Analysis and Compensation of I/Q Imbalance Effects for Uplink Multiuser Systems

Transcription

1 Institute for Integrated Signa Processing Systems (ISS) RWTH Aachen University Prof. Dr.-Ing. Gerd Ascheid Master Thesis : M 042 Anaysis and Compensation of I/Q Imbaance Effects for Upink Mutiuser Systems by Pranav Krishna Sakukar Matr.-No Juy 2013 Supervisors: Prof. Dr.-Ing. Gerd Ascheid M.Sc. Aamir Ishaque This document is for interna use ony. A copyrights are controed by the supervising chair. Pubications of any kind are ony authorized with permission of the chair.

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5 I assure that this project was accompished by me, without any foreign assistance except the officia support of the chair. The used iterature is fuy indicated in the bibiography. Aachen, 22 Juy 2013 (Pranav Krishna Sakukar)

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7 FIXME: Pacehoder Pease repace this sheet by the officia thesis tite/task document signed by the professor and you.

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9 Abstract One of the imiting issues in impementing high data rate wireess systems is the impairment associated with anaog processing due to the hardware imperfections. In upink transmission of the mutiuser systems, a major source of such impairment is in-phase (I) and quadrature-phase (Q) imbaance introduced at the mutipe transmitters. In this report, we dea with the singe-carrier frequencydomain mutipe access (SC-FDMA) system which has been adapted as the physica ayer transmission scheme in 3GPP Long Term Evoution (LTE). The report focuses on muti-user SC-FDMA systems in the presence of frequency-dependent I/Q imbaance. We anayze the capacity performance of the conventiona equaizer and frequency domain pairwise equaizer. Capacity expressions indicate the substantia deterioration in performance of conventiona equaizer in the presence of I/Q imbaance and motivate the need for its compensation. Capacity anaysis of pairwise equaizer shows the near perfect compensation of I/Q imbaance. In order to use the pairwise equaizer for I/Q imbaance compensation, I/Q parameters and the channe have to be estimated at the receiver. In the presence of frequency-dependent transmitter-side I/Q imbaance, the received signa can be modeed as the sum of signa components from two different channes known as the direct and image channes. In this report, we consider data aided approach in time-domain to estimate the direct and image channes. We derive piot design conditions for muti-user system to minimize the MSE of the channe estimators. In the presence of guard bands, the channe estimation MSE increases near the band edges. It is known that the equi-powered piots are no onger optima in the presence of guard bands. We redesign the piots to minimize the MSE of channe estimates using overa MSE and per-subcarrier MSE criteria. We propose a muti-user piot position optimization agorithm to aocate the subcarriers to piots of different users. Performance of the proposed piot aocation techniques is anayzed through simuations.

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11 i Contents Symbos and Notations 1 1 Introduction Motivation Previous works Organization of the thesis System Mode OFDMA vs. SC-FDMA Subcarrier Mapping Mathematica Notations I/Q Imbaance Sources of I/Q Imbaance Mode of I/Q Imbaance Effect of I/Q Imbaance in Frequency Domain MIMO Systems Simuation Parameters Mutipath Fading Modeing Tx - I/Q Imbaances Anaysis of I/Q Imbaance Affected Systems Capacity Evauation for Idea OFDMA System Signa Mode Capacity Infuence of I/Q Imbaance on OFDMA Modified Signa Mode Per Subcarrier Signa-to-Interference Ratio (SIR) Capacity Capacity Evauation for Idea SC-FDMA System Signa Mode Per symbo SINR Capacity

12 ii CONTENTS 3.4 Infuence of I/Q Imbaance on SC-FDMA Systems Modified Signa Mode Capacity Computation Frequency Domain Pairwise Equaizer Numerica Resuts Idea OFDMA and SC-FDMA System Infuence of I/Q Imbaance on SC-FDMA System Frequency Domain Pairwise Equaizer Summary Channe Estimation for Pairwise Equaizer Various Schemes for I/Q Imbaance Estimation and Compensation Time Domain Channe Estimation System Mode Piot Design Criteria Subcarrier Assignment Method MMSE Estimator Estimating Variabe Number of Taps Presence of Guard Band Muti-user MIMO System System Mode Piot Design Criteria Subcarrier Assignment Methods Numerica Resuts Comparison of LS and MMSE Estimators Estimating Variabe Number of Taps Effect of Guard Band on MSE Effect of Guard Band on MSE Comparison with Other Estimation Schemes Summary Muti-user Piot Power Aocation and Position Optimization Singe User Piot Power Aocation Sum MSE Minimization Probem Max. MSE Minimization Probem Muti-user Piot Power Aocation Sum MSE Minimization Probem Max. MSE Minimization Probem Power Optimization with MSE Threshod Piot Position Optimization

13 CONTENTS iii 5.5 Numerica Resuts Singe User Power Optimization Muti-user Piot Power Optimization Piot Position Optimization Summary Appication to LTE Long Term Evoution (LTE) Upink Frame Structure Time Varying Channe Numerica Resuts Summary Concusion Summary Outook A Appendices 71 A.1 Capacity of OFDMA System without I/Q Imbaance A.2 Capacity of OFDMA System with I/Q Imbaance A.3 Per Symbo SINR for SC-FDMA A.4 ZF Equaizer: SINR for SC-FDMA A.5 MMSE Equaizer: SINR for SC-FDMA A.6 MMSE Equaizer: Capacity Upper Bound A.7 Capacity Comparison: ZF and MMSE Equaizers A.8 Per Symbo SINR for SC-FDMA with I/Q Imbaance List of Figures 80 List of Tabes 81 List of Listings 83 References 85

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15 1 Symbos and Notations Ô Õ Ô Õ H Ô Õ T Compex conjugate of Ô Õ Hermitian of Ô Õ Transpose of Ô Õ Distance between adjacent piots 0 M N M N a-zero matrix F N I M M k p q circøxù diagøxù EÖ Σ k σn 2 E P K L L ½ L C M N N p N Rx N Tx N N DFT matrix M M identity matrix Subcarrier mapping matrix of kth user Direct channe for th user Image channe for th user circuar matrix with first row given by the vector x diagona matrix of diagona eements from vector x Expectation operator Frequency domain equaizer for kth user Variance of the AWGN Tota power of the piots Tota number of users Effective ength of the channe Length of the estimated channe Actua ength of the channe Number of subcarriers per user Tota number of subcarriers Number of piots per user Number of receiver antennas Number of transmitter antennas

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17 3 1 Introduction Wireess communication has experienced a huge growth in recent years. It needs much higher data rates to support the new services ike video streaming. The data rates depend argey on the kind of architecture used at the transceivers. Whie each of those architecture have their pros and cons, their usage in the systems depend on the characteristics requirements set by the standards. A directconversion (or zero-if) receiver demoduates the incoming radio signa using a oca osciator whose frequency is approximatey equa to the carrier frequency of the signa. This does not have the cassica image band probem seen in super-heterodyne architectures due to the intermediate frequency (IF). However, it suffers from many probems due front-end imperfections [1]: Spurious oca osciator (LO) eakage DC offset Carrier frequency offset (CFO) 1/f or fiker noise I/Q imbaance. One major competing architecture is the ow-if architecture, in which the RF signa is mixed down to a non-zero ow intermediate frequency, typicay a few megahertz. Low-IF receiver avoids the DC offset and 1/f noise probems. The use of a non-zero IF re-introduces the image issue. By carefuy designing the fiters of the transceivers, the image issue can be overcome. Newer generation of wireess systems needed to support high data rates. Hence, the signa processing capabiity of the end device needed to grow. This propeed the advancements in semiconductor technoogy. The current trend of semiconductor downscaing has resuted in sma and cheap devices with high signa processing power. This resuts in aggravation of the device characteristics variations. Most current up-conversion architectures operate in quadrature mode. Any mismatch between the gains and phase difference between the inphase (I) and quadrature (Q) branches resut in mismatched I/Q processing. This probem is caed as I/Q imbaance probem. 1.1 Motivation I/Q imbaance causes desired subcarrier to interfere with the so-caed image subcarrier at the negative frequency index for the muti-carrier based systems ike OFDMA and SC-FDMA, giving rise to inter carrier interference (ICI). Amount of interference caused depends on the eve of mismatch at the transceivers. When mutipe users share the subcarrier space in SC-FDMA, the image subcarrier coud beong to a different user. Hence, the interference experienced at a subcarrier due to I/Q imbaance may come from other users, causing the inter user interference (IUI). Different users have varying degree of mismatch and hence, varying degree of I/Q imbaance. In this case, the performance degradation of a particuar user depends not ony on its own I/Q parameters but aso on the mismatch of the other users. This is especiay unfair for the users with high precision of matching at their own transmitters. For wideband OFDM/SC-FDMA systems, I/Q imbaances can be frequency-dependent, and can exist

18 4 Chapter 1: Introduction at both the transmitter and the receiver sides. For the ow cost mobie users, the transmitter side I/Q imbaance pays a major roe, as the base station is expected to maintain a higher precision matching eve. I/Q imbaance causes the channe seen at the receiver to spit into two separate effective channes, namey, direct channe and image channe. Estimation of the effective channes is essentia for the frequency-dependent I/Q imbaance compensation. We consider muti-user SC-FDMA system affected by frequency dependent I/Q imbaance and anayze the effect of I/Q imbaance on the capacity of the system. We reaize the need for compensation of the I/Q imbaance using frequency dependent pairwise equaizer. We estimate the effective channes at the receiver and expoit this information to compensate for the effects of I/Q imbaance to enhance the system performance. For the estimation of I/Q imbaanced channe, we use data-aided time-domain estimation technique. To suppress the effect of the guard bands, we design optima piots using different piot power optimization probems. For highy varying channes, piot are further optimized by using an agorithm for seecting the positions of piots for different users. 1.2 Previous works The probem of I/Q imbaance has been studied in the past for various systems ike OFDM in [2], [3], [4], [5]; OFDMA in [6], [7]. [2] addresses effective channe estimation in the presence frequency dependent I/Q imbaance for OFDM system. An LS-estimation based time domain channe estimator is discussed and a piot design criteria is designed for a singe user system with data subcarriers and piot subcarriers in the same OFDM bock. Whie migrating to muti-user SC-FDMA systems, piot symbos need to have mutiuser orthogonaity property. In [4], piot designs for MIMO-OFDM systems are considered. It discusses various techniques for mutipexing piot symbos from various antennas. However a singe user system was considered in the paper. [8] proposes a method to estimate the frequency-independent I/Q imbaance and DC offset jointy. The estimation technique was restricted to the receiver side I/Q imbaance. In the presence of transmitter side I/Q imbaance, interference due to I/Q imbaance cannot be canceed at the receiver. A frequency domain estimation of the effective channe was proposed for OFDMA system in [6]. The technique can be appied ony to the ocaized data subcarrier aocations, as it assumes constant channe for adjacent subcarriers. Through our work, we try to overcome these probems. We work on a channe estimation scheme which considers muti-user system with frequency dependent I/Q imbaance and works for a kinds of data subcarrier aocations. Whie deriving optima piots for effective channe estimation in time domain, effects of the presence of guard bands need to be considered to minimize the channe estimation mean squared error (MSE). An equispaced and equipowered piot aocation scheme for OFDM is proposed in [9]. the channe estimation MSE for the subcarriers near the guard band is very high for the proposed piot scheme. In [10], [11], [12], piot aocations for a singe user OFDM system are discussed. [10] describes a power aocation minimizing the overa channe estimation MSE for a user. In [11], a power aocation minimizing the maximum MSE for each subcarrier in the data band is discussed. [12] proposes a piot position assignment scheme based on a cubic poynomia to minimize the MSE. In a of the works mentioned above, piot aocation is studied for ony the singe user case. In muti-user case, the usabe data subcarriers are shared by mutipe users and each user gets a portion of the data band. The piot aocation probem becomes a muti-user optimization of the MSEs over the specific subcarrier sets. In this report, we anayze the muti-user piot power and position optimization probem based on MSE minimization criteria.

19 1.3: Organization of the thesis Organization of the thesis In chapter 2, we introduce the signa mode, the mode of I/Q imbaance and the effect of I/Q imbaance on muti-carrier systems. We anayze the effect of I/Q imbaance on igna-to-noise-ratio (SINR) and the capacity of OFDM and SC-FDMA system in chapter 3. We aso anayze the effect of frequency domain pairwise equaizer on the system capacity. Chapter 4 discusses data aided time domain channe estimation technique for muti-user SC-FDMA in the presence of I/Q imbaance. Various issues ike estimation of variabe number of channe taps, the presence of guard band are aso anayzed in the context of the time domain estimation scheme. Various optimization probems for piot power and position aocation for the I/Q imbaance affected systems are discussed in chapter 5. Chapter 6 presents simuations for LTE upink system in MATLAB and chapter 7 concudes the work.

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21 7 2 System Mode 2.1 OFDMA vs. SC-FDMA One of the key areas of concern for a mobies is that of the battery ife. It is necessary to ensure that the mobies use as itte battery power as possibe. With the RF power ampifier that transmits the radio frequency signa to the base station, consuming high power within the mobie, it is necessary that it operates efficienty. This can be significanty affected by moduation and the waveform used for transmission. Signas, that have a high peak to average ratio and need inear ampification, requires the use of RF power ampifiers in a non-efficient way. As a resut it is necessary to empoy a mode of transmission that has as near a constant enveope. OFDM has a high peak to average ratio [13]. Whie this is not a probem for the base station where power is not a particuar probem, it is unacceptabe for the mobie. As a resut, LTE uses a moduation scheme known as Singe Carrier Frequency Division Mutipe Access (SC-FDMA) which is a hybrid scheme. This combines the ow peak to average ratio offered by singe-carrier systems with the mutipath interference resiience and fexibe subcarrier frequency aocation that OFDM provides. Figure 2.1 and 2.2 show the bock diagrams of OFDMA transmitter and receiver respectivey. In case of SC-FDMA, a DFT bock at the transmitter before the subcarrier mapping and an IDFT bock at the receiver are added as shown. 2.2 Subcarrier Mapping Two of the main subcarrier mapping methods are ocaized an intereaved mapping. In ocaized mapping, consecutive subcarriers are aocated to a user. So, the spectrum has bocks of subcarriers corresponding to different users. This scheme is advantageous for channe dependent scheduing (CDS). In this scheme, base station aocates the channes to the users as per the channe quaity experienced by the transmitting users. In intereaved mapping, subcarriers assigned to one user are equay distributed across the spectrum of usabe subcarriers. This intereaving of carriers has advantage in frequency seective fading scenarios. When the subcarriers are spread out in the spectrum range and independent fading is reaized among the subcarries of a user, it causes esser distortions and hence, esser errors. Mod. Data Source S/P M point FFT Subcarrier Mapping M N point IFFT P/S Add CP DAC /RF Insert Piots Figure 2.1: Bock diagram of the transmitter

22 8 Chapter 2: System Mode Demod. Data P/S Source M point IFFT Frequency Domain Equaizer Subcarrier N Demapping point M H FFT S/P Rem. CP ADC /RF FD Ch. Est. TD Ch. Est. Figure 2.2: Bock diagram of the receiver Intereaved subcarrier mapping User User 2 Locaized subcarrier mapping User User 4 Figure 2.3: Intereaved and ocaized subcarrier mapping Figure 2.3 shows the difference between two subcarrier mappings. With 64 subcarriers being aocated to 4 users, it is the choice of those 16 subcarriers that changes capacity, SINR (signa to interference pus noise ratio) and many other properties. We examine the effect of those in ater chapters. 2.3 Mathematica Notations Before introducing the signa mode and the mode of I/Q imbaance, we expain the notations used in this thesis. The usua bod capita etters are used to denote the coumn vectors or matrices and Ô Õ, Ô Õ T and Ô Õ H are compex conjugate, transpose, and Hermitian of Ô Õ respectivey. An M M identity matrix is denoted byi M and0 M N is an a-zero matrix of sizem N. We aso usediagøxù as a diagona matrix whose diagona eements consist of vectorx,circøxù denotes the circuar matrix whose first row is given by the vector x,ð Ð denotes the Eucidean norm of vector, and EÖ is the expectation operator. Let the number of users bek and numbers of subcarriers assigned per user bem. So, the tota number of subcarriers is N KM. Now, et us introduce some matrices and their properties which wi be used frequenty in this report. A N N matrix F N represents N-point DFT matrix and its Ôm,nÕ eement is given by F N Ôm,nÕ 1 Ôm 1ÕÔn 1Õ j2π e N, (2.1) N where m,n 1,,N. The corresponding IDFT matrix is, thus, F H N. Let us consider the square of the DFT matrix F 2 1 Ôm n 1 orm N n 2Õ NÔm,nÕ, (2.2) 0 ÔotherwiseÕ

23 2.4: I/Q Imbaance 9 where m,n 1,,N. Thus, F 2 N is a permutation matrix. The assignment of the data symbos to the user specific set of M subcarriers can be represented by N M mapping matrix M for the th user. The mapping matrixm has a unique eement of one in each coumn and has properties of M T M k 0 M M Ô kõ Ô kõ I M (2.3) and M M T k 0 N N Ô kõ diagøm k Ù Ô kõ, (2.4) where then 1 vector m k denotes the pacement of the occupied subcarriers by thekth user, whose eements consist of M 1 s sand N 0 s. 2.4 I/Q Imbaance Sources of I/Q Imbaance Before transmission, user signas from both the I/Q branches are sent through digita-to-anaog converters (DACs), ampifiers and puse shaping fiters. In practica systems, especiay in the ow-cost mobie terminas, the perfect matching of the fiters in I and Q branches is not possibe. This gives rise to frequency-dependent or frequency-seective I/Q imbaance. The user signa is then unconverted to the desired passband using a oca osciator (LO). Ideay, the I and Q branches shoud have same ampitudes and phase difference of90. Due to imperfect matching of the ampitude gains and the phase difference of the oca osciator, frequency-independent I/Q imbaance is introduced Mode of I/Q Imbaance During the upink transmission, user equipments (UE) act as transmitters and base station (BTS) acts as a receiver. Due to mass production of the sma and cheap UEs, they are more prone to I/Q imbaance. BTS, usuay, have a much higher precision in terms of matching of the I and Q branches due to the use of better quaity semiconductors. So, in our work, we assume the BTS i.e. the receiver in the upink transmission scheme to be idea. Hereafter, we ony consider I/Q imbaance caused at the transmitter. The bandpass system is modeed by its equivaent owpass system. Let Øa I,,a Q, Ù and Øθ I,,θ Q, Ù represent the frequency-independent gains and phase offsets of the I and Q channes for the th user. The puse shaping fiters at the transmitter are represented byg I, ÔtÕ andg Q, ÔtÕ. The transmit system with I/Q imbaance can be modeed as the summation of two systems namey direct system whose input is the same as the transmitter input signa sôtõ s I ÔtÕ js Q ÔtÕ and mirror system whose input iss ÔtÕ. The bandpass system mode with frequency-dependent and frequency-independent I/Q imbaance is shown in figure 2.4, whereas figure 2.5 shows its equivaent owpass mode. The impuse responses of the direct and mirror systems at the transmitter are denoted by g D, ÔtÕ and g M, ÔtÕ. They are reated to the I/Q imbaance parameters and the equivaent puse shaping fiters as g D, ÔtÕ 1 2 Öa I,e jθ I, g I, ÔtÕ a Q, e jθ Q, g Q, ÔtÕ (2.5) g M, ÔtÕ 1 2 Öa I,e jθ I, g I, ÔtÕ a Q, e jθ Q, g Q, ÔtÕ. (2.6)

24 10 Chapter 2: System Mode s I, [n]δ(t nt) n g I, (t) a I, cos(2πf c t+θ I, ) ÇØÖ Ù Ö ³ ÒÐ y bp,k (t) k r I [n] cos(2πf c t) s Q, [n]δ(t nt) n a Q, sin(2πf c t+θ Q, ) g Q, (t) 90 o +θ Q, θ I, Channe h bp, (t) y bp, (t) ÏÆ w bp (t) 90 o sin(2πf c t) r Q [n] Figure 2.4: Bandpass SISO system mode with I/Q imbaance s [n]δ(t nt) n g D, (t) Other users signas y k (t) k Channe h (t) y (t) r[n] ( ) g M, (t) ÏÆ w(t) Figure 2.5: Equivaent owpass SISO system mode with I/Q imbaance So, the I/Q imbaance affected user signa, which is transmitted over the channe, can be modeed as x ÔtÕ s ÔtÕ g D, ÔtÕ s ÔtÕ g M,ÔtÕ. (2.7) Let this x ÔtÕ be transmitted over a channe with impuse response h ÔtÕ. So, the received signa can be modeed as rôtõ ô ô x ÔtÕ h ÔtÕ wôtõ (2.8) Ös ÔtÕ g D, ÔtÕ s ÔtÕ g M,ÔtÕ h ÔtÕ wôtõ, (2.9) wherewôtõ represents the additive white Gaussian noise (AWGN). The equivaent direct channe and mirror channes (incuding the transmit fiters as we as the channe hôtõ) can be written as Using these notations, the received signa gets simpified to p ÔtÕ g D, ÔtÕ h ÔtÕ (2.10) q ÔtÕ g M, ÔtÕ h ÔtÕ. (2.11) r Ô tõ ô s ÔtÕ p ÔtÕ s ÔtÕ q ÔtÕ wôtõ. (2.12) Effect of I/Q Imbaance in Frequency Domain Consider an OFDM system with N subcarriers, where the cycic prefix is chosen to be onger than the maximum ength L of the overa channe impuse response. Further, assume the transmitter and

25 2.5: MIMO Systems 11 S n I/Q imbaance X n wireess channe R n n n n Figure 2.6: Image subcarrier interference due to transmitter I/Q imbaance receiver to be perfecty synchronized in frequency and time. LetX n be a compex vaued symbo that is transmitted at subcarrier n. Then, the transmission of each X n over a frequency seective channe can be modeed in baseband by Y n H n X n W n, (2.13) whereh n andw n represent the associated frequency domain channe coefficient and AWGN sampe, respectivey. In case of an AWGN channe, n : H n 1. Here, each X n may in genera differ from the origina transmit symbo at subcarrier n, which wi be denoted bys n. Ony in the case of an idea transmitter X n S n hods. Simiary, each receive symbo may in genera differ from the actuay received symbo R n, where R n X n ony hods in case of an idea receiver. As we are anayzing upink case, we assume idea receiver at the base station (BST). So, for further anaysis R n Y n is assumed. Considering the upink transmission in OFDM system with I/Q imbaance of the the mobie termina, I/Q imbaance at the transmitter front-end affects the transmitted symbos before the corruption by the wireess channe according to (2.7). So, the origina transmit symbo S n at subcarrier n wi be interfered by the compex conjugated transmit symbo S n at subcarrier n and vice versa. Here, n th and n th subcarriers represent subcarriers ocated symmetricay to the DC carrier. So, in DFT domain, the impact of I/Q imbaance on the upink transmission can be modeed by X n G D,n S n G M,n S n, (2.14) where the compex vaued weighing factors G D and G M are the DFTs of g D ÔtÕ and g M ÔtÕ respectivey. Figure 2.6 shows the effect of transmit I/Q imbaance in frequency domain. We get the received symbos of the upink transmission as R n H n ÔG D,n S n G M,n S nõ W n. (2.15) The amount of I/Q imbaance can be quantified in terms of a subcarrier dependent image-eakage-ratio ILR n which is given by 2 G D,n ILR n G M,n, (2.16) where the idea case of no I/Q imbaance corresponds tog M,n 0 and G D,n 1, i.e. n : ILR n db. The practica vaues of ILR vary from -25 db to -40 db [14] over the subcarriers. 2.5 MIMO Systems Mutipe-Input-Mutipe-Output (MIMO) is a promising technoogy that improves the communication performance by having mutipe transmit and receive antennas. In contrast to SISO (Singe Input and Singe Output), MIMO is shown to offer reativey huge spectra efficiencies cose to Shannons bound. The capacity of MIMO system is inear to the number of antennas, providing arge performance

26 12 Chapter 2: System Mode h 11 h 12 n 1 x 1 h1n R r 1 MIMO x 2 n 2 r 2 MIMO Transmitter Receiver n NRx x NTx r NRx Parameter Figure 2.7: MIMO system mode Specification FFT/IFFT size 256 Number of users 3 Channe 7-tap Rayeigh faded Piots per user 8 Data subcarriers per user 64 Tabe 2.1: Simuation parameters increases. Because of the attractive features, MIMO has entered many wireess standards, such as WiMAX, WLAN, LTE. However, such advantages of MIMO systems require the intensive compex signa processing. To better evauate the performance of MIMO systems, the systems are often modeed as the quasistatic fat-fading, shown in figure 2.7. The channe matrix H is constant over the burst transmission of severa symbos and varies independenty from burst to burst. Under the condition that the spatia pacement of the antennas is arge enough, the path gains are aso independent. So the received signa at time instant t can be given as RÔtÕ HXÔtÕ nôtõ, (2.17) where RÔtÕ : Ör 1 r 2 r NRx T, XÔtÕ : Öx 1 x 2 x NRx T, H : N Rx N Tx channe matrix. The noise n for each stream is assumed to be independent. For the MIMO SC-FDMA systems, effective channe estimation in time domain is studied in chapter Simuation Parameters In Tabe 2.1, we present the parameters used for simuations presented in this report. These parameters are used throughout the report uness stated otherwise.

27 2.6: Simuation Parameters 13 P h [n] ½ ¾ Ä n Figure 2.8: Exponentia power deay profie (EPDP) Mutipath Fading When a signa is transmitted from the transmitter, it experiences refection, diffraction, scattering before it reaches the desired receiver. Due to these effects, the receiver receives mutipe copies of the transmitted signa with different deays and attenuation. These effects can be modeed by mutipath propagation. The effects of mutipath incude constructive and destructive interference, and phase shifting of the signa. Destructive interference causes fading. Where the magnitudes of the signas arriving by the various paths have a distribution known as the Rayeigh distribution, this is known as Rayeigh fading. The power deay profie (PDP) gives the intensity of a signa received through a mutipath channe as a function of time deay. The time deay is the difference in trave time between mutipath arrivas. In this report, we consider two different types of power deay profies, namey, uniform power deay profie (UPDP) and exponentia power deay profie (EPDP). UPDP considers the case where a the channe taps have equa power and EPDP considers the case where the power of channe taps decreases exponentiay with each tap. The power deay profies for a channe with ength L can be given as P UPDP h Ön 1 L 1 n L, (2.18) P EPDP h Ön 1 P 0 e cn 1 n L, (2.19) wherep 0 is the normaization power,c 0 is the decay factor of the exponentia distribution. Figures 2.8 shows the exponentia power deay profie. EPDP modes the mutipath effects reaisticay, as the power of channe taps decreases with each refection. EPDP is compatibe with the channe modes specified in [15] for the simuations of LTE systems. 3GPP specifies the channe modes based on COST-259 channe mode for channe simuations. For LTE reated simuations in chapter 7, we used the reduced compexity mode for Typica Urban environment from [15]. It uses one exponentia custer consisting of N P Rayeigh fading paths with cassica Dopper spectrum and decay of -2 db between the adjacent taps. N P 7 was chosen for the simuations for consistency.

28 14 Chapter 2: System Mode Figure 2.9: Comparison of subcarrier dependent ILRs for different δ Modeing Tx - I/Q Imbaances Let us consider an exampe based on above I/Q imbaance mode. Let the parameters of the gain imbaance α a I a Q 0.4 db and the phase imbaance θ θ I θ Q 4. To be abe to have frequency dependent I/Q imbaance, fiter responses shoud ateast have 2 fiter taps. Let the fiters be Ö1,δ and Ö1, δ, where δ decides the eve of frequency dependence. In figure 2.9, we consider δ È Ö0, 0.04 and pot the ILR vaues for the subcarriers. We see that the ILR vaues fuctuate around a DC vaue which is determined by the frequency independent I/Q parameters. The variation of I/Q imbaance over the subcarriers is considerabe. This makes the study of frequency dependent I/Q imbaance significant. In this report, we use the vaue ofδ 0.02 for the simuations uness expicity specified.

29 15 3 Anaysis of I/Q Imbaance Affected Systems In this chapter, we anayze the effect of I/Q imbaance on signa-to-interference ratio (SIR) and capacity for OFDMA and SC-FDMA systems. Channe capacity is the upper bound on the rate of information that can be reiaby transmitted over a communications channe. Due to impairments ike I/Q imbaance, capacity of SC-FDMA system reduces. We derive the expressions for SIR and capacity of systems with and without I/Q imbaance in this chapter. Simuations are performed to compare the performances. The conditions for maximizing capacity are discussed. 3.1 Capacity Evauation for Idea OFDMA System In this section, we anayze the muti-user OFDMA system in the absence of I/Q imbaance. A signa mode based on the transreceiver structure from figures 2.1 and 2.2 wi be discussed. The capacity expressions are derived and anayzed Signa Mode Lets andx denote the origina user signa and the signa sent over the channe for userrespectivey. We can see from the OFDMA transceiver bock diagram, that the moduated data of kth user wi be first mapped to its corresponding subcarriers by a mapping matrixm k, foowed by then point IFFT matrix F H N. So, the transmitted signa x k can be expressed as x k F H NM k s k. (3.1) At the base station, signas from a the different users are received. The received signa r, at the base station after CP remova, can be described as a sum of circuar convoutions of the transmitted signa x k with corresponding channe h k. Mathematicay, r Kô k1 F H N H k x k n K ô k1 Λ k M k s k «n, (3.2) where Λ k : diagøf N h k Ù, H k circøh k Ù. Here, Λ k denotes the frequency domain channe between kth user and the receiver. This received signa is passed through the DFT bock and inverse subcarrier mapper. After demapping, received signa corresponding the different users get separated as r M T F Nr Kô k1 M T Λ km k s k M T F Nn (3.3)

30 16 Chapter 3: Anaysis of I/Q Imbaance Affected Systems Since, Λ k is a diagona matrix, M T Λ km k 0, Đ k. (3.4) This excusiveness property shows that, interference terms from other users are bocked by the demapping procedure. So, (3.3) simpifies to r Λ s n, (3.5) where Λ M T Λ M and n M T F Nn, is the noise term of th user. Now, Λ signifies the frequency domain effective channe for the th user. This received signa is affected by the channe. We perform equaization to eiminate the effect of channe and get the data signa sent by the users. The equaizers are designed with some knowedge of the channe. To obtain the knowedge about the channe, certain known piots or preambe signas are sent. This information is used to estimate the channe in time or frequency domain. Let the equaizer for theth user be Σ. So, the equaized signa is ŝ Σ r Σ Λ s Σ n. (3.6) For the frequency domain (FD) equaization without joint subcarrier processing, Σ M T Σ M, (3.7) where Σ is an N N diagona matrix representing the frequency domain equaizer for th user s channe. Linear equaizers ike zero forcing and MMSE can be used for the equaization. For the zero forcing (ZF) equaizer, Σ Λ H Λ 1Λ H. (3.8) Simiary for the MMSE equaizer, where γ is the signa-to-noise ratio (SNR) of the th user. Σ Λ H Λ γ 1 I N 1Λ H, (3.9) Capacity The instantaneous per-subcarrier capacity for OFDMA is found to be C 1 M 1 M Mô i1 Mô i1 og 2 1 σs 2 σn 2 Λ ÔiÕ 2 Å og 2 Ô1 γ j, Õ, (3.10) where γ j, σ2 s Λ σn 2 ÔjÕ 2 represents the SNR of the jth subcarrier of the th user. We notice that, when the instantaneous channe gains are high, the instantaneous SNRs woud be high giving a higher capacity. Refer to appendix A.1 for a detaied proof. Thus, the ergodic capacity of th subcarrier of the OFDMA system is found out to be C OFDMA E ÖC 1 M Mô i1 E Öog 2 Ô1 γ j, Õ. (3.11)

31 3.2: Infuence of I/Q Imbaance on OFDMA 17 For the case of Rayeigh fading channes, we obtain the cosed form expression for the ergodic capacity as Å Å C OFDMA nô2õ exp 2σ 2 Ei γ 0 2σ 2, (3.12) γ 0 where σ is the Rayeigh parameter, γ 0 σ2 s and Ei is the exponentia integra function defined as σn 2 EiÔxÕ x e t t dt. (3.13) Hence, we observe that the ergodic capacity of OFDMA system without I/Q imbaance increases with with increase in SNR. 3.2 Infuence of I/Q Imbaance on OFDMA In this section, we anayze the muti-user OFDMA system with I/Q imbaance. The SIR, per subcarrier SINR and capacity expressions are derived and anayzed Modified Signa Mode Being an OFDMA system, the transmitted signa x k can be expressed as x k F H NM k s k. (3.14) But due to frequency-dependent (FD) I/Q imbaance at the transmitter, the actua transmitted signa for the kth user x k becomes x k F H N G D k M ks k G M k F2 N M ks k, (3.15) where the parameters G D k and GM k represent the I/Q mismatch for the direct and mirror channes respectivey. The transmitted signa experiences the fading channe and additive noise. The received signa r, at the base station after CP remova, can be described as r Kô k1 F H N k1 H k x k n Kô Λ k G D k M ks k Λ k G M k F2 NM k s k n, (3.16) where Λ k : diagøf N h k Ù. Here, we can observe that the first term inside summation is the contribution of the signas through the direct channe. The second term shows the interference caused due to I/Q imbaance as the signa experiences the mirror channe. We observe a matrix F 2 N in the second term. This matrix is a permutation matrix giving image subcarriers corresponding to the user s subcarriers. Passing this signa through the FFT and subcarrier demapping bocks, we get the received signa of the th user as r M T F Nr Kô M T Λ kg D k M ks k M T Λ kg M k F2 N M ks k k1 M T F Nn (3.17)

32 18 Chapter 3: Anaysis of I/Q Imbaance Affected Systems Since, Λ k G D k is a diagona matrix, M T Λ kg D k M k 0, Đ k. (3.18) It shows that interference cannot come from direct channes of other users no matter what the channe statistics or I/Q imbaance is. (3.17) simpifies the signa to a sum of user signa, interference from the mirror channes of a the users and the noise term as where r Λ G D s Kô k1 Λ k G M k P ks k n, (3.19) Λ k M T Λ km (3.20) G D M T GD M (3.21) G M k M T GM k M (3.22) P k M T F2 NM k, (3.23) with n M T F Nn, is the noise term of th user. P k is a decision matrix containing just 0s and 1s. Ôm,nÕth eement of P k decides whether nth subcarrier of user k has its image into mth subcarrier of user depending upon its vaue. This signa is now passed through an equaizer. Let the equaizer for th user be Σ. When the receiver does not take into account the effect of I/Q imbaance and uses the conventiona equaizers, the equaized signa can be written as ŝ Σ r Σ Λ G D s Kô k1 For the conventiona frequency domain (FD) equaization, Σ Λ k G M k P ks k Σ n. (3.24) Σ M T Σ M, (3.25) where Σ is an N N diagona matrix. We can anayze the effect of I/Q imbaance on a system unaware of I/Q imbaance. We can derive the expressions for capacity and SINR assuming the singe tap equaizer Per Subcarrier Signa-to-Interference Ratio(SIR) In this subsection, we compute the per subcarrier SIR which is a measure of the infuence of interference due to I/Q imbaance on a noise free system. The SIR of the mth subcarrier Ôm 1,s,MÕ of the th user s received signa is given by SIR ÔmÕ KÔsÕ Ôm, mõ K ÔiÕ Ôm,mÕ, (3.26) wherek ÔsÕ andk ÔiÕ denote the covariance matrices of the signa part and the interference part of the received signa respectivey and can be cacuated as K ÔsÕ E r ÔsÕ r ÔsÕ H σ 2 s Λ Λ H G D ÔG D ÕH (3.27)

33 3.3: Capacity Evauation for Idea SC-FDMA System 19 and K ÔiÕ E K ô k1 r ÔiÕ r ÔiÕ H σ 2 sλ k Λ H k GM k ÔGM k ÕH m ½ k, (3.28) where r ÔsÕ and r ÔiÕ represent the singa and interference parts of r respectivey. Thus, the SIR can be rewritten as SIR ÔmÕ G D ÔmÕ 2 G M ÔmÕ 2 m ½ ÔmÕ K Λ k. (3.29) ÔmÕ 2 G M k1ôkđõ Λ ÔmÕ 2 k ÔmÕ 2 m ½ k ÔmÕ In the denominator of (3.29), the first term corresponds to the interference due to the th user s own Tx I/Q imbaance known as inter carrier interference (ICI), whereas the second term corresponds to the interference caused by other users I/Q imbaance known as inter user interference (IUI). Note that ony one of these terms wi be non-zero depending on the subcarrier assignment to the users Capacity Per-subcarrier capacity for a user in the presence of I/Q imbaance is found to be C 1 M Mô i1 og 2 1 σ 2 s σ 2 n σ 2 s K k1 «G D ÔiÕ 2 Λ ÔiÕ 2 G M k ÔiÕ 2 Λ k ÔiÕ 2 m ½ k ÔiÕ. (3.30) Detaied proof in appendix A.2. We can observe some fundamenta differences between expression and that without I/Q imbaance. The major difference is one extra additive term in denominator of the instantaneous SINR. This is the effect of interference term seen in the received signa due to I/Q imbaance. Of the K terms in the interference summation, ony one term wi be non-zero. This term comes from the user occupying the image subcarrier of the particuar subcarrier at hand. This is taken care of by the vector m ½ k containing just 0s and 1s. We observe that - When the I/Q imbaance is not that severe, i.e. when G M k ÔiÕ 2 is negigibe, the additive term in the denominator vanishes and the capacity reduces to the cassica OFDMA capacity expression In the ow SNR regions, the effect of the interference term wi not be severe. So, the divergence form the cassica capacity wi be ess. As the SNR increases, the effect of interference term becomes higher and the capacity decreases. 3.3 Capacity Evauation for Idea SC-FDMA System In this section, we anayze the effect of subcarrier aocation on capacity of idea SC-FDMA system Signa Mode In SC-FDMA systems, there is an additiona M-point FFT bock at the transmitter as shown in figure 2.1. The transmitted signa mode for an SC-FDMA system can be expressed as x k F H NM k F M s k. (3.31)

34 20 Chapter 3: Anaysis of I/Q Imbaance Affected Systems After the channe effects, the received signa r, can be described as r Kô k1 F H N H k x k n K ô k1 Λ k M k F M s k «n. (3.32) Passing this signa through FFT, subcarrier demapping and IDFT bocks, we get the received signa for theth user as r F H MM T F Nr F H M K ô k1 M T Λ km k s k «F H M MT F Nn. (3.33) Using (3.4), we can remove terms corresponding to transmitted symbos of other users and simpify (3.33) to r F H MΛ F M s n, (3.34) whereλ M T Λ M andn F H M MT F Nn, is the noise term ofth user. The difference between the received signas for OFDMA and SC-FDMA is that for OFDMA matrix mutipyings is diagona. So, the received signa at the ith symbo just has the contributions from signa sent at the jth subcarrier with a noise term. But in the case of SC-FDMA, the matrix F H M Λ F M is a circuant matrix. Hence, there are contributions from the signas of other streams. In SC-FDMA systems the equaizer structure aso has the ower order DFT matrix term. The equaizer in the case of SC-FDMA forth user isf H M Σ F M. So, the equaized signa is ŝ F H MΣ F M r F H MΣ Λ F M s F H MΣ F M n. (3.35) For the singe tap frequency domain (FD) equaizer, Σ is same as (3.7) in OFDMA. We observe that, using the ZF equaizer where Σ Λ I M, we can eiminate the interference terms from other streams of the user. Whie reducing the interference terms by equaizing, the equaizer which is nondiagona matrix in this case adds noise contributions from other symbos. Hence, whie cacuating the interference and noise terms, these factors aso have to be taken into account. Effect of the choice of equaizer can be seen on the capacity expressions. We now go on to derive the expressions of SINR and capacity and anayze the effect of subcarrier mapping on these aspects Per symbo SINR Instantaneous per symbo SINR for an SC-FDMA system can be written as γ σ2 s 1 M M j1 Σ ÔjÕ 2 ÔΛ ÔjÕ 2 σn 2 σ 2 Õ s 1 M M j1 Σ ÔjÕΛ ÔjÕ 2 1 σ 2 s 1. (3.36) See appendix A.3 for the detaied derivation. Using this expression, we can now compare the SINR performance of different equaizers. The SINR expressions for ZF and MMSE equaizers are as foows γ ZF γ MMSE 1 M Mô j1 1 1 M M j1 1 1, (3.37) γ j, γ j, γ j, (3.38)

35 3.4: Infuence of I/Q Imbaance on SC-FDMA Systems 21 The derivations of (3.37) and (3.38) can be found in appendices A.4 and A.5 respectivey. Per symbo SINR of SC-FDMA system with ZF equaizer can be seen as the Harmonic Mean (HM) of the the SNRs experienced on the aocated subcarriers. This is due to scaing of the noise by channe inversion foowed by the IDFT. We notice that the SINRs for a the symbos of a user are same. They do not soey depend on the channe statistics for the corresponding subcarrier, but on a the subcarriers aocated to that user unike that in OFDMA (see (3.29)) Capacity The choice of equaizer is one of the most important decisions for an SC-FDMA system designer as it directy affects the SINR and hence, the capacity of the system. The instantaneous capacity corresponding to the th user can be cacuated using instantaneous SINR,γ as Now, the ergodic capacity can be cacuated as foows C og 2 Ô1 γ Õ. (3.39) C E Öog 2 Ô1 γ Õ. (3.40) Using (3.39) and the SINR reationships (3.37) and (3.38), we get the instantaneous capacities for ZF and MMSE equaizers. They are reated as C ZF C MMSE, (3.41) where the equaity hods when a the Λ ÔjÕ are equa. For proof of this inequaity, refer to appendix A.7. Thus, the same reation hods for the ergodic capacities too, i.e. C ZF C MMSE. (3.42) The capacity of SC-FDMA using MMSE equaizer is bounded by the capacity of OFDMA with same subcarrier aocation (proof in appendix A.6). C MMSE C OFDMA, (3.43) where the equaity hods when a the γ j, are equa, which in turn means when a the Λ ÔjÕ are equa. The ergodic capacity aso foows the same reation. So, for achieving equaity in (3.42) and (3.43), channe has to be a fat fading channe. When the channe has ony a few taps i.e. when we have a arge coherence bandwidth and the subcarriers have ocaized mapping, we can get quite cose to the OFDMA capacity. 3.4 Infuence of I/Q Imbaance on SC-FDMA Systems In this section, we anayze the muti-user SC-FDMA system with of I/Q imbaance. We derive the SINR expressions and anayze the effect of I/Q imbaance on capacity of the system Modified Signa Mode Being SC-FDMA system, the transmitted signa x k can be expressed as x k F H NM k F M s k. (3.44)

36 22 Chapter 3: Anaysis of I/Q Imbaance Affected Systems But due to frequency-dependent (FD) I/Q imbaance, the actua transmitted signa for thekth user x k becomes x k F H N G D k M kf M s k G M k F2 NM k F H Ms k, (3.45) represent the I/Q mismatch for direct and mirror channe respec- where the parameters G D k and GM k tivey. The received signa r at the base station, is r Kô k1 H k x k Kô F H N k1 n Λ k G D k M kf M s k Λ k G M k F2 N M kf H M s k n, (3.46) where Λ k : diagøf N h k Ù. So, the received signa of the th user after passing through the DFT bock, subcarrier demapping and IFFT bock can be written as r F H M MT F Nr Kô M T Λ kg D k M kf M s k F H M k1 M T Λ kg M k F2 N M kf H M s k F H M MT F Nn. (3.47) Using (3.18) simpifies equation (3.47) to r F H M Λ G D F Ms Kô k1 F H M Λ kg M k P kf H M s k n, (3.48) where Λ k, G D, GM k and P k are defined in equations (3.20), (3.21), (3.22) and (3.23) respectivey and n F H M MT F Nn, is the noise term of th user. In SC-FDMA systems without I/Q imbaance, we observed the interference due to other streams of the same user. Here, another kind of interference is introduced due to the I/Q imbaance caused by other users i.e. IUI. This signa needs to be equaized before being sent to the signa demoduator. When the conventiona equaizers are used without the knowedge of I/Q imbaance, then after equaization we have interferences from other users. Here, we anayze the effect of I/Q imbaance on these equaizers. The equaizer in the case of SC-FDMA for th user is Σ F H M Σ F M. The equaized signa can be written as ŝ F H M Σ F M r F H M Σ Λ G D F M s Kô k1 F H M Σ Λ k G M k P kf H M s k FH M Σ F M n. (3.49) For the frequency domain (FD) equaization without joint subcarrier processing Σ is same as (3.7) in OFDMA. The residua interferences and the noise are used to cacuate the per symbo SINR of a user which is a measure of the system performance Capacity Computation The SINR of a user depends on the tota power of the signa partsp ÔtÕ ÔiÕ, noise powerp ÔnÕ ÔiÕ and the power of the desired signa part P ÔdÕ ÔiÕ. Here, the tota signa power has three contributors, namey

37 3.5: Frequency Domain Pairwise Equaizer 23 power of the desired user P Ôt,1Õ, cross power between desired user P Ôt,2Õ and other users and the power of the interfering users P Ôt,3Õ. We can cacuate the SINR for the th user i.e. γ ÔiÕ as where γ ÔiÕ P Ôt,1Õ P Ôt,1Õ ÔiÕ P Ôt,2Õ P ÔdÕ ÔiÕ ÔiÕ σs 2 1 M P Ôt,2Õ Mô j1 ÔiÕ P Ôt,3Õ ÔiÕ P ÔnÕ ÔiÕ 1 1, (3.50) Σ ÔjÕ 2 Λ ÔjÕ 2 G D ÔjÕ2, (3.51) ÔiÕ 0, (3.52) P Ôt,3Õ ÔiÕ σ2 s P ÔdÕ ÔiÕ σ 2 s ÔiÕ σn 2 1 M P ÔnÕ ÔΣ Σ H ÕΛ kλ H k GM k ÔGM k ÕH m ½ k ô K M Tr k1 1 Mô 2 Σ ÔjÕΛ ÔjÕG D M ÔjÕ j1 Mô j1 «, (3.53), (3.54) Σ ÔjÕ 2. (3.55) See appendix A.8 for the derivation. These equations wi be anayzed to observe the effect on I/Q imbaance parameters on the system. In the absence of I/Q imbaance, i.e. when G D ÔjÕ2 1 and G M ÔjÕ2 0, we getp Ôt,3Õ ÔiÕ 0 and SINR reduces to the expression without I/Q imbaance from previous chapter. Here the equaizer Σ ÔiÕ takes fowing vaues ZF Equaization : Σ ÔiÕ Λ 1 ÔiÕ (3.56) MMSE Equaization : Σ ÔiÕ ÔΛ ÔiÕ 2 γ 0 1 Λ ÔiÕ (3.57) These expressions are used to cacuate the capacity of the system and the effects of I/Q imbaance on it. The instantaneous capacity corresponding to the th user can be cacuated using instantaneous SINR, γ as C og 2 Ô1 γ Õ. (3.58) Now, the ergodic capacity can be cacuated as foows C E Öog 2 Ô1 γ Õ. (3.59) 3.5 Frequency Domain Pairwise Equaizer In the previous sections, we anayzed the effect of I/Q imbaance on system unaware of I/Q imbaance. Conventiona equaizers performance deteriorate due to the presence of I/Q imbaance. Aong with compensating for the channe effects, we shoud compensate for I/Q imbaance as we. Due to the I/Q imbaance, the data signa on subcarrier k spis into its image subcarrier i.e.n k. The received signa at the kth subcarrier consists of the contributions from the kth and kth subcarriers. In the presence of I/Q imbaance, we have to compensate for the effects of channe and I/Q jointy for kth and kth subcarriers. In this section, we discuss frequency domain pairwise equaizer to mitigate the effects of I/Q imbaance. We anayze the performance of pairwise equaizers and compare their performance with their conventiona counterparts.