk +1 (cid:1) k + 1 (63) which under fading conditions is expressed as follows P s, FSK = M − X k =1 ( − k +1 k + 1 (cid:18) M − k (cid:19) M γ IQI (cid:18) − x kk + 1 (cid:19) . (64)Therefore, substituting the derived MGF expressions in (64) yields the average SER in thepresence of IQI, which for the case of multi-carrier systems with RX IQI only is given by P s, FSK = M − X k =1 ( − k +1 k + 1 (cid:18) M − k (cid:19)" k | µ r | Ei (cid:16) − | ν r | + | µ r | γ | ν r | − k | µ r | ( k +1) | ν r | (cid:17) e − | νr | | µr | γ | νr | − k | µr | k +1) | νr | ( k + 1) | ν r | . (65)To the best of the authors’ knowledge, the derived analytic expressions have not been previ-ously reported in the open technical literature.V. A SYMPTOTIC A NALYSIS
In this section, we analyze the performance of both single-carrier and multi-carrier systems inthe asymptotic regime by deriving SER upper bounds. Moreover, since IQI results in interferencefrom either the signal’s conjugate or the signal at the image subcarrier, increasing the transmitSNR also increases the interference. Hence, we study the asymptotic behaviors of the derivedbounds which provides useful insights into the system behavior.
A. Single-Carrier Systems
We first provide simple upper bounds to the SER of single-carrier-systems for M -PSK and M -DPSK modulation with IQI at the TX and/or RX.
1) M-ary PSK:
It is recalled that the SER of single-carrier systems is given in (59). It isevident that by setting θ = π/ , the SER is upper bounded by P s, PSK ≤ ˜ M e − g PSK αβ + Aβγ Γ (cid:18) , Aγβ ; − g PSK αAβ γ (cid:19) (66)where ˜ M = ( M − /M , which for high SNR levels simplifies to P s, PSK ≤ ˜ M e − g PSK αβ . (67)This upper bound provides insights into the asymptotic behavior of the considered system.For instance, assuming TX or RX IQI only, αβ = IRR t / r ; hence, as IRR t / r approaches ∞ , P s,P SK → . On the contrary, as IRR t / r approaches , P s,P SK → M − M e − g PSK which is directlyproportional to M . Hence, a higher modulation order implies a higher error floor. Moreover, itis evident that the asymptotic behavior of the SER depends on both the modulation index andthe IQI parameters.
2) M-ary DPSK:
Assuming noncoherent M -ary DPSK, the SER is obtained by substituting(30) in (61). Hence, by setting θ = 0 , the SER can be upper bounded as follows P s, DPSK ≤ ˜ M e − αβ g PSK1+ ρ + Aβγ Γ (cid:18) , Aγβ ; − g PSK αA (1 + ρ ) β γ (cid:19) (68)which for high SNR values, since Γ (1 , ,
0) = 1 , simplifies to P s, DPSK ≤ ˜ M e − αβ g PSK1+ ρ . (69)We observe that the exponential function argument in (69) is similar to the argument in (67) butdivided by ρ > . Hence, from the derived upper bound, we can conclude that for a fixed M , the SER of DPSK is asymptotically greater than the SER of PSK. B. Multi-Carrier Systems
In this subsection, upper bounds and asymptotic expressions are derived for the SER of theconsidered modulation schemes for multi-carrier systems with joint TX and/or RX IQI.
1) M-ary PSK:
Assuming coherent M -ary PSK, the SER of multi-carrier systems with TXIQI only, RX IQI only and joint TX/RX IQI is obtained by substituting (49), (55) and (40) − (42)in (58), respectively. • TX IQI and ideal RX:
Based on the above and setting θ = π/ , one obtains P s, PSK ≤ ˜ M e − g PSK | µt | | νt | + | νt | γ Γ (cid:18) , Aγ | ν t | ; − g PSK | µ t | | ν t | γ (cid:19) (70)which for high SNR values reduces to the following simple closed-form upper-bound P s, PSK ≤ ˜ M e − g PSK | µt | | νt | . (71)It is noticed that (70) and (71) are similar to (66) and (67) when α = | µ t | and β = | ν t | .Importantly, this implies that under TX IQI only, single-carrier and multi-carrier systemsexhibit similar behaviors. • RX IQI and ideal TX:
For RX IQI only, the SER is upper bounded by P s, PSK ≤ ˜ M (cid:18) g PSK | µ r | | ν r | e γ + | µr | γ | νr | + g PSK | µr | | νr | Ei (cid:18) − γ − | µ r | γ | ν r | − g PSK | µ r | | ν r | (cid:19)(cid:19) . (72)It is evident that for asymptotic SNR values, the above inequality simplifies to P s, PSK ≤ ˜ M (cid:18) g PSK | µ r | | ν r | e g PSK | µr | | νr | Ei (cid:18) − g PSK | µ r | | ν r | (cid:19)(cid:19) . (73) Also, as
IRR r = | µ r | / | ν r | approaches ∞ , the exponential integral function can be approx-imated by Ei ( − z ) ≈ − e − z z (cid:0) − z + z − . . . (cid:1) [31] and hence P s, PSK → . Likewise, as IRR r approaches unity, one obtains P s, PSK → P s, PSK ≤ ˜ M (1 + g PSK e g PSK Ei ( − g PSK )) . (74)Moreover, it is noted that ∀ x ≥ and y = 1 + xe x Ei ( x ) , we have x ∝ /y . As a result, itfollows that P s, PSK ∝ / IRR r and P s, PSK ∝ M . • Joint TX/RX IQI:
Finally, for joint TX/RX IQI with | ξ | = | ξ | i.e. IRR t = IRR r thecorresponding SER is upper bounded by P s, PSK ≤ ˜ M C − | ξ | e − g PSK | ξ | | ξ | + | µr | | νr | | ξ | γ g PSK ( | ξ | − | ξ | ) γ (cid:18) , − g PSK | ξ | | ξ | ; − g PSK | ξ | ( | µ r | + | ν r | ) | ξ | γ (cid:19) ! (75)which for asymptotic SNR values simplifies to P s, PSK ≤ ˜ M C − | ξ | e − g PSK | ξ | | ξ | g PSK ( | ξ | − | ξ | ) γ (cid:18) , − g PSK | ξ | | ξ | (cid:19) (76)where γ ( a, x ) = R x t a − e − t d t is the lower incomplete gamma function [32]. It is notedthat ∀ x ≥ and y = 1 − e − x x γ (2 , − x ) , we have x ∝ /y and thus, P s, PSK ∝ M . Moreover,since | ξ | / | ξ | = IRR t = IRR r , it follows that P s, PSK ∝ / IRR t / r .
2) M-ary DPSK:
Assuming noncoherent M -ary DPSK, the SER of multi-carrier systemswith TX IQI only, RX IQI only and joint TX/RX IQI is obtained by substituting (49), (55) and(40) − (42) in (61), respectively. • TX IQI and ideal RX:
Based on the above and by setting θ = 0 , it follows that P s, DPSK ≤ ˜ M e − g PSK | µt | ρ ) | νt | + | νt | γ Γ (cid:18) , Aγ | ν t | ; − g PSK | µ t | (1 + ρ ) | ν t | γ (cid:19) (77)which for high SNR values reduces to the following simple bound P s, DPSK ≤ ˜ M e − g PSK | µt | ρ ) | νt | . (78)It is noted that (77) and (78) are similar to (68) and (69) when α = | µ t | and β = | ν t | .Therefore, in the case of TX IQI only, M -DPSK based single-carrier and multi-carriersystems show similar behaviors. • RX IQI and ideal TX:
For RX IQI only, the SER is upper bounded by P s, DPSK ≤ ˜ M g PSK | µ r | (1 + ρ ) | ν r | e γ + | µr | γ | νr | + g PSK | µr | ρ ) | νr | Ei (cid:18) − γ − | µ r | γ | ν r | − g P PSK | µ r | (1 + ρ ) | ν r | (cid:19) ! (79) which for asymptotic SNR values simplifies to P s, DPSK ≤ ˜ M (cid:18) g PSK | µ r | (1 + ρ ) | ν r | e g PSK | µr | ρ ) | νr | Ei (cid:18) − g PSK | µ r | (1 + ρ ) | ν r | (cid:19)(cid:19) . (80)Notably, since g PSK | µ r | | ν r | > g PSK | µ r | (1+ ρ ) | ν r | , we can conclude that for a fixed M , the SER of DPSKis asymptotically greater than the SER of PSK. • Joint TX/RX IQI:
Finally, for the case of joint TX/RX IQI with | ξ | = | ξ | , the SER isupper bounded by P s, DPSK ≤ ˜ M C − (1 + ρ ) | ξ | e − g PSK1+ ρ | ξ | | ξ | + | µr | | νr | | ξ | γ g PSK ( | ξ | − | ξ | ) × γ (cid:18) , − g PSK (1 + ρ ) | ξ | | ξ | ; − g PSK (1 + ρ ) | ξ | ( | µ r | + | ν r | ) | ξ | γ (cid:19) ! (81)which for asymptotic SNR values simplifies to P s, DPSK ≤ ˜ M C − (1 + ρ ) | ξ | e − g PSK | ξ | ρ ) | ξ | g PSK ( | ξ | − | ξ | ) γ (cid:18) , − g PSK | ξ | (1 + ρ ) | ξ | (cid:19) . (82)It is also shown, for this case, that for a fixed M , the SER of DPSK is asymptoticallygreater than the SER of its PSK counterpart.VI. N UMERICAL AND S IMULATION R ESULTS
In this section, we quantify the effects of IQI on the performance of single-carrier and multi-carrier based M -PSK, M -DPSK and M -FSK systems over flat Rayleigh fading channels in termsof the corresponding average SER. For a fair comparison, we assume that the transmit powerlevel is always fixed. This implies that the transmitted signal is normalized by | µ t | + | ν t | forTX IQI, by | µ r | + | ν r | for RX IQI and by ( | µ t | + | ν t | ) ( | µ r | + | ν r | ) for joint TX/RX IQI.To this end, Figs. 1 − − M -PSK, M -DPSK and M -FSK forsingle-carrier systems and multi-carrier systems, respectively. Assuming IRR t = IRR r = 20dB ,all possible combinations of ideal/impaired TX/RX are presented. It is noted that the numericalresults are shown with continuous lines, whereas markers are used to illustrate the respectivecomputer simulation results. For both single-carrier and multi-carrier systems, it is noticedthat the derived expressions characterize accurately the simulated SER performance for all the This demonstrates that our assumption of uncorrelated carrier and its image does not affect the accuracy of the SER analysisin multi-carrier systems. s /N (dB)10 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQI
M=16M=4 M=8
Fig. 1: Single-carrier system average SER as a function of the normalized E s /N for M -PSKwhen IRR t = IRR r = 20dB and φ = 3 °.considered modulation schemes in the presence of IQI. Specifically, it is first observed that RXIQI has more detrimental impact on the system performance than TX IQI. This result is expectedsince RX IQI affects both the signal and the noise while TX IQI impairs the information signalonly. Second, it is noticed that IQI exhibits different effects on the different modulation schemesconsidered. For example, it can be drawn from Fig.3 that the effects of IQI on single-carrier FSKare rather limited irrespective of the modulation order. This can be explained by the fact thatthe tone spacing in FSK is constant regardless of the modulation order. Hence, unlike PSK andDPSK, the IQI effects on FSK do not depend on the modulation order for both single-carrier andmulti-carrier systems. However, the cost of increasing M for FSK is an increased transmissionbandwidth. This is not the case for the other two modulation schemes where the angle separationdepends on the modulation order. For instance, the effects of IQI can be considered acceptablei.e., no error floor observed for the considered SNR range, only for M ≤ and M ≤ forthe cases of PSK and DPSK modulations, respectively. In fact, for single-carrier systems, when M = 16 , an error floor is observed at around when PSK modulation suffers from joint S /N (dB)10 -5 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQIM=2 M=8 M=16
Fig. 2: Single-carrier system average SER as a function of the normalized E s /N for M -DPSKwhen IRR t = IRR r = 20dB and φ = 3 °.TX/RX IQI, while for DPSK this error floor appears at around for all the consideredimpairment scenarios. It is also worth mentioning that for the joint TX/RX IQI case, this errorfloor is around × − for PSK versus × − for DPSK. Hence for a fixed M , the errorfloor is higher for DPSK than PSK, which confirms our observations in Section V.Even though the effects of IQI on the different modulation schemes follow the same trend inmulti-carrier systems as in single-carrier systems, it is observed that IQI affects the former moreseverely than the latter. This is because IQI in multi-carrier systems causes interference fromthe image subcarrier, which can have higher SNR than the desired signal, while single-carrierIQI causes interference from the signal’s own complex conjugate. An interesting example is thecase of M -FSK constellation, where in single-carrier systems the effects of IQI are negligible,while in multi-carrier systems an error floor is observed in Fig. 6, regardless of the modulationorder, for the RX IQI only as well as for joint TX/RX IQI cases. In the same context, the errorfloor for binary FSK appears at around . This error floor is observed for PSK and DPSK aswell with binary PSK being the most robust to IQI among the considered modulations, since the s /N (dB)10 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQIM=2 M=32
Fig. 3: Single-carrier system average SER as a function of the normalized E s /N for M -FSKwhen IRR t = IRR r = 20dB and φ = 3 °.error floor appears at around . It is also noted that unlike single-carrier systems where insome cases IQI could be neglected, for the considered IRR values, in multi-carrier systems theeffects of IQI at the RX should be compensated in order to achieve a reliable communicationlink, even in the case of the relatively simple binary modulation schemes.For multi-carrier systems, Fig. 7 compares the derived upper bound to the exact SER of M -PSK when IRR t = IRR r = 27dB . In this scenario, we consider the cases of TX IQI only, RXIQI only and joint TX/RX IQI for M = 2 and M = 32 . The solid lines correspond to the exactSER while the dashed lines represent the respective bound. It is noticed that although the boundis not particularly tight, it exhibits the same behavior as the exact SER curves and hence canprovide useful insights into the system performance.Finally, Fig. 8 and Fig. 9 demonstrate the effects of the IRR on the SER of the differentconsidered modulation schemes for multi-carrier systems, when
SNR = 25dB and
SNR = 40dB ,respectively. It is assumed that both TX and RX are IQI-impaired and that
IRR t = IRR r . Thephase imbalance assumed is ° in Fig. 8 and ° in Fig. 9. It is also noted that the continuous lines s /N (dB)10 -5 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQIM=2 M=16M=8
Fig. 4: Multi-carrier system average SER as a function of the normalized E s /N for M -PSKwhen IRR t = IRR r = 20dB and φ = 3 °.and dashed lines correspond to the IQI-impaired and ideal cases, respectively. For moderate SNRvalues, one can see that IQI affects the different modulations schemes in a different manner.For instance, joint TX/ RX IQI exhibits a constant loss in the SER performance of M -FSKregardless of the modulation order, which is not the case when considering phase modulation.Moreover, it is noticed that for lower SNR values, the effects of IQI vanish when the IRR isincreased; however, for higher SNR values and given that IQI effects dominate noise effectsat high SNR, there is a noticeable performance degradation even when considering high
IRR values. VII. C
ONCLUSION
We developed a general framework for the SER performance analysis of different M -arycoherent and non-coherent modulation schemes over Rayleigh fading channels in the presenceof IQI at the RF front end. The realistic cases of TX IQI only, RX IQI only and joint TX/RXIQI were considered and the corresponding average SER expression of the underlying schemes s /N (dB)10 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQIM=2 M=8 M=16
Fig. 5: Multi-carrier system average SER as a function of the normalized E s /N for M -DPSKwhen IRR t = IRR r = 20dB and φ = 3 °.was derived both in exact and in asymptotic form providing useful insights into the overallsystem behavior. The derived analytic results were corroborated with respective results fromcomputer simulations. It was shown that the performance degradation caused by IQI depends onthe considered modulation scheme with M -DPSK being the most sensitive modulation schemeto IQI. Moreover, for coherent and noncoherent phase modulation, increasing the modulationorder increases the impact of IQI on the system, while for frequency modulation the performancedegradation observed is constant regardless of the modulation order and single carrier frequencymodulation is the most robust scheme to IQI effects.A PPENDIX AD ERIVATION OF
MGF
FOR M ULTI -C ARRIER S YSTEMS I MPAIRED BY J OINT
TX/RX IQIFrom (31) and (39), taking u = e sγ and dv = f γ ( γ ) and integrating by parts, one obtains M γ IQI ( s ) = C + s Z | ξ | | ξ | | ξ | − γ | ξ | | ξ | − | ξ | + x ( | ξ | − | ξ | ) e sx e − xγ (cid:18) | µR | | νR | | ξ | − x | ξ | (cid:19) d x (83) s /N (dB)10 -5 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQI M=32M=2
Fig. 6: Multi-carrier system average SER as a function of the normalized E s /N for M -FSKwhen IRR t = IRR r = 20dB and φ = 3 °.where C is given in (43). For the case of | ξ | = | ξ | and setting z = | ξ | − x | ξ | , equation(83) simplifies to M γ IQI ( s ) = C + s | ξ | ( | ξ | − | ξ | ) e s | ξ | | ξ | + | µr | | νr | | ξ | γ Z | ξ | ze − z s | ξ | − | ξ | ( | µr | | νr | ) γ | ξ | z d z (84)which, considering the change of variable y = zs | ξ | , is equivalent to (40). On the contrary, for | ξ | = | ξ | and setting z = | ξ | − xa | ξ | , equation (83) becomes M γ IQI ( s ) = | ξ | | ξ | − | ξ | + s e | µR | | νR | | ξ | γ + s | ξ | | ξ | | ξ | − | ξ | Z | ξ | z e − | ξ | ( | µR | | νR | ) | ξ | γz − s z | ξ | d | ξ | −| ξ | + z d z (85)where d is given in (44). For the case of (cid:12)(cid:12)(cid:12) | ξ | | ξ | −| ξ | | ξ | | ξ | −| ξ | (cid:12)(cid:12)(cid:12) < | ξ | , we expand the involvedbinomial which yields M γ IQI ( s ) = | ξ | | ξ | − | ξ | + ∞ X k =0 ( − k s d k e | µR | | νR | | ξ | γ + s | ξ | | ξ | ( | ξ | − | ξ | ) k +1 Z | ξ | z − k e − | ξ | ( | µR | | νR | ) | ξ | γz − s z | ξ | d z (86) s /N (dB)10 -7 -6 -5 -4 -3 -2 -1 S y m bo l E rr o r R a t e Ideal TX/RXIQI TX and Ideal RXIQI RX and Ideal TXJOINT TX/RX IQIM=32M=2
Fig. 7: Multi-carrier system average SER (solid line) and derived upper bound (dashed line) asa function of the normalized E s /N for M -PSK when IRR t = IRR r = 27dB and φ = 1 °.By setting once more y = xs/ | ξ | , equation (41)) is deduced. Meanwhile for (cid:12)(cid:12)(cid:12) | ξ | | ξ | −| ξ | | ξ | | ξ | −| ξ | (cid:12)(cid:12)(cid:12) > | ξ | , and expanding the binomial in (85), one obtains the following analytic expression M γ IQI ( s ) = | ξ | | ξ | − | ξ | + ∞ X k =0 ( − k s e | µR | | νR | | ξ | γ + s | ξ | | ξ | ( | ξ | − | ξ | ) k d k +1 × Z | ξ | z k +1 e − | ξ | ( | µR | | νR | ) | ξ | γz − s z | ξ | d z (87)Finally, equation (42) is obtained by taking y = sz/ | ξ | .R EFERENCES [1] S. Mirabbasi and K. Martin, “Classical and modern receiver architectures,”
IEEE Commun. Mag. , vol. 38, no. 11, pp.132–139, Nov 2000.[2] S. Bernard, “Digital communications fundamentals and applications,”
Prentice Hall, USA , 2001.[3] O. Ozdemir, R. Hamila, and N. Al-Dhahir, “I/Q imbalance in multiple beamforming OFDM transceivers: SINR analysisand digital baseband compensation,”
IEEE Trans. Commun. , vol. 61, no. 5, pp. 1914–1925, May 2013.[4] B. Selim, P. C. Sofotasios, S. Muhaidat, and G. K. Karagiannidis, “The effects of I/Q imbalance on wireless communica-tions: A survey,” in
IEEE MWSCAS’16 , Oct 2016, pp. 1–4.
20 25 30 35IRR (dB)10 -3 -2 -1 S y m bo l E rr o r R a t e PSKDPSKFSKM=32 M=2
Fig. 8: Multi-carrier system average SER as a function of the
IRR for M -PSK, M -DPSK and M -FSK, with RX IQI only, when E s /N = 25dB and φ = 2 °. [5] R. Hamila, O. Ozdemir, and N. Al-Dhahir, “Beamforming OFDM performance under joint phase noise and I/Q imbalance,” IEEE Trans. Vehicular Technol. , vol. 65, no. 5, pp. 2978–2989, May 2016.[6] O. Ozdemir, R. Hamila, and N. Al-Dhahir, “Exact average OFDM subcarrier SINR analysis under joint transmit receiveI/Q imbalance,”
IEEE Trans. Veh. Technol. , vol. 63, no. 8, pp. 4125–4130, Oct 2014.[7] M. Lupupa and J. Qi, “I/Q imbalance in generalized frequency division multiplexing under Weibull fading,” in
IEEEPIMRC ’15 , Aug 2015, pp. 471–476.[8] S. Krone and G. Fettweis, “On the capacity of OFDM systems with receiver I/Q imbalance,” in
IEEE ICC’08 , May 2008,pp. 1317–1321.[9] A. Ishaque, P. Sakulkar, and G. Ascheid, “Capacity analysis of uplink multi-user SC-FDMA system with frequency-dependent I/Q imbalance,” in
Allerton’13 , Oct 2013, pp. 1067–1074.[10] B. Selim, P. Sofotasios, G. K. S. Muhaidat, and B. Sharif, “Performance of differential modulation under RF impairments,”in
IEEE ICC’17 .[11] Y. Zou, M. Valkama, N. Y. Ermolova, and O. Tirkkonen, “Analytical performance of OFDM radio link under RX I/Qimbalance and frequency-selective rayleigh fading channel,” in
IEEE SPAWC’11 , June 2011, pp. 251–255.[12] A. A. A. Boulogeorgos, P. C. Sofotasios, B. Selim, S. Muhaidat, G. K. Karagiannidis, and M. Valkama, “Effects of RFimpairments in communications over cascaded fading channels,”
IEEE Trans. Veh. Technol. , vol. 65, no. 11, pp. 8878–8894,Nov 2016.[13] C. Zhu, J. Cheng, and N. Al-Dhahir, “Error rate analysis of subcarrier QPSK with receiver I/Q imbalances over Gamma-
20 25 30 35 40IRR (dB)10 -5 -4 -3 -2 -1 S y m bo l E rr o r R a t e PSKDPSKM=2M=32
Fig. 9: Multi-carrier system average SER as a function of the
IRR for M -PSK and M -DPSK,with RX IQI only, when E s /N = 40dB and φ = 1 °. Gamma fading channels,” in
ICNC’17 , Jan 2017, pp. 88–94.[14] L. Chen, A. G. Helmy, G. Yue, S. Li, and N. Al-Dhahir, “Performance and compensation of I/Q imbalance in differentialSTBC-OFDM,” in
IEEE Globecom’16 , Dec 2016, pp. 1–7.[15] L. Chen, A. Helmy, G. R. Yue, S. Li, and N. Al-Dhahir, “Performance analysis and compensation of joint TX/RX I/Qimbalance in differential STBC-OFDM,”
IEEE Trans. Veh. Technol. , vol. PP, no. 99, pp. 1–1, 2016.[16] A. Gouissem, R. Hamila, and M. O. Hasna, “Outage performance of cooperative systems under IQ imbalance,”
IEEETrans. on Commun. , vol. 62, no. 5, pp. 1480–1489, May 2014.[17] M. Mokhtar, N. Al-Dhahir, and R. Hamila, “OFDM full-duplex DF relaying under I/Q imbalance and loopback self-interference,”
IEEE Trans. Vehicular Technol. , vol. 65, no. 8, pp. 6737–6741, Aug 2016.[18] L. Samara, M. Mokhtar, O. Ozdemir, R. Hamila, and T. Khattab, “Residual self-interference analysis for full-duplex OFDMtransceivers under phase noise and I/Q imbalance,”
IEEE Commun. Lett. , vol. 21, no. 2, pp. 314–317, Feb 2017.[19] J. Li, M. Matthaiou, and T. Svensson, “I/Q imbalance in AF dual-hop relaying: Performance analysis in Nakagami- m fading,” IEEE Trans. Commun. , vol. 62, no. 3, pp. 836–847, March 2014.[20] ——, “I/Q imbalance in two-way AF relaying,”
IEEE Trans. Commun. , vol. 62, no. 7, pp. 2271–2285, July 2014.[21] X. Zhang, M. Matthaiou, M. Coldrey, and E. Bjrnson, “Impact of residual transmit RF impairments on training-basedMIMO systems,”
IEEE Trans. Commun. , vol. 63, no. 8, pp. 2899–2911, Aug 2015.[22] N. Kolomvakis, M. Matthaiou, and M. Coldrey, “IQ imbalance in multiuser systems: Channel estimation and compensation,”
IEEE Trans. Commun. , vol. 64, no. 7, pp. 3039–3051, July 2016. [23] J. Abouei, K. N. Plataniotis, and S. Pasupathy, “Green modulations in energy-constrained wireless sensor networks,” IETcommun. , vol. 5, no. 2, pp. 240–251, 2011.[24] B. Natarajan, C. R. Nassar, and S. Shattil, “CI/FSK: bandwidth-efficient multicarrier FSK for high performance, highthroughput, and enhanced applicability,”
IEEE Trans. Commun. , vol. 52, no. 3, pp. 362–367, March 2004.[25] F. F. Digham, M. S. Alouini, and S. Arora, “Variable-rate variable-power non-coherent M-FSK scheme for power limitedsystems,”
IEEE Trans. Wireless Commun. , vol. 5, no. 6, pp. 1306–1312, June 2006.[26] M. K. Simon and M.-S. Alouini,
Digital communication over fading channels . John Wiley & Sons, 2005.[27] T. Schenk,
RF Imperfections in High-Rate Wireless Systems . The Netherlands: Springer, 2008.[28] P. Y. et al. , “Single-carrier SM-MIMO: A promising design for broadband large-scale antenna systems,”
IEEE Commun.Surveys Tutorials , vol. 18, no. 3, pp. 1687–1716, 2016.[29] M. A. Chaudhry and S. M. Zubair,
On a class of incomplete gamma functions with applications . CRC press, 2001.[30] I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series, and Products , 6th ed. New York: Academic, 2000.[31] M. Abramowitz, I. A. Stegun et al. , “Handbook of mathematical functions,”
Applied mathematics series , vol. 55, no. 62,p. 39, 1966.[32] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,
Inegrals and Series, Vol. 2: Special Functions . Gordon and BreachScience Publishers, 1992. r X i v : . [ ee ss . SP ] A ug o f M IM O O F DM S ystems w ith P hase N oise a t T ransmit a nd R eceive A ntennas, T itle = P erf ormanceof M IM O − OF DM SystemswithP haseN oiseatT ransmitandReceiveAntennas, Author = LiJu − huandW uW ei − ling, Booktitle =7 thInt.Conf.onW irelessCommunications, N etworkingandM obileComputing, Y ear =2011 , P ages =1 − , o f M IM O O F DM S ystems w ith P hase N oise a t T ransmit a nd R eceive A ntennas.jsptp = arnumber =6040094 : , ISSN =2161 − , Keywords = M IM Ocommunication ; OF DM modulation ; Rayleighchannels ; errorstatistics ; intercarrierinterf erence ; interf erencesuppression ; phasenoise ; receivingantennas ; statisticaldistributions ; transmittingantennas ; M IM O ; OF DM systems ; Rayleighf adingchannels ; W ishartdistribution ; biterrorrate ; f requencyselectivechannels ; intercarrierinterf erence ; phasenoise ; receivingantennas ; signal − to − interf erenceplusnoiseratio ; transmittingantennas ; zero − f orcingdetectors ; AW GN ; Biterrorrate ; F ading ; OF DM ; P hasenoise ; Receivers ; Signaltonoiseratio, Owner = Administrator b ased s pecrum s ensing o f O F DM I QI, T itle = Likelihood − basedspectrumsensingof OF DM signalsinthepresenceof T x/RxI/Qimbalance, Author = AhmedElSamadouny, andAhmadGomaa, andN aof alAl − Dhahir, Booktitle = IEEEGlobalCommunicationsConf., Y ear =2012 , M onth = Dec, P ages =3616 − , P HN, T itle = M ulti − channelenergydetectionunderphasenoise : analysisandmitigation, Author = AhmetGokceoglu, andY aningZou, andM ikkoV alkama, andP aschalisC.Sof otasios, Journal = M obileN etworksandApplications, Y ear =2014 , S pectrum A ccess i n C R N etworks U nder I mperf ect S pectrum S ensing, T itle = OpportunisticSpectrumAccessinCognitiveRadioN etworksU nderImperf ectSpectrumSensing, Author = O.AltradandS.M uhaidaandA.Al − DweikandA.ShamiandP.D.Y oo, Journal = IEEE
J V
T, Y ear =2014 , K avehT W C , T itle = ExactSymbolErrorP robabilityof aCooperativeN etworkinaRayleigh − F adingEnvironment, Author = P aulA.AnghelandM ostaf aKaveh, Journal = IEEE
J W
COM, Y ear =2004 , N umber =5 , P ages =1416 − , V olume =3 , K avehT W C .jsptp = arnumber =1343871 : , Owner = Administrator S ymbol E rror P robability o f aC ooperative N etwork i n aR ayleigh F ading E nvironment, T itle = ExactSymbolErrorP robabilityof aCooperativeN etworkinaRayleigh − F adingEnvironment, Author = P aulA.AnghelandM ostaf aKaveh, Journal = IEEE
J W
COM, Y ear =2004 , S ymbol E rror P robability o f aC ooperative N etwork i n aR ayleigh F ading E nvironment.jsptp = arnumber =1343871 : , Owner = Administrator, T imestamp =2013 . . C ooperative S pectrum S ensing i n C R, T itle = EnergyDetectionBasedCooperativeSpectrumSensinginCognitiveRadioN etworks, Author = SamanAtapattuandChinthaT ellamburaandHaiJiang, Journal = IEEE
J C
OM, Y ear =2011 , n onlinearity, T itle = Ef f ectsof HP Anonlinearityonf requencymultiplexedOF DMsignals, Author = P aoloBanelliandGiuseppeBaruf f aandSaverioCacopardi, Journal = IEEE
J B
C, Y ear =2001 , N umber =2 , P ages =123 − , V olume =47 , n onlinearity.jsptp = arnumber =948265 : , ISSN =0018 − , Keywords = AW GN channels ; OF DM modulation ; cellularradio ; digitalvideobroadcasting ; errorstatistics ; f requencydivisionmultiplexing ; interactivesystems ; multimediacommunication ; nonlineardistortion ; poweramplif iers ; radioaccessnetworks ; radiolinks ; radiof requencyamplif iers ; AW GN channels ; BERdegradation ; CABSIN ET project ; DV B − T signal ; DV B − T standard ; HP Anonlinearity ; LM DSsystem ; biterrorrate ; broadbandaccesssystem ; computersimulations ; downlinkchannel ; f requencymultiplexedOF DM signals ; highpoweramplif ier ; macro − cells ; multimediainteractiveservicesystem ; nonlineardistortions ; perf ormanceanalysis ; predistortion ; spectralregrowth ; systemperf ormance ; Biterrorrate ; Digitalvideobroadcasting ; Downlink ; F requencydivisionmultiplexing ; Inf ormationanalysis ; M ultimediasystems ; N onlineardistortion ; OF DM ; P erf ormanceanalysis ; Signalanalysis, Owner = Administrator f or s election r elay n etworks, T itle = Distributedswitchandstaycombiningf orselectionrelaynetworks, Author = V oN guyenQuocBaoandHyung − Y unKong, Journal = IEEE
J C
OM L, Y ear =2009 , o f C oupling o n M ultiple A ntenna C apacity i n C orrelated F ast F ading E nvironments, T itle = Impactof CouplingonM ultiple − AntennaCapacityinCorrelatedF ast − F adingEnvironments, Author = N icolasW.BikhaziandM ichaelA.Jensen, Journal = IEEE
J V
T, Y ear =2009 , N umber =3 , P ages =1595 − , V olume =58 , o f C oupling o n M ultiple A ntenna C apacity i n C orrelated F ast F ading E nvironments.jsparnumber =04531123 : , ISSN =0018 − , Keywords = antennaarrays ; channelcapacity ; correlationmethods ; covarianceanalysis ; electromagneticcoupling ; f adingchannels ; channelcovarianceinf ormation ; correlatedf ast − f adingenvironment ; electromagneticconsideration ; multiple − antennacapacitycoupling ; signalspatialcorrelation ; M IM Osystems ; M ultiple − inputmultiple − output ( M IM O ) systems ; M utualcoupling ; mutualcoupling, Owner = Administrator n ew l ook a t d ual h op r elaying p erf ormance l imits w ith h w i mpairments, T itle = AN ewLookatDual − HopRelaying : P erf ormanceLimitswithHardwareImpairments, Author = EmilBjornsonandM ichailM atthaiouandM erouaneDebbah, Journal = IEEE
J C
OM, Y ear =2013 , o n A F r elaying, T itle = Ontheimpactof transceiverimpairmentsonAF relaying, Author = EmilBjornsonandAgisilaosP apadogiannisandM ichailM atthaiouandM erouaneDebbah, Booktitle = IEEEInt.Conf.onAcoustics, SpeechandSignalP rocessing, Y ear =2013 , M onth = M ay, P ages =4948 − , s imple C ooperative d iversity m ethod b ased o n n etwork p ath s election, T itle = AsimpleCooperativediversitymethodbasedonnetworkpathselection, Author = AggelosBletsasandAshishKhistiandDavidP.ReedandAndrewLippman, Journal = IEEE
J J
SAC, Y ear =2006 , N umber =3 , P ages =659 − , V olume =24 , s imple C ooperative d iversity m ethod b ased o n n etwork p ath s election.jsptp = arnumberarnumber