A Robust Nonlinear RLS Type Adaptive Filter for Second-Order-Intermodulation Distortion Cancellation in FDD LTE and 5G Direct Conversion Transceivers
Andreas Gebhard, Oliver Lang, Michael Lunglmayr, Christian Motz, Ram Sunil Kanumalli, Christina Auer, Thomas Paireder, Matthias Wagner, Harald Pretl, Mario Huemer
AA Robust Nonlinear RLS Type Adaptive Filter forSecond-Order-Intermodulation DistortionCancellation in FDD LTE and 5G Direct ConversionTransceivers
Andreas Gebhard ∗ , Oliver Lang ‡ , Michael Lunglmayr ‡ , Christian Motz ∗ Ram Sunil Kanumalli † , Christina Auer ∗ , Thomas Paireder ∗ , Matthias Wagner ∗ ,Harald Pretl §† and Mario Huemer ∗ ∗ Christian Doppler Laboratory for DigitallyAssisted RF Transceivers for Future Mobile Communications,Institute of Signal Processing, Johannes Kepler University, Linz, Austria ‡ Institute of Signal Processing, Johannes Kepler University, Linz, Austria † Danube Mobile Communications Engineering GmbH & Co KG, Freist¨adterStraße 400, 4040 Linz, Austria § Institute for Integrated Circuits, Johannes KeplerUniversity, Linz, Austria Email: [email protected]
Abstract
Transceivers operating in frequency division duplex experience a transmitter leakage (TxL) signalinto the receiver due to the limited duplexer stop-band isolation. This TxL signal in combination withthe second-order nonlinearity of the receive mixer may lead to a baseband (BB) second-order intermod-ulation distortion (IMD2) with twice the transmit signal bandwidth. In direct conversion receivers, thisnonlinear IMD2 interference may cause a severe signal-to-interference-plus-noise ratio degradation ofthe wanted receive signal. This contribution presents a nonlinear Wiener model recursive-least-squares(RLS) type adaptive filter for the cancellation of the IMD2 interference in the digital BB. The includedchannel-select-, and DC-notch filter at the output of the proposed adaptive filter ensure that the providedIMD2 replica includes the receiver front-end filtering. A second, robust version of the nonlinear RLSalgorithm is derived which provides numerical stability for highly correlated input signals which arise ine.g. LTE-A intra-band multi-cluster transmission scenarios. The performance of the proposed algorithmsis evaluated by numerical simulations and by measurement data. a r X i v : . [ ee ss . SP ] J u l ndex terms — second-order intermodulation, self-interference, adaptive filters, interference can-cellation, LTE-A, 5G, RLS I. I
NTRODUCTION
Modern radio frequency (RF) transceivers are enhanced by digital signal processing to mitigatenon-idealities in the analog front-end. One of the main reasons of receiver desensitization infrequency division duplex (FDD) transceivers is the limited duplexer isolation between thetransmitter and the receiver which is around 50 dB to 55 dB [1,2]. The resulting transmitterleakage (TxL) signal can be identified as the root cause of several receiver baseband (BB)interferences. Especially in carrier aggregation (CA) receivers multiple clock sources are neededto cover the different CA scenarios and band combinations. Due to cross-talk between thereceivers on the chip and device nonlinearities, spurs appear in the receiver front-end. - - x BB [ n ] y RxRF ( t ) d Q [ n ] d I [ n ] w I [ n −
1] ˆ y I ˆ y AC,I [ n ] = ˆ y IMD2BB,I [ n ] e Q [ n ] = = (cid:8) ˆ y RxBB [ n ] (cid:9) + = { v BB [ n ] } e I [ n ] | | LNAPA ¯ h s [ n ]¯ h s [ n ] ¯ h s [ n ] CSF + DC − CSF + DC − CSF + DC − h TxLRF ( t ) A DA DA D nonlinearIM2RLS
Tx leakage f Tx f Tx f Tx f Tx f Rx f Rx ◦ ◦ α Q α I f f f RxTx
Fig. 1. Block diagram depicting an RF transceiver operating in FDD mode which experiences a second-order intermodulationdistortion in the receiver due to the transmitter leakage signal and the Rx mixer RF-to-LO terminal coupling. A nonlinearRLS-type adaptive filters is used to estimate the I-path IMD2 interference. The Q-path IMD2 interference is estimated with alinear 1-tap RLS adaptive filter which uses the estimated I-path IMD2 replica as reference input. f such a spur falls near the actual transmit (Tx) frequency, then the TxL signal is down-converted into the Rx BB where it causes a signal-to-interference-plus-noise ratio (SINR) degra-dation of the wanted receive signal. The cancellation of this so called modulated spurs withadaptive filtering is demonstrated in [3,4].Another prominent interference caused by the TxL signal and the second-order nonlinearity ofthe receiver is the second-order intermodulation distortion (IMD2). This second-order nonlineardistortion is caused by e.g. a coupling between the RF- and LO-ports in the I-, and Q-path ofthe Rx IQ-mixer as indicated in Fig. 1 [5]. An interesting fact of this nonlinear interferenceis, that one part of the generated second-order intermodulation products always falls aroundzero-frequency independent of the Tx-to-Rx frequency offset (duplexing distance). In case ofdirect-conversion receiver architectures, this leads to a degradation of the wanted receive signal.The mathematical modeling in [6,7] shows that the BB IMD2 interference contains the squaredenvelope of the BB equivalent TxL signal. The resulting BB IMD2 interference has twice theTx signal bandwidth and contains a DC due to the envelope-squaring. In the receiver front-end,the overall DC arising from a number of sources is canceled by a mixed-signal cancellation toprevent the analog-to-digital converter (ADC) from saturation. In the digital domain, the signalis filtered by a channel-select filter (CSF) to reduce its bandwidth to the Long Term Evolution(LTE) signal bandwidth.In the existing literature, the authors of [8]–[10] discussed adaptive least-mean-squares (LMS)type IMD2 interference cancellation algorithms for frequency-flat duplexer stop-bands. In [11]a Volterra kernel based least-squares (LS) approach for frequency-selective Tx-Rx responses isproposed. The authors in [7] presented a two-step LS approach for the IMD2 cancellation andconsidered a static 3rd-order power amplifier (PA) nonlinearity and IQ-imbalance in the transmitmixer. In [12] a Tx CA transceiver is considered where the transmit signal of both transmittersleaks through a diplexer into one unpaired CA receiver. The diplexer stop-band is modeled asa first-order finite impulse response (FIR) system which states a nearly frequency-flat response.The authors incorporated a fourth-order nonlinearity without memory into the estimation process,which results in an LS problem with four unknown coefficients.This contribution presents a nonlinear Wiener model RLS type adaptive filter (IM2RLS) withexponential forgetting factor which is suitable for highly frequency selective duplexer stop-band frequency responses like indicated in Fig. 2. It targets the digital IMD2 cancellation forhigh performance cellular base stations and mobile phones. The Wiener model uses a staticonlinearity at the output of the adaptive filter which has the advantage that less coefficients areneeded in the estimation process compared to a Volterra kernel based adaptive filter [13].An additional version of the proposed algorithm is presented which enhances the algorithmby a DC-notch filter to cancel the DC in the interference replica. This is needed because direct-conversion receivers employ a DC cancellation to suppress the DC in order to prevent the ADCfrom saturation. The DC in the received signal is time-variant and has many sources like e.g.LO-LO self mixing [5], and therefore must not be related explicitly to the DC which is generatedby the IMD2 interference. Consequently, the IMD2 interference related DC is removed from thereceived signal which complicates the IMD2 replica estimation. This DC removal is consideredin [6,11], and neglected in [7]–[9,14].The derived IM2RLS with DC-notch filter is extended by a regularization (R-IM2RLS) whichmakes the algorithm applicable for highly correlated BB transmit signals where the autocorre-lation matrix can be close to singular. A high correlation in the transmit signal can be due tooversampling which happens e.g. in the case of multi-cluster transmissions (introduced in 3GPPLTE-A Release 11) where only a part of the available resource blocks (RBs) are allocated. Thepresented IM2RLS algorithm is an extension to the nonlinear LMS type adaptive filter derivedin [6] with improved steady-state cancellation and convergence speed.The structure of the presented work is as follows: Section II explains the second-order inputintercept point (IIP2) characterization and demonstrates the degradation of the Rx performancedue to the IMD2 interference. Section III provides a detailed IMD2 interference model whichmotivates the proposed structure of the nonlinear adaptive filter. In Section IV, the IM2RLSalgorithm is derived and the impact of adding a DC-notch filter to the algorithm is evaluated.The R-IM2RLS alrorithm is derived in section V which is robust against highly correlated inputsignals as they occur in intra-band multi-cluster transmissions. Finally, in the sections VI andVII, the performance of the R-IM2RLS algorithm is evaluated with simulations and measureddata using RF components. II. P
ROBLEM STATEMENT
The receiver IIP2 is characterized by using two cosine signals with the frequencies f and f ofequal amplitude and the total power P in,2t at the input of the nonlinear mixer. The resulting totalIMD2 power generated at DC, f + f and f − f at the output of the mixer can be calculatedby P Tot,2tIM2 = 2 P in,2t − IIP2 [15], where IIP2 is the two-tone IIP2 value in dBm. Here, half ofhe total IMD2 power falls to DC, and one quarter each to f + f and f − f . To characterizethe IIP2 in a zero-IF receiver, the frequencies f and f are chosen such that f − f falls withinthe CSF bandwidth. Thereby the power at f − f is measured and the IIP2 is determined byIIP2 = 2 P in,2t − P f − f IM2 − dB.For modulated signals, the BB IMD2 power is modulation dependent and further reduced bythe CSF. This is considered by a correction-factor which corrects the IMD2 power calculatedby the two-tone formula [16,17].Although the DC-, and channel-select filtering in the receiver reduces the IMD2 BB inter-ference power by 6 dB in the two-tone signal case [15], and by about 13.4 dB [6,16,17] in thecase of modulated Tx signals, the left-over IMD2 interference may lead to a severe signal-to-noise ratio (SNR) degradation of the wanted Rx signal in reference sensitivity cases [18].Assuming a transmitter power of 23 dBm at the antenna, and an average Tx-to-Rx duplexerisolation at the transmit frequency of 50 dB, the TxL signal power at the input of the receiveris P TxLRF = 23 dBm − dB = − dBm. After amplification with the low noise amplifier (LNA)gain which is assumed as 20 dB, the RF TxL signal power increases to P TxLRF = − dBm at theinput of the nonlinear mixer.The two-tone IIP2 value of typical RF mixers is between 50 dBm and 70 dBm [19,20].Assuming an IIP2 of 60 dBm, the resulting BB IMD2 power with a full allocated LTE10 QPSKmodulated transmission and the determined correction factor of CF = 13 . dB is P CSF,LTEIM2 = 2 P TxLRF − IIP2 − CF = − . dBm [6]. In an LTE10 reference sensitivity case, thewanted signal power at the antenna can be as low as -97 dBm [18]. The thermal noise powerwithin 10 MHz bandwidth is -104.5 dBm and the assumed receiver noise figure (NF) is 4.5 dBwhich results in a receiver noise floor at -100 dBm. After amplification with 20 dB LNA gain,the wanted signal power is -77 dBm and the noise floor at -80 dBm corresponding to an RxSNR of 3 dB. The SNR drops from 3 dB to an SINR of 2.27 dB due to the IMD2 interferenceassuming an IIP2 of +60 dBm. In case of an reduced IIP2 of 55 dBm / 50 dBm, the SINR dropseven further to 1 dB / -1.4 dB, respectively. Fig. 2 depicts the spectrum of the frequency selectiveBB equivalent TxL signal y TxLBB which generates the complex valued IMD2 interference y IMD2BB by a coupling between the RF-to-LO terminals of the I-, and Q-path mixer. The total receivedsignal y TotBB contains the wanted Rx signal y RxBB which is degraded by the IMD2 interference andthe noise. − − − − − − − − − − − f [MHz] PS D [ d B m / H z ] y TxLBB y TotBB y RxBB y IMD2BB
Noise
Fig. 2. Equivalent BB spectrum of the frequency-selective Tx leakage signal y TxLBB (the corresponding passband signal is locatedat f Tx ) and the total received signal y TotBB after amplification with 20 dB LNA gain. The wanted Rx signal with SNR = 3 dB, andthe receiver noise floor after amplification with 20 dB LNA gain are at -77 dBm and -80 dBm (cid:98) = -108.2 dBm/15 kHz respectively.The total received signal contains the DC-, and channel-select filtered IMD2 interference with P Tx = 23 dBm at an assumed IIP2of 50 dBm. III. S
YSTEM MODEL
A. IMD2 Interference Model
Based on the block diagram in Fig. 1 depicting an RF transceiver operating in FDD mode,a detailed IMD2 interference model is derived. The used mathematical operators ( . ) ∗ , ( . ) T , ( . ) H , and ∗ denote the complex conjugate, transpose, Hermitian transpose, and convolution,respectively. The complex BB transmit signal x BB ( t ) = x I ( t ) + jx Q ( t ) is up-converted to thepassband and amplified by the linearly assumed PA with gain A PA resulting in the RF transmitsignal x RF ( t ) = A PA (cid:60) (cid:8) x BB ( t ) e j πf Tx t (cid:9) . (1)This signal leaks through the duplexer RF stop-band impulse response h TxLRF ( t ) = 2 (cid:60) (cid:8) h TxLBB ( t ) e j πf Tx t (cid:9) , (2)which is modeled by the BB equivalent duplexer impulse response h TxLBB ( t ) into the receiver,thereby creating the TxL signal y TxLRF ( t ) = x RF ( t ) ∗ h TxLRF ( t )= A PA (cid:60) (cid:8)(cid:2) x BB ( t ) ∗ h TxLBB ( t ) (cid:3) e j πf Tx t (cid:9) . (3)he received signal at the output of the LNA with gain A LNA y TotRF,LNA ( t ) = A LNA (cid:2) y TxLRF ( t ) + y RxRF ( t ) + v RF ( t ) (cid:3) , (4)is composed by the amplified TxL signal, the wanted Rx signal y RxRF ( t ) and the noise signal v RF ( t ) . The output signal of the I-, and Q-path mixer is combined into the complex valuedsignal y TotRF,mixer ( t ) (5). It contains the wanted signal which is down-converted with the linear gain α = α I + jα Q , and the second order interference with the mixer RF-to-LO terminal couplingcoefficient α = α I + jα Q . y TotRF,mixer ( t ) = y TotRF,LNA ( t ) α I cos (2 πf Rx t )+ y TotRF,LNA ( t ) (cid:2) α I y TotRF,LNA ( t ) (cid:3) − jy TotRF,LNA ( t ) α Q sin (2 πf Rx t )+ jy TotRF,LNA ( t ) (cid:104) α Q y TotRF,LNA ( t ) (cid:105) = y TotRF,LNA ( t ) α e − j πf Rx t + α y TotRF,LNA ( t ) (5)Assuming a direct conversion receiver, and using the identity (cid:60) { ηe jκ } = ( ηe jκ + η ∗ e − jκ ) , thetotal mixer output signal by neglecting the signal content which falls outside the BB bandwidthbecomes y TotRF,mixer ( t ) = α A LNA y RxBB ( t ) + α A LNA v BB ( t )+ α · (cid:18)(cid:12)(cid:12) A LNA A PA x BB ( t ) ∗ h TxLBB ( t ) (cid:12)(cid:12) + 12 (cid:12)(cid:12) y RxBB ( t ) (cid:12)(cid:12) + (cid:60) (cid:8) y RxBB ( t ) v ∗ BB ( t ) (cid:9) + 12 | v BB ( t ) | (cid:19) . (6)As | α | << , the three last terms in (6) may be neglected [6,7]. The total received discrete-timeBB signal including the DC-cancellation and channel-select filtering becomes y TotBB [ n ] = α A LNA y RxBB [ n ] ∗ ¯ h s [ n ] + α A LNA v BB [ n ] ∗ ¯ h s [ n ]+ α (cid:12)(cid:12) A LNA A PA x BB [ n ] ∗ h TxLBB [ n ] (cid:12)(cid:12) ∗ ¯ h s [ n ] (cid:124) (cid:123)(cid:122) (cid:125) y IMD2BB [ n ] , (7)where the DC-, and CSF are combined in the impulse response ¯ h s [ n ] = h DC [ n ] ∗ h s [ n ] . Here, h TxLBB [ n ] = T s h TxLBB ( t ) (cid:12)(cid:12) t = nT s is the impulse invariant [21,22], scaled and sampled version of thecontinuous-time BB duplexer impulse response h TxLBB ( t ) . . Interference Replica Model For the adaptive filter development to cancel the IMD2 interference in the digital BB, theinterference model (7) is rewritten to the form y TotBB [ n ] = α I (cid:12)(cid:12) A LNA A PA x BB [ n ] ∗ h TxLBB [ n ] (cid:12)(cid:12) ∗ ¯ h s [ n ] (cid:124) (cid:123)(cid:122) (cid:125) y IMD2,IBB [ n ] + j α Q (cid:12)(cid:12) A LNA A PA x BB [ n ] ∗ h TxLBB [ n ] (cid:12)(cid:12) ∗ ¯ h s [ n ] (cid:124) (cid:123)(cid:122) (cid:125) y IMD2,QBB [ n ] + v (cid:48) BB [ n ] (8)where the complex valued wanted signal and the noise signal are combined in v (cid:48) BB [ n ] . Assuming α I > , and approximating the duplexer impulse response h TxLBB [ n ] by the FIR impulse responsevector h TxLBB of length N w , we can rewrite the model (8) further to y TotBB [ n ] = (cid:12)(cid:12) x T [ n ] h I (cid:12)(cid:12) ∗ ¯ h s [ n ] + j (cid:12)(cid:12) x T [ n ] h Q (cid:12)(cid:12) ∗ ¯ h s [ n ] + v (cid:48) BB [ n ]= y IMD2,IBB [ n ] + j(cid:15) y IMD2,IBB [ n ] + v (cid:48) BB [ n ] , (9)where h I and h Q are incorporating h TxLBB and all scalar scaling factors in the I-, and Q-pathrespectively. The used vector x [ n ] is the complex valued tapped delay-line input signal vector x [ n ] = [ x BB [ n ] , x BB [ n − , . . . , x BB [ n − N w + 1]] T , and the real valued scaling factor (cid:15) showsthat the Q-path IMD2 interference may be modeled as a scaled version of the I-path interference.Motivated by the model (9) we propose the I-path IMD2 interference replica model ˆ y AC,I [ n ] = (cid:12)(cid:12) x T [ n ] w I [ n ] (cid:12)(cid:12) ∗ ¯ h s [ n ] , (10)using the adaptive filter coefficient vector w I [ n ] . The index AC indicates the DC cancellationin the IMD2 replica generation. The replica model comprises an adaptive Wiener model FIRfilter where the output signal is DC-, and channel-select filtered. The Q-path IMD2 interferenceis generated by estimating the scaling parameter (cid:15) by a linear single-tap RLS algorithm whichuses the estimated I-path IMD2 interference as reference input. This model is used to derive theadaptive filter structure shown in Fig. 1 to cancel the IMD2 interference in the digital BB. Forthe case if α I < , the sign of the desired signal in the I-path d I and the replica signal of theadaptive filter need to be changed.IV. N ONLINEAR R ECURSIVE L EAST -S QUARES ALGORITHM
In this section, a nonlinear Wiener model RLS type adaptive filter to estimate the channel-select filtered I-path IMD2 interference is developed. In a first step the IM2RLS algorithmithout DC-notch filter, which implies that the received signal contains the DC, is developed.Therefore, the replica model (10) without DC cancellation ˆ y I [ n ] = (cid:12)(cid:12) x T [ n ] w I [ n ] (cid:12)(cid:12) ∗ h s [ n ]= x T [ n ] w I [ n ] x H [ n ] w ∗ I [ n ] ∗ h s [ n ] (11)is used. The LS cost function up to the time index n with the exponential forgetting factor << λ ≤ is J LS [ n ] = n (cid:88) i =0 λ n − i (cid:12)(cid:12) d I [ i ] − x T [ i ] w I [ n ] x H [ i ] w ∗ I [ n ] ∗ h s [ i ] (cid:12)(cid:12) . (12)This cost function is visualized in Fig. 3 for an example impulse response h I = [1 , . T and λ = 1 where the estimated coefficients w I,0 and w I,1 are constrained to be real valued. Twoequivalent global minimum points and a local maximum at the origin w I = can be observed.The two solutions w I,1 = [1 , . T , and w I,2 = [ − , − . T minimize the cost function which canbe explained with the absolute-squaring nature of the IMD2 interference. Both solutions lead tothe same IMD2 replica signal. Assuming real valued CSF impulse response coefficients h s [ n ] , − − . . − − . . · w I , w I , J L S ( w I , , w I , ) Fig. 3. Shape of the cost function (12) for white Gaussian input signals with λ = 1 and for the real valued coefficient vector h I = [1 , . T when the desired signal d I [ n ] and the IMD2 replica are containing the DC. At the origin w I = , a local maximumcan be observed. and observing that d I [ i ] is the desired signal in the I-path, and therefore real valued, the gradientf the cost function (12) may be derived. The gradient of the cost function with respect to theconjugate coefficient vector w ∗ I using the Wirtinger calculus [23]–[25] becomes ∇ w ∗ I J LS = (cid:20) ∂J LS [ n ] ∂ w ∗ I [ n ] (cid:21) T = n (cid:88) i =0 λ n − i (cid:2) − d I [ i ] x T [ i ] w I [ n ] x ∗ [ i ] ∗ h s [ i ]+2 (cid:0) x T [ i ] w I [ n ] x ∗ [ i ] ∗ h s [ i ] (cid:1) · (cid:0) x H [ i ] w ∗ I [ n ] x T [ i ] ∗ h s [ i ] (cid:1) w I [ n ] (cid:3) . (13)By setting the gradient to zero, the Wiener Filter equation is obtained by ˜ R ( w I [ n ]) w I [ n ] = ˜ r ( w I [ n ]) , (14)where it can be observed that the autocorrelation matrix ˜ R and the cross-correlation vector ˜ r are functions of the unknown coefficient vector w I [ n ] . In a slowly varying or nearly stationarysystem environment it can be assumed that x T [ i ] w [ n ] ≈ x T [ i ] w [ i − when the index i is closeto n [26,27]. If the index i << n , the approximation introduces an error which is howeverattenuated by the forgetting factor. Defining the new cost function J (cid:48) LS [ n ] = n (cid:88) i =0 λ n − i (cid:12)(cid:12) d I [ i ] − x T [ i ] w I [ i − x H [ i ] w ∗ I [ n ] ∗ h s [ i ] (cid:12)(cid:12) = n (cid:88) i =0 λ n − i (cid:12)(cid:12) d I [ i ] − z T [ i ] w ∗ I [ n ] ∗ h s [ i ] (cid:12)(cid:12) = n (cid:88) i =0 λ n − i | e I [ i ] | (15)and introducing the new input vector z [ i ] = x T [ i ] w I [ i − x ∗ [ i ] , we can overcome this limitation.Following the traditional RLS derivation [28], the IM2RLS algorithm to estimate the I-path IMD2interference in the digital BB becomes (16)-(20): ˆ y I [ n ] = z T [ n ] w ∗ I [ n − ∗ h s [ n ] (16) e I [ n ] = d I [ n ] − ˆ y I [ n ] (17) k [ n ] = P [ n − z f [ n ] λ + z H f [ n ] P [ n − z f [ n ] (18) P [ n ] = 1 λ (cid:2) P [ n − − k [ n ] z H f [ n ] P [ n − (cid:3) (19) I [ n ] = w I [ n −
1] + e I [ n ] k [ n ] (20)To avoid the channel-select filtering of each element in the vector z f [ n ] = z [ n ] ∗ h s [ n ] whichis mainly necessary to align the signals due to the CSF group delay, we introduce the signals x f [ n ] = x [ n ] ∗ h s [ n ] and y (cid:48) I [ n ] = x T [ n ] w I [ n − . Using the delay line vector x f [ n ] = [ x f [ n ] , x f [ n − , . . . , x f [ n − M + 1]] T , the vector z f [ n ] may be approximated by z f [ n ] ≈ ( y (cid:48) I [ n ] ∗ h s [ n ]) x ∗ f [ n ] . With this formulation, a fractionaland non-constant group delay of the CSF may be incorporated. In case if the group delay τ g isconstant, and an integer multiple of the sampling time (as e.g. in linear phase FIR filters), theCSF may be approximated by delaying the signal by z f [ n ] ≈ x T [ n − τ g ] w I [ n − − τ g ] x ∗ [ n − τ g ] .In both approximations, the band-limiting effect of the CSF on z f [ n ] is ignored. However, thismay be tolerated because due to the envelope-squaring operation in (11) which doubles thesignal bandwidth, anyhow an oversampling factor (OSF) of 2 is mandatory to avoid aliasing.Due to the fact, that the I-, and Q-path IMD2 interference differ only by a real valued scalingfactor (cid:15) as derived in (8), the estimated I-path IMD2 replica may be used as a reference toestimate the Q-path IMD2 replica. This may be done by a linear 1-tap RLS algorithm whichuses the estimated I-path replica as reference input signal to estimate the Q-path IMD2 replica.In this case, the 1-tap RLS estimates also a possible sign difference between the I-, and Q-pathIMD2 interference. Consequently, only the sign of α I has to be detected during calibration of thereceiver which may be done by correlation. The replica signal generation (16) is channel-selectfiltered which reduces the bandwidth of the replica signal to the bandwidth of the received LTEsignal. A. Second-Order Condition
The complex Hessian [24,29] of the cost function (12) at the coefficient value w I = becomes H I = ∂∂ w I (cid:20) ∂J LS ∂ w ∗ I (cid:21) T | w I = = n (cid:88) i =0 λ n − i (cid:2) − d I [ i ] x ∗ [ i ] x T [ i ] ∗ h s [ i ] (cid:3) . (21)If the desired signal d I [ n ] contains the DC (when the receiver has no DC filtering), then E { d I [ n ] } ≥ and the Hessian matrix becomes negative semi-definite like depicted with thelocal maximum in Fig. 3. The usual choice of the zero-vector as initialization of w I [ − resultsin a zero-gain vector k [ n ] for all n . This is reasoned in the cost function (12) depicted in Fig. which has a local maximum at w I = and therefore a vanishing gradient. Consequently, thealgorithm is initialized with w I [ − (cid:54) = and the parameters << λ ≤ , and P [ −
1] = ν I with ν > . B. DC Cancellation
To employ an IMD2 interference replica without DC, the replica signal (16) is filtered bythe DC-notch filter (23). The new error signal e AC,I [ n ] = d AC,I [ n ] − ˆ y AC,I [ n ] with the DC-filteredsignals is used in the update equation (27). Here, the introduced index AC indicates the DCfiltered signals. The IM2RLS algorithm with DC-suppression can be summarized as (22)-(27): ˆ y I [ n ] = z T [ n ] w ∗ I [ n − ∗ h s [ n ] (22) ˆ y AC,I [ n ] = a ˆ y AC,I [ n −
1] + ˆ y I [ n ] − ˆ y I [ n − . (23) e AC,I [ n ] = d AC,I [ n ] − ˆ y AC,I [ n ] (24) k [ n ] = P [ n − z f [ n ] λ + z H f [ n ] P [ n − z f [ n ] (25) P [ n ] = 1 λ (cid:2) P [ n − − k [ n ] z H f [ n ] P [ n − (cid:3) (26) w I [ n ] = w I [ n −
1] + e AC,I [ n ] k [ n ] (27)The parameter << a < in (23) determines the sharpness of the DC-notch filter and is chosenas a = 0 . . In case of DC filtering in the main receiver E { d I [ n ] } = 0 , and the Hessianmatrix (21) at w I = is not positive semi-definite anymore. In this case, the local maximumbecomes a saddle-point like depicted in Fig. 4. Using N CSF as the number of coefficients of theCSF impulse response, the computational complexity of the IM2RLS with DC-notch filter is N w + 5 N CSF + 20 N w + 1 real multiplications and N w real divisions per iteration. C. Multiple Solutions of the IM2RLS Algorithm
In the cost function shapes depicted in Fig. 3 and Fig. 4, the estimated impulse responsecoefficients w and w (omitting the index I for the I-path) are constrained to be real valued. Itcan be observed that the two solutions w = [1 , . T , and w = [ − , − . T minimize the costfunction. The existence of multiple solutions can be explained by the absolute-squaring natureof the IMD2 interference. − . . − − . . · w I , w I , J L S ( w I , , w I , ) Fig. 4. Shape of the cost function (12) for white Gaussian input signals with λ = 1 and for the two real valued coefficients h = [1 , . T . The local maximum at w I = (with DC) changed to a saddle-point because the DC filtering is applied. If the coefficients are allowed to be complex valued, all coefficient pairs { w , w } convergeto (cid:12)(cid:12) w end (cid:12)(cid:12) = | h | and (cid:12)(cid:12) w end (cid:12)(cid:12) = | h | . This scenario is visualized in Fig. 5 where the convergenceof the coefficients with the ten different initializations w i [ −
1] = [1 e − , T exp ( j π/ i ) for i = 0 ... is depicted. Furthermore, each of the estimated coefficient vectors w end i = (cid:2) w end ,i , w end ,i (cid:3) T after convergence reach the group delay of the real system impulse response h . D. Performance of the IM2RLS with DC Suppression
In this section, the performance of the IM2RLS w/o and w/ DC cancellation is compared.In the first case, the receiver and the IMD2 replica generation of the IM2RLS do not use aDC cancellation. In this hypothetical example it is assumed that the IMD2 interference is theonly DC source. In the second case, the receiver uses a DC suppression, and the IM2RLSthe DC-notch filter. Both cases are compared within an FDD scenario with full allocated LTEsignals using 10 MHz bandwidth, QPSK modulation, short cyclic prefix, and an OSF of 2. Thefrequency-selective duplexer stop-band impulse response shown in Fig. 6 is used in (7) for theIMD2 interference generation. It is modeled with an FIR system which has 15 complex valuedcoefficients (on the native LTE10 sampling rate of 15.36 MHz) and a mean Tx-to-Rx isolationof 50 dB [1]. The resulting TxL signal has a strong frequency-selectivity like indicated in Fig. 2.The wanted Rx signal power is at reference sensitivity level P Rx = − dBm and the thermal − . . − − . . (cid:60) { w } , (cid:60) { w } (cid:61) { w } , (cid:61) { w } h h | h | w ,i [ n ] traj. | h | w ,i [ n ] traj. w end ,i w end ,i Fig. 5. Illustration of the initialization-dependent multiple solutions where the true coefficient values are h = [1 , . T . Theinitial coefficient w [ − is initialized in a 10-point grid around a circle with radius e − . The initial value of h [ − isalways zero. With each initialization, the coefficients converge to the correct absolute value. All ten resulting estimated impulseresponse vectors w end i maintain the same group delay as h . noise floor is -104.5 dBm within 10 MHz bandwidth. The receiver NF is 4.5 dB which resultsin an receiver noise floor of -100 dBm. The LNA gain is 20 dB, and the two-tone mixer IIP2 is50 dBm. This results in an desensitization of the wanted Rx signal from an SNR = 3 dB to anSINR of -1.4 dB at P Tx = 23 dBm. The I-path IMD2 interference is estimated by the IM2RLSusing 15 taps, running at the sampling frequency of 30.72 MHz (OSF = 2). This means, theadaptive filter has less taps than the duplexer stop-band impulse response which has 30 complexvalued coefficients at OSF = 2. The Q-path IMD2 replica is estimated by a linear 1-tap RLS(running at 30.72 MHz sampling rate) which uses the I-path IMD2 replica as reference input.The IM2RLS algorithm uses the forgetting-factor λ = 0 . and P [ −
1] = 100 I as suggestedin [30]. The 1-tap RLS in the Q-path uses the same forgetting factor and the initial coeffi-cient p [ −
1] = 1 e . The coefficient vector of the I-path IM2RLS algorithm is initialized with w I [ −
1] = [1 e − , , , ..., T , and the 1-tap RLS with zero. Fig. 7. shows the steady state SINRimprovement at different transmit power levels for an IIP2 of +50 dBm. It can be observed, thatin both cases (w/o and w/ DC cancellation) the SINR is improved nearly up to the Rx SNR of3 dB. The convergence behavior at the transmit power of 23 dBm is depicted in Fig. 8. For the − − − · − n (cid:60) (cid:8) h T x L BB [ n ] (cid:9) , (cid:61) (cid:8) h T x L BB [ n ] (cid:9) (cid:60) (cid:8) h TxLBB [ n ] (cid:9) (cid:61) (cid:8) h TxLBB [ n ] (cid:9) Fig. 6. Real and imaginary part of the 15-tap complex valued duplexer impulse response. − − SINRimprovementRx SNR = 3 dBTx Power [dBm] R x S I N R [ d B ] w/o cancellation, IIP2=50dBmProposed IM2RLS w/o DC-notch filterProposed IM2RLS w/ DC-notch filter Fig. 7. Improvement of the Rx SINR with the proposed IMD2 cancellation algorithms w/o and w/ using the DC-notch filter atdifferent transmitter power levels. The mixer IIP2 is 50 dBm and the wanted signal at the antenna has a power of P Rx = -97 dBmand a SNR of 3 dB. hypothetical case that the receiver and the IM2RLS are using no DC suppression, the IM2RLSconverges faster than with DC suppression. This is reasoned in the additional DC-IMD2 powerwhich supports the algorithm to converge faster. The IIP2 improvement by the digital cancellation ABLE IIIP2
IMPROVEMENT BY DIGITAL CANCELLATION
IM2RLS Algorithm P CSFIMD2 before P CSFIMD2 after IIP2 after canc.w/o DC cancellation -77.5 dBm -95.8 dBm 68.4 dBmw/ DC cancellation -77.5 dBm -94.5 dBm 67 dBm is summarized in Table I and may be calculated for the IM2RLS with DC-notch filter viaIIP2 after canc. = 2 P TxLRF − P CSF,LTEIM2, after canc. − . dB = 2 · (23 dBm − dB + 20 dB )+ 94 . dBm − . dB = 67 dBm . (28)The IIP2 is improved from +50 dBm to 68.4 dBm and 67 dBm by the digital cancellationwith the IM2RLS w/o and w/ DC suppression, respectively. The correction factor of 13.4 dBcorrects the IMD2 power calculated with the 2-tone formula, to the channel-select, and DC-filtered in-band IMD2 power for the LTE10 full allocation case [6]. For the calculation of theIIP2 improvement, the IMD2 power without DC is used in both cases. The derived IM2RLSalgorithm with included DC-notch filter shows an excellent cancellation performance for a fullallocated LTE10 transmit signal. However, for small bandwidth allocations like e.g. used inmulti-cluster transmissions, the RLS-type algorithm suffers from numerical instability due tothe badly-conditioned autocorrelation matrix ˜ R . To overcome this limitation, the regularizedIM2RLS (R-IM2RLS) is derived in the next section.V. T IKHONOV R EGULARIZATION OF THE NONLINEAR
RLSTo reduce the spectral out-of-band (OOB) emission of the LTE signals, not all availablesubcarriers are allocated. A portion of the subcarriers at the band-edges (guard-band) are forcedto zero which introduces correlation in the transmit BB samples. E.g. in a 10 MHz LTE signal amaximum of 600 out of 1024 subcarriers may be occupied by data [31]. This correlation in the TxBB signal x BB [ n ] leads to an badly-conditioned autocorrelation matrix R = E (cid:8) x BB [ n ] x H BB [ n ] (cid:9) and respectively ˜ R = E (cid:8) z f [ n ] z H f [ n ] (cid:9) . Algorithms which need the estimation of the autocorre-lation matrix or its inverse P = R − to estimate the system coefficients either iteratively or inbatch-mode, are sensitive to the condition number of R and may suffer from numerical instabilityif R is badly-conditioned. Because of this reason, a regularized version of the IM2RLS algorithm(R-IM2RLS) is derived in this section. 2 4 6 8 10 12 14 16 18 20 − − − LTE10 slots with OSF=2 N M S E [ d B ] Proposed IM2RLS w/o DC-notch filterProposed IM2RLS w/ DC-notch filter
Fig. 8. Convergence of the IM2RLS w/o and w/ DC-notch filter for an LTE transmit signal with 10 MHz bandwidth, OSF of 2and P Tx = 23 dBm. The wanted Rx signal power at the antenna input is P Rx = -97 dBm and the Rx SNR = 3 dB. The mixer IIP2is +50 dBm which corresponds to an Rx SNR desense of 4.4 dB. A common method to overcome the problem of badly-conditioned autocorrelation matricesis regularization [28]. Adding a positive definite matrix to the estimated auto-correlation matrixin each iteration of the RLS algorithm guarantees that the regularized autocorrelation matrix ˜ R (cid:48) stays positive definite and maintains therefore the necessary condition for convergence andexistence of P = ˜ R (cid:48)− [32].This method is commonly known as Tikhonov-regularization where a matrix L is used for theregularization [33]. By including a regularization term in the cost function (15), the new costfunction J (cid:48) R [ n ] = n (cid:88) i =0 λ n − i (cid:2) | e I [ i ] | + σ (cid:107) Lw I [ n ] (cid:107) (cid:3) = n (cid:88) i =0 λ n − i (cid:2) | e I [ i ] | + σ w T I [ n ] L T Lw ∗ I [ n ] (cid:3) (29)is defined where e I [ i ] = d I [ i ] − z T [ i ] w ∗ I [ n ] ∗ h s [ i ] . The regularization parameter σ ≥ is usedto adjust the regularization amount and the real valued matrix L is typically chosen as L = I standard Tikhonov regularization), L = upperbidiag (1 , − (first order derivative), or L = − − − . . . . . . . . . − (30)(second order derivative) [33]. Using the Wirtinger calculus [23] to obtain the gradient of thecost function (29), and setting the gradient to zero results in (cid:34) n (cid:88) i =0 λ n − i (cid:0) z f [ i ] z H f [ i ] + σ L T L (cid:1)(cid:35)(cid:124) (cid:123)(cid:122) (cid:125) ˜ R (cid:48) [ n ] w I [ n ] = n (cid:88) i =0 λ n − i d I [ i ] z f [ i ] (cid:124) (cid:123)(cid:122) (cid:125) ˜ r [ n ] . (31)Reformulating the above equation leads to w I [ n ] = ˜ R (cid:48)− [ n ]˜ r [ n ] = P [ n ]˜ r [ n ] which is solvedrecursively using the RLS algorithm. By expressing the cross-correlation vector ˜ r [ n ] by itsprevious estimate ˜ r [ n − , a recursive estimation of the form ˜ r [ n ] = λ ˜ r [ n −
1] + d I [ n ] z f [ n ] (32)may be formulated. Similarly, a recursive estimation of the regularized autocorrelation matrix isobtained by ˜ R (cid:48) [ n ] = λ n − (cid:88) i =0 λ n − i − (cid:0) z f [ i ] z H f [ i ] + σ L T L (cid:1) + z f [ n ] z H f [ n ] + σ L T L = λ ˜ R (cid:48) [ n −
1] + σ L T L + z f [ n ] z H f [ n ] . (33)Substituting Ω [ n ] − = λ ˜ R (cid:48) [ n −
1] + σ L T L into (33), the matrix P [ n ] = ˜ R (cid:48)− [ n ] becomes P [ n ] = (cid:2) Ω [ n ] − + z f [ n ] z H f [ n ] (cid:3) − . (34)After applying the matrix inversion lemma ( A + BCD ) − = A − − A − B (cid:0) C − + DA − B (cid:1) − DA − (35)to avoid the matrix inversion, (34) may be formulated as P [ n ] = Ω [ n ] − k [ n ] z H f [ n ] Ω [ n ] (36)sing the gain vector k [ n ] = Ω [ n ] z f [ n ]1 + z H f [ n ] Ω [ n ] z f [ n ] . (37)For the inversion Ω [ n ] = (cid:2) λ P − [ n −
1] + σ L T L (cid:3) − , (38)again the matrix inversion lemma is applyied which yields Ω [ n ] = 1 λ ( P [ n − − Σ [ n ] LP [ n − (39)where the substitution Σ [ n ] = σ P [ n − L T (cid:2) λ I + σ LP [ n − L T (cid:3) − (40)is used. After rearranging (40), the expression Σ [ n ] = σλ ( P [ n − − Σ [ n ] LP [ n − L T = σ Ω [ n ] L T (41)is obtained. Unfortunately, the calculation of Σ [ n ] in (40) and therefore Ω [ n ] still includes amatrix inversion after applying the matrix inversion lemma. However, by decomposing the matrix L T L in (38) into a sum of V dyads [34] Ω [ n ] = (cid:34) λ P − [ n −
1] + σ V (cid:88) k =1 p k, p Tk, (cid:35) − , (42)applying the matrix inversion lemma results in the recursive calculation of (42) via Ω k [ n ] = Ω k − [ n ] − Ω k − [ n ] p k, σ + p Tk, Ω k − [ n ] p k, p Tk, Ω k − [ n ] (43)for k = 1 . . . V in each iteration n and Ω [ n ] = λ P [ n − . Reformulating (37) yields k [ n ] = P [ n ] z f [ n ] . (44)The recursive update of the coefficient vector w I [ n ] is obtained by inserting (36), (32), (44), (39)and (41) into w I [ n ] = P [ n ]˜ r [ n ] . The final nonlinear R-IM2RLS algorithm to estimate the I-pathIMD2 interference is summarized by (45)-(51): ˆ y I [ n ] = z T [ n ] w ∗ I [ n − ∗ h s [ n ] (45) e I [ n ] = d I [ n ] − ˆ y I [ n ] (46) Ω k [ n ] = Ω k − [ n ] − Ω k − [ n ] p k, σ + p Tk, Ω k − [ n ] p k, p Tk, Ω k − [ n ] (47) [ n ] = Ω V [ n ] z f [ n ]1 + z H f [ n ] Ω V [ n ] z f [ n ] . (48) P [ n ] = Ω V [ n ] − k [ n ] z H f [ n ] Ω V [ n ] (49) Σ [ n ] = σ Ω V [ n ] L T (50) w I [ n ] = (cid:2) I − (cid:0) I − k [ n ] z H f [ n ] (cid:1) Σ [ n ] L (cid:3) w I [ n −
1] + k [ n ] e I [ n ] (51)The proposed algorithm is initialized with w I [ − (cid:54) = , << λ ≤ and P [ −
1] = ν I with ν > .When the DC suppression is used, then the R-IM2RLS update equations become (52)-(59): ˆ y I [ n ] = z T [ n ] w ∗ I [ n − ∗ h s [ n ] (52) ˆ y AC,I [ n ] = 0 .
998 ˆ y AC,I [ n −
1] + ˆ y I [ n ] − ˆ y I [ n − (53) e AC,I [ n ] = d AC,I [ n ] − ˆ y AC,I [ n ] (54) Ω k [ n ] = Ω k − [ n ] − Ω k − [ n ] p k, σ + p Tk, Ω k − [ n ] p k, p Tk, Ω k − [ n ] (55) k [ n ] = Ω V [ n ] z f [ n ]1 + z H f [ n ] Ω V [ n ] z f [ n ] . (56) P [ n ] = Ω V [ n ] − k [ n ] z H f [ n ] Ω V [ n ] (57) Σ [ n ] = σ Ω V [ n ] L T (58) w I [ n ] = (cid:2) I − (cid:0) I − k [ n ] z H f [ n ] (cid:1) Σ [ n ] L (cid:3) w I [ n − k [ n ] e AC,I [ n ] (59)The DC-notch filter (53) is used to remove the DC from the IMD2 replica (52). The complexityof the R-IM2RLS with DC-notch filter and L = σ I is N w + 21 N w + 5 N CSF + 18 N w + 1 realmultiplications and N w + 2 N w real divisions per iteration.I. S IMULATION ENVIRONMENT
The performance of the R-IM2RLS algorithm with the three above mentioned regularizationmatrices L is evaluated with an FDD scenario using an LTE10 multi-cluster intra-band Tx signalwhich has a native sampling frequency of f s = 15 . MHz, QPSK modulation and short cyclicprefix. The IMD2 interference in the I-path is estimated by the R-IM2RLS, while the Q-pathIMD2 is estimated by a linear 1-tap RLS which uses the I-path IMD2 replica as reference input.The resulting multi-cluster TxL signal has a strong frequency-selectivity like indicated in Fig. 9.The R-IM2RLS in the I-path has 15 taps and runs on the higher sampling rate of 30.72 MHz dueto the OSF of 2. This means, the adaptive filter has less taps than the impulse response whichis estimated. The linear 1-tap Q-path RLS runs also on the sampling rate of 30.72 MHz. Thereceived signal d [ n ] is DC filtered and the proposed algorithm is using the DC-notch filter tosuppress the DC of the IMD2 replica signal. The wanted Rx signal has a power of P Rx = -97 dBmat the antenna with an SNR of 3 dB. The assumed Rx mixer IIP2 is +60 dBm which correspondsto an Rx SNR desense of 1 dB for the specific intra-band multi-cluster transmit signal at 23 dBmpower level. The thermal noise floor of the receiver is assumed at -104.5 dBm per 10 MHz andthe receiver NF is 4.5 dB. The resulting receiver noise floor and Rx power with 20 dB LNAgain is at -80 dBm (cid:98) = -108.2 dBm/15 kHz and -77 dBm respectively. The spectrum of the signalsat P Tx = 23 dBm is depicted in Fig. 9. It can be observed, that the resulting IMD2 interference y IMD2BB is mostly below the receiver noise floor but still leads to an SNR degradation of 1 dB. Thedepicted interference replica is estimated by the R-IM2RLS with the regularization L = 3 e − I .The multi-cluster LTE10 Tx signal uses 21/50 RBs (252 subcarriers from 1024), which meanshat 3.78 MHz of the available 9.015 MHz are allocated. With an OSF of 2 this correspondsto an allocated bandwidth-to-sampling-rate ratio of 3.78/30.72 = 0.12 which introduces a highcorrelation in the transmit BB samples. The resulting condition number cond ( ˜ R ) of the × dimensional autocorrelation matrix ˜ R = E (cid:8) z f z H f (cid:9) is in the order of which results in abad conditioned estimation, and may lead to numerical problems. The regularization of the R-IM2RLS improves numerical estimation of the matrix P [ n ] by lowering the condition numberof the regularized matrix ˜ R (cid:48) . A. IMD2 Self-Interference of a Multi-Cluster Tx Signal
For the estimation of the resulting IMD2 interference bandwidth, the bandwidth between theminimum and maximum allocated subcarrier in the multi-cluster Tx signal is of interest. In the − − − − − − − − − − − − − f [MHz] PS D [ d B m / H z ] y TxLBB y TotBB y RxBB y IMD2BB ˆ y IMD2BB
Noise
Fig. 9. Equivalent BB spectrum of the frequency-selective Tx leakage signal y TxLBB (the corresponding passband signal is locatedat f Tx ) and the total received signal y TotBB after amplification with 20 dB LNA gain. The wanted Rx signal with SNR = 3 dB, andthe receiver noise floor after amplification with 20 dB LNA gain are at -77 dBm and -80 dBm (cid:98) = -108.2 dBm/15 kHz respectively.The total received signal contains the DC-, and channel-select filtered IMD2 interference at P Tx = 23 dBm and the IIP2 is 60 dBm. used clustered LTE10 transmit signal the allocated RBs are { − , − } with a numberingfrom left to right and the total number of 50 RBs. For the IMD2 bandwidth estimation theresulting bandwidth between the lowest allocated subcarrier (RB 9) and the upper edge (RB46) of the allocated RBs is (3 + 17 + 18) · · kHz = 6 . MHz. Each RB has 12 subcarriersand 15 kHz subcarrier spacing. The resulting IMD2 interference bandwidth is × . MHz =13 . MHz which means that a small portion of the IMD2 interference is suppressed by theCSF. The full IMD2 interference including the DC, the IMD2 interference after the CSF andDC-removal, and the estimated IMD2 replica are visualized in Fig. 10. It can be observed, thatthe R-IM2RLS is able to estimate the IMD2 interference down to 20 dB below the receiver noisefloor.
B. Numerical Simulation Results
In the following simulation results, the IMD2 self-interference cancellation performance incase of an intra-band multi-cluster Tx signal, using the R-IM2RLS algorithm (52)-(59) using theDC-notch filter with different regularization matrices is evaluated. The forgetting factor of theR-IM2RLS is chosen as λ = 0 . , P [ −
1] = 100 I , and the regularization constant σ = 3 e − .The 1-tap RLS in the Q-path uses the same forgetting factor but the initial coefficient p [ −
1] = − − − − − − − f [MHz] PS D [ d B m / H z ] full IMD2 with DC y IMD2BB ˆ y IMD2BB
Noise
Fig. 10. Generated IMD2 interference with the bandwidth of 13.68 MHz at P Tx = 23 dBm. The resulting in-band BB IMD2interference y IMD2BB after the CSF and DC-removal is below the receiver noise floor. The R-IM2RLS estimates the IMD2interference down to 20 dB below the noise floor. e . The coefficient vector of the R-IM2RLS is initialized with w I [ −
1] = [1 e − , , , ..., T forthe I-path, and the 1-tap Q-path RLS is initialized with zero. The performance is evaluated for thedifferent regularization matrices L = 3 e − I (Tikhonov regularization), L = 3 e − upperbidiag (1 , − (first order derivative smoothing matrix), and L = 3 e − diag (1 , − , (second order derivativesmoothing matrix). The IM2RLS without regularization is not included in the comparison dueto numerical instability reasoned by the extremely high condition number of ˜ R which is in theorder of . The performance of the R-IM2RLS is compared with the recently published LMS-type algorithm (IM2LMS) [6]. The IM2LMS uses the step-size µ = 0 . , the regularizationparameter γ = 0 . , and the initial coefficient vector ˆ w I [ −
1] = [1 e − , , , ..., T . The Q-pathIMD2 replica is estimated by a linear normalized 1-tap LMS which uses the I-path IMD2 replicaestimated by the IM2LMS as reference input. The normalized 1-tap LMS uses a step-size of 1,the regularization parameter is set to 1e-7 and the initial coefficient is set to zero. The value of thestep-size is set to the best compromise between steady-state cancellation and convergence time.The convergence of the algorithms is compared using the ensemble normalized mean-square-error(NMSE), and the steady-state cancellation by the SINR. The SINR improvement of the Rx signalfor the different algorithms and regularizations is depicted in Fig. 11. The convergence behaviourof the algorithms is depicted in Fig. 12. The R-IM2RLS shows a faster initial convergence thanthe IM2LMS algorithm which takes about twice as long to reach an NMSE of -10 dB. The
12 15 18 21 2322 . . . SINRimprovementRx SNR = 3 dBTx Power [dBm] R x S I N R [ d B ] w/o cancellationR-IM2RLS, L = 3 e − I R-IM2RLS, L = 3 e − upperbidiag (1 , − R-IM2RLS using (30) and σ = 3 e − IM2LMS [6] with µ = 0 . , γ = 0 . Fig. 11. Improvement of the Rx SINR at different transmitter power levels and an Rx mixer IIP2 of +60 dBm. The algorithmsare using the DC-filtered receive signal, and the R-IM2RLS/IM2LMS algorithms are using the DC-notch filter to remove theDC. The wanted signal at the antenna has the power P Rx = -97 dBm and a SNR of 3 dB. − − LTE10 slots with OSF=2 N M S E [ d B ] R-IM2RLS, L = 3 e − I R-IM2RLS, L = 3 e − upperbidiag (1 , − R-IM2RLS using (30) and σ = 3 e − IM2LMS [6] with µ = 0 . , γ = 0 . Fig. 12. Convergence of the R-IM2RLS with different regularization matrices and the IM2LMS algorithm at the transmit powerlevel of P Tx = 23 dBm. The algorithms are using the DC-notch filter to suppress the DC. evolution of the condition number of ˜ R (cid:48) [ n ] = P [ n ] − is illustrated in Fig. 13. The conditionnumber of ˜ R estimated by the IM2RLS without regularization drastically increases up to valuesbetween and . In contrast to that, the condition number of ˜ R (cid:48) estimated by the R-IM2RLSwith different regularization matrices L stays below 400 for the specific clustered Tx example.The achieved IIP2 after the digital IMD2 cancellation is summarized in Table II. The R-IM2RLS 2 4 6 8 10 12 , LTE10 slots with OSF=2 C ond iti onnu m b e r o f ˜ R (cid:48) IM2RLS w/o regularizationR-IM2RLS, L = 3 e − I R-IM2RLS, L = 3 e − upperbidiag (1 , − R-IM2RLS using (30) and σ = 3 e − Fig. 13. Evolution of the condition number of ˜ R (cid:48) [ n ] = P − [ n ] for a clustered allocation like depicted in Fig. 9 and 23 dBmtransmit power. The condition number of ˜ R = E (cid:8) zz H (cid:9) without regularization is in the order of to . and IM2LMS algorithms are improving the IIP2 from 60 dBm to about 77 dBm and 73 dBm,respectively. TABLE IIIIP2
IMPROVEMENT BY DIGITAL CANCELLATION FOR THE CLUSTERED T X SIGNAL
Algorithm IIP2 after canc.R-IM2RLS, L = 3 e − I L = 3 e − upperbidiag (1 , − σ = 3 e − VII. V
ERIFICATION OF THE DERIVED ALGORITHM WITH MEASUREMENT DATA
The proposed R-IM2RLS algorithm is evaluated with measurement data and Matlab post-processing. The measurement setup (A) depicted in Fig. 14 includes the LTE band 2 duplexermodel B8663 from TDK, the LNA ZX60-2534MA+ with 41.3 dB gain and 2.6 dB NF and theZAM-42 Level 7 mixer which has 25 dB RF-to-LO terminal isolation. The measurement is carriedout for the I-path mixer and a full allocated LTE-A transmit signal with 10 MHz bandwidth,QPSK modulation and short cyclic prefix. The transmit frequency is set to f Tx = 1 . GHz andthe mixer LO frequency is f Rx = 1 . GHz (80 MHz duplexing distance). The LTE transmitsignal is generated with the R&S SMW 200A signal generator (B), and the TxL signal whichleaks into the receiver with 80 MHz frequency offset to the LO signal is amplified by the LNAain. This amplified TxL signal generates the BB IMD2 interference at the output of the I-pathmixer which is measured with the real-time oscilloscope RTO 1044 (C). The TxL signal afterthe LNA is measured by the R&S FSW26 spectrum analyzer (D), and the LO signal with 7 dBmfor the ZAM-42 mixer is generated by the R&S SMB 100A signal generator (E). The transmit (E) (B) (C)(A) (D)
Fig. 14. Measurement setup including the DUT (A) with the LNA ZX60-2534MA+, the mixer ZAM-42 from Mini Circuitsand the LTE band 2 duplexer B8663. The signal generator R&S SMW 200A (B) generates the LTE transmit signal and the R&Sreal-time oscilloscope RTO 1044 (C) is used to measure the BB signal after the mixer. The R&S FSW26 spectrum analyzer (D)is used to measure the TxL signal, and the signal generator R&S SMB 100A (E) generates the mixer LO signal. power is set to P TxRF = 19 . dBm, which leads in combination with the duplexer attenuation of67.6 dB (at f Tx = 1 . GHz) and the LNA gain of 41.3 dB to the typical TxL signal power of P TxLRF = 19 . dBm − . dB + 41 . dB = − dBm. The measured I-path mixer BB output datastream and the complex valued BB transmit samples are used for the Matlab post-processing.The spectrum of the signals before and after digital cancellation with the R-IM2RLS usinga Tikhonov regularization and the parameters P [ −
1] = 10 I , λ = 0 . and L = 1 e − I aredepicted in Fig. 15. The Matlab post-cancellation showed that 10 taps were sufficient to cancelthe IMD2 interference by 2.2 dB down to the noise floor. The coefficient vector was initializedwith w I [ −
1] = [1 e − , , , ..., T , and the convergence of the coefficients is shown in Fig. 16which indicates that the coefficients converged after about 5 LTE symbols. − − − − − − f [MHz] PS D [ d B m / H z ] y TxLBB y TotBB ˆ y IMD2BB remaining IMD2+noise
Fig. 15. Spectrum of the measured TxL signal y TxLBB and the receive signal y TotBB including noise and the IMD2 interference. TheBB equivalent TxL signal shows a strong frequency selectivity. Also shown are the spectrum of the estimated IMD2 replica ˆ y IMD2BB and the remaining IMD2 and noise after the cancellation. · − LTE10 slots with OSF=2 | w i | | w i | Fig. 16. Evolution of the estimated coefficients by the R-IM2RLS.
VIII. C
ONCLUSION
This paper presented a novel nonlinear RLS type adaptive filter (IM2RLS) and its robust ver-sion (R-IM2RLS) for the digital IMD2 self-interference cancellation in LTE FDD RF transceivers.The R-IM2RLS provides stability and numerical tractability for highly correlated transmit signalshich may result in an ill-conditioned autocorrelation matrix. The proposed R-IM2RLS is ableto cancel the IMD2 interference generated by a highly frequency-selective Tx leakage signal,and its performance is evaluated with different regularization matrices. Typical RF receivers usea DC cancellation to prevent the ADC form saturation and a CSF to limit the signal bandwidth.Therefore the IMD2 interference which is generated by the second-order nonlinearity in themixer is DC filtered and its bandwidth is reduced to the LTE signal bandwidth. Consequently,the adaptive filter needs to provide a DC-filtered in-band IMD2 replica. This contribution showsthat the IM2RLS/R-IM2RLS adaptive filter is able to reproduce the in-band IMD2 interferencewithout DC by including the CSF and a DC-notch filter within the algorithm. It is shown, thatthe proposed algorithm may have multiple solutions of the estimated coefficient vector becauseof the envelope-squaring nature of the IMD2 interference. The algorithm converges within aview LTE symbols and the steady-state Rx SNR degradation by the IMD2 self-interference incase of an multi-cluster transmit signal is improved in simulation from 1 dB to less than 0.05 dB.The performance of the R-IM2RLS is proved in an LTE measurement scenario with discrete RFcomponents. The IMD2 interference in the received signal is canceled to the noise floor and aconvergence of the coefficients within 5 LTE symbols is achieved.A
CKNOWLEDGMENT
The authors wish to acknowledge DMCE GmbH & Co KG, an Intel subsidiary for supportingthis work carried out at the Christian Doppler Laboratory for Digitally Assisted RF Transceiversfor Future Mobile Communications. The financial support by the Austrian Federal Ministry ofScience, Research and Economy and the National Foundation for Research, Technology andDevelopment is gratefully acknowledged.R
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