A Stochastic Gradient LMS Algorithm for digital Compensation of Tx ...

A Stochastic Gradient LMS Algorithm for digital Compensation of Tx Leakage in Zero-IF-Receivers Andreas Frotzscher and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Technische Universit¨at Dresden, D-01062 Dresden, Germany Email: {frotzsch, fettweis}@ifn.et.tu-dresden.de

I. I NTRODUCTION Wireless communication systems employing the frequency division duplex (FDD) scheme use two separate frequency bands for uplink (UL) and downlink (DL) communication. In the terminal a duplexer connects the transmit (Tx) and receive (Rx) branch with the antenna. Since these two branches are operating simultaneously, the duplexer has to provide a high Tx-Rx isolation. Modern duplexers using surface acoustic wave (SAW) technology have much smaller size than their predecessors, ceramic duplexers, but achieving good Tx-Rx isolation is more challenging. Thus, a significant part of the transmit signal is leaking through the duplexer in the receive branch and can cause in difficult receive conditions (e.g. at cell edges) severe interference to the desired receive signal. In the past some research has been done in the area of Tx Leakage. Several publications use analytical results of the impact of Tx Leakage to derive design requirements for the receive branch, e.g. linearity [1], [2]. Others focus on the mitigation of Tx Leakage by employing an adaptive filter in parallel to the duplexer. This provides an additional attenuation in the receive branch at the transmit frequency of around 10 - 20 dB [3], [4]. These approaches aim at the mitigation of Tx Leakage before it causes severe interference. However, their limited reconfigurability makes them difficult to use in frequency agile multi standard platforms. Furthermore their mitigation performance suffers from the significant group delay introduced by the SAW duplexer. Compensating the This research was supported by the German Ministry of Education and Research within the project Multi Access System in Packet Radio (MARIO) under grant 01 BU 570 in cooperatation with EPCOS AG. The authors would like to thank Ernesto Zimmermann and Steffen Bittner for valuable ideas and discussions.

Tx Leakage in the digital baseband of the receive branch after it has caused severe interference, is one possibility that provides the required flexibility. However, to the authors best knowledge, so far there has not been published an approach on the digital compensation of Tx Leakage. In the front end design of multi standard receivers, the Zero-IF (or direct conversion) principle, is the preferred choice due to its flexibility, reconfigurability etc. However, this principle suffers from some analog impairments, e.g. I/Q-imbalance, intermodulation products 2nd order (IM2). In presence of Tx Leakage the nonlinearity of the I/Q down converter generates IM2 of the leaking transmit signal, which partly lies around DC and thus provokes severe interference to the desired, down converted receive signal. This contribution focuses on the compensation of Tx Leakage in the digital baseband of the receive branch. It uses an adaptive Stochastic Gradient Least Mean Square (SG-LMS) algorithm, based on a linear approximation of the system model. The outline of the paper is as follows. The next section presents the system model. Section III discusses some properties of the duplexer’s Tx-Rx isolation. In Section IV, the proposed compensation algorithm is derived and a genie version of it is given, to verify the maximum achievable performance using the linear approximation. In Section V simulation results can be found, showing the impact of Tx Leakage on the system performance as well as the improvement offered by the SG-LMS and genie algorithm. Section VI concludes this study. II. S YSTEM MODEL The problem of Tx Leakage occurs in FDD transmission schemes in base stations as well as in terminals. This paper focuses on the Tx Leakage (TxL) present at the terminal, as depicted in Fig. 1. The terminal employs the direct conversion DL signal

LODL

LNA Duplexer

Abstract— In frequency division duplex transceivers the transmit signal is leaking through the duplexer into the receive branch. For the case of Zero-IF receivers the nonlinearity of the I/Q down converter generates 2nd order intermodulation products of the Transmitter Leakage, causing severe interference to the desired receive signal in the baseband. In this paper we present a Stochastic Gradient Least Mean Square algorithm based on a linear approximation to compensate digitally the interference caused by Tx Leakage in the baseband of the receive branch. Simulation results are used to evaluate the impact of Tx Leakage on the system performance. Furthermore they point out, that the Least Mean Square compensation algorithm significantly improves the system performance and gets very close to the maximum achievable performance using this linear approximation.

Tx Leakage

A TxL- Channel select IM2 Filter

PA

LOUL

A

D

D

sBB [k]

+

Matched Filter

TxL s [k] Estimation CMP sUL[k]

Baseband processing

Impulse shaping Filter

Fig. 1. Block diagram of a FDD transceiver in a terminal, employing the direct conversion principle and compensating the Tx Leakage impact

principle to meet the demanded reconfigurability and flexibility. The I/Q down converter in the Rx branch generates intermodulation products 2nd order of the leaking UL signal (TxL-IM2), which arise partly around DC and thus cause severe interference to the received, down converted DL signal. Due to imperfections, the local oscillator signal (LO) is mixed with it self in the down converter, generating a strong static DC offset. Assuming in this contribution the use of an analog, static DC offset cancellation, the TxL IM2 is not affected by such a cancellation. Furthermore this contribution assumes the perfect I/Q match of the down converter and the further signal processing. Thus the problem of I/Q-Imbalance [5] is not considered here. After passing through the channel select filter and the analog-to-digital converter (ADC) the TxL-IM2 is estimated and compensated by the gray high lighted TxL-Estimation block in Fig. 1. From this diagram the following equivalent, time discrete baseband model can be derived [6]:
2 c2 c1 (1) sBB [k] = sDL [k] + hT xL ∗ sU L [k] +n[k] , |2 {z } |2 {z }

Usually in the transceivers the impulse shaping filter and matched filter are implemented in digital domain. Therefore, the DAC and ADC employ an oversampling with the factor osr = 2...8 and the 3 dB cut-off frequency of the channel select filter, is chosen according to the DL signal bandwidth and the oversampling ratio. As a consequence the IM2 of the leaking UL signal is lying in the passband of the the channel select filter and therefore is not significantly attenuated. The graphical representation of the baseband model in (1) is given in Fig. 2. The upper left branch in figure represents the direct down conversion of the DL signal into baseband. The lower branch models the IM2 generation of the leaking UL signal.

where ∗ denotes the discrete convolution operator. This model describes the impact of the DL signal and the leaking UL signal on the discrete signal sBB [k] provided by the ADC in the Rx branch. Eq. (1) consists of three terms, which will be explained in the following. Starting with A, the desired DL signal at the input of the I/Q down converter is denoted by its equivalent discrete baseband signal sDL [k]. The direct conversion gain factor c1 and IM2 gain factor c2 in (1) are derived from the power conversion gain Gmix and the input referred intercept point 2nd order (IIP 2mix ) of the down converter: r p Gmix c1 = 2Gmix , c2 = (1 + j) . (2) IIP 2mix

In order to evaluate the impact of Tx Leakage on the DL signal in different signal power conditions, the power ratio between the part of the desired DL signal, being of interest for this specific terminal and the IM2 of the leaking UL signal at the ADC in the Rx branch is calculated, describing the Tx Leakage signal-to-interference ratio SIRT xL : PDL,i SIRT xL = . (3) PT xL

A

n[k] sDL [k]

Fig. 2.

c1 2

sBB [k] c2 2

(.)*

hTxL[k]

sUL [k]

Baseband model of Tx Leakage in direct conversion transceivers

B

Since the exact I/Q match is assumed, the IM2 generated in the I- and Q-branch are exactly the same. Therefore the factor 1 + j is included in the calculation of the IM2 gain factor in (2). Thus, the term A in (1) describes the direct down converted DL signal. Since the exact channel model does not effect the estimation and compensation of Tx Leakage, the DL transmission channel is assumed to be an AWGN channel. Considering Term B, the discrete UL signal, provided for the digital-to-analog converter (DAC) in the Tx branch, is represented by sU L [k]. This signal partly leaks in the Rx branch, passing the power amplifier (PA) in the Tx branch, the Tx-Rx isolation of the duplexer and the LNA in the Rx branch. The combination of this filtering and amplification to one filter operation can be interpreted as Tx Leakage channel (TxL channel) with the equivalent discrete baseband impulse response hT xL [k]. Thus, term B in (1) represents the TxL-IM2 at baseband after the I/Q down converter. The last term n[k] in (1) is a gaussian noise term [6], containing the equivalent discrete AWGN channel noise at baseband, being narrowed by the channel select filter.

ADC

III. T X -R X I SOLATION OF THE D UPLEXER In order to develop an appropriate compensation algorithm, it is important to analysis the TxL channel coefficients hT xL [k] in more detail. The shape of these coefficients are determined by the frequency selectivity of the TxL channel. As defined in section II hT xL [k] describes the entire filtering of the leaking UL signal between the DAC in the Tx branch and the I/Q down converter in the Rx branch. The Tx branch can be considered to be frequency flat within the UL band. In contrast the Rx branch must not necessarily be frequency flat around the UL signal bandwidth. However, in this paper it is assumed to be frequency flat. Thus, the frequency selectivity of the TxL channel is determined by the Tx-Rx isolation of the duplexer, which therefore has to be considered in more detail. The duplexer consists of two bandpass filters, adjusted to the UL and DL frequency bands, respectively. These filters are connected by a phase shifting and matching network to the antenna port as depicted in Fig. 3. Because objects (e.g. hands, head etc.) in the nearfield of the antenna effect its impedance, the matching condition at the antenna port as well as the Rx port of the duplexer are influenced by the nearfield distortion. Since impedance mismatch simply represents an additional filtering, the time varying nearfield distortion leads to a time variant Tx-Rx isolation. Fortunately, the objects in

phase shifting and matching network

ANT

bandwidth, hD [k] contains only one very strong coefficient and L − 1 much weaker coefficients. Due to the frequency flat assumption of the Tx and Rx branch, the TxL channel is approximately frequency flat and the TxL channel coefficients hT xL [k] are proportional to hD [k].

RX RX-BP

TX TX-BP

Fig. 3.

IV. C OMPENSATION OF T X L EAKAGE

Block diagram of a duplexer

the nearfield usually move with low speed of a few m/s and so the Tx-Rx isolation will only change slowly in time. In order to evaluate the influence of nearfield objects on the Tx-Rx isolation, measurements have been carried out, using the SAW duplexer B7632 from EPCOS AG, which is specified for FDD-UMTS terminals. The antenna port of the duplexer was connected to a monopol antenna matched at 2 GHz. Measuring the Tx-Rx isolation a metallic plate with 2 cm×2 cm edge length facing the antenna is moved within the distance 0.036 m < d < 0.303 m of the antenna to distort its nearfield. Fig. 4 depicts the Tx-Rx isolation. For better clarity only curves representing the mean and maximum / minimum values of the measured Tx-Rx isolation are shown.

Tx−Rx isolation [dB]

−50

UL−Band

DL−Band

−55

−60 B −65 mean min / max

−70 1.9

1.95

2

2.05 2.1 Frequenz [GHz]

2.15

2.2

Fig. 4. Measurement results of the Tx-Rx isolation using the FDD-UMTS duplexer B7632 of EPCOS AG

As it can clearly be seen the Tx-Rx isolation varies due to the changing nearfield distortion. A detailed analyse of the measurement results points out, that the maximum attenuation difference in the UL band of the Tx-Rx isolation within the UMTS signal bandwidth B = 5 MHz is approx. 2 dB. This is not shown in the figure. Since the cut-off frequency of a filter is usually defined as a 3 dB attenuation difference in the passband, the Tx-Rx isolation can be considered as frequency flat for the UL signal. This fact is important for section IV. Extracting the part within the current UL signal band from the Tx-Rx isolation and transferring it to baseband leads to an equivalent Tx-Rx isolation filter at baseband with the discrete impulse response hD [k] of the length L. Due to the frequency flat characteristic of the Tx-Rx isolation within the UL signal

In order to cancel out the impact of Tx Leakage, the TxL channel coefficients hT xL [k] and the IM2 gain factor c2 of the I/Q down converter must be estimated out of (1). It is obvious that the best compensation signal would be
2 c2 (4) sCM P [k] = hT xL ∗ sU L [k] . 2 Subtracting sCM P [k] from sBB [k] would result in a complete cancellation of the Tx Leakage impact. Unfortunately, estimating c2 hT xL [k] with the knowledge of sBB [k] and sU L [k] is a nonlinear, ambiguous estimation problem. Linearizing the system model in (1) significantly simplifies the estimation problem. One possible linear approximation could be:
1+j ˆ c1 h ∗ |sU L |2 [k] + n[k] , sBB [k] ∼ = sDL [k] + 2 2

(5)

ˆ where h[k], k = 0, ..., L − 1 denotes the unknown filter coefficients, assuming perfect knowledge of the filter length ˆ L of hT xL [k]. Note that h[k] is real valued, because the same TxL-IM2 is generated in the I- and Q-branch of the receiver. The approximation in (5) is equal to (1), if the TxL channel coefficients hT xL [k] are perfectly frequency flat and thus contain only one coefficient unequal to zero, while all other L−1 coefficients remain zero. As shown in section III hT xL [k] ˆ is approximately frequency flat and thus h[k] too. Using the linear approximation and the fact of almost frequency flat ˆ channel coefficients, only the strongest coefficient in h[k] is of interest for the compensation. Based on (5), the compensation signal from (4) can now be rewritten as the following vector product: 1 + j ˆT h · sUL,IM2 [k] , (6) 2   ˆ ˆ − 1] T denotes the L coefficients to ˆ = h[0], where h ..., h[L be estimated. The current last L values of the UL signal IM2 T are denoted by sUL,IM2 [k] = |sU L [k]|2 , ..., |sU L [k−L+1]|2 . In the following bold letters represents vectors. In order to solve the linear estimation problem in (5) the adaptive least mean square (LMS) filter approach [7] is employed, which can in general estimate frequency selective filter coefficients in linear problem statements. This capability ˆ because not only the value is important for the estimation of h, of the strongest filter coefficient, but also the corresponding index is needed. The index represents the time delay due to the UL signal propagation delay from the DAC in the Tx branch up to the ADC in the Rx branch. Therefore, the LMS filter is estimating L filter coefficients. Using (6) the estimation error can be written as: sCM P [k] =

e[k] = sBB [k] − sCM P [k] 1 + j ˆT = sBB [k] − · h sUL,IM2 [k] . (7) 2 The LMS filter aims to minimize the mean square estimation error. It originates from the method of steepest descent, which ˆ will be explained first. This method iteratively calculates h[k] using the deterministic gradient:   ∗ ∂ E e[k] e[k] ˆ + 1] = h[k] ˆ +α , (8) h[k ˆ ∂h where E[·] denotes the expectation operator and α is a step size parameter determining the convergence and the remaining estimation error. Note that the time index k is appended ˆ indicating its time dependency due to the iterative to h calculation. Inserting (7) in (8) and exchanging the expectation and derivation the filter coefficients can be calculated by: h  i ˆ + 1] = h[k] ˆ + 2α E sUL,IM2 [k] · < (1 − j) e[k] . (9) h[k The block LMS algorithm approximates the expectation value by averaging over the last M values of its argument: 2α ˆ ˆ h[k+1] = h[k]− M

M −1 X

Since the linear approximation assumes hTxL to be freˆ SG is used quency flat, only the strongest filter coefficient of h for the compensation. Its index is determined by:   hSG [m] . mSG = argmax ˆ

(12)

m

Now the compensation signal of the SG-LMS algorithm can be formed: 2 1+j ˆ scmp [k] = hSG [mSG ] sU L [k − mSG ] . (13) 2 In order to determine the maximum achievable performance gain of this linear compensation approach, a genie algorithm is used, which means that c2 and hTxL are perfectly known at the receiver. Following the linear compensation approach the mean attenuation of hTxL is required. This can be done by averaging out the magnitude and phase of the transfer function of the TxL channel within the UL signal bandwidth and afterwards performing the inverse fourier transform. The resulting filter impulse response hgen contains only one coefficient unequal to zero with the index:   mgen = argmax hT xL [m] . (14) m

Thus the corresponding compensation signal will be  sUL,IM2 [k−m]·< (1−j) e[k−m] .

m=0

(10) Note that now e[k − m], m = 0, ..., M − 1 is computed ˆ based on the last estimation h[k]. To ensure the convergence of the estimation, the step size must be chosen 0 < α <  
2/ L · E sU L [k]|2 [7]. Adapting the signal model given in [2] to the UMTS system, the DL signal sDL [k], the AWGN channel noise n[k], the UL signal sU L [k] and, thus, also the IM2 of the UL signal sUL,IM2 [k] can be considered to be ergodic random processes. Furthermore, since sDL [k] and n[k] are uncorrelated with sUL,IM2 [k], the averaging in (10) is a suitable approximation of the expectation for large M . In order to lower the computational complexity, in most applications the block size is set to M = 1. As a consequence the LMS algorithm employs a stochastic gradient, giving the algorithm its name, Stochastic Gradient LMS. The random ˆ nature of the gradient causes the estimates h[k] to move randomly around the minimum of the mean square error surface. hTxL can be considered as constant during its coherence time, which results from the time variant antenna nearfield ˆ to be estimated is also constant within distortion. Thus, the h this time range, denoted by N . Averaging the estimated filter ˆ within the time range N : coefficients h[k] N −1 X ˆ SG = 1 ˆ − m] h h[k N m=0

(11)

is a suitable way of getting closer to the minimum of the error-performance surface. Therefore, a blockwise estimation is employed. In each block of the length N , the SG-LMS ˆ algorithm calculates iteratively h[k]. Averaging the estimates ˆ SG of the current block. at the end of the block leads to h

scmp,gen [k] =

2 1 + j c2 hgen [mgen ] sU L [k − mgen ] . (15) 2 V. S IMULATION R ESULTS

The simulation chain used in this contribution is based on a wideband code division multiple access (WCDMA) transmission scheme according to the FDD UMTS standard. The DL transmitter is constructed according the 3GPP base station conformance test model 1 [8], employing the spreading factor 128 to the Dedicated Physical Channel (DPCH). The UL signal contains only one Dedicated Physical Data Channel (DPDCH) and Dedicated Physical Control Channel (DPCCH) with the spreading factor 4 and 256, respectively. An oversampling factor osr = 2 is applied in the DAC and ADC. Furthermore, the exact I/Q match of the down converter is assumed. The TxL channel coefficients hT xL [k] are constructed by using the measurement results of the Tx-Rx isolation, presented in Section III, choosing randomly the position of the distortion object, and assuming a frequency flat Tx and Rx branch. Since the nearfield distortion slowly varies with time, hT xL [k] is assumed to be constant during one UMTS radio frame (10 ms). Therefore, the SG-LMS algorithms is executed blockwise for every radio frame and the averaging in (11) is carried out over each radio frame. The simulation results in Figs. 5 and 6 clearly show the impact of Tx Leakage on the system performance. For SIRT xL = −10 dB in Fig. 5 the impact of Tx Leakage is weak. Considering the uncoded BER performance = 10−2 , being of interest for channel coding, the Tx Leakage impact leads to a 2 dB SNR performance loss. The compensation with the SG-LMS algorithm or the genie approach achieves the same performance as without any Tx Leakage. With

0

8

10

7 −1

6 ∆SNR [dB]

uncoded BER

10

−2

10

−3

10

5 w/o TxL−IM2 compensation

4 3

SG−LMS

2 −4

10

0

1

TxL IM2 SG−LMS Genie w/o TxL 2

Fig. 5.

0 −25 4

6

8 Eb/N0 [dB]

10

12

14

16

Genie

−20

−15 SIR

TxL

−10 [dB]

−5

0

Fig. 7. SNR loss at BER 10−2 for the SG-LMS and genie compensation algorithm

System performance for SIRT xL = −10 dB

0

to be developed, aiming at the estimation of c2 and hTxL .

10

VI. C ONCLUSION −1

uncoded BER

10

−2

10

−3

10

−4

10

0

TxL IM2 SG−LMS Genie w/o TxL 2

4

6

8 E /N [dB] b

Fig. 6.

10

12

14

16

0

System performance for SIRT xL = −20 dB

decreasing SIRT xL the impact of Tx Leakage becomes more severe and the compensation algorithms achieve higher SNR gains. In order to characterize the sensitivity of WCDMA ZeroIF receivers to Tx Leakage and the performance improvement achieved by the compensation algorithms, the SNR loss w.r.t. no Tx Leakage at a target uncoded BER 10−2 is shown in Fig. 7 as a function of SIRT xL . As it can be seen the SG-LMS algorithm efficiently compensates the TxL-IM2 distortion, leading to a significant lower SNR loss than in the case without any compensation. Thus, more Tx Leakage interference can be allowed by compensating with the proposed SG-LMS algorithm. Furthermore, this compensation algorithm achieves slightly worst results than the genie approach. As a consequence, the SG-LMS can not be improved significantly, since it is limited by approximation errors entailed by the linearization. In order to achieve better performance results, nonlinear approaches have

An adaptive Stochastic-Gradient LMS algorithm, based on a linear approximation of the system model, has been presented. It is used to compensate the Tx Leakage in Zero-IF receivers at the digital baseband. Additionally, a genie algorithm was derived to evaluate the maximum achievable performance of this linear approach. The simulation results show that the LMS algorithm significantly improves the system performances and achieves almost the same performance as the genie approach. However, since the true system model is nonlinear, the approximation errors entailed in the linear compensation approach degrade increasingly the system performance for stronger Tx Leakage. Thus, nonlinear compensation approaches must be investigated to achieve further improvements. R EFERENCES [1] Marc Recouly Norm Swanber, James Pheips. WCDMA Cross Modulation Effects and Implications for Receiver Linearity Requirements. In IEEE Radio and Wireless Conference, 2002. [2] Vladimir Aparin and Lawrence E. Larson. Analysis and Reduction of Cross-Modulation Distortion in CDMA Receivers. In IEEE Transactions on Microwave Theory and Techniques, volume 51 of 5, May 2003. [3] Vladimir Aparin, Gary J. Ballantyne, Charles J. Persico, and Alberto Cicalini. An Integrated LMS Adaptive Filter of TX Leakage for CDMA Receiver Front Ends. In IEEE Journal of Solid-State Circuits, volume 41 of 5, May 2006. [4] Toms O’Sullivan, Robert A. York, Bud Noren, and Peter M. Asbeck. Adaptive Duplexer Implemented Using Single-Path and Multipath Feedforward Techniques With BST Phase Shifters. In IEEE Transactions on Microwave Theory and Techniques, volume 53 of 1, January 2005. [5] Marcus Windisch and Gerhard Fettweis. Performance Degradation due to I/Q Imbalance in Multi-Carrier Direct Conversion Receivers: A Theoretical Analysis. In IEEE International Conference on Communications (ICC’06), June 2006. [6] Andreas Frotzscher and Gerhard Fettweis. Baseband Analysis of Tx Leakage in WCDMA Zero-IF Receivers. In IEEE International Symposium on Control, Communications and Signal Processing (ISCCSP’08), March 2008. Invited paper. [7] Simon Haykin. Adaptive Filter Theory. Prentice Hall Inc., 1996. [8] 3GPP. Technical Specification Group Radio Access Network. Base Station (BS) conformance testing (FDD), Release 7, 2005.