I Introduction
Maximum Doppler spread measures the coherence time, related to the rate of change, of wireless communication channels. Its knowledge is important to design efficient wireless communication systems for highspeed vehicles [1, 2, 3]. In particular, accurate estimation of the MDS is required for the design of adaptive transceivers, as well as in cellular and smart antenna systems [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. For example, in the context of adaptive transceivers, system parameters such as coding, modulation, and power are adapted to the changes in the channel [4, 5, 6, 7]. In cellular systems, handoff is dictated by the velocity of the mobile station, which is also directly obtained from the Doppler information. Knowledge of the rate of the channel change is also employed to reduce unnecessary handoff; the handoff is initiated based on the received power at the mobile station, and the optimum window size for power estimation depends on the MDS [7, 8, 9, 10]. In the context of smart antenna systems, the MDS is used in the design of the maximum likelihood (ML) spacetime transceivers [11, 12]. In addition, knowledge of MDS is required for channel tracking and equalization, as well as for the selection of the optimal interleaving length in wireless communication systems [13].
In general, parameter estimators can be categorized as: i) DA, where the estimation relies on a pilot or preamble sequence [14, 15, 16, 17, 18], ii) NDA, where the estimation is performed with no a priori knowledge about the transmitted symbols [19, 20, 21, 22, 23], and iii) codeaided (CA), where the decoding gain is used via iterative feedback to enhance the estimation performance of the desired parameters [24, 25, 26, 27, 28, 29].
With regard to the MDS estimation, the DA approach often provides accurate estimates for slowlyvarying channels by employing a reduced number of pilot symbols, whereas this does not hold for fastvarying channels. In the latter case, the details of the channel variations cannot be captured accurately, and more pilots are required, which results in increased overhead and reduced system capacity.
There are five major classes of MDS estimators: MLbased, power spectral density (PSD)based, levelcrossingrate (LCR)based, covariancebased, and cyclostationaritybased estimators. The MLbased estimator maximizes the likelihood function, and, in general, is asymptotically unbiased, achieving the CRLB [30, 31, 32]. However, MLE for MDS suffers from significant computational complexity. Hence, different modified lowcomplexity MLEs for MDS in singleinput singleoutput (SISO) flatfading channel were developed [33, 34]. With the PSDbased estimators, some unique features from the Doppler spectrum are obtained through the sample periodogram of the received signal [35]. Covariancebased estimators extract the Doppler information which exists in the sample autocovariance of the received signal [36, 37, 38]. LCRbased estimators rely on the number of level crossings of the received signal statistics, which is proportional to the MDS[39]. The cyclostationaritybased estimators exploit the cyclostationarity of the received signal [40]. Comparing with other MDS estimators, the advantage of the cyclostationaritybased estimators is the robustness to stationary noise and interference.
While the problem of MDS estimation in SISO flatfading channel has been extensively investigated in the literature [38, 33, 34, 31, 32, 37, 36, 30, 40, 35, 39], the MDS estimation in MIMO frequencyselective or in MIMO flatfading channel has not been considerably explored. Furthermore, DAMDS estimation has mainly been studied in the literature. To the best of our knowledge, only a few works have addressed MDS etimation in conjunction with multiple antenna systems. In [32], the authors derived an asymptotic DAMLE and DACRLB for joint MDS
and noise variance estimation in
MIMO flatfading channel. In [40], the CC of linearly modulated signals is exploited for the MDS estimation for single transmit antenna scenarios. While both DA and NDA estimators are studied in [40], only frequencyflat fading and single transmit antenna are considered.In this paper, we investigate the problem of MDS estimation in MIMO frequencyselective fading channel for both DA and NDA scenarios. The DACRLB, NDACRLB, DAMLE, and NDAMLE in MIMO frequencyselective fading channel are derived. In addition, a lowcomplexity NDAMBE is proposed. The proposed MBE relies on the second and fourthorder moments of the received signal along with the leastsquare (LS) curvefitting optimization technique to estimate the normalized squared autocorrelation function (AF) and MDS of the fading channel. Since the proposed MBE is NDA, it removes the need of pilots and preambles used for DAMDS estimation, and thus, it results in increased system capacity. The NDAMBE outperforms the derived DAMLE in the presence of imperfect timefrequency synchronization. Also, the MBE outperforms the NDA estimator (CCE) in [40] and the DA lowcomplexity MLE in [33, 34] in SISO systems and under flat fading channels and in the presence of perfect timefrequency synchronization.
Ia Contributions
This paper brings the following original contributions:

The DA and NDACRLBs for MDS estimation in MIMO frequencyselective fading channel are derived;

The DA and NDAMLEs for MDS in MIMO frequencyselective fading channel are derived;

A lowcomplexity NDAMBE is proposed. The proposed estimator exhibits the following advantages:

lower computational complexity compared to the MLEs;

does not require time synchronization;

is robust to the carrier frequency offset;

increases system capacity;

does not require a priori knowledge of noise power, signal power, and channel delay profile;

does not require a priori knowledge of the number of transmit antennas;

removes the need of joint parameter estimation, such as carrier frequency offset, signal power, noise power, and channel delay profile estimation;


The optimal combining method for the NDAMBE in case of multiple receive antennas is derived through the bootstrap technique.
IB Notations
Notation.Random variables are displayed in sans serif, upright fonts; their realizations in serif, italic fonts. Vectors and matrices are denoted by bold lowercase and uppercase letters, respectively. For example, a random variable and its realization are denoted by and ; a random vector and its realization are denoted by and
; a random matrix and its realization are denoted by
and , respectively. Throughout the paper, is used for the complex conjugate, is used for transpose, represents the absolute value operator, is the floor function, denotes the Kronecker delta function, is the factorial of , is the statistical expectation, is an estimate of , and denotes the determinant of the matrix .The rest of the paper is organized as follows: Section II describes the system model; Section III obtains the DA and NDACRLBs for MDS estimation in MIMO frequencyselective fading; Section IV derives the DA and NDAMLEs for MDS in MIMO frequencyselective fading channel; Section V introduces the proposed NDAMBE for MDS; Section VI evaluates the computational complexity of the derived estimators; Section VII presents numerical results; and Section VIII concludes the paper.
Ii System model
Let us consider a MIMO wireless communication system with transmit antennas and receive antennas, where the received signals are affected by timevarying frequencyselective Rayleigh fading and are corrupted by additive white Gaussian noise. The discretetime complexvalued baseband signal at the th receive antenna is expressed as [41]
(1) 
where is the number of observation symbols, is the length of the channel impulse response, is the symbol transmitted from the th antenna at time , satisfying , with being the transmit power of the th antenna, is the complexvalued additive white Gaussian noise at the th receive antenna at time , whose variance is , and denotes the zeromean complexvalued Gaussian fading process between the th transmit and th receive antennas for the th tap of the fading channel and at time . It is considered that the channels for different antennas are independent, with the crosscorrelation of the and taps given by^{1}^{1}1Here we consider the Jakes channel; it is worth noting that different parametric channel models can be also considered.
(2) 
where is the zeroorder Bessel function of the first kind, is the variance of the th tap between the th transmit and th receive antennas, denotes the symbol period, and represents the MDS in Hz, with as the relative speed between the transmitter and receiver, as the wavelength, as the carrier frequency, and as the speed of light.
Iii CRLB for MDS Estimation
In this section, the DA and NDACRLB for MDS estimation in MIMO frequencyselective fading channel are derived.
Iiia DaCrlb
Let us consider , , , as employed pilots for DAMDS estimation. The received signal at th receive antenna in (1) can be written as
(3) 
where , , , , , and .
Let us define
(4) 
and
(5) 
The elements of the vector , , are linear combinations of the correlated Gaussian random variables as in (IIIA). Thus, , is a Gaussian random vector with probability density function (PDF) given by
(6) 
where , , , and is the parameter vector, with
(7a)  
(7b)  
(7c)  
Since and , , are uncorrelated random vectors, i.e. , the covariance matrix of , , is block diagonal as
(8) 
where . By employing (2), (IIIA), and (4), using the fact the real and imaginary part of the fading tap are independent random variables with , and after some algebra, the elements of the covariance matrix , , are obtained as
(9a)  
(9b)  
The Fisher information matrix of the parameter vector , , for the zeromean Gaussian observation vector in (6) is obtained as
(10)  
For the MDS, , , , and one obtains
(11)  
where is obtained by replacing with in , where is the Bessel function of the first kind.
Finally, by employing (11), the DACRLB for MDS estimation in MIMO frequencyselective fading channel is obtained as
(12) 
(19) 
IiiB NdaCrlb
Let us consider that the symbols transmitted by each antenna are selected from a constellation with elements , where . The PDF of the received vector for NDAMDS estimation is expressed as
(13) 
where is the constellation vector as , , is the constellation point of the th transmit antenna at time , and with , and and are given in (7).
By employing the chain rule of probability and using
, , one can write (13) as(14) 
where represents the th possible constellation vector at the transmitside.
Similar to the DACRLB, is Gaussian and
(15) 
where is the covariance matrix of the received vector given the constellation vector is , . The covariance matrix is block diagonal as in (8), where its diagonal elements, i.e., , , are obtained as
(16a)  
(16b)  
By substituting (15) into (IIIB), one obtains
(17) 
Finally, by employing (17), the NDACRLB for MDS estimation in MIMO frequencyselective fading channel is expressed as
(18) 
where is given in (19) on the top of this page, and . As seen, there is no an explicit expression for (19), and thus, for the CRLB in (18). Therefore, numerical methods are used to solve (19) and (18).
(28) 
Iv ML estimation for MDS
In this section, we derive the DA and NDAMLEs for MDS in MIMO frequencyselective fading channel.
Iva DaMle for Mds
The DAMLE for is obtained as
(20) 
where is given in (6). Since is a differentiable function, the DAMLE for is obtained from
(21) 
By substituting (6) into (21) and after some mathematical manipulations, one obtains
(22)  
As seen in (22), there is no closedform solution for (21). Thus, numerical methods need to be used to obtain solution. By employing the Fisherscoring method [42],^{2}^{2}2The Fisherscoring method replaces the Hessian matrix in the NewtownRaphson method with the negative of the Fisher information matrix [43]. the solution of (22) can be iteratively obtained as
(23) 
IvB NdaMle for MDS
Similar to the DAMLE, the NDAMLE for MDS is obtained from
(24) 
where is given in (17). Since is a linear combination of differentiable functions, the NDAMLE for is obtained from
(25) 
By substituting (17) into (25) and after some algebra, one obtains
(26) 
Similar to the DAMLE, there is no closedform solution for (IVB); thus, numerical methods are used to solve (IVB).
V Nda momentbased (Mb) estimation of Mds
In this section, we propose an NDAMB MDS estimator for multiple input single output (MISO) systems under frequencyselective Rayleigh fading channel by employing the fourthorder moment of the received signal. Then, an extension of the proposed estimator to the MIMO systems is provided.
Va NdaMbe for Mds in Miso Systems
Let us assume that the parameter vector is unknown at the receiveside. The statistical MB approach enables us to propose an NDAMBE to estimate without any priori knowledge of , , and . Let us consider the fourthorder twoconjugate moment of the received signal at the th receive antenna, defined as
(27) 
With the transmitted symbols, , being independent, drawn from symmetric complexvalued constellation points,^{3}^{3}3 for ary phaseshiftkeying (PSK) and quadrature amplitude modulation (QAM), [44]. and with , is expressed as in (IIIB) at the top of this page (see Appendix A for proof).
By employing the firstorder autoregressive model of the Rayleigh fading channel, one can write
[45, 46](29) 
where and is a zeromean complexvalued Gaussian white process with variance , which is independent of .
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