PM Nonlinearities in OFDM Systems
Blind Estimation of Memoryless AM/PM Nonlinearities in OFDM Systems Jan Dohl and Gerhard Fettweis Vodafone Chair Mobile Communication Systems Technische Universit¨at Dresden, D-01062 Dresden, Germany {jan.dohl,fettweis}@ifn.et.tu-dresden.de http://www.ifn.et.tu-dresden.de/MNS
Abstract—Nonlinear distortions by analog frontend components are becoming a growing problem. Not only do stringent linearity requirements that are crucial for modern modulation schemes like OFDM boost the costs in development and fabrication, but highly linear amplifiers are usually very energy inefficient. After looking at the theoretical and practical aspects of estimation of memoryless AM/AM nonlinearities [1] [2], the focus of this paper is on AM/PM nonlinearities. First, we will derive the maximum likelihood estimator for a parameterized AM/PM nonlinearity. Then, we point out the complexity issues in practical implementations, propose suboptimal methods and compare them in terms of complexity and estimation performance.
I. I NTRODUCTION When amplifiers are driven close to saturation, they introduce nonlinear distortions. OFDM signals are particularly affected because their time domain signals exhibit a normal distribution yielding a very high peak to average power ratio (PAPR). If power peaks in the signal are subject to clipping, severe inter carrier interference is induced [3]. Since this affects the system performance in a very negative way, amplifiers in OFDM systems are usually operated with a large back-off. In turn, this requires amplifiers with a linear characteristic over a large input range which is expensive to develop and produce. Furthermore, amplifiers operated with back-off are usually not energy efficient [4] which can significantly impair the running time of battery powered devices. Amplifier predistortion is an approach that has drawn a lot of research attention [5]. Using knowledge of the amplifier characteristic, the transmitter predistorts the signal so that after the nonlinear amplifier, the signal is distortion free. To capture the effects of temperature, aging and production tolerances, this architecture needs additional calibration circuitry or specially designed pilots. If only a static model of the amplifier is available, residual nonlinearity effects will remain after predistortion. Similarly, many models and algorithms to equalize nonlinearities in the receiver have been developed [6]–[8]. However, most of them require knowledge about the nonlinearity characteristic and often perfect knowledge is assumed. In order to obtain knowledge about the nonlinearity characteristics without additional pilots or calibration circuitry, in [1] we presented a low-complexity maximum likelihood based method for blind estimation of the parameters of a memoryless AM/AM nonlinearity and showed that there is no
visible performance difference between perfect and estimated knowledge. In [2] we presented an implementation of the method on a low-cost Software Defined Radio platform. By driving the amplifiers of the system in saturation, we showed that the method provides significant performance gains even if used with real, not necessarily memoryless power amplifiers. In this paper, we expand the scope to memoryless AM/PM nonlinearities. The rest of the paper is structured as follows. In Section II we present the system model and in Section III we derive the maximum likelihood estimator. In section IV we propose several suboptimal solutions with reasonable complexity. The results are presented in Section V and the paper is concluded in Section VI. II. S YSTEM M ODEL The system model is shown in figure 1. The source vector
s0
H
s
g(sk , θ)
x
y n
Fig. 1. System Model
s0 consists of N elements s0k , which are mutually independent and follow a complex normal distribution with normalized power per subcarrier, i.e. s0k ∼ CN (0, 1). H is a N × N matrix that describes the effects of oversampling or frequency selective channels on the system and generally introduces correlation between the random samples. To that end, s exhibits a zero-mean multivariate complex normal distribution with the N × N covariance matrix Σ = HH H , i.e. s ∼ CN (0, Σ). Since the time domain representation of OFDM signals with sufficiently many subcarriers exhibit a normal distribution [9], s0 describes the time-domain signal of a N -subcarrier OFDM system without guardband. Depending on H, s describes the time-domain signal of a N -subcarrier OFDM system with guardband or other frequency selective effects. In order for AM/PM-estimation to work, the elements sk must not be independent of one another (i.e. Σ must have some non-zero off-diagonal elements). The proof follows intuitively from the PDF and will be shown later. The correlation introduced by common OFDM guardbands is usually sufficient. g(sk , θ) is
a memoryless nonlinearity which can be decomposed into an AM/AM and AM/PM part as follows: g(sk , θ) = sk · gam (|sk |, θ) exp [j · gpm (|sk |, θ)]
(1)
Since this publication covers the AM/PM estimation, we assume gam (|sk |, θ) = 1. Furthermore, we assume that the characteristic of the nonlinearity is known despite a parameter vector θ which is to be estimated blindly. Finally, n is an AWGN vector so the elements nk are i.i.d. zero-mean complex normal distributed with variance σn2 . The vector y is the observation vector that is available to the estimator. While we keep the derivations generic and independent of a concrete nonlinearity, in the simulations, we use the wellknown Travelling Wave Tube Amplifier (TWTA) model [10] which is defined as follows: αA (2) gam (sk , (αA , βA )) = 1 + βA |sk |2 αφ |sk |2 . (3) gpm (sk , (αφ , βφ )) = 1 + βφ |sk |2 With αA = 1 and βA = 0, gam (sk , (1, 0)) = 1 so that the earlier assumption of only AM/PM distortion is fulfilled. Joint estimation of AM/AM and AM/PM distortions will be subject to a later publication.
First, we derive p(x|θ), the PDF of the signal after the nonlinearity. It is well-known that the PDF of the zero-mean multivariate complex normal distributed random vector s is given as: H −1 1 s Σ s ps (s) = exp − . (6) N 1 2 (2π) 2 |Σ| 2 Since the nonlinearities are usually defined in terms of amplitudes and phases, the distribution (6) is first transformed to polar coordinates. With the amplitude vector sA = |s| and phase vector sφ = ]s, the resulting PDF in terms of ps (s) is given as: psAφ (sA , sφ ) = ps (sA · cos(sφ ) + jsA · sin(sφ )) ·
N −1 Y
An important question is how to choose H so that the resulting random vector s matches an actual OFDM system. One way that covers frequency selective channels and systems with guardband is to design H to be a circular convolution matrix. Take a column vector h = (h0 , ..., hN −1 ) of length N and set the elements hk = 0 if the k-th subcarrier belongs to the guardband and hk = 1 if it is a data carrier. Then, with the N × N DFT matrix W : 1 W H diag(h)W . N
(4)
Choosing h differently allows to model the effects of persubcarrier power allocation or other frequency selective influences. After defining the system, in the next section we will derive the ML-estimator for θ. III. M AXIMUM L IKELIHOOD E STIMATOR With M independently observed OFDM symbols, the MLestimator is defined in a straight forward way as ˆ = arg max θ θ
M Y
p (y m |θ)
(5)
m=0
ˆ can be interpreted as the parameters and the ML-estimate θ under which the observations become most plausible [11]. The main issue in implementing this method is that p (y|θ) needs to be known. Deriving this PDF is the goal of this section.
sA;k .
k=0
(7) It can be easily derived from (1) that the relationship between s and x in polar coordinates is given as xA = sA xφ = sφ + gpm (sA , θ) .
(8) (9)
Solving (9) for sA and sφ gives sA = xA sφ = xφ − gpm (xA , θ) .
A. Covariance Matrix
H=
A. PDF after the nonlinearity
(10) (11)
The Jacobian determinant of this transformation is J = 1 and using the rules for transformation of densities [12], the PDF of x in polar coordinates is given as: pxAφ (xA , xφ |θ) = psAφ (xA , xφ − gpm (xA , θ)) .
(12)
It can be seen that due to the special structure of the problem, the requirements on gpm are very low. Neither does it need to be invertible nor differentiable as otherwise often required in random variable transformations. Transforming (12) back to Cartesian coordinates then leads to the final PDF of x as follows: 1 px (x|θ) = N/2 (2π) |Σ|1/2 x exp [−j · gpm (|x|, θ)] Σ−1 xH exp [j · gpm (|x|, θ)] exp − . 2 (13) (13) is equal to (6) with the exception of the two exponential terms that are used to reverse the phase rotations induced by the AM/PM nonlinearity. Furthermore, from (13) it becomes clear that AM/PM estimation with this method does not work if the samples are independent, i.e. Σ is only nonzero on the main diagonal. In case of independent samples, the two exponential terms cancel out completely making (13) indistinguishable from (6) and independent of θ so that MLestimation does not work anymore. A shorter notation for (13) is px (x|θ) = ps (x exp (−jgpm (|x|, θ))) . (14)
B. PDF after Additive White Gaussian Noise Since y = x + n and x and n are mutually independent, it is well-known that the PDF py (y|θ) of the sum is given by the convolution of the PDFs of both random vectors [12]. This results in py (y|θ) = px (x|θ) ∗ pn (n) Z = px (y − n|θ) · pn (n)dn
(15)
CN
which is a 2N -dimensional convolution. A general solution to (15) is not possible since it depends on gpm . Furthermore, the authors have not been able to find an analytical expression under the assumption that gpm is a TWTA nonlinearity. Solving this convolution numerically is a difficult problem in itself and is analyzed in the next section. IV. S UBOPTIMAL A LGORITHMS When implementing the ML-estimator presented in the last section, the computational efforts required for the convolution of px (x|θ) and pn (n) proved to be prohibitively high even if the number of subcarriers was very small. Therefore, three suboptimal algorithms are proposed in this section. A. Ignoring the noise In the high SNR region, where the influence of the nonlinearity dominates, the distortion caused by AWGN is insignificant so that y = x + n ≈ x. (16) Obviously, in the lower to medium SNR region (16) does not hold and estimation errors are to be expected. Furthermore, when evaluating the full PDF of x, special care must be taken to choose a nonsingular H. Otherwise the covariance matrix will be singular as well and the PDF cannot be evaluated since it requires Σ−1 . When designing H from a frequency response h as described in section II-A, the vector h contains the Eigenvalues of H. If any hk = 0, this will lead to singular H and therefore a singular Σ. The authors solved this problem by setting hk = 0.001 for guardband carriers. That way, the power of the guardband carriers will be 60dB lower than the data carriers and the problem of a singular covariance matrix is avoided. Evaluating px (x|θ) using this method has a computational complexity of O(N 2 ) which might still be too complex for some implementations. An even less complex algorithm is presented in the next section. B. Two-carrier decomposition From the derivations in section III it can be seen that the smallest system that allows ML-estimation of an AM/PM nonlinearity consists of two subcarriers. The idea of this algorithm is to decompose the large random vector s into L random vectors of size N = 2 which are assumed independent. This transforms each observation of a N -subcarrier OFDM symbol into L independent observations of two-subcarrier OFDM symbols.
Building the two-subcarrier random vectors works as follows. Firstly, two distinct elements si , sj of s with highest correlation are selected and form the random vector sij = (si , sj ). The resulting PDF p(sij ) is calculated by marginalization of all other sk . Since s exhibits a zeromean complex multivariate normal distributon, the resulting distribution is also zero-mean complex bivariate normal, i.e. sij ∼ CN (0, Σij ) with σ(i,i) σ(i,j) . (17) Σij = σ(j,i) σ(j,j) The process is repeated until the L two-carrier random vectors with the highest correlations are extracted. Similar to (14), the PDF of xij is given as: pxij (xij |θ) = psij (xij exp (−jgpm (|xij |, θ))
(18)
Using this method, calculating the PDF of one N -carrier OFDM symbol is split up into the calculation of L two-carrier OFDM symbols. While the former has complexity O(N 2 ), the latter has complexity O(L), yielding a significant performance boost. If H is a circular convolution matrix, Σ exhibits a Hermetian Toeplitz structure so that there always are N pairs of samples with the same (highest) correlation, so that at least N two-carrier random vectors can be extracted from a N subcarrier OFDM symbol. This is the case for frequency selective channels (assuming appropriate cyclic prefix is employed) and OFDM systems with guardband. Furthermore, since the covariance matrix Σij for the two-carrier random vectors is usually invertible (except for academic cases) even if Σ itself is not, no singularity problems arise in systems with guardband carriers. C. Two-carrier decomposition & convolution approximation Decomposing the problem into multiple independent problems with two subcarriers each reduces the amount of dimensions of the convolution in (15) to four. While still a significant computational effort, approximating the convolution in four dimensions is actually an option. Since the noise is white, the noise samples ni and nj are mutually independent and each exhibits circular symmetry in the PDF. For that reason it is viable to sample each dimension equally. Assuming Q samples per dimension with equal distance d between the samples, let q be the vector whose elements qk hold the positions of the samples. Then, the convolution integral can be approximated as follows: Z pyij (y ij |θ) = pxij (y ij − nij |θ) · pnij (nij )dnij C2 2
≈d
X
···
X
pxij (y ij − nij |θ) · pnij (nij )
Abstract—Nonlinear distortions by analog frontend components are becoming a growing problem. Not only do stringent linearity requirements that are crucial for modern modulation schemes like OFDM boost the costs in development and fabrication, but highly linear amplifiers are usually very energy inefficient. After looking at the theoretical and practical aspects of estimation of memoryless AM/AM nonlinearities [1] [2], the focus of this paper is on AM/PM nonlinearities. First, we will derive the maximum likelihood estimator for a parameterized AM/PM nonlinearity. Then, we point out the complexity issues in practical implementations, propose suboptimal methods and compare them in terms of complexity and estimation performance.
I. I NTRODUCTION When amplifiers are driven close to saturation, they introduce nonlinear distortions. OFDM signals are particularly affected because their time domain signals exhibit a normal distribution yielding a very high peak to average power ratio (PAPR). If power peaks in the signal are subject to clipping, severe inter carrier interference is induced [3]. Since this affects the system performance in a very negative way, amplifiers in OFDM systems are usually operated with a large back-off. In turn, this requires amplifiers with a linear characteristic over a large input range which is expensive to develop and produce. Furthermore, amplifiers operated with back-off are usually not energy efficient [4] which can significantly impair the running time of battery powered devices. Amplifier predistortion is an approach that has drawn a lot of research attention [5]. Using knowledge of the amplifier characteristic, the transmitter predistorts the signal so that after the nonlinear amplifier, the signal is distortion free. To capture the effects of temperature, aging and production tolerances, this architecture needs additional calibration circuitry or specially designed pilots. If only a static model of the amplifier is available, residual nonlinearity effects will remain after predistortion. Similarly, many models and algorithms to equalize nonlinearities in the receiver have been developed [6]–[8]. However, most of them require knowledge about the nonlinearity characteristic and often perfect knowledge is assumed. In order to obtain knowledge about the nonlinearity characteristics without additional pilots or calibration circuitry, in [1] we presented a low-complexity maximum likelihood based method for blind estimation of the parameters of a memoryless AM/AM nonlinearity and showed that there is no
visible performance difference between perfect and estimated knowledge. In [2] we presented an implementation of the method on a low-cost Software Defined Radio platform. By driving the amplifiers of the system in saturation, we showed that the method provides significant performance gains even if used with real, not necessarily memoryless power amplifiers. In this paper, we expand the scope to memoryless AM/PM nonlinearities. The rest of the paper is structured as follows. In Section II we present the system model and in Section III we derive the maximum likelihood estimator. In section IV we propose several suboptimal solutions with reasonable complexity. The results are presented in Section V and the paper is concluded in Section VI. II. S YSTEM M ODEL The system model is shown in figure 1. The source vector
s0
H
s
g(sk , θ)
x
y n
Fig. 1. System Model
s0 consists of N elements s0k , which are mutually independent and follow a complex normal distribution with normalized power per subcarrier, i.e. s0k ∼ CN (0, 1). H is a N × N matrix that describes the effects of oversampling or frequency selective channels on the system and generally introduces correlation between the random samples. To that end, s exhibits a zero-mean multivariate complex normal distribution with the N × N covariance matrix Σ = HH H , i.e. s ∼ CN (0, Σ). Since the time domain representation of OFDM signals with sufficiently many subcarriers exhibit a normal distribution [9], s0 describes the time-domain signal of a N -subcarrier OFDM system without guardband. Depending on H, s describes the time-domain signal of a N -subcarrier OFDM system with guardband or other frequency selective effects. In order for AM/PM-estimation to work, the elements sk must not be independent of one another (i.e. Σ must have some non-zero off-diagonal elements). The proof follows intuitively from the PDF and will be shown later. The correlation introduced by common OFDM guardbands is usually sufficient. g(sk , θ) is
a memoryless nonlinearity which can be decomposed into an AM/AM and AM/PM part as follows: g(sk , θ) = sk · gam (|sk |, θ) exp [j · gpm (|sk |, θ)]
(1)
Since this publication covers the AM/PM estimation, we assume gam (|sk |, θ) = 1. Furthermore, we assume that the characteristic of the nonlinearity is known despite a parameter vector θ which is to be estimated blindly. Finally, n is an AWGN vector so the elements nk are i.i.d. zero-mean complex normal distributed with variance σn2 . The vector y is the observation vector that is available to the estimator. While we keep the derivations generic and independent of a concrete nonlinearity, in the simulations, we use the wellknown Travelling Wave Tube Amplifier (TWTA) model [10] which is defined as follows: αA (2) gam (sk , (αA , βA )) = 1 + βA |sk |2 αφ |sk |2 . (3) gpm (sk , (αφ , βφ )) = 1 + βφ |sk |2 With αA = 1 and βA = 0, gam (sk , (1, 0)) = 1 so that the earlier assumption of only AM/PM distortion is fulfilled. Joint estimation of AM/AM and AM/PM distortions will be subject to a later publication.
First, we derive p(x|θ), the PDF of the signal after the nonlinearity. It is well-known that the PDF of the zero-mean multivariate complex normal distributed random vector s is given as: H −1 1 s Σ s ps (s) = exp − . (6) N 1 2 (2π) 2 |Σ| 2 Since the nonlinearities are usually defined in terms of amplitudes and phases, the distribution (6) is first transformed to polar coordinates. With the amplitude vector sA = |s| and phase vector sφ = ]s, the resulting PDF in terms of ps (s) is given as: psAφ (sA , sφ ) = ps (sA · cos(sφ ) + jsA · sin(sφ )) ·
N −1 Y
An important question is how to choose H so that the resulting random vector s matches an actual OFDM system. One way that covers frequency selective channels and systems with guardband is to design H to be a circular convolution matrix. Take a column vector h = (h0 , ..., hN −1 ) of length N and set the elements hk = 0 if the k-th subcarrier belongs to the guardband and hk = 1 if it is a data carrier. Then, with the N × N DFT matrix W : 1 W H diag(h)W . N
(4)
Choosing h differently allows to model the effects of persubcarrier power allocation or other frequency selective influences. After defining the system, in the next section we will derive the ML-estimator for θ. III. M AXIMUM L IKELIHOOD E STIMATOR With M independently observed OFDM symbols, the MLestimator is defined in a straight forward way as ˆ = arg max θ θ
M Y
p (y m |θ)
(5)
m=0
ˆ can be interpreted as the parameters and the ML-estimate θ under which the observations become most plausible [11]. The main issue in implementing this method is that p (y|θ) needs to be known. Deriving this PDF is the goal of this section.
sA;k .
k=0
(7) It can be easily derived from (1) that the relationship between s and x in polar coordinates is given as xA = sA xφ = sφ + gpm (sA , θ) .
(8) (9)
Solving (9) for sA and sφ gives sA = xA sφ = xφ − gpm (xA , θ) .
A. Covariance Matrix
H=
A. PDF after the nonlinearity
(10) (11)
The Jacobian determinant of this transformation is J = 1 and using the rules for transformation of densities [12], the PDF of x in polar coordinates is given as: pxAφ (xA , xφ |θ) = psAφ (xA , xφ − gpm (xA , θ)) .
(12)
It can be seen that due to the special structure of the problem, the requirements on gpm are very low. Neither does it need to be invertible nor differentiable as otherwise often required in random variable transformations. Transforming (12) back to Cartesian coordinates then leads to the final PDF of x as follows: 1 px (x|θ) = N/2 (2π) |Σ|1/2 x exp [−j · gpm (|x|, θ)] Σ−1 xH exp [j · gpm (|x|, θ)] exp − . 2 (13) (13) is equal to (6) with the exception of the two exponential terms that are used to reverse the phase rotations induced by the AM/PM nonlinearity. Furthermore, from (13) it becomes clear that AM/PM estimation with this method does not work if the samples are independent, i.e. Σ is only nonzero on the main diagonal. In case of independent samples, the two exponential terms cancel out completely making (13) indistinguishable from (6) and independent of θ so that MLestimation does not work anymore. A shorter notation for (13) is px (x|θ) = ps (x exp (−jgpm (|x|, θ))) . (14)
B. PDF after Additive White Gaussian Noise Since y = x + n and x and n are mutually independent, it is well-known that the PDF py (y|θ) of the sum is given by the convolution of the PDFs of both random vectors [12]. This results in py (y|θ) = px (x|θ) ∗ pn (n) Z = px (y − n|θ) · pn (n)dn
(15)
CN
which is a 2N -dimensional convolution. A general solution to (15) is not possible since it depends on gpm . Furthermore, the authors have not been able to find an analytical expression under the assumption that gpm is a TWTA nonlinearity. Solving this convolution numerically is a difficult problem in itself and is analyzed in the next section. IV. S UBOPTIMAL A LGORITHMS When implementing the ML-estimator presented in the last section, the computational efforts required for the convolution of px (x|θ) and pn (n) proved to be prohibitively high even if the number of subcarriers was very small. Therefore, three suboptimal algorithms are proposed in this section. A. Ignoring the noise In the high SNR region, where the influence of the nonlinearity dominates, the distortion caused by AWGN is insignificant so that y = x + n ≈ x. (16) Obviously, in the lower to medium SNR region (16) does not hold and estimation errors are to be expected. Furthermore, when evaluating the full PDF of x, special care must be taken to choose a nonsingular H. Otherwise the covariance matrix will be singular as well and the PDF cannot be evaluated since it requires Σ−1 . When designing H from a frequency response h as described in section II-A, the vector h contains the Eigenvalues of H. If any hk = 0, this will lead to singular H and therefore a singular Σ. The authors solved this problem by setting hk = 0.001 for guardband carriers. That way, the power of the guardband carriers will be 60dB lower than the data carriers and the problem of a singular covariance matrix is avoided. Evaluating px (x|θ) using this method has a computational complexity of O(N 2 ) which might still be too complex for some implementations. An even less complex algorithm is presented in the next section. B. Two-carrier decomposition From the derivations in section III it can be seen that the smallest system that allows ML-estimation of an AM/PM nonlinearity consists of two subcarriers. The idea of this algorithm is to decompose the large random vector s into L random vectors of size N = 2 which are assumed independent. This transforms each observation of a N -subcarrier OFDM symbol into L independent observations of two-subcarrier OFDM symbols.
Building the two-subcarrier random vectors works as follows. Firstly, two distinct elements si , sj of s with highest correlation are selected and form the random vector sij = (si , sj ). The resulting PDF p(sij ) is calculated by marginalization of all other sk . Since s exhibits a zeromean complex multivariate normal distributon, the resulting distribution is also zero-mean complex bivariate normal, i.e. sij ∼ CN (0, Σij ) with σ(i,i) σ(i,j) . (17) Σij = σ(j,i) σ(j,j) The process is repeated until the L two-carrier random vectors with the highest correlations are extracted. Similar to (14), the PDF of xij is given as: pxij (xij |θ) = psij (xij exp (−jgpm (|xij |, θ))
(18)
Using this method, calculating the PDF of one N -carrier OFDM symbol is split up into the calculation of L two-carrier OFDM symbols. While the former has complexity O(N 2 ), the latter has complexity O(L), yielding a significant performance boost. If H is a circular convolution matrix, Σ exhibits a Hermetian Toeplitz structure so that there always are N pairs of samples with the same (highest) correlation, so that at least N two-carrier random vectors can be extracted from a N subcarrier OFDM symbol. This is the case for frequency selective channels (assuming appropriate cyclic prefix is employed) and OFDM systems with guardband. Furthermore, since the covariance matrix Σij for the two-carrier random vectors is usually invertible (except for academic cases) even if Σ itself is not, no singularity problems arise in systems with guardband carriers. C. Two-carrier decomposition & convolution approximation Decomposing the problem into multiple independent problems with two subcarriers each reduces the amount of dimensions of the convolution in (15) to four. While still a significant computational effort, approximating the convolution in four dimensions is actually an option. Since the noise is white, the noise samples ni and nj are mutually independent and each exhibits circular symmetry in the PDF. For that reason it is viable to sample each dimension equally. Assuming Q samples per dimension with equal distance d between the samples, let q be the vector whose elements qk hold the positions of the samples. Then, the convolution integral can be approximated as follows: Z pyij (y ij |θ) = pxij (y ij − nij |θ) · pnij (nij )dnij C2 2
≈d
X
···
X
pxij (y ij − nij |θ) · pnij (nij )
