Optimal Pilot Power Allocation for OFDM Systems with Transmitter and Receiver IQ Imbalances

Transcription

1 Optimal Pilot Power Allocation for OFDM Systems with Transmitter Receiver mbalances V. K. Varma Gottumukkala Hlaing Minn Department of Electrical Engineering University of Texas at Dallas Richardson, Texas {vkg07000, Abstract n this paper, we derive optimal pilot power allocation for OFDM systems suffering from in-phase quadraturephase () imbalances. Existing works in literature on imbalances optimize for pilot spacings pilot designs. However, in all these works, optimal power allocation between pilot data symbols has not been considered. Using a lower bound on the average channel capacity as a metric, we optimize for the pilot data power allocations. Simulations show that the resulting optimal pilot power allocation increases the channel capacity along with lowering the bit error rate (BER). We further show that the power allocation is flexible in the sense that several power allocation choices exist that improve capacity compared to the equal power allocation scenario.. NTRODUCTON Orthogonal frequency division multiplexing (OFDM) systems are widely adopted in many current future wireless systems (e.g. EEE 80.a/g/n, 80.6a/e, LTE) [] [6]. The main advantage is that it helps cope with the frequency selectivity of the channel without the need for complex equalization it provides better resource granularity adaptability. However, due to analog imperfections at the transmitter receiver, there exist mismatch in amplitude phase between the in-phase quadrature-phase () carriers. The effect of imbalances has been dealt in detail in literature. The effect of the imbalance on the OFDM system is that it creates a mirror channel that sees the conjugate of the input symbol as its input in addition to the direct channel. This poses a new challenge in designing the receiver. Several compensation algorithms exist (e.g. [7] [0]) that compensate for the imbalances if the parameters are known. Pilots are typically used to learn these parameters along with the actual channel taps before compensation can be done. Several pilot designs have been proposed for estimating / imbalance or combined effect of / imbalance channel [7], [] [8]. n [8] pilot designs were proposed to efficiently estimate these imbalance parameters by turning the problem of estimating the channel taps along with the imbalance parameters into a problem of estimating two parallel channels. However, in all of these pilot designs, the question of what is the optimal pilot power allocation in the presence of imbalance hasn t been dealt with. n [9], the problem of optimal power allocation in OFDM systems the channel is learnt through training was considered. However, it did not consider the effect of imbalance in the OFDM system. n this work, we consider a more general case of [9] the OFDM system suffers from imbalances the imbalance parameters are estimated in the form of the channel taps for both direct mirror channels. When there are no amplitude phase mismatches between the carriers, the mirror channel becomes zero, the problem reduces to the problem as considered in [9]. Simulations show the improved capacity reduced bit error rate (BER) as a result of using the optimal power allocation that we derived. We focus on the impact of / imbalance while assuming ideal conditions for other impairments such as carrier frequency offset (CFO). n practical systems, the mobile station can perform frequency synchronization from the downlink received signal, after which the residual CFO for the uplink transmission would be quite small. n this scenario, neglecting the CFO as is considered in this paper is relevant. Similarly, for the downlink reception, after proper synchronization, the effects of synchronization errors would be quite negligible; hence the following data transmission phase would fit our considered system setup well. n brief, with proper synchronization as always required in practice, the proposed approach can be applied in practical systems. The rest of the paper is organized as follows: Section discusses the background material the system model to formulate our problem. Section presents the capacity lower bound as applied to our system model. n Section V, we optimize this capacity lower bound derive the optimal power allocation. Simulations results are discussed in Section V we conclude in Section V. The following notations are used throughout this paper: (i) bold font lower case denotes vector, (ii) bold font capital case denotes matrix, (iii) regular font denotes scalar, (iv) ( ) T denotes transpose, (v) ( ) denotes complex conjugate, (vi) ( ) H denotes conjugate transpose, (vii) N denotes the identity matrix of size N, (viii) Tr(X) denotes trace of the matrix X, (ix) x denotes norm of vector x.. BACKGROUND AND SYSTEM MODEL First we present the system model that captures the effects of imbalance. Consider a single antenna system using

2 OFDM modulation. The frequency-independent imbalance gains phase offsets are denoted by {a t,a t } {θt,θ t }. The equivalent pulse-shaping filters (i.e. overall impulse responses including D/A converters, amplifier, pulse shaping frequency-independent imbalances) for the branches of the transmitter are denoted by gt (t) g t (t). The corresponding receiver side parameters are denoted by a r,a r,θr,θ r,gr (t) gr (t). The low-pass equivalent channel is h(t). The transmit system receive system with imbalance can each be viewed as the summation of two systems namely, the direct system a mirror system. The impulse responses of the direct mirror systems at the transmitter are denoted by gt D(t),gM T (t) those at receiver are denoted by gr D(t),gM R (t). These are related to the imbalance parameters by: g D T (t) = [a t e jθ t g t (t)+a t e jθ t g t (t)] () g M T (t) = [a t e jθ t g t (t) a t e jθ t g t (t)] () g D R (t) = [a re jθ r g r (t)+a r e jθ r g r (t)] (3) g M R (t) = [a re jθ r g r (t) a r e jθ r g r (t)]. (4) The overall system can be expressed as the sum of direct mirror channels p(t) q(t) respectively, as shown in Fig., they are given by: p(t) =gt D (t) h(t) gr D (t)+(gt M (t)) h (t) gr M (t) (5) q(t) =gt M (t) h(t) gr D (t)+(gt D (t)) h (t) gr M (t). (6) Now, consider a discrete-time OFDM system with N subcarriers. The discrete-time versions of the channels p(t) q(t) are denoted by p q respectively they consist of maximum of L taps each. f the direct channel sees an input s(t), the mirror channel sees an input s (t). The time-domain training signal is denoted by s(k),k [ N CP,N ] N CP is the number of cyclic prefix samples. The time-domain data signals are denoted by x(k),k [ N CP,N ]. The time-domain N received signal vector r for one OFDM symbol after cyclic prefix removal is given by: r =(S + X)p +(S + X )q + n (7) S denotes the training signal convolution matrix (circulant) of size N L. The elements of S are given by S(m, k) =s(m k),m [0,N ],k [0,L ]. The data matrix X of size N L is defined similar to S with the elements given by X(m, k) =x(m k),m [0,N ],k [0,L ]. The complex Gaussian noise vector n is given by n = gr Dw + gm R w w consists of independent identically distributed (iid) circularly-symmetric complex Gaussian rom variables with variance σw. Thus, n consists of complex Gaussian rom variables with variance denoted by σn. We consider a pilot design in [8] called frequency domain nulling (FDN). n particular, non-zero pilots are placed in equally spaced L tones while zero-pilots are placed in the L n s ( n) ( t nt) AWGN w(t) ( )* p(t) q(t) (t) g D R ( )* g M R (t) Fig.. System Model n[k ] r[k] mirror tones corresponding to the L tones which contain the non-zero pilots. The data signals are placed in the remaining N L tones. This arrangement provides for interference avoidance between data pilots in the frequency domain. The channel taps h l are assumed to be complex Gaussian distributed with zero-mean variance σh l, they are independent of each other. Frequency-independent imbalance is mainly caused by imperfection in the mixers hence is prevalent in all direct-conversion radios. Frequency-dependent imbalance is typically caused by imperfection in filters/amplifiers in verywideb systems, its frequency selectivity is typically quite mild. Hence, apart from low-cost very-wideb radios, imbalance in typical systems can be well modelled as frequency-independent. We therefore consider frequencyindependent imbalance in this work. This causes the imbalance pulse shaping filters to have single taps each. The discrete-time versions of direct mirror channels can then be written as p = a h + a h (8) q = b h + b h (9) a = gt D gr D (0) a = (gt M ) gr M () b = gt M gr D () b = (gt D ) gr M (3) are fixed constants. This makes the channels p q iid complex Gaussian when h is iid complex Gaussian. Thus, we have p l CN(0,σ p l ) q l CN(0,σ q l ), CN denotes complex Gaussian σ p l =(a a + a a )σ h l (4) σ q l =(b b + b b )σ h l. (5) We consider least-squares estimation of channels p q

3 as follows: Then, their MSEs are given as, ˆp = (S H S) S H r (6) ˆq = (S T S ) S T r. (7) MSE p := σ Δp = σ ntr[(s H S) ] (8) MSE q := σ Δq = σ ntr[(s H S) ]. (9) For the pilot design considered in this paper, the above can be simplified further. We have S H S = P s L, P s = s is the total power in the training signal. Therefore, we get, σδp = σnl/p s (0) σδq = σnl/p s. () The frequency domain taps can then be obtained as follows: [ ˆP 0, ˆP,..., ˆP N ] T = NF Lˆp () [ ˆ 0, ˆ,..., ˆ N ] T = NF Lˆq (3) F L consists of the first L columns of the unitary N N DFT matrix F whose (m, n)th element is given by F (m, n) = N e jπmn N,m [0,N ] n [0,N ]. We also have [P 0,P,..., P N ] T = NF L p (4) [ 0,,..., N ] T = NF L q. (5) Note that P i for any i [0,N ] is CN(0,σ P ) σ P := L l=0 σ p l. Similarly, i for any i [0,N ] is CN(0,σ ) σ := L l=0 σ q l.let denote the index set consisting of the M data tones J denote the index set consisting of the K pilot tones. Let N = N + NCP be the total number of samples in one OFDM symbol including the cyclic prefix. n the following, we consider cyclic prefix of length L, i.e. N CP = L.. CAPACTY LOWER BOUND To enhance the capacity of our system, we find the optimal pilot power allocation using a lower bound on the channel capacity. Let the total transmit power be P = P x + P s, P x = E{ x } P s = s. Under the total power constraint, following stard steps (as in [0] [3]), we have the channel capacity (normalized per transmit symbol) averaged over the channel matrix as, C ideal = M N E[log( + ρ ideal g )] (6) ρ ideal := (σ P + σ )P x Mσn. (7) ncorporating the channel estimation error, we apply a lower bound (as found in [0], [4]), for joint detection of X i X i, after summing across subcarriers, we obtain for the case of frequency-independent imbalances, C C := (8) { ( E log + i X i + ˆ )} i X i N E{ ΔP i X i +Δ i X i i } + Mσn (9) We then define: g = ˆP i X i + ˆ i X i. (30) E{ ˆP i X i + ˆ i X i } Substituting the above g in (9), we obtain { ( C = N E log + g σ P + σ Δp + σ + )} σ Δq σ i Δp + σ Δq + Mσ n/p x (3) wehaveused We can verify that [9] E{ ˆP i } = σ P + E{ ΔP i } (3) E{ ˆ i } = σ + E{ Δ i }. (33) E{ ΔP i } = σ Δp i (34) E{ Δ i } = σ Δq i. (35) Using the above, we can write (3) as C = M N E{log( + ρ g )} (36) ρ = σ P + σ Δp + σ + σ Δq σδp + σ Δq +. (37) Mσ n P x V. OPTMAL PLOT POWER ALLOCATON We define the ratio of the data power to total power to be α with 0 <α<. Then, P x = αp P s =( α)p P is the total power in one OFDM symbol. From (36), we can see that once M N = N + L are fixed, the capacity lower bound becomes a function of ρ only. Since log( ) is an increasing function, for fixed M K, Cis maximized when ρ is maximized hence the optimal α can be found out by maximizing the expression for ρ. The optimal α can be found numerically by finding the value of α that maximizes (37) over a discrete set of values. For fixed M K = N M, a closed-form solution for α can be obtained for high SNR regime. At high SNR, channel estimation error is very small, hence the error variance terms σδp σ Δq approach zero we have σp + σ Δp = σp σ + σ Δq = σ.also,σ Δp = σntr[(s H S) ]=σnl/p s σδq = σ ntr[(s H S) ]= σnl/p s. Substituting these approximations P x = αp P s =( α)p in (37), we obtain ρ = σ P + σ σ nl/p s + σ nl/p s + Mσ n/p x (38)

4 i.e., σp ρ = + σ σnl/p s + Mσn/P x = σ P + σ σn L/P s + M/P x = σ P + σ σn = P (σ P + σ ) σ n L ( α)p + M αp L ( α) + M α = M P (σ P + σ ) α( α) Mσn Lα + M( α) α( α) = Mρ snr Lα + M( α) ρ snr = P (σ P + σ ) Mσ n (39) (40) is the output SNR. The above value of ρ is maximized by differentiating it with respect to α, setting it equal to zero solving the resulting equation. The optimal α for high SNR regime is then given by α opt = + L M. (4) Note that this solution resembles the solution found in [9] but with a channel length of L instead of the length L due to the mirror channel introduced by the imbalance. We can see that neither the transmitter nor the receiver need to know apriori the imbalance parameters. The receiver learns the parameters by using pilot symbols. The optimal power allocation at high SNR depends only on the number of pilot tones the number of data tones. Thus the result of our power allocation does not require knowledge of imbalance parameters. The benefits of the derived results are reflected in the improvement in capacity in Figs. 4 BER in Fig. 3. V. SMULATONS AND RESULTS We use simulations to compare the performance of our system under the optimal power allocation derived above equal power allocation of α equal = +( L M ). For simulation, we use N =64subcarriers we consider L =8time domain channel taps for both p q channels. We use K =L =6 pilot tones M = N L =48data tones within one OFDM symbol. Fig. shows comparison of capacity for four different cases: ) when channel state is perfectly known no training is required, ) optimal power allocation with imperfect channel estimation, 3) optimal power allocation with perfect channel estimation, 4) equal power allocation α = α equal. From the results, we can see that optimal power allocation improves the channel capacity compared to equal power allocation case, while there is a significant loss due to imperfect channel estimation also compared to the case when channel state information (CS) is known. Fig. 3 shows the BER performance when the optimal power allocation is done. We can see that there is slight performance improvement compared to equal power allocation case. And significant difference exists compared to perfect channel estimation case. Fig. 4 shows the variation of capacity as a function of α. As expected, the maximum is seen at α =0.63 compared to the calculated value of α opt = + L M = Alsothe relatively flat peak of the capacity curve shows that several choices exist that give similar capacity performance. This also validates the results seen in Fig. Fig. 3 that there is only a marginal improvement in capacity BER from the equal =0.75) which lies close to the flat peak of the capacity curve. However, when the pilot power is significantly boosted, for e.g. α =0., a degradation of about % decrease in capacity is observed at SNR = 0dB. power case (α equal = + L M Capacity Lower Bound (bits/s/hz) Equal Powers Optimal Powers optimal w/o error known CS SNR (db) Fig.. Capacity Lower Bound versus SNR V. CONCLUSONS n this paper, we considered an OFDM system suffering from frequency-independent in-phase quadrature-phase imbalances derived an optimal power allocation between pilots data symbols for a fixed total transmit power constraint. Using a lower bound on the average channel capacity as a metric, we optimized for the power allocation between pilot data symbols. We obtained a closed form solution for high SNR scenario. Simulation results show that the resulting power allocation leads to increased channel capacity also reduced BER compared to equal power allocation. Also, the slight increase in performance of the optimal power allocation

5 BER Capacity Lower Bound (bits/s/hz) Equal Powers Optimal Powers Optimal Powers w/o Error SNR (db) db 0dB 0dB Fig. 3. BER versus SNR α Fig. 4. Capacity Lower Bound versus α = Px P compared to the equal power allocation shows that equal power allocation is close to optimal power allocation. REFERENCES [4] Mobile Broadb Wireless Access: EEE MAN stard, EEE LAN/MAN Stards Committee EEE 80.0 Std. [5] 3GPP, Long-Term Evolution (LTE), Physical Channels Modulation (Release 8), 3GPP TS 36. v..0, June 007. [6] 3GPP, Physical Layer form Ultra Mobile Broadb (UMB) Air nterface Specification,, 3GPP C.S v3.0, August 008. [7] A. Tarighat, R. Bagheri, A. H. Sayed, Compensation schemes performance analysis of imbalances in OFDM receivers, EEE Trans. Signal Processing, vol. 53, no. 8, pp , Aug [8] M. Valkama, M. Renfors, V. Koivunen, Compensation of frequency-selective / imbalances in wideb receivers: models algorithms, in Proc. EEE Third Workshop on Signal Processing Advances in Wireless Communications (SPAWC 0), 0 3 March 00, pp [9] J. Tubbax, A. Fort, L. Van der Perre, S. Donnay, M. Engels, M. Moonen, H. De Man, Joint compensation of imbalance frequency offset in OFDM systems, in EEE Global Telecommunications Conference, 003. GLOBECOM 03, vol. 4, 003. [0] D. Tur M. Moonen, Joint Adaptive Compensation of Transmitter Receiver mbalance Under Carrier Frequency Offset in OFDM-Based Systems, EEE Trans. Signal Processing, vol. 55, no., pp , Nov [] W. Kirkl K. Teo, / distortion correction for OFDM direct conversion receiver, Electronics Letters, vol. 39, no., pp. 3 33, 003. [] L. Brotje, S. Vogeler, K. Kammeyer, R. Rueckriem, S. Fechtel, Estimation Correction of transmitter-caused / mbalance in OFDM Systems, in Proc. 7th nternational OFDM Workshop, 00, pp [3] Y. Egashira, Y. Tanabe, K. Sato, A Novel mbalance Compensation Method with Pilot-Signals for OFDM System, 008. [4] R. Chrabieh S. Soliman, mbalance Mitigation via Unbiased Training Sequences, in EEE Global Telecommunications Conference, 007. GLOBECOM 07, 007, pp [5] E. Lopez-Estraviz, S. De Rore, F. Horlin, L. Van der Perre, Optimal training sequences for joint channel frequency-dependent imbalance estimation in OFDM-based receivers, in EEE nternational Conference on Communications, 006. CC 06, vol. 0, 006. [6] E. Lopez-Estraviz, S. De Rore, F. Horlin, A. Bourdoux, Pilot design for Joint Channel Frequency-Dependent Transmit/Receive mbalance Estimation Compensation in OFDM-Based Transceivers, in EEE nternational Conference on Communications, 007. CC 07, 007, pp [7] T. Schenk, P. Smulders, E. Fledderus, Estimation compensation of TX RX imbalance in OFDM based MMO systems, in Proc. EEE Radio Wireless Symposium (RWS 006, 006, pp [8] H. Minn D. Munoz, Pilot designs for channel estimation of OFDM systems with frequency dependent / imbalances, accepted in EEE WCNC, Apr [9] S. Ohno G. B. Giannakis, Capacity maximizing MMSE-optimal pilots for wireless OFDM over frequency-selective block Rayleighfading channels, EEE Trans. nform. Theory, vol. 50, no. 9, pp , Sept [0] E. Biglieri, J. Proakis, S. Shamai, Fading channels: informationtheoretic communications aspects, EEE Trans. nform. Theory, vol. 44, no. 6, pp , Oct [] T. Cover, J. Thomas, J. Wiley, W. nterscience, Elements of information theory. Wiley New York, 99. [] T. L. Marzetta B. M. Hochwald, Capacity of a mobile multipleantenna communication link in Rayleigh flat fading, EEE Trans. nform. Theory, vol. 45, no., pp , Jan [3]. Telatar, Capacity of multi-antenna Gaussian channels, European transactions on telecommunications, 999. [4] M. Medard, The effect upon channel capacity in wireless communications of perfect imperfect knowledge of the channel, EEE Trans. nform. Theory, vol. 46, no. 3, pp , May 000. [] Wireless LAN Medium Access Control (MAC) Physical Layer (PHY) Specifications: High-Speed Physical Layer in the 5 GHz B, EEE Stard 80.a, 999. [] Broadb Wireless Access: EEE MAN stard, EEE LAN/MAN Stards Committee EEE 80.6a, 003. [3] Broadb Wireless Access: EEE MAN Stard, EEE LAN/MAN stards committee EEE 80.6e, 005.