In order to design suitable ... density (PSD) of the TxL interference is derived. In Sec. IV .... by first calculating the discrete power spectral density (PSD). ΦB,ηN ...
On the performance of OFDM in zero-IF receivers impaired by Tx Leakage Andreas Frotzscher, Marco Krondorf and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Technische Universit¨at Dresden, D-01062 Dresden, Germany Email: {frotzscher, krondorf, fettweis}@ifn.et.tu-dresden.de
II. S YSTEM MODEL This paper focuses on the Tx Leakage (TxL) present at the mobile terminal, as depicted in Fig. 1. After passing the power DL signal ...
LO DL DC-1
LNA Tx Leakage
...
3
6
1
...
CP-1
CP
sBB[k]
sUL[k]
A D ...
LOUL
...
5 ∆k
2
D A
TxL CSF IM2
PA
7
TxL IM2
ηN-FFT
DL signal
ηN-IFFT
I. I NTRODUCTION In wireless handset devices, operating in the frequency division duplex (FDD) mode, a duplexer connects the transmit (Tx) and receive (Rx) chain with the antenna. Since both chains are operating simultaneously, the duplexer has to provide a high Tx-Rx isolation. However, achieving a sufficient Tx-Rx isolation is very challenging if low cost implementation is targeted and thus, a significant part of the transmitted Uplink (UL) signal is leaking through the duplexer in the Rx chain, causing severe interference to the received Downlink (DL) signal. Tx Leakage (TxL) is of considerable importance especially in difficult receive conditions (e.g. at cell edges), where the demodulation of the weak DL signal is difficult, because the TxL significantly reduces the receiver sensitivity. Focusing on zero-IF receivers, the nonlinearity of the I/Q down converter generates 2nd order intermodulation products (IM2) of the leaking UL signal, which are partly located around DC and thus severely interfere the down converted DL signal [1]. Currently, tight requirements in terms of linearity and Tx-Rx isolation are set on the analog font end in order to avoid severe TxL interference. However, meeting these requirements is very challenging for modern terminals, since several communication standards must be supported in parallel at even reduced terminal sizes. Compensating the TxL interference in the digital domain [2] relaxes the requirements on the analog front end and thereby reduces development and production costs. In order to determine the conditions under which a digital TxL compensation is required, the performance degradation caused by TxL must be derived. In this contribution we therefore analyze the statistical properties of the TxL interference, considering an OFDM transmission in the UL and DL. Based on the obtained results, we evaluate the impact of Tx Leakage on the performance of OFDM DL communication over the frequency selective fading channel. Specifically, we will show
that the TxL interference on the OFDM subcarriers can be modeled as a Gaussian noise process, whose variance can be calculated for each subcarrier by a closed form expression. The outline of the paper is as follows. The next section presents the system model. In Sec. III the power spectral density (PSD) of the TxL interference is derived. In Sec. IV the performance of the DL OFDM transmission impaired by TxL is derived in terms of bit error probability and mutual information. In Sec. V the analytical results are verified with simulation results and in Sec. VI conclusions are drawn.
Duplexer
Abstract—Transmitter Leakage has a significant impact on the system performance in mobile devices using zero-IF receivers. In this contribution, the statistical properties of the Tx Leakage impact in zero-IF receivers are derived, in the context of Uplink and Downlink OFDM transmission. In order to design suitable algorithms for the digital compensation of Tx Leakage in OFDM communication systems, an accurate modeling of the corresponding performance loss is required. We therefore provide a statistical description of Tx Leakage effects on OFDM signals, which is used to compute bit error rates and mutual information.
1
2
3
Ch. EQ
OFDM Demodulator
OFDM Modulator
...
CP OFDM symb.
Fig. 1. Block diagram of a FDD zero-IF transceiver in a terminal impaired by Tx Leakage
amplifier (PA), the up converted UL signal partly leaks into the Rx chain. The compound amplification and filtering, which the leaking UL signal undergoes between the digital analog converter (DAC) in the Tx chain and the down converter in the Rx chain can be considered as an effective TxL channel. The I/Q down converter in the Rx chain shifts the received signal to baseband, but it also generates intermodulation products 2nd order of the leaking UL signal (TxL IM2). These intermodulation products arise partly around DC and thus introduce severe interference to the down converted DL signal. Due to the unavoidable I/Q mismatch of the down converter, the TxL IM2 in the I- and Q branch are different from each other. The down converter is followed by the channel select filter (CSF) and a DC offset cancellation (DC−1 ). Finally, the analog-to-digital converter (ADC) delivers the digital baseband signal sBB [k]. In practice the DAC and ADC operate with an oversampling of usually 2, ..., 8. In this study an oversampling factor η = 2 is assumed. The reduction of the effective ADC resolution for the DL signal, due to the TxL interference, is
not considered. However, further studies have shown, that the SNR loss due to a reduced ADC resolution is ≤ 1 dB for TxL interference power levels allowable in practice. Moreover, any other impact of the ADC such as quantization and distortion noise effects are not considered in our contribution, as they have been studied in [3]. In [1] the following baseband model of Tx Leakage in zero-IF receivers was derived: 2 sBB [k] = hDL ∗ sDL [k] + c hCSF ∗ hT xL ∗ sU L ) [k] | {z } | {z } a[k]
b[k]
+ w[k] + aDC ,
(1)
describing the impact of the DL signal and the leaking oversampled UL signal sU L [k] on the discrete signal sBB [k], provided by the ADC. Term a[k] in (1) describes the oversampled, received DL signal sDL [k], being distorted by the DL transmission channel hDL [k]. The discrete channel impulse response (CIR) hDL [k] is assumed to have L independent Rayleigh faded taps. OFDM systems usually use a the cyclic prefix extension to avoid inter symbol interference (ISI) in the received signal. In the following, the CIR length L is assumed to be shorter than the η times oversampled cyclic prefix, ensuring an ISI free DL transmission. The average channel gain per subcarrier, i.e. averaged over all fading states, 2 and is identical for all subcarriers. is denoted by σH Term b[k] describes the TxL IM2 interference, where hT xL [k] and hCSF [k] denotes the impulse responses of the TxL channel and CSF, respectively and c is a complex scaling factor, representing the I/Q mismatch of TxL IM2. Finally w[k] is a zero-mean, complex, Gaussian noise signal, representing the several noise sources in the receiver and aDC denotes the counter signal value of the DC offset compensation. The UL signal is assumed to be a continuous OFDM data symbol sequence. Due to the TxL propagation delay, the UL signal frame structure can be asynchronous to the DL signal frame structure, resulting in a sample offset ∆k (Fig. 1). In cellular networks the timing advance of the UL transmission can enlarge ∆k significantly. Thus, ∆k must be taken into account in the derivation of the TxL impact on the DL communication as detailed in Sec.III. Let γT xL denote the TxL signal-to-interference ratio (SIR), describing the power ratio between the received DL signal and the TxL IM2 within the band of interest, excluding DC: o n 2 E |a[k]| o γT xL = n (2) 2 E |b[k]|
For OFDM demodulation, the cyclic prefix is removed from the received discrete baseband signal sBB [k], where a perfect DL time synchronization is assumed in the following. Subsequently, sBB [k] is fed into a FFT unit which yields to the frequency domain signal SBB [n] on subcarrier n. Using the notation of (1), SBB [n] can be detailed as follows: SBB [n] = HDL [n]SDL [n] + B[n] + W [n] + aDC δ[n] ,
(3)
where δ[n] denotes the Kronecker delta function. The AWGN 2 W [n] has a variance of σW . The term B[n] arises from size-N FFT processing of the TxL signal: n 2 o (4) B[n] = FFTN c hCSF ∗ hT xL ∗ sU L ) [k] .
Due to the central limit theorem, B[n] can be well approximated as zero-mean complex Gaussian random variable on all carriers n, except for the DC carrier which is not modulated anyway. The TxL signal power is subcarrier specific, i.e. it is not white, and is derived in the following section. Within this paper, data symbols SDL [n] are assumed to have the same power σS2 on all data subcarriers. Let the SNR γ averaged over the channel fading, be defined as 2 σS2 σH σ 2 E{|HDL |2 } = . (5) γ= S 2 2 σW σW ˜ [n] of variance σ 2˜ [n], Finally, we define the ’effective noise’ W W consisting of AWGN and TxL interference: ˜ [n] = B[n] + W [n] . W
(6)
For coherent demodulation, the received signal SBB [n] is ˆ DL [n] as equalized using the estimated channel coefficient H follows: SBB [n] Z[n] = , (7) ˆ DL [n] H where Z[n] denotes the decision variable, which is subsequently fed in the detector/decoder stage. Pilot aided channel estimation is performed, where we assume that one entire OFDM pilot symbol is used to estimate the channel coefficients among all modulated carries using the following least squares method: ˆ DL [n] = SBB,P [n] , H SDL,P [n]
(8)
where SBB,P [n] denotes the received OFDM pilot symbol on subcarrier n which is corrupted by AWGN and TxL as well: ˜ [n]. SBB,P [n] = HDL [n]SDL,P [n] + W
(9)
Here, the pilot power is identical on all subcarriers and equal to the signal power σS2 , i.e. |SDL,P [n]|2 = σS2 for all n. Thus, it is reasonable to model the channel estimate as the true channel coefficient which is corrupted by an additive Gaussian error term: 2 2 ˆ DL [n] = HDL [n] + ν[n] with ν[n] ∈ CN 0, σW H ˜ [n]/σS . (10) 2 Finally, using (10) we define the variance σH [n] to be: ˆ 2 σ 2˜ [n] ˆ 2 2 (11) σH + W2 . HDL [N ] = σH ˆ [n] = E σS
III. P OWER S PECTRAL D ENSITY OF T X L IM2 The TxL IM2 interference after the ADC in the Rx chain can be written as: 2 b[k] = c hCSF ∗ hT xL ∗ sU L [k] ,
where sU L [k] denotes the oversampled UL signal, being an OFDM data symbol sequence. For ease of readability the index U L is skipped in the following. In this analysis we assume the CSF being an ideal low pass filter. Therefore, its band limiting effect on the TxL IM2 can be taken into account by first calculating the discrete power spectral density (PSD) ΦB,ηN [n], −ηN/2 + 1 ≤ n ≤ ηN/2 of the oversampled TxL signal b[k], neglecting the CSF. Afterwards only the subcarriers of ΦB,ηN [n] within the band of interest must be considered, because all other subcarriers are canceled by the CSF. In the following the set of the Nd data and Np pilot subcarriers are denoted by Nd and Np , respectively. The oversampled time index is indicated by k. Additionally, this analysis assumes a frequency flat TxL channel, hT xL [k] = cT xL δ[k − kT xL ] with the attenuation factor cT xL and the group delay kT xL . Since ΦB,ηN [n] is affected by the UL/DL frame asynchronism, it must be calculated depending on ∆k. The group delay kT xL of the TxL channel influences ∆k and thus, is already involved in the analysis. For calculating ΦB,ηN [n], two subsequent UL OFDM data symbols and the cyclic prefix between them must be considered, as sketched in Fig. 1. Implementing the oversampling by zero padding in frequency domain (FD), the ith OFDM symbol in time domain (TD) can be written as: si [k] = √
1 ηN
X
nk
Si [n] e−j2π ηN ,
n∈Nd ∪Np
0 ≤ k < ηN ,
(12) where Si [n] denotes the data and pilot subcarrier values of the ith OFDM symbol, respectively. The data subcarriers are modulated by uniformly distributed, uncorrelated M -QAM symbols and the pilot subcarriers are modulated by uniformly distributed, uncorrelated BPSK symbols. Asynchronism case 1 (0 ≤ ∆k < ηNCP + 1): The block s[k] of the UL signal to be considered, consists of the last N1 = ηN − ∆k samples of the first OFDM symbol s1 [k] and the first ∆k samples of the CP of s2 [k]: ( s1 [k + ηN − N1 ] 0 ≤ k < N1 s[k] = (13) s2 [k + η(N − NCP ) − N1 ] N1 ≤ k < ηN Calculating first the discrete PSD of |s[k]|2 results in: ΦIM 2,ηN [n, ∆k]=
1 ηN
ηN −1 X
k1 −k2 E |s[k1 ]|2 |s[k2 ]|2 e−j2πn ηN
k1 ,k2 =0
(14) P where E[·] is the expectation operator and k1 ,k2 denotes a double sum with equal limits. Using (13) and exploiting the statistical properties mentioned above, (14) results to:
ΦIM 2,ηN [n, ∆k] =
Gv 2 X X
Gv X
C+
v=1 k1 =Fv k2 =Fv
−1 NX 1 −1 ηN X
D (15)
k1 =0 k2 =N1
where PU2L h (2a − 3/2)Nd − Np + (Nd + Np )2 + (16) (ηN )3 (n − n )(k − k ) X 1 2 1 2 + cos 2π ηN n1 ,n2 ∈Nd ∪Np (n −n )(k +k ) i X k1 −k2 1 2 1 2 cos 2πn cos 2π ηN ηN n1 ,n2 ∈Np 2(Nd + Np )2 PU2L k1 − k2 D= , (17) cos 2πn (ηN )3 ηN C=
and the term a denotes the 4th order moment of the real part of the M -QAM data symbols, resulting to: √ P√M −1 √ 4 M m=0 (− M + 1 + 2m) a= (18) P√ 2 √ 4 M −1 2 m=0 (− M + 1 + 2m)
and PU L denotes the signal power of s[k]. The summation limits in (15) are summarized in Tab. I.
Asynchronism case 2 (ηNCP + 1 ≤ ∆k < ηN ): In this case the considered signal block s[k] is composed of: 0 ≤ k < N1 s1 [k + ηN − N1 ] s[k]= s2 [k+η(N −NCP )−N1 ] N1 ≤k
