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 Open Access
Performance of fractional delay estimation in joint estimation algorithm dedicated to digital Tx leakage compensation in FDD transceivers
 Robin Gerzaguet^{1}Email author,
 Laurent Ros^{2},
 Fabrice Belvèze^{3} and
 JeanMarc Brossier^{2}
https://doi.org/10.1186/s1363801709223
© The Author(s) 2017
Received: 27 October 2016
Accepted: 26 July 2017
Published: 25 August 2017
Abstract
This paper deals with the performance of the fractional delay estimator in the joint complex amplitude/delay estimation algorithm dedicated to digital Tx leakage compensation in FDD transceivers. Such transceivers suffer from transmitterreceiver signal leakage. Combined with non linearity of components in the receiver path, the baseband received signal is impaired by a baseband polluting signal. This baseband polluting term depends on the equivalent Tx leakage channel which models leakages of the receiver path. In (Gerzaguet et al., Digit. Signal Proc 51:35–46, 2016), we have proposed a Tx leakage interference cancellation algorithm based on joint estimation of the complex gain and the fractional delay of the equivalent Tx leakage channel and we have derived the asymptotic performance of the complex gain estimator, that showed the necessity of the fractional delay estimation. In this paper, we propose a comprehensive study of the fractional delay estimation algorithm and its analytic performance. The study is based on the analysis of the Scurve and loop noise variance of the timing error detector, from which an approximation of the asymptotic performance of the joint estimation algorithm is derived.
Keywords
 Tx leakage
 FDD transceiver
 Digital compensation
 Leastmeansquare algorithm
 Joint estimation
 Scurve
1 Introduction
Cognitive radios offer the possibility to improve the spectrum use and to adapt the transmission scheme to optimize the sharing of the available bandwidth, which leads to important constraints on the hardware components, located on the physical layer [1]. These constraints thus lead to performance limitations [2], and on this paper we focus on a hardware impairment that may occur in compact wireless transceivers when the radio follows a Frequency Division Duplexing (FDD) scheme. In FDD framework, transmission and reception are done simultaneously, using two different carrier frequencies [3].
However, due to the non linearity and imperfections of components in the analog Rx stage, especially the low noise amplifier (LNA) [6] and the demodulator [7], intermodulation products can lead to a pollution in baseband. This polluting term corresponds to the square component of the Tx leakage signal, shifted to baseband [8]. As a consequence, this polluting signal will impair the received signal and can severely degrade the performance. The pollution is potentially detrimental in the cell edge context (i.e., when the receiver is far from the base station), where the power of the received signal is low, and the power of the transmitted signal is strong [9] leading to a strong polluting signal and thus a low signal to interference ratio.
To avoid this pollution, passive methods based on analog filtering can be implemented. Such mitigation methods consist in adding a band pass filter to attenuate the leaked transmitted signal in the Rx stage before or after the LNA [10]. As radio frequency (RF) transceivers contain more and more digital parts, and as signalprocessing techniques are becoming an area of interest for RF impairments problematic [11], several digital compensation methods have been investigated in the past few years for the Tx leakage (TxL) compensation [5, 9, 12].
In this paper, we complete the performance analysis of our previously proposed joint estimation (JE) algorithm dedicated to Tx leakage compensation [12, 13]. This algorithm estimates the equivalent TxL channel (approximated by a time varying complex gain and a fractional delay (FD)). It works with two referencebased leastmeansquare (LMS) algorithms. As [13] and [12] mainly focus on the performance of the complex gain estimator (with and without a priori knowledge of the FD), we focus in this paper on the FD estimation part. The two main objectives, in addition to propose a comprehensive study, are (i) to prove that the FD estimation algorithm can lock around the desired FD value (ii) and to derive an approximation of the asymptotic performance of the JE algorithm.
The FD estimation algorithm has similitudes with dataaided algorithm that were designed for phase and timing synchronization as well as for automatic gain control or channel complex gain estimation [14]. However in the proposed estimation process, the received signal is considered as noise, and an image of the Tx signal is considered as a pilot sequence (used as reference). In this paper, as we focus on the FD estimator, we assume a constant complex channel gain and we derive analytic formulae of the Scurve of the FD detector [14] (defined as the expectation of the socalled error signal that updates the FD estimation). A linear approximation of the estimation error variance of the FD stage is established as a function of the slope of the Scurve, which represents the gain of the equivalent timingerror detector. This result is finally used to derive an approximation of the asymptotic performance of the JE algorithm, in terms of signal to interference ratio (SIR), using the primary results obtained in [13].
This paper is organized as follows. We give the baseband polluting model in Section 2. We recall the joint estimation of the complex gain and the fractional delay in Section 3. We derive the performance of the fractional delay estimation algorithm in Section 4. Section 5 validates our method and theoretical results through simulations.
2 Baseband model and issues
where n is the time index, x(n) is the databearing signal, which is assumed to be white and zeromean with variance \(\sigma _{x}^{2}\), b(n) is the white additive Gaussian noise, of variance \(\sigma _{b}^{2}\), s _{ TxL }(n) is the TxL polluting signal and x _{ b }(n)=x(n)+b(n) is then the desired (or expected) noisy received signal without interference, with variance \(\sigma ^{2}_{xb} = \sigma ^{2}_{x} + \sigma ^{2}_{b}\).
One can see that the reference signal u(n) corresponds to the polluting signal with δ=0 and β _{TxL}=1. It should also be noted that, in practice, additional processing such as an upsampler, a LPF, and a resampler may be required for the reference generation.
3 TxL compensation based on joint estimation algorithm
where e(n) is the compensated output, \(\hat {\beta }_{\text {TxL}}(n)\) the complex gain estimation, μ the stepsize of the complex gain estimator, \(L(n) = \partial \hat {\beta }_{\text {TxL}}(n) / \partial \delta \) and ν, the constant step size of the fractional delay estimator. It can be seen that this algorithm is recursive, online (as it provides a compensated output e(n) at each iteration), and with low complexity.
4 Performance of the FD estimation

Since the fractional delay is constant, the variance of the fractional delay estimation converges to zero when the stepsize ν tends toward to zero. In practice, decreasing the stepsize would decrease the estimation variance but would also lead to a longer transient state before convergence.

The variance is proportional to the power of the noisy received signal \(\sigma _{xb}^{2}=\sigma _{x}^{2} + \sigma _{b}^{2}\). This has to be taken into account when using the algorithm in practical case. In the simulation part (see Section 5), the stepsize ν is thus normalized by the power of the noisy signal. The presence of the noisy signal acts as a perturbation for the fractional delay estimation. If the entry signal would only the Tx polluting term, the fractional delay estimation would be straightforward.

A stability condition can be deduced from (12) \( \nu < 1/ \left (\sigma ^{4}_{\text {Tx}} \vert \beta _{\text {TxL}} \vert ^{2} \right) \).

The first one is related to the complex gain estimation stage. If the stepsize μ of the complex gain algorithm is small, it can be approximated as \(\mu \sigma _{u}^{2} /2\).

The second and third term are due to the fractional delay estimation stage. The second term is due to the residual fractional delay due to the fractional delay estimation error (see the estimation error expression in (54) [12]).
The third term models the impact of the fractional delay when fed back in the complex gain estimator (see the misalignment expression in (56) [12]). Both terms show the impact of the residual fractional delay on the performance and the overall performance depends on the algorithm stepsizes μ, ν and on the pollution statistics \(\left (\!{\vphantom {\sigma _{x_{b}}^{2}}}\text {channel amplitude} \vert \beta _{\text {TxL}} \vert ^{2}\right.\left. \text {and noisy signal variance} \sigma _{x_{b}}^{2}\right)\).
5 Simulations

On the Fig. 4, we first plot the theoretical, linearized theoretical, and simulated Scurve versus the estimation error of the FD estimator in open loop. It shows that the theory is corroborated and that the Scurve can be linearized when the estimation error is low (between −10 and 10 percent of the sampling time). It also demonstrates that the FD estimation loop will lock and is unbiased as S _{ c }(0)=0.

On Fig. 5, we represent the theoretical and simulated loop noise zerodelay autocorrelation \(\Gamma _{N_{T}}[0]\) versus the FD estimation error. When the FD estimation error is low, it can be seen that the assumption of a white loop noise with variance expressed in (11) leads to accurate results.

On Fig. 6, the simulated delay error variance \(\sigma ^{2}_{\epsilon _{\delta }}\) is compared to the theory derived in (12). The curves are obtained for different fractional delay values and different normalized fractional delay estimator stepsize values. One can see that the delay error variance is independent from the initial FD value and linear with respect to the FD stepsize (for small stepsize ν); which is aligned with the theoretical equation derived in (12).

On Fig. 7, we finally consider the performance of the JE algorithm considering the impact of the complex gain stepsize. We consider that the data bearing signal is a white Gaussian signal polluted by a TxL signal that follows (4). The polluting SIR (i.e., the ratio between the power of the TxL polluting signal and \(\sigma _{x_{b}}^{2}\)) is set to 0 dB and the power of the desired noisy signal is \(\sigma _{x}^{2}+\sigma _{b}^{2}=10^{8}\). The noisy data signal has thus a low power whereas the transmitted signal (of unitary variance, \(\sigma ^{2}_{\text {Tx}}=1\)) is strong (cell edge context). At the output of the demodulator, the polluting signal has the same power as the desired noisy signal (initial SIR of 0dB) due to the attenuation induced by the pollution. The complex gain algorithm is driven by a non interpolated reference u(n) defined with (5) and the JE algorithm uses a Farrow structure to apply the estimated FD to u(n). We also assume a constant TxL channel, and we represent the performance of the complex gain estimator and the JE algorithm for several values of FD. The FD estimation stepsize is set to \(\nu =10^{6} / \sigma _{xb}^{2}\).
The main objectives of this Figure are (i) show the impact of a non compensated fractional delay (ii) show the potential benefits of the FD estimation (iii) demonstrates that the theoretical performance (13) corroborates with the simulated results. We indeed show that the performance can be dramatically reduced if the FD estimation algorithm is not enabled. Besides, the JE algorithm greatly improves the asymptotic performance. If there is no FD and the FD estimation stage is disabled, the performance becomes linear w.r.t the complex gain stepsize μ (for small μ) as, in this case, (13) becomes \( 10 \log \left (\mu \sigma _{u}^{2} / (2\mu \sigma _{u}^{2}) \right)\).
If there is no FD but the JE scheme is enabled, one can see a performance floor at around 60 dB when the complex gain estimation stepsize tends toward to zero. This is due to the residual error introduced with the FD estimation stage. From (12) and (13), with μ=0, the asymptotic SIR for the JE becomes
We can finally conclude that the asymptotic performance of the structure can be well approximated and interpreted with (13); leading to a direct link between a desired asymptotic level and a stepsize value for the FD estimator ν.$$\mathrm{SIR_{comp}} =  10 \log \left(\frac{\nu \vert \beta_{\text{TxL}} \vert^{2}} {4 \left(1  \nu \sigma^{4}_{\text{Tx}} \vert \beta_{\text{TxL}} \vert^{2}\right)}\right) ~. $$ 
To better stress the impact of the FD estimation stage on the complex gain stage, we finally represent the performance of the complex gain algorithm and the joint estimation algorithm with a constant gain stepsize and a varying FD stepsize on Fig. 8. The same polluting model as in Fig. 7 have been used. The stepsize of the complex gain algorithm is set to μ=10^{−5} have been used. The red curves (which represents the gain estimator algorithm without taking into account the presence of a fractional delay) are constant (as the FD stage is deactivated, and as the TxL channel is constant). The higher the FD, the poorer the performance as a non compensated FD severally degrades the performance (see (17) of [13] for the theoretical expression of the performance loss). It is thus shown that the performance can be dramatically reduced if the FD estimation algorithm is not enabled. The blue curves represent the performance when the FD stage is activated, and the corresponding theoretical expression derived in (13) is also represented. Is is shown that the asymptotic performance of the structure can be approximated with good accuracy with (13) leading to a direct link between a desired asymptotic performance level and stepsize value for the FD estimator ν. If the stepsize of the FD stage is high, the performance can be low as the variance of the stage (12) is proportional to \(\nu \sigma _{xb}^{2}\). In this case, the FD stage can be seen as a disruptive element for the complex gain stage. With a low stepsize ν, the performance is greatly enhanced (at the price of a longer transient state) and the performance converges to the performance of the complex gain without FD (first term of the theoretical expression derived in (13) as \(\mathrm {SIR_{comp}} \approx 10 \text {log}_{10}\left [{\mu \sigma _{u}^{2}}/{\left (2  \mu \sigma _{u}^{2}\right)}\right ]\)).
6 Conclusions
This paper deals with the performance of a joint estimation algorithm dedicated to the compensation of the digital Tx leakage in RF transceivers. The strong constraints that apply to cognitive radios lead to a potential detrimental loss of performance due to hardware impairments and we focus here on a pollution that occurs in FDD transceivers, related to the pollution of an image of the transmitter stage on the receiver stage. Based on the initial algorithm proposed in [13], in this paper we focus on the fractional delay estimation algorithm that can be considered as a firstorder tracking loop synchronisation algorithm piloted by an interpolated reference signal. We have firstly derived the analytical Scurve, defined as the conditional expectation of the error signal that controls the loop, and we have deduced an approximation of the asymptotic estimation error variance of the FD estimation algorithm. As the proposed approximated theoretical performance formula of the whole JE algorithm shows good accordance with simulations, this result can be used to properly tune the delay estimation algorithm stepsize.
Notes
Declarations
Funding
The authors declare that they have no funding sources.
Contributions
The two main objectives of the proposed paper, in addition to propose a comprehensive study on the Tx Leakage digital compensation, are (i) to prove that the initially proposed FD estimation algorithm can lock around the desired FD value (ii) and to derive an approximation of the asymptotic performance of the JE algorithm. This leads to a direct link between a desired asymptotic performance level for the compensation structure and a stepsize value for the FD estimator
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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