Joint Data-Aided Carrier Frequency Offset, Phase Offset, Amplitude and SNR Estimation for Millimeter-Wave MIMO Systems
Javier Rodríguez-Fernández, Nuria González-Prelcic, Robert W. Heath Jr
11 Joint Data-Aided Carrier Frequency Offset,Phase Offset, Amplitude and SNR Estimationfor Millimeter-Wave MIMO Systems
Javier Rodr´ıguez-Fern´andez † , Nuria Gonz´alez-Prelcic † , and Robert W. Heath Jr. ‡† Universidade de Vigo, Email: { jrodriguez,nuria } @gts.uvigo.es ‡ The University of Texas at Austin, Email: { kiranv,rheath } @utexas.edu Abstract
This work is devoted to solve the problem of estimating the carrier frequency offset, phase offset,amplitude, and SNR between two mmWave transceivers.The Cram´er-Rao Lower Bound (CRLB) for the different parameters is provided first, as well asthe condition for the CRLB to exist, known as Regularity Condition. Thereafter, the problem of findingsuitable estimators for the parameters is adressed, for which the proposed solution is the MaximumLikelihood estimator (ML).
Notation : here we describe the notation to be used throughout this document: bold uppercase A is used to denote matrices, bold lowercase a denotes a column vector and non-bold lowercase a denotes a scalar value. We use A to denote a set. Further, (cid:107) A (cid:107) F is the Frobenius norm and A ∗ , A C , A T and A † denote the conjugate transpose, conjugate, transpose and Moore-Penrosepseudoinverse of a matrix A , respectively. The ( i, j ) -th entry of a matrix A is denoted using [ A ] i,j . Similarly, the i -th entry of a column vector a is denoted as [ a ] i . The identity matrix oforder N is denoted as I N . If A and B are two matrices, A ◦ B is the Khatri-Rao product of A and B and A ⊗ B is their Kronecker product. We use N ( µ , C ) to denote a circularly symmetriccomplex Gaussian random vector with mean µ and covariance matrix C . We use E to denoteexpectation. Discrete-time signals are represented as x [ n ] . a r X i v : . [ c s . I T ] D ec . . . Baseband Combiner W BB RF chain RF chain . . . . . . . . . . . . . . . . . . . . .
RF chain RF chain . . . . . . . . . . . . . . . . . . . . . . . . N s N RF N t N r N RF N s Baseband Precoder F BB H Fig. 1: Illustration of the structure of a hybrid MIMO architecture, which include analog anddigital precoders and combiners. I. S
YSTEM M ODEL
We consider a mmWave MIMO link to send N s data streams using a transmitter with N t antennas and a receiver with N r antennas. Both the transmitter and the receiver use a fully-connected hybrid MIMO architecture as shown in Fig. 1, with L r and L t RF chains. A hybridprecoder is used, with F = F RF F BB ∈ C N t × N s , where F RF ∈ C N t × L t is the analog precoder and F BB ∈ C L t × N s the digital one. The RF precoder and combiner are implemented using a fullyconnected network of phase shifters, as described in [1].The MIMO channel between the transmitter and the receiver is modeled as a N r × N t matrixdenoted as H , which is assumed to be a sum of the contributions of C spatial clusters, eachcontributing with R c rays. This matrix is given by [1] H = (cid:115) N r N t ρ L (cid:80) Cc =1 R c C (cid:88) c =1 R c (cid:88) r =1 α c,r a R ( φ c,r ) a ∗ T ( θ c,r ) , (1)where ρ L denotes the path loss, C is the number of scattering clusters, R c is the number of raysfor c -th cluster, α c,r ∈ C is the complex gain of the r -th ray within c -th cluster, φ c,r ∈ [0 , π ) and θ c,r ∈ [0 , π ) are the angles of arrival and departure (AoA/AoD), respectively of the r -th raywithin c -th cluster, and a R ( φ c,r ) ∈ C N r × and a T ( θ c,r ) ∈ C N t × are the array steering vectorsfor the receive and transmit antennas respectively. This matrix can be written in a more compactway as H = A R GA ∗ T , (2) where G ∈ C (cid:80) Cc =1 R c × (cid:80) Cc =1 R c is diagonal with non-zero complex entries, and A R ∈ C N r × (cid:80) Cc =1 R c and A T ∈ C N t × (cid:80) Cc =1 R c contain the receive and transmit array steering vectors a R ( φ c,r ) and a T ( θ c,r ) , respectively. H d can be approximated using the extended virtual channel representation[ ? ] as H ≈ ˜ A R G v ˜ A ∗ T (3)where G v ∈ C G r × G t is a sparse matrix which contains the path gains of the quantized spatialfrequencies in the non zero elements. The dictionary matrices ˜ A T and ˜ A R contain the transmitterand receiver array response vectors evaluated on grids of sizes G t and G r . Assuming that thereceiver applies a hybrid combiner W = W RF W BB ∈ C N r × L r , with W RF ∈ C N r × L r the analogcombiner, and W BB ∈ C L r × N s the baseband combiner, the received signal at discrete time instant n can be written as r [ n ] = W ∗ HFs [ n ] e j π ∆ fn + v [ n ] , (4)for n = 0 , . . . , N − .The signal s [ n ] ∈ C L t × , ≤ n ≤ N − is a training sequence known tothe receiver, ∆ f is the unknown carrier frequency offset (CFO), and v [ n ] ∼ N (0 , σ W ∗ W ) isthe circularly symmetric complex Gaussian distributed additive noise vector.We consider that the training sequence can be expressed as s [ n ] = q t [ n ] , with q ∈ C L t × aspatial filter consisting of normalized QPSK symbols. Let us define α (cid:44) D −∗ w W ∗ HFq , α = (cid:104) α e jβ α e jβ . . . α L r e jβ L r (cid:105) T and Ω n = e j π ∆ fn I L r , where D w ∈ C L r × L r is the Choleskyfactor of C w = W ∗ W , i.e., C w = D ∗ w D w . Then, if we stack the N samples of the receivedsignal in (4) we obtain the signal model r (cid:122) (cid:125)(cid:124) (cid:123) r [0] r [1] ... r [ N − = M ( α , ∆ f ) (cid:122) (cid:125)(cid:124) (cid:123) Ω α Ω α . . . Ω N − α t (cid:122) (cid:125)(cid:124) (cid:123) t [0] t [1] ...t [ N − + v (cid:122) (cid:125)(cid:124) (cid:123) v [0] v [1] ... v [ N − , (5)such that r ∼ N ( M ( α , ∆ f ) t , σ I ML r ) . Then, the vector of parameters to be estimated is ξ = (cid:104) α . . . α L r β . . . β L r ∆ f σ (cid:105) T .We define the Signal to Noise Ratio (SNR) for the i -th signal [ r [ n ]] i , i = 1 , . . . , L r as γ i = α i σ . (6) Furthermore, the average Signal to Noise Ratio at digital level can be defined as well. Let usdefine Λ ∈ C L r × L r as Λ = diag { α e jβ , α e jβ , . . . , α L r e jβ L r } . Therefore, if the SNR at basebandlevel is denoted as SNR , it can be written as
SNR = 1 L r L r (cid:88) i =1 γ i = trace { ΛΛ ∗ } L r σ , (7)which is just the average of the post-combining γ i at each RF chain.II. R EGULARITY C ONDITION
For the CRLB to exist, the regularity condition must be fulfilled by the probability densityfunction (pdf) of the data. This condition states [2] E (cid:26) ∂ ln p ( r ; ξ ) ∂ξ i (cid:27) = 0 , for all ξ i ∈ ξ . (8)The pdf of the vector r is written as p ( r ; ξ ) = 1 π NL r det (cid:0) σ I NL r (cid:1) e − σ ( r − M ( α , ∆ f ) t ) ∗ ( r − M ( α , ∆ f ) t ) , (9)and the log-likelihood function (LLF) as ln p ( r ; ξ ) = − N L r ln πσ − σ (cid:18) r ∗ r − { t ∗ M ∗ ( α , ∆ f ) r } ++ t ∗ M ∗ ( α , ∆ f ) M ( α , ∆ f ) t (cid:19) . (10)Before computing the gradient of the LLF, it is important to note that t ∗ M ( α , ∆ f ) M ( α , ∆ f ) t = N − (cid:88) n =0 s C [ n ] α ∗ Ω ∗ n Ω n α s [ n ]= N − (cid:88) n =0 s C [ n ] trace { P } s [ n ]= N trace { P } , (11)where P = ΛΛ ∗ .Therefore, for α i , ∂ ln p ( r ; ξ ) ∂α i = − σ (cid:18) − (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂α i r (cid:27) + N L r ∂ trace { P } ∂α i (cid:19) = − σ (cid:18) − N − (cid:88) n =0 Re (cid:8) s C [ n ] e − jβ i e Ti Ω ∗ n r [ n ] (cid:9) + 2 α i N (cid:19) = − σ (cid:18) − N − (cid:88) n =0 Re (cid:8) s C [ n ] e − jβ i e Ti Ω ∗ n Ω n α s [ n ] (cid:9) + 2 α i N (cid:19) = E (cid:26) ∂ ln p ( r ; ξ ) ∂α i (cid:27) = − σ (cid:18) − N − (cid:88) n =0 Re { α i } + 2 α i N (cid:19) = 0 . (12)Now, for σ , ∂ ln p ( r ; ξ ) ∂σ = − N L r σ + 1 σ (cid:107) r − M ( α , ∆ f ) t (cid:107) = − N L r σ + 1 σ N − (cid:88) n =0 E (cid:8) (cid:107) w n [ n ] (cid:107) (cid:9) = − N L r σ + 1 σ N − (cid:88) n =0 L r (cid:88) i =1 (cid:0) σ (cid:122) (cid:125)(cid:124) (cid:123) E { w I,i [ n ] } + E { w Q,i [ n ] (cid:1) } = 0 . (13)For ∆ f , ∂ ln p ( r ; ξ ) ∂ ∆ f = − σ (cid:18) − (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂ ∆ f r (cid:27) (cid:19) = − σ (cid:18) − N − (cid:88) n =0 Re (cid:8) s C [ n ] α ∗ ( − j πn ) Ω ∗ n Ω n α s [ n ] (cid:9)(cid:19) = − σ (cid:18) − N − (cid:88) n =0 Re { trace { P } ( − j πn ) } (cid:19) = 0 (14)Finally, for β i , ∂ ln p ( r ; ξ ) ∂β i = − σ (cid:18) − (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂β i r (cid:27) (cid:19) = − σ (cid:18) − N − (cid:88) n =0 Re (cid:8) s C [ n ] α i ( − j ) e − jβ i e Ti Ω ∗ n Ω n α s [ n ] (cid:9)(cid:19) = − σ (cid:18) − N − (cid:88) n =0 Re { α i ( − j ) } (cid:19) = 0 , (15) since the term inside brackets is also purely imaginary.Thus, since the regularity condition holds, the CRLB is to be found in the next section.III. C RAM ´ ER -R AO L OWER B OUND
Since the model for the received signal is Gaussian, the Slepian-Bangs formula can be usedto find the elements in the Fisher Information Matrix. This formula is given by [2] F i,j ( ξ ) = 2 Re (cid:26) ∂ µ ∗ ( ξ ) ∂ξ i C − ( ξ ) ∂ µ ( ξ ) ∂ξ j (cid:27) + trace (cid:26) C − ( ξ ) ∂ C ( ξ ) ∂ξ i C − ( ξ ) ∂ C ( ξ ) ∂ξ j (cid:27) . (16)Thus, the elements in the FIM can be found as F α i ,α i ( ξ ) = 2 σ Re (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂α i ∂ M ( α , ∆ f ) ∂α i t (cid:27) . (17)The partial derivative of M ( α , ∆ f ) is given by ∂ M ∗ ( α , ∆ f ) ∂α i ∂ M ( α , ∆ f ) ∂α i = e Ti e i e Ti e i . . . e Ti e i = I N , (18)being e i ∈ R L r a vector of zeros with a single one in the i -th position. Therefore, F α i ,α i ( ξ ) = 2 σ N − (cid:88) n =0 Re {| s [ n ] | } = 2 Nσ . (19)For the phase offset parameter, F β i ,β i ( ξ ) = 2 σ Re (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂β i ∂ M ( α , ∆ f ) ∂β j t (cid:27) , (20)where ∂ M ∗ ( α , ∆ f ) ∂β i ∂ M ( α , ∆ f ) ∂β i = α i e Ti e i α i e Ti e i . . . α i e Ti e i = α i I N . (21) Then, the Fisher Information for β i yields F β i ,β i ( ξ ) = 2 σ N − (cid:88) n =0 Re { α i | s [ n ] | } = 2 N α i σ . (22)For the carrier frequency offset parameter, F ∆ f, ∆ f ( ξ ) = 2 σ Re (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂ ∆ f ∂ M ( α , ∆ f ) ∂ ∆ f t (cid:27) , (23)where ∂ M ∗ ( α , ∆ f ) ∂ ∆ f ∂ M ( α , ∆ f ) ∂ ∆ f = trace { P } (2 π (2 π . . . (2 π ( N − . (24)Therefore, F ∆ f, ∆ f ( ξ ) = 2 σ trace { P } N − (cid:88) n =0 (2 πn ) . (25)For the noise variance, F σ ,σ ( ξ ) = 1 σ trace { I NL r } = N L r σ . (26)For the non-diagonal elements in the FIM, it can be checked that all of them are zero-valuedexcept for F ∆ f,β i ( ξ ) = 2 σ Re (cid:26) t ∗ ∂ M ∗ ( α , ∆ f ) ∂ ∆ f ∂ M ( α , ∆ f ) ∂β i t (cid:27) , (27)where ∂ M ∗ ( α , ∆ f ) ∂ ∆ f ∂ M ( α , ∆ f ) ∂β i = α i e jβ i (2 π α ∗ e i (2 π α ∗ e i . . . (2 π ( N − α ∗ e i . (28) Therefore, F ∆ f,β i ( ξ ) = 2 σ N − (cid:88) n =0 Re { (2 πn ) α i } = 2 α i σ N − (cid:88) n =0 (2 πn ) . (29)Then, the FIM is found to be F ( ξ ) = F ( ξ )
00 F ( ξ ) , (30)with F ( ξ ) = diag (cid:26) Nσ TL r , L r Nσ (cid:27) (31) F ( ξ ) = { P } σ (cid:80) N − n =0 (2 πn ) σ (cid:80) N − n =0 πn TL r P σ (cid:80) N − n =0 πn P1 L r Nσ P . (32)Finally, upon inverting the block diagonal matrix F ( ξ ) , the CRLB for the parameters is givenby var { ˆ α i } ≥ σ N (33) var { (cid:98) σ } ≥ σ L r N (34) var { (cid:99) ∆ f } ≥ σ · N L r (cid:0) N ( N − N − / (cid:1) trace { P } N L r − (cid:80) L r k =1 α k (cid:0) N ( N + 1) / (cid:1) N L r (35) var { ˆ β i } ≥ σ · (cid:0) ( N − N − / (cid:1) trace { P } N L r (cid:81) L r j =1 j (cid:54) = i α j − (cid:80) L r k =1 k (cid:54) = i α k (cid:0) ( N + 1) / (cid:1) N L r (cid:81) L r s =1 s (cid:54) = k,i α s (cid:0) N ( N − N − / (cid:1) trace { P } N L r − (cid:80) L r k =1 α k (cid:0) N ( N + 1) / (cid:1) N L r (36) Finally, the
SNR in the i -th RF chain can be distinguished from the average of the individual SNR s of the different signals involved in the problem. If the
SNR of the i -th signal is denotedby γ i , then γ i = α i σ (37) SNR = trace { P } L r σ , (38) for which, according to the Transformed Parameters Property , the CRLB can be found to be var { ˆ γ i } ≥ γ i N + γ i N (39) var { (cid:100) SNR } ≥ L r N + SNR L r N . (40)The last formulas provide a clear insight on how the estimation of γ i actually works. Forlower values of γ i , the CRLB is dominated by the term growing linearly with the parameter,whereas for higher values the CRLB is dominated by the second term, making the estimationof the parameter much harder. The same happens for (cid:100) SNR , although it actually depends on thedifferent γ i for the different streams.Henceforth, our effort is focused on searching for suitable estimators of these parameters.Owing to the non-linear dependence between the data and the parameters, an efficient estimatorcannot be found, in general. The practical approach to follow is to seek the ML estimators forthese parameters, for which the next subsection is devoted.IV. M AXIMUM L IKELIHOOD E STIMATION
The problem of finding the ML Estimator for the parameters in ξ can be formalized as (cid:110) ˆ α i, ML , ˆ σ ML , ˆ∆ f ML , ˆ β i, ML (cid:111) i =1 ,...,L r = arg max ξ ln p ( r ; ξ ) , (41)which involves a joint maximization over L r + 1) scalar variables. This problem can be solvedin four different steps by splitting the original problem into four interconnected optimizationproblems. A. MLE for the Phase Offset
Recall that the LLF is given by ln p ( r ; ξ ) = − N L r ln πσ − σ (cid:18) r ∗ r − { t ∗ M ∗ ( α , ∆ f ) r } + N trace { P } (cid:19) , (42)such that the term that depends on β i is the second variable inside brackets. Therefore, the value ˆ β i that maximizes the function above is found from ˆ β i = arg max β i Re { t ∗ M ∗ ( α , ∆ f ) r } . (43) The first derivative of the function to maximize yields ∂ ln p ( r ; ξ ) ∂β = N − (cid:88) n =0 Re { s C [ n ] α i ( − j ) e − jβ i e Ti Ω n r [ n ] } (44) ∂ ln p ( r ; ξ ) ∂β = N − (cid:88) n =0 Im { s C [ n ] α i e − jβ i e − j π ∆ fn r i [ n ] } (45)Now the following result can be applied. Lemma . Given two complex numbers, z , z , the imaginary part of their product is definedas Im { z z } = Re { z } Im { z } + Im { z } Re { z } . Therefore, setting the previous derivative to zero allows us to obtain N − (cid:88) n =0 Im { s C [ n ] r i [ n ] e − j π ∆ fn } Re { α i e − jβ i } = N − (cid:88) n =0 Re { s C [ n ] r i [ n ] e − j π ∆ fn } Im { α i e − jβ i } (46) ˆ β i = tan − (cid:26) (cid:80) N − n =0 Im { s C [ n ] r i [ n ] e − j π ∆ fn } (cid:80) N − n =0 Re { s C [ n ] r i [ n ] e − j π ∆ fn } (cid:27) , (47)which can be interpreted as a matched-filtering operation with the training sequence s [ n ] afterthe carrier frequency offset has been corrected. Notice that the ML estimator of β i requires theknowledge of the true value ∆ f . Since it is impossible to know the exact value of ∆ f , the MLestimator of ∆ f is to be substituted in (47) instead, such that the final estimator can be appliedin a practical scenario. B. MLE for the Amplitude
In this subsection we provide a closed-form expression for the amplitude parameters, whichhave already been denoted as α i . By inspection of the LLF ln p ( r ; ξ ) = − N L r ln πσ − σ (cid:18) r ∗ r − { t ∗ M ∗ ( α , ∆ f ) r } + N trace { P } (cid:19) , (48)it can be seen that the terms that depend on this parameter are the two last within brackets.Therefore, the problem of finding α i is formalized as ˆ α i = arg max α i { t ∗ M ∗ ( α , ∆ f ) r } − N trace { P } . (49)Taking the first derivative of the objective function yields ∂f ( α i ) ∂α i = 2 N − (cid:88) n =0 Re { r i [ n ] s C [ n ] e − jβ i e − j π ∆ fn } − α i N, (50) such that the value of α i that maximizes the function above is ˆ α i = 1 N N − (cid:88) n =0 Re { r i [ n ] s C [ n ] e − jβ i e − j π ∆ fn } . (51)From (51), it is clear that the statistic ˆ α i depends on ∆ f and β i , similarly as with the estimatorof β i . Therefore, the estimators of these parameters are to be substituted in (51) to find the MLestimator. C. MLE for the Carrier Frequency Offset
In this subsection we will find the ML estimator of the carrier frequency offset. In (48), theonly term that depends on ∆ f is the second one. Therefore, the problem of finding the MLestimator for this parameter can be stated as (cid:99) ∆ f = arg max ∆ f Re { t ∗ M ∗ ( α , ∆ f ) r } . (52)We can express the objective function in (52) as f (∆ f ) = Re (cid:26) N − (cid:88) n =0 L r (cid:88) i =1 α i r i [ n ] s C [ n ] e − jβ i e − j π ∆ fn (cid:27) , (53)in which the statistics ˆ α i and ˆ β i are to be substituted. These are given by ˆ β i = ∠ (cid:26) N − (cid:88) n =0 r i [ n ] s C [ n ] e − j π ∆ fn (cid:27) (54) ˆ α i = 1 N (cid:12)(cid:12)(cid:12)(cid:12) r i [ n ] s C [ n ] e − j π ∆ fn (cid:12)(cid:12)(cid:12)(cid:12) , (55)where the statistic ˆ α i follows from substituting (54) into (51). Therefore, (54) and (55) can besubstituted into (53) to yield the value of ∆ f that maximizes the objective function. This canbe written as ˆ∆ f = arg max ∆ f N L r (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) n =0 r i [ n ] s C [ n ] e − j π ∆ fn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (56)Accordingly, the ML estimator for the carrier frequency offset can be understood as the bincorresponding to the maximum of the sum of the squared-periodograms of the received signals,after a temporal matched filter. Since the periodogram is defined as the Fourier Transform ofthe autocorrelation, it is necessary to find its maximum numerically, by using the Fast - FourierTransform (FFT). The FFT yields a sampled version of the DTFT, such that there is no guarantee that the carrierfrequency offset falls within a specfic integer bin of the FFT. Thus, the three largest points inthe FFT can be found and parabolic interpolation can be performed thereafter. The motivationbehind this approach is trying to find the real maximum of the function. The actual maximumis not generally found, but an accurate estimate can be found, instead.
D. Quadratic Interpolation of Spectral Peaks
This subsection is meant to explaining how parabolic interpolation can be applied to obtain abetter estimate of the carrier frequency offset. In general, assume that the shape of the EnergySpectral Density (ESD) of the received stochastic process is the one pictured in Figure 2. Then,the equation of a parabola is given by y ( x ) (cid:44) a ( x − p ) + b, (57)where p is the interpolated peak location (in FFT bins). At the three nearest samples, the imagesof the parabola at these points are y ( −
1) = α y (0) = β y (1) = γ. (58)Then, expressing the three samples in terms of the interpolating parabola, it is found that α = ap + 2 ap + a + bβ = ap + bγ = ap − ap + a + b. (59)Therefore, the interpolated peak location in bins is given by p = 12 α − γα − β + γ ∈ (cid:2) − , (cid:3) , (60)such that the amplitude at that bin is y ( p ) = β −
14 ( α − γ ) p. (61)Finally, by using (60), an estimate of the interpolated peak location can be obtained. If k ∗ denotesthe FFT bin that yields the maximum value, then ( k ∗ + p ) yields the absolute position of theinterpolated bin. Then, the analog frequency estimate is expressed as ˜ ω = k ∗ + pN f s , (62)where f s is the sampling frequency in Hertz and N is the number of points in the FFT. Fig. 2: Illustration of parabolic peak interpolation using the three samples closest to the peak.
E. MLE for the Noise Variance
This subsection is devoted to find the MLE for the noise variance parameter, σ . Recallingthe Log-Likelihood Function ln p ( r ; ξ ) = − N L r ln πσ − σ (cid:107) r − M ( α , ∆ f ) t (cid:107) , (63)the MLE for the noise variance can be easily found from ∂ ln p ( r ; ξ ) ∂σ = − N L r σ + 1 σ (cid:107) r − M ( α , ∆ f ) t (cid:107) . (64)The condition for the derivative to vanish is that (cid:98) σ = 1 N L r (cid:107) r − ˆ M ( α , ∆ f ) t (cid:107) , (65)where (cid:99) M ( α , ∆ f ) is the estimate of the matrix M ( α , ∆ f ) that models how the signal energy istransferred from the transmitter to the receiver. The estimate (cid:99) M ( α , ∆ f ) is computed using theML estimators of the previous parameters. F. MLE for the Signal to Noise Ratio
This last subsection is devoted to provide the ML estimator for the
SNR metric. To find thisstatistic, the following property of ML estimation can be applied [2].
Lemma 1:
Invariance Property of the MLE : The MLE of the parameter α = g ( ξ ) , wherethe pdf p ( x ; ξ ) is parameterized by ξ , is given by ˆ α = g ( ˆ ξ ) , (66)where ˆ ξ is the MLE of ξ .This nice property allows us to find the MLE for SNR = α i σ , which is simply ˆ γ i = ˆ α i (cid:98) σ , (67)where ˆ α i and (cid:98) σ are the MLEs for α i and σ provided in the previous subsections. As a summary,the ML estimators for the parameters are (cid:99) ∆ f ML = arg max ∆ f L r (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) n =0 s C [ n ] r i [ n ] e − j π ∆ fn (cid:12)(cid:12)(cid:12)(cid:12) ˆ β i, ML = ∠ { N − (cid:88) n =0 s C [ n ] r i [ n ] e − j π ∆ fn } ˆ α i, ML = 1 N | N − (cid:88) n =0 s C [ n ] r i [ n ] e − j π ∆ fn | (cid:98) σ ML = 1 L r N || r − ˆ M ( α , ∆ f ) t || ˆ γ i, ML = ˆ α i (cid:98) σ (cid:100) SNR ML = trace { (cid:98) P } L r (cid:98) σ , (68)To assess the performance of the different estimators, some results are presented hereafter.In the simulations, a range from − dB to dB has been considered for SNR , which is atypical range in millimeter wave communications. The amplitude parameters are generated froma U [0 , distribution. The phase-offsets are generated from a U (0 , π ) and the carrier frequencyoffset is generated as U [ − / , / . The number of antennas is set to N t = 32 and N r = 32 ,and the number of RF chains employed at both transmitter and receiver are set to L t = 4 and L r = 4 . The training sequence t [ n ] , ≤ n ≤ N − contains normalized QPSK symbols, with N = 64 time-domain samples. The results have been averaged over N MC = 1000 Monte Carlorealizations.The normalized sample bias of the the different parameters is shown in Figure 4. As predictedby the Estimation Theory, at low
SNR regime the sample bias of the parameters is non zerowhereas it decreases to zero asymptotically with
SNR . The estimators are, thereby, asymptoticallyunbiased. Fig. 3: Normalized sample bias for the amplitude and
SNR parameters.Now, the efficiency of the estimators is evaluated through the normalized sample variance.The normalized sample variances for the different estimators is shown in Figure 5, Figure 6,Figure 7 and Figure 8, along with their corresponding normalized CRLBs. It can be noticedthat the normalized sample variance of the estimators does not lie within the NCRLB forlow values of
SNR , as it is expected. Nevertheless, for high values of
SNR , the estimatorspresent a normalized sample variance that lies within the NCRLB. Therefore, the estimators areasymptotically efficient. Fig. 4: Normalized sample bias for the angular parameters β i , i = 1 , . . . , and ∆ f .Fig. 5: Normalized sample variance and NCRLB for the amplitude parameters. Fig. 6: Normalized sample variance and NCRLB for
SNR .Fig. 7: Normalized sampled variance and NCRLB for the phase offsets β i , i = 1 , . . . , . Fig. 8: Normalized sample variance and NCRLB obtained for the carrier frequency offsetparameter. R
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Proc. IEEE Globecom , 2014.[2] S. M. Kay,