New Expression for the Functional Transformation of the Vector Cramér-Rao Lower Bound

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1 New Expression for the Functional Transformation of the Vector Cramér-Rao Lower Bound Ali A. Nasir, Hani Mehrpouan, and Rodne A. Kenned, Research School of Information Sciences and Engineering, the Australian National Universit, Australia. Department of Electrical and Computer Engineering and Computer Science, California State Universit, Baersfield, USA. and Abstract Assume that it is desired to estimate α = fθ), where f ) is an r-dimensional function. This paper derives the general expression for the functional transformation of the vector Cramér-Rao lower bound CRLB). The derived bound is a tight lower bound on the estimation of uncoupled parameters, i.e., parameters that can be estimated separatel. Unlie previous results in the literature, this new expression is not dependent on the inverse of the Fisher s information matrix FIM) of the untransformed parameters, θ. Thus, it can be applied to scenarios where the FIM for θ is ill-conditioned or singular. Finall, as an application, the derived transformation is applied to determine the exact CRLB for estimation of channel parameters in amplif-and-forward relaing networs. Index Terms Cramér-Rao lower bound CRLB), Fisher information matrix FIM), channel estimation. I. INTRODUCTION The Cramer-Rao lower bound CRLB) provides a lower bound on the variance of unbiased estimators, and is widel used in the fields of communication and signal processing [1]. It frequentl occurs in practice that the parameters that need to be estimated are a function of some more fundamental sstem parameters, θ. We refer to those required estimation parameters as transformed sstem parameters, α = fθ). The final Cramér-Rao lower bound CRLB) expression for the transformed sstem parameters has been well studied for real [1] and complex parameters []. Subsequentl, the CRLB results in [] were simplified in [3], and extended for the estimation of constrained parameters in [4] and for a combination of constrained and unconstrained parameters in [5]. Recentl, the asmptotic behavior high signal-to-noise ratio SNR) and/or large number of samples) of the CRLB was investigated in [6]. Note that the final expression of the CRLB for transformed sstem parameters given in [1] [6] depends on the inverse of the Fisher information matrix FIM) of the fundamental sstem parameters, θ. However, it ma happen that the FIM of these parameters is ran deficient or ill-conditioned impling that the CRLB expression is infinit or numericall unstable [7], even when an unbiased estimator and CRLB both exist. In such situations, it has been proposed to tae the reciprocal of the corresponding This research was supported under Australian Research Council s Discover Projects funding scheme project number DP ). diagonal elements of FIM [8]. However, the resulting bound is either loose for estimating untransformed sstem parameters [8] or incorrect for the estimation of transformed sstem parameters, as shown in this paper. Later, the use of pseudo inverse of the FIM has been proposed for such problems [7]. However, the calculation of pseudo inverse ma be numericall unstable and computationall prohibitive as the dimension of an estimation problem increases. In order to address this deficienc in the existing CRLB expressions, this letter derives a new CRLB expression for the estimation of transformed sstem parameters, which does not depend on an tpe of matrix inverse and is valid even when the FIM of the fundamental sstem parameters is ran deficient. The application and usefulness of the proposed result is demonstrated through its use in channel estimation for amplif-and-forward AF) cooperative relaing networs. More specificall our problem concerns the estimation of an overall channel gain vector, α from source-to-relasto-destination transformed sstem parameters), which is a function of individual channels from source-to-relas, h, and from relas-to-destination, g. These component channels are the fundamental sstem parameters or vector, i.e., θ [h, g] T. In context of this channel estimation problem, the existing CRLB results have ambiguit about formulating the vector of parameters of interest, i.e., whether the derivative of g with respect to α is zero or not as can be observed in [9] or [10], respectivel). Thus, the evaluation of direct CRLB is not obvious. Moreover, in the considered estimation problem, the FIM of fundamental sstem parameters, θ, is ran deficient. Hence in this situation no conclusions can be drawn regarding whether there exists an unbiased estimator and CRLB for estimation of the overall channel gains α = fθ). In contrast, as corroborated using a minimum variance unbiased MVU) estimator, the CRLB expression in this letter, is numericall stable since it does not depend on an tpe of matrix inverse. Notation: Superscripts ), ) T,, and ) denote the conjugate, the transpose, and the conjugate transpose, and Moore-Penrose pseudoinverse operators, respectivel. and are used for element-wise product and elementwise division of two vectors, respectivel. E{ } denotes the expectation operator. R{ } and I{ } denote the real and

2 imaginar parts of a complex quantit. ˆx, represents the estimated value of x. CN µ, σ ) denotes a complex Gaussian distributions with mean µ and variance σ. Boldface small letters, x, and boldface capital letters, X, are used for vectors and matrices, respectivel. [x] represents the th element of vector x and [X] x, represents the entr in row x and column of X. I X and 0 X X denote the X X identit and all zero matrices. Tr{X} denotes the trace of X. diagx) is used to denote a diagonal matrix, where its diagonal elements are given b the vector x. Finall, a set of complex x- dimensional vectors is denoted b C x and f : C x C is used to represent that the function f taes as its argument a complex x-dimensional vector and returns a complex - dimensional vector. II. FUNCTIONAL TRANSFORMATION OF VECTOR CRLB Assume that it is desired to estimate α = fθ), where θ C p, α C r, and f : C p C r. Using functional transformation of the vector CRLB in [], [7], the CRLB for the estimation of α = [fθ)], for = 1,..., r, is given b CRLB α = [fθ)] F 1 [fθ)] θ 1a) θ T [fθ)] 1b) CRLB α = [fθ)] F θ where F θ is the FIM of the untransformed parameter, θ. If F θ is singular or ill-conditioned, the bound in 1a) results in an invalid CRLB for estimation of α. Moreover, the bound in 1b) can also result in an invalid CRLB due to numerical instabilit, e.g., due to quantization error or can be computationall prohibitive to evaluate, e.g., if the dimension of θ is significantl larger than α. To address these shortcomings, here, a new expression for calculating the CRLB of transformed parameter, α, that does not require the evaluation of an form of matrix inverse is derived. Let us assume that the lielihood function, p; θ), satisfies the regularit condition [1], i.e., { } ln p; θ) E = 0 p 1, θ, ) θ where p; θ) is the lielihood function of the complex observation vector,, parameterized b θ and expectation is taen with respect to p; θ). For an unbiased estimator, E { ˆα} = α = fθ). 3) Using ) and 3), we have ln p; θ) ˆα α) p; θ) d = fθ). 4) Pre- and post-multipling 4) b a T and b, respectivel, where a and b are arbitrar r 1 and p 1 vectors, we have a T ln p; θ) ˆα α) p; θ)b d = a T fθ) b. 5) θt Appling Cauch-Schwarz inequalit [1, p. 71] m)n)o)d m) n) d m) o) d. 6) where m) p; θ), n) a T ˆα α), and o) ln p;θ) b. Thus, 5) can be written as 7) given at the bottom of this page, where Σ ˆα is the covariance of ˆα. Let b = fθ) a. hen, 7) can be written as T fθ) fθ) a at a T Σ ˆα a a H fθ) fθ) a. F θ 8) In order to find the lower bound on the estimation α = [fθ)] for = 1,..., r, let a = [0 1 1, 1, 0 1 r ] T. Thus, 8) can be written as [fθ)] [fθ)] σ ˆα [fθ)] [fθ)] F θ 9) where σ ˆα = [Σ ˆα ], is the variance of ˆα. Subsequentl, σ ˆα is given b [fθ)] [fθ)] σ ˆα θ T [fθ)] [fθ)], = 1,..., r. F θ 10) Using 9), the CRLB for the estimation of all the elements of the transformed parameter vector α can be obtained. Remar 1: Unlie the CRLB expressions for the transformed parameters in 1) that rel on the inverse or pseudo inverse of FIM of the untransformed parameters, the transformed CRLB in 10) does not depend on an tpe of inverse of the FIM of untransformed parameters, θ. Thus, the proposed functional transformation of vector CRLB in 10) results in a valid lower bound for α regardless of whether the FIM of the untransformed parameter is singular or not see Section III-C for more details). It is also important to note that following 10), we are able to find the diagonal a T fθ) b p; θ)a T ˆα α) ˆα α a d = a T Σ ˆα a b H F θ b. p; θ)b H ln p; θ) ln p; θ) b d 7)

3 elements of the CRLB. Thus, the derived bound is a tight lower bound for the estimation of uncoupled parameters, i.e., parameters that do not need to be jointl estimated. A practical application of the derived CRLB for lower bounding channel estimation accurac in AF cooperative relaing networs is presented in the following section. III. APPLICATION OF THE PROPOSED CRLB In this section, as an application, the derived CRLB expression in 10) is used to obtain the lower bond on the channel estimation variance in a single-input-single-output SISO) multi-rela cooperative networ. The networ consists of one source node, K relas, and a single destination node. Quasi-static and frequenc flat-fading channel parameters from source to the th rela and th rela to destination are denoted b h and g, respectivel, for = {1,..., K}. It is assumed that unit-amplitude phase shift eing PSK) training signals TSs) are transmitted from source to the th rela and from the th rela to the destination,. The received complex baseband signal at the th rela, r [r 0),..., r L 1)] T, is given b [11] r = h t [s] + u, 11) where L is the length of observation vector, t [s] [t [s] 0),..., t [s] L 1)] T is the training signal from source to relas, h is the unnown flat fading channel parameter, u [u 0),..., u L 1)] T, and u i) for i = 0,..., L 1, denotes the zero-mean complex additive white Gaussian noise AWGN) at the ith sample of the received signal at the th rela, i.e., u i) CN 0, σu ). Each th rela applies distinct training signal t [r] [t[r] 0),..., t[r] L 1)]T to the received signal [10]. The received signal at the destination, [0),..., L 1)] T, is given b [11] where = Tα + Nβ + w, 1) T [t 1,..., t K ] is an L K matrix, t = t [s] t [r],, is combined training from source and th rela, N [n 1,..., n K ] is an L K matrix, n = u t [r],, is the noise signal affected b the training signal from the th rela, such that the noise statistics of n and u are same, i.e., n i) CN 0, σu ), for i = 0,..., L 1, due to unit amplitude PSK training signals, α [α 1,..., α K ] T is the overall channel gain from source to rela to destination such that α = ζ h g, ζ = 1/ σu + σh is the th rela s power constraint, β [β 1,..., β K ] T such that β = ζ g, w [w0),..., wl 1)] T, and wi) for i = 0,..., L 1, denotes the zero-mean complex AWGN at the destination, i.e., wi) CN 0, σw). The following subsection evaluates the CRLB expression for estimating transformed parameter α, which depends on the fundamental sstem parameters θ [h T, g T ] T A. CRLB Evaluation Based on the assumptions in Section III, the AWGN at the relas, u,, and destination, w in 1), are mutuall independent. Accordingl, the received training signal at the destination, in 1), is a circularl smmetric complex Gaussian random variable, CN µ, Σ ), with mean µ = Tα and covariance matrix Σ = K =1 σ u β + σw) IL. In order to find CRLB for the estimation of overall channel gain α = fθ), we can use the proposed CRLB expression in 10), where α = [ diagζ g) diagζ h) ], 13) where ζ [ζ 1,..., ζ K ] T, and FIM for the untransformed parameter, θ [h T, g T ] T, can be calculated using following expression [1], [1] µ H F θ = θ Σ 1 µ ) { + Tr Σ 1 Σ θ Σ 1 Σ }. 14) Using the definition of µ and Σ, defined above 13), F θ is given b F θ = 1 [ ] G H T H TG G H T H TH, 15) ρ H H T H TG Σ 1 }ɛɛ H H H T H TH+Tr{Σ 1 where G diagβ), H diagζ h), ρ K =1 σ u β +σw, ɛ [ɛ 1,..., ɛ K ] T, and ɛ σu ζ g. Remar : The FIM, F θ in 15) is ran-deficient. Particularl, the ran of K K matrix F θ is K +1. Thus, there is a ran deficienc of K 1 degrees. The ran of the matrix F θ can be found b evaluating reduced row echelon form U of F θ, which is determined as U = I K diagh g) 0 1 K x T 0 K 1 K 0 K 1 K, 16) where x [x 1,..., x K ] T and x = σ u ζ g. The K 1 σu ζ 1 1 g 1 rows of zeros in U indicates the ran deficienc of F θ b K 1 degrees. Thus, the existing CRLB expression for the evaluation of transformed sstem parameter in 1) results in an invalid bound. However, using 13) and 15), our proposed expression in 10) is able to calculate the CRLB of transformed parameter, α, accuratel see Section III-C). B. MVU Estimation In this subsection, the best linear unbiased estimator BLUE) for the estimation of α is presented. Given that the received signal in 1) is Gaussian distributed, the BLUE can also be shown to be the MVU estimator [1, p. 133]. Following the approach given in [1, ch. 6], the BLUE or MVU estimate of α is given b ˆα = T H T) 1 T H, 17) where the estimation covariance matrix of the linear unbiased estimate ˆα is T H Σ 1 T) 1.

4 10 0 MVU MSE, K= MVU MSE, K=4 CRLB [8], K= 10 1 CRLB [8], K=4 CRLB via Eq. 10), K= CRLB via Eq. 10), K=4 10 CRLB Fig. 1: MSE, proposed CRLB in 10), and the CRLB in [8], for the estimation of transformed parameter, α = fθ) for training length L = 16 and different number of relas, K = and K = 4. C. Numerical Simulation In this subsection, we numericall simulate the proposed CRLB in 10) and mean-square error MSE) for the estimation of overall channel gain α. Specific channels are used to evaluate the CRLB and MSE of MVU estimator, i.e., h = [ j, j, i, i] T and g = [ j, j, i, i] T, similar to [13] and [10]. Note that the first two and all four entries of h and g are used for K = and K = 4 relas, respectivel. Distinct unit-amplitude PSK training signals are transmitted from source and different relas with training length L = 16. It is assumed that the noise at all relas have the same variance, i.e., σu = σu 1 = = σu K. Moreover, without loss of generalit, it is assumed that σw = σu = 1/SNR. The estimation MSE is numericall calculated as 1 R R r=1 α 1 ˆα 1 r)), where R = is the total number of simulations, ˆα 1 r) is the estimate of α 1 during rth simulation for r = 1,..., R. Without loss of generalit, CRLB and estimation MSE for the first rela are plotted however similar results are obtained for all the relas. Fig. 1 plots the CRLB and MSE for the estimation of α versus SNR for K = and 4 relas. It can be observed from Fig. 1 that estimation MSE of MVU estimator is close to the proposed CRLB for the whole range of considered SNR values. The results in Fig. 1 also verifies the proposed CRLB expression in 10). On the other hand, the existing CRLB expressions in 1a) depends on the inverse of F θ, which is undefined for singular FIM. Fig. 1 also plots the CRLB in [8], where it has been proposed to replace F 1 θ in 1a) with a diagonal matrix, whose diagonal elements are the reciprocal of the corresponding diagonal elements of F θ. It bit quantized CRLB [7] 16 bit quantized CRLB [7] 3 bit quantized CRLB [7] 8 bit quantized CRLB via Eq. 10) True CRLB SNR db) Fig. : Comparison of the proposed CRLB in 10) with [7] see 1b)) for different levels of quantization. Other parameters are set to K = 4 and L = 16 can be observed from Fig. 1 that such bound is above the MSE of the proposed MVU estimator and is not a valid lower bound. 1 Fig. compares the proposed CRLB in 10) with [7] see 1b)) for different levels of quantization. In order to investigate the numerical stabilit of the proposed expression in 10) over the existing result in 1b), 8, 16, and 3 bit quantization is applied. It can be observed from Fig. that our proposed expression is not sensitive to quantization error and even produce the true CRLB with 8-bit quantization. However, the CRLB expression in 1b) is ver sensitive to quantization error. This is because the CRLB expression in 1b) depends on the pseudo inverse of F θ, which is numericall unstable. IV. CONCLUSION This letter derives the general CRLB expression for the estimation of transformed sstem parameters, α = fθ) which are a function of some fundamental sstem parameters, θ. Unlie the previous results in the literature, the final CRLB expression, derived in this letter, does not depend on the inverse or pseudo inverse of the FIM of the untransformed parameters, θ. Thus, it can be applied to scenarios where the FIM of θ is ill-conditioned/singular or when the pseudo inverse of the FIM of θ is numericall unstable or computationall prohibitive to evaluate. Finall, as an example the proposed bound is applied in the context of channel estimation for in amplif-and-forward cooperative relaing 1 According to [8], their CRLB should be loose bound for untransformed parameter estimation. However, for the estimation of transformed parameters, the CRLB in [8] is incorrect as demonstrated in Fig. 1. Similar results as true CRLB are obtained for 16 and 3 bit quantization with the proposed expression.

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