RECENTLY, the 2G standard GSM was enhanced by

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1 274 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 5, NO. 2, FEBRUARY 2006 The Training Sequence Coe Depenence of EDGE Receivers using Zero IF Sampling Martin Krueger, Member, IEEE, Robert Denk, an Bin Yang, Member, IEEE Abstract In the mobile communication stanar GSM/EDGE, the base station can select one of eight training sequence coes as the miamble of the ownlink transmitte bursts. If the receiving mobile station uses zero intermeiate frequency sampling, the channel estimation is sensitive to the DC offset an IQ gain/phase imbalance of the RF transceiver. This letter shows for a common class of channel estimators that the sensitivity epens on the selecte training sequence coe. This sensitivity can become significant for 8PSK moulation. Inex Terms GSM/EDGE receiver, channel estimation, IQ gain/phase imbalance, DC offset, 8PSK moulation. I. INTRODUCTION RECENTY, the 2G stanar GSM was enhance by EDGE. In [1] a goo overview on EDGE equalization concepts an suggeste reaing on the EDGE stanar are given. The focus of this letter is the training sequence coe (TSC) base channel estimation which is one before equalization. We restrict ourselves to zero IF sampling receivers, which are most common for EDGE mobile stations. GSM originally use GMSK moulation an robust channel coing only. To achieve higher ata rates, GPRS an later EDGE were introuce. In contrast to GPRS, EDGE aitionally uses 8PSK moulation. Both apply coing schemes with various levels of reunancy which are chosen aaptively by the network. In case of channels with high SINR (up to 30 B), sensitive moulation an coing schemes with high ata rates are chosen. Here RF impairments like IQ gain/phase imbalance an DC offset play an important role in aition to noise. In this letter we show how these impairments compromise the channel estimate which is an important step before emoulation of the receive burst. The importance of DC offset for EDGE equalizers has alreay been pointe out in [2] an [3]. The first joint channel an DC estimation algorithm was presente in [2]. A first analysis of the epenence of DC estimation error on training sequence coe can be foun in [4], whereas IQ imbalance is not aresse. The authors of [4] propose to use a small training sequence coe epenent intermeiate frequency (IF, approximately 10 khz) instea of zero IF. As most EDGE receivers are not prepare for IF operation, we treat zero IF receivers in the present letter. We show that IQ imbalance can Manuscript receive October 28, 2003; revise June 15th, 2004 an January 17th, 2005; accepte May 12th, The associate eitor coorinating the review of this letter an approving it for publication was J. Cavers. M. Krueger is with Infineon Technologies AG, Munich, Germany ( martin.krueger@infineon.com). R. Denk is with the Department of Mathematics an Statistics, University of Konstanz, Germany ( robert.enk@uni-konstanz.e). B. Yang is with the Chair of System Theory an Signal Processing, University of Stuttgart, Germany ( bin.yang@ss.uni-stuttgart.e). Digital Object Ientifier /TWC /06$20.00 c 2006 IEEE lea to a significant loss of receiver performance in aition to the loss cause by the DC offset. In the following section we escribe a moel of the receive EDGurst that inclues two major RF impairments: DC offset an IQ gain/phase imbalance. In Section III we analyze a linear joint channel an DC estimator. In contrast to [2], we erive separate explicit equations for the estimation of the DC an the channel impulse response. In Section IV we erive the error terms of this estimator from noise an IQ imbalance. In GSM/EDGE systems, the base station can select one of eight training sequence coes as miamble of the ownlink transmitte bursts. We show that the selection of the training sequence has a significant influence on the channel estimation quality. The impact of these error terms on the bit error rate is shown by simulations. In Section V conclusions are rawn base on the results an analysis presente in the letter. II. MODEING OF THE RECEIVED EDGE BURST Deriving a complete transmission moel for an EDGurst is not within the scope of this letter. A goo explanation can be foun, e. g., in [1]. We focus on that part of the receive signal that is use for a non-blin channel estimation in the mobile station for ownlink reception. Fig. 1(a) shows the principal blocks of the EDGE transmission system: transmitter, channel, raio frequency receiver, an baseban processor. To keep the moel simple, the channel impulse response inclues faing, pulse shaping, an all (igital an analog) filters in the transmit an receive path. The orer of the FIR filter representing the channel impulse response is enote by. Concerning faing, we assume that the resulting channel impulse response is constant for the short perio that is use for the channel estimation of one burst. Fig. 1(b) shows a signal moel for each block. All raio frequency signals are replace by their baseban equivalent an all analog signals are replace by their samples. We assume that the sampling rate is ientical to the GSM/EDGE symbol rate f T =13MHz/48. Any interference outsie the Nyquist interval [ f T /2,f T /2] is assume to be sufficiently suppresse by linear filters. Interference within the Nyquist interval is approximate by aitive white Gaussian noise w k. Moreover, we assume that the symbol-by-symbol rotation of φ = π/2 (GMSK approximate by rotate BPSK) or φ = 3π/8 (8PSK) of the base station transmitter is compensate by e-rotation in the igital part of the receiver. Only those receive samples that are use for channel estimation are consiere. Those samples are a function of the N =26 training symbols that are summarize into a coe wor [t 0,...,t N 1 ]. Eight ifferent training sequence coes are efine in the GSM/EDGE stanar [5]. They are enumerate

2 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 5, NO. 2, FEBRUARY (a) transmitter channel raio frequency receiver baseban processor (b) rotation channel noise IQ gain an phase imbalance DC erotation v x ~ k k * =e jko ~ w k conjugate signal =e -jko (c) x k () ~ = w k = w~ k * x k w k (e) 2 x k n k =w k + 2 w k * 2 Fig. 1. Block iagram to erive the signal moel. from 0 to 7 an we use the same enumeration throughout this letter. The iscrete time is enote by k. Accoring to our moel the input signal to the RF receiver is the convolution of the rotate training symbols with the overall channel impulse response plus aitive white Gaussian noise w k with variance N 0. We obtain = h l l e j(k l)φ + w k. (1) l=0 IQ gain/phase errors are ae to. When replacing the complex-value samples with two-imensional real-value vectors, these can be moelle in a straightforwar way [6], e. g. by applying ifferent gains on I an Q component. To keep the benefits of the complex notation, we choose aing a mirror signal ηu k instea, as suggeste e. g. in [7] (p. 339, Eq. (4)), i. e. = + ηu k. Besies IQ imbalance the RF receiver as a DC offset. Eventually, the signal is e-rotate by = e jkφ, yieling emoulate output: x k = ( + )= +. (2) In Fig. 1, (c)+() show some rearrangements leaing to an equivalent block iagram (e). The corresponing equation is ( ) x k = h l l + ηa 2 k h l l + + n k. (3) where l=0 l=0 n k = w k + ηa 2 kw k (4) Since in w k real an imaginary part are uncorrelate it is straightforwar to show that the variance of n k is (1+ η 2 )N 0. However, the real an imaginary part of n k are uncorrelate for η =0only. In vector notation, Equations (3) an (4) are an x = Th + ηa 2 Th + a + n, (5) n = w + ηa 2 w, (6) respectively. The real-value (N ) ( +1)-matrix is efine by t... t 0 t t 1 T =... t N 1... t N 1 The complex-value column vectors x, h, h,a an the complex-value iagonal (N ) (N )-matrix A are efine by x =[x... x N 1 ] T, h =[h 0... h ] T, h =[h 0... h ] H, a =[e j φ... e j(n 1) φ ] T A =iag{a} where ( ) T enotes transposition an ( ) H enotes conjugation plus transposition.

3 276 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 5, NO. 2, FEBRUARY 2006 III. CHANNE ESTIMATION BASED ON THE TRAINING SEQUENCE CODE Base on Equation (5) a maximum likelihoo channel estimator can be erive. A maximum likelihoo channel estimator aresses DC offset an IQ imbalance an minimizes the error ue to noise [6], [7]. Maximum likelihoo joint estimation of the unknown imbalance parameter η an the unknown channel impulse response h is a nonlinear problem an requires iterative processing. In frequency hopping channels, the RF impairments of the receiver can vary burst-byburst, so that the estimation has to be base on the N samples of the current burst an has to be complete before equalization. Consequently, a maximum likelihoo estimator seems not feasible for most GSM/EDGE mobile stations. To have an estimator that can be implemente with reasonable effort, we restrict ourselves to linear estimators. In the following, we consier the maximum likelihoo channel estimator for the case η =0. In this case n k is white Gaussian noise with variance N 0. Real an imaginary part of the noise have the same variance N 0 /2 an are uncorrelate. Hence, the joint maximum likelihoo estimate for h an is given as the solution of the least squares problem (ĥ ) ( ) =argmin h ˆ h, (T a) 2 x. (7) The solution is (see also [4]) (ĥ ) =(R ˆ H R) 1 R H x (8) where R =(T a). To unerstan the training sequence epenence of EDGE channel estimation, separate explicit expressions for the channel an DC estimation are useful. This separation can be obtaine using the matrix inversion lemma. The result is ˆ = b H x (9) for the DC estimation where b = P T a/ P T a 2, P T = I TT +,ant + =(T T T) 1 T T. The channel impulse response is given by ĥ = T + (x a ˆ). (10) Please note that Equations (9) an (10) have the following properties: 1) They are mathematically equivalent to Equation (8). 2) They require fewer real-by-real multiplications than (8). Since T an T + are real-value matrices, most complex-by-complex multiplications in (8) can be replace by complex-by-real multiplications in (10). 3) They nee less memory storage. Our metho requires the storage of the real (N ) (+1) matrix T + an the complex (N ) 1 vector b. The irect estimation requires the storage of the complex (N ) ( +2) matrix R +. Consequently, our DC an channel estimation metho is more efficient than (8). Moreover, we have separate explicit equations for the estimation of h an. In the next section we stuy the sensitivity of the joint DC an channel estimator with respect to noise an IQ gain/phase errors. IV. ESTIMATION ERRORS In the previous section we escribe the maximum likelihoo DC an channel estimator for a signal without IQ gain/phase error, i. e. η =0. In this section we analyze the error of this estimator for a signal with aitive white Gaussian noise an with IQ gain/phase error, i. e. η 0.Inotherwors: while IQ imbalance is not aresse by the estimator, the error terms o consier IQ imbalance of the signal. A. DC Estimation Errors DC estimation errors have impact on the channel estimation. Moreover, for the equalization the DC-compensate samples x a ˆ are use. They are irectly biase by an error of the DC estimation. Therefore, we first iscuss the DC estimation errors. From Equations (5) an (9) as well as from P T T = 0 we erive the DC estimation error ˆ = η ah P T A2 Th P + ah P T n T }{{ a 2 P. (11) T }}{{ a 2 } eˆ,η eˆ,n First we look at the DC estimation error ue to noise e ˆ,n. Using Equation (6) an because of E [ ww H] = N 0 I an E [ ww T] =0for complex Gaussian noise, we obtain after a few calculations E [ nn H] = ( 1+ η 2) N 0 I. Consequently, the energy of this error term is given by [ Eˆ,n = E e ˆ,n 2] = N 0(1 + η 2 ) P T a 2 an in case of no IQ-imbalance (η =0) it is equivalent to the corresponing expression in [4]. The resulting noise suppression Sˆ,n = N 0 /Eˆ,n of the DC estimator for ifferent training sequence coes (TSCs) is liste in Table I, (a)+(b). Here an in the following, noise suppression values of an estimator are the ratio of the noise energy to the estimation error energy cause by noise (see also Sˆ,n in Fig. 2). Next we have a look at the sensitivity to IQ gain/phase imbalance. The error energy is Eˆ,η = η 2 a H P T A2 Th 2 P T a 4 an is a function of the impulse response h. To quantify the error energy without assumptions on h, we replace the error energy by the maximum error energy using the Schwarz inequality. The maximum error energy is E max ˆ,η = η 2 h 2 P T a 4 a H P TA 2 T 2 Eˆ,η To characterize the sensitivity to IQ imbalance the minimum imbalance suppression S min ˆ,η = η 2 h 2 /E max ˆ,η is use. Table I, (c)+(), summarize the minimum suppression values. Similar to noise suppression, the imbalance suppression is efine as the ratio of the imbalance error energy

4 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 5, NO. 2, FEBRUARY TABE I SUPPRESSION PARAMETERS OF THE DC AND CHANNE ESTIMATION E s = h 2 tsc (a) Sˆ,n [B] (GMSK) (b) Sˆ,n [B] (8PSK) (c) S min [B] (GMSK) ˆ,η () S min ˆ,η [B] (8PSK) (e) Sĥ,n [B] (GMSK) (f) Sĥ,n [B] (8PSK) (g) S min [B] (GMSK) (h) S min [B] (8PSK) η 2 h 2 to the estimation error energy cause by the IQ imbalance Eˆ,η (see also Sˆ,η in Fig. 2). We enote the energy of the error ae to the receive signal by IQ imbalance, i. e. the energy of the mirror signal, as imbalance error energy. We see that moulation types an training sequences with small noise suppression Sˆ,n also have a small IQ imbalance suppression Sˆ,η. The same is true for large suppression values. However, the orer is not exactly the same. Moreover, the values for the noise suppression Sˆ,n are always several B larger than for the IQ imbalance suppression Sˆ,η min. Consequently, if the DC offset an the energy of the mirror signal ηth are of the same orer of magnitue, the DC estimation error ue to IQ imbalance is ominant. N 0 E,n S,n E s E h,n S h,n 2 E s E, 2 h 2 S, E h, -2 Fig. 2. evel iagram to explain the contribution of the ifferent error an suppression terms. B. Channel Estimation Errors The channel estimation is biase by the DC estimation error accoring to Equation (10). This bias can be split into two error terms. The error ue to noise is given by eĥ,n = T + ae ˆ,n an the error ue to IQ imbalance is given by e = T + ae ˆ,η. The noise suppression Sĥ,n an the minimum imbalance suppression S min of the channel estimator are calculate in the same way as for the DC estimator: S h, Sĥ,n =N 0 (1 + η 2 )/Eĥ,n, S min = η 2 h 2 /E max. The results are presente in Table I, (e) (h). We can see that the noise suppression is excellent for GMSK (more than 17 B). However, for 8PSK the noise suppression for training sequence numbers 2 7 is roughly 4 B weaker than for training sequence numbers 0 an 1. For GMSK the IQ imbalance suppression is always more than 7.5 B so that the error ue to IQ imbalance is negligible. Besies these two error terms, IQ imbalance an noise a errors to the channel estimators even in case of an error-free DC estimator. Fig. 2 shows all error an suppression terms in a level iagram. With the values in Table I an values of E s as well as η, the ifferent contributions can be compare for any zero IF receiver using joint channel an DC estimation. For example, a receiver with an input signal level of E s = 90 Bm, a noise level N 0 = 110 Bm (equivalent noise banwith 270 khz) an η =0.025(1 + j) (corresponing to 0.44 B gain imbalance an 2.9 phase error), i. e. η 2 E s = 119 Bm, has the following aitional error contributions for TSC 0 an TSC 7, respectively. Eˆ,n / Bm = 121.9, / Bm = 126.4, ĥ,n Eˆ,η / Bm = 123.3, / Bm = 127.8, E max E max

5 278 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 5, NO. 2, FEBRUARY 2006 (a)η=0, =0 (b)η=0.025(1+j), =0 [B] [B] (c)η=0, 0 ()η=0.025(1+j), 0 [B] tsc 0 tsc 1 tsc 2 tsc 3 tsc 4 tsc 5 tsc 6 tsc 7 [B] Fig. 3. Bit error rate as a function of for training sequences 0 to 7. The numbers suggest that the loss for training sequence coe number 7 shoul be more than 0.5 B higher than for training sequence coe number 0. In the next subsection, bit error rate simulations confirm this suggestion. C. Bit Error Rates To illustrate the effects of the erive DC an channel estimation errors, bit error rate simulations were performe. For each simulation 100,000 GSM/EDGE compliant 8PSK bursts with ranom ata bits were generate. Noise, DC offset, an IQ imbalance were ae an ifferent training sequences were use. The receiver is very similar to the one presente in [1], except that no noise whitening prefilter was use. The simulate receiver is fully compliant with the GSM/EDGE stanar for all training sequence coes. A static channel was chosen to have a constant channel impulse response. Note, that although the physical channel impulse response is a Dirac pulse the overall channel impulse response is sprea over several taps ue to pulse shaping an filtering. In GSM/EDGE the pulse shape is not esigne for Nyquist filtering. Fig. 3 shows the resulting bit error rate curves. The ratio of signal energy per bit an the white noise Energy N 0 were use as the abscissa. Note that for 8PSK the energy per bit is 4.8 B below the signal energy per symbol. In the first simulation, the DC offset an IQ imbalance were set to zero. Also the receiver assumes a zero DC offset. The resulting bit error rate curves are shown in Fig. 3(a). Introucing an IQ imbalance with η =0.025(1 + j) corresponing to 0.44 B gain imbalance an 2.9 phase error leas to Fig. 3(b). In Fig. 3(c), the receiver estimates an compensates the non-zero DC offset but the IQ imbalance is zero. In Fig. 3() both, IQ imbalance an DC offset are present. To compare the ifferent bit error rate curves, those values for that are require to have a bit error rate of 10 3 an 10 4, respectively, were calculate by log-linear interpolation. The values are summarize in Table II. Without DC an IQ imbalance, the bit error performance is almost ientical for all training sequence coes. The same hols for the case of no DC but with IQ imbalance although a loss of up to 1 B can be seen for a bit error rate as low as If the IQ imbalance is zero but DC estimation is introuce, a loss of up to 1.1 B arises. As was expecte from the estimate error energy an suppression values, the loss is ifferent for ifferent training sequences. The mutual ifference has a maximum value of 0.5 B. Aing IQ imbalance leas to a maximum ifference of 1.5 B. The loss in sensitivity for a bit error rate as low as 10 4 varies from 1.5 B to 3.0 B. In the last two cases, training sequence numbers 0 an 1 are significantly better than numbers 2 to 7, whereas number 7 is the worst. The bit error results match well with the suppression values erive in the previous two subsections. V. CONCUSION We analyze a maximum likelihoo joint DC an channel estimator for GSM/EDGE irect conversion receivers. We erive estimation errors cause by noise an IQ gain/phase

6 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 5, NO. 2, FEBRUARY TABE II REQUIRED TO HAVE A BIT ERROR RATE OF 10 3 AND 10 4 WITH THE SIMUATED 8PSK RECEIVER, RESPECTIVEY tsc (a) η =0, ˆ = = (b) η =0.025(1 + j), (c) η =0, () η =0.025(1 + j), for most training sequence coes. Consequently, a rotation by φ = π/2 for both, GMSK an 8PSK woul allow a better 8PSK receiver performance. However, in EDGE the receiver has to etect the moulation without aitional signalling. This woul be more ifficult, if the rotation angle is the same for GMSK an 8PSK. Nevertheless, choices like φ = π/4 woul ecrease the estimation errors as can be easily shown using the erive expressions. For 8PSK training sequence numbers 0 an 1 lea to better etection results than other training sequences. This means that bit/block error rate simulations an measurements with bursts using one training sequence o not accurately represent simulations an measurements with bursts using other sequences. EDGE receiver simulation results without information about the training sequence have to be questione critically. REFERENCES imbalance. It turns out that the selection of the training sequence has a significant influence on these errors. The influence of those error terms on the bit error rate of the receiver was emonstrate by Monte-Carlo receiver simulations. Differences of up to 1.5 B for a targete bit error rate of 10 4 in a static (but non-zero orer) channel were shown. Bit error rate results can only be etermine for specific RF transceivers an specific equalizer algorithms. Some equalizers can be more sensitive to DC estimation errors ue to noise while others may be more sensitive to channel estimation errors ue to IQ imbalance. Nevertheless, we can erive the following rough conclusions: A symbol-by-symbol rotation by φ = 3π/8 leas to significant DC estimation, channel estimation, an bit errors [1] W. H. Gerstacker an R. Schober, Equalization concepts for EDGE, IEEE Trans. Wireless Commun., vol. 1, no. 1, pp , Jan [2] B. inoff, Using a irect conversion receiver in EDGE terminals: a new DC offset compensation algorithm, in Proc. PIMRC 2000, pp , Sept [3] B. inoff an P. Malm, performance analysis of a irect conversion receiver, IEEE Trans. Commun., vol. 50, no. 5, pp , May [4] D. Hui, B. inoff, an K. Zangi, Enhance DC estimation via sequence-specific frequency offset, in 2002 IEEE 56th Vehicular Technology Conference Proceeings. VTC 2002, pp , Sept [5] 3r Generation Partnership Project (3GPP), TSG GSM/EDGE Raio Access Network, 45 series (Raio aspects), [6] J. K. Cavers an M. W. iao, Aaptive compensation for imbalance an offset losses in irect conversion transceivers, IEEE Trans. Veh. Technol., vol. 42, no. 4, pp , Nov [7] G. T. Gil, I. H. Sohn, J. K. Park, an Y. H. ee, Joint M estimation of carrier frequency, channel, I/Q mismatch, an DC offset in communication receivers, IEEE Trans. Veh. Technol., vol. 54, no. 1, pp , Jan