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 high-speed 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) space-time 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 .
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) code-aided (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 slowly-varying channels by employing a reduced number of pilot symbols, whereas this does not hold for fast-varying 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: ML-based, power spectral density (PSD)-based, level-crossing-rate (LCR)-based, covariance-based, and cyclostationarity-based estimators. The ML-based 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 low-complexity MLEs for MDS in single-input single-output (SISO) flat-fading channel were developed [33, 34]. With the PSD-based estimators, some unique features from the Doppler spectrum are obtained through the sample periodogram of the received signal . Covariance-based estimators extract the Doppler information which exists in the sample auto-covariance of the received signal [36, 37, 38]. LCR-based estimators rely on the number of level crossings of the received signal statistics, which is proportional to the MDS. The cyclostationarity-based estimators exploit the cyclostationarity of the received signal . Comparing with other MDS estimators, the advantage of the cyclostationarity-based estimators is the robustness to stationary noise and interference.
While the problem of MDS estimation in SISO flat-fading channel has been extensively investigated in the literature [38, 33, 34, 31, 32, 37, 36, 30, 40, 35, 39], the MDS estimation in MIMO frequency-selective or in MIMO flat-fading channel has not been considerably explored. Furthermore, DA-MDS 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 , the authors derived an asymptotic DA-MLE and DA-CRLB for joint MDS
and noise variance estimation inMIMO flat-fading channel. In , 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 , only frequency-flat fading and single transmit antenna are considered.
In this paper, we investigate the problem of MDS estimation in MIMO frequency-selective fading channel for both DA and NDA scenarios. The DA-CRLB, NDA-CRLB, DA-MLE, and NDA-MLE in MIMO frequency-selective fading channel are derived. In addition, a low-complexity NDA-MBE is proposed. The proposed MBE relies on the second- and fourth-order moments of the received signal along with the least-square (LS) curve-fitting 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 DA-MDS estimation, and thus, it results in increased system capacity. The NDA-MBE outperforms the derived DA-MLE in the presence of imperfect time-frequency synchronization. Also, the MBE outperforms the NDA- estimator (CCE) in  and the DA low-complexity MLE in [33, 34] in SISO systems and under flat fading channels and in the presence of perfect time-frequency synchronization.
This paper brings the following original contributions:
The DA- and NDA-CRLBs for MDS estimation in MIMO frequency-selective fading channel are derived;
The DA- and NDA-MLEs for MDS in MIMO frequency-selective fading channel are derived;
A low-complexity NDA-MBE 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 NDA-MBE in case of multiple receive antennas is derived through the bootstrap technique.
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 byand , 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 NDA-CRLBs for MDS estimation in MIMO frequency-selective fading; Section IV derives the DA- and NDA-MLEs for MDS in MIMO frequency-selective fading channel; Section V introduces the proposed NDA-MBE 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 time-varying frequency-selective Rayleigh fading and are corrupted by additive white Gaussian noise. The discrete-time complex-valued baseband signal at the th receive antenna is expressed as 
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 complex-valued additive white Gaussian noise at the th receive antenna at time , whose variance is , and denotes the zero-mean complex-valued 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 cross-correlation of the and taps given by111Here we consider the Jakes channel; it is worth noting that different parametric channel models can be also considered.
where is the zero-order 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 NDA-CRLB for MDS estimation in MIMO frequency-selective fading channel are derived.
Let us consider , , , as employed pilots for DA-MDS estimation. The received signal at th receive antenna in (1) can be written as
where , , , , , and .
Let us define
where , , , and is the parameter vector, with
Since and , , are uncorrelated random vectors, i.e. , the covariance matrix of , , is block diagonal as
where . By employing (2), (III-A), 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
The Fisher information matrix of the parameter vector , , for the zero-mean Gaussian observation vector in (6) is obtained as
For the MDS, , , , and one obtains
where is obtained by replacing with in , where is the Bessel function of the first kind.
Finally, by employing (11), the DA-CRLB for MDS estimation in MIMO frequency-selective fading channel is obtained as
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 NDA-MDS estimation is expressed as
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
where represents the th possible constellation vector at the transmit-side.
Similar to the DA-CRLB, is Gaussian and
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
Finally, by employing (17), the NDA-CRLB for MDS estimation in MIMO frequency-selective fading channel is expressed as
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).
Iv ML estimation for MDS
In this section, we derive the DA- and NDA-MLEs for MDS in MIMO frequency-selective fading channel.
Iv-a Da-Mle for Mds
The DA-MLE for is obtained as
where is given in (6). Since is a differentiable function, the DA-MLE for is obtained from
As seen in (22), there is no closed-form solution for (21). Thus, numerical methods need to be used to obtain solution. By employing the Fisher-scoring method ,222The Fisher-scoring method replaces the Hessian matrix in the Newtown-Raphson method with the negative of the Fisher information matrix . the solution of (22) can be iteratively obtained as
Iv-B Nda-Mle for MDS
Similar to the DA-MLE, the NDA-MLE for MDS is obtained from
where is given in (17). Since is a linear combination of differentiable functions, the NDA-MLE for is obtained from
V Nda- moment-based (Mb) estimation of Mds
In this section, we propose an NDA-MB MDS estimator for multiple input single output (MISO) systems under frequency-selective Rayleigh fading channel by employing the fourth-order moment of the received signal. Then, an extension of the proposed estimator to the MIMO systems is provided.
V-a Nda-Mbe for Mds in Miso Systems
Let us assume that the parameter vector is unknown at the receive-side. The statistical MB approach enables us to propose an NDA-MBE to estimate without any priori knowledge of , , and . Let us consider the fourth-order two-conjugate moment of the received signal at the th receive antenna, defined as
With the transmitted symbols, , being independent, drawn from symmetric complex-valued constellation points,333 for -ary phase-shift-keying (PSK) and quadrature amplitude modulation (QAM), . and with , is expressed as in (III-B) at the top of this page (see Appendix A for proof).
By employing the first-order autoregressive model of the Rayleigh fading channel, one can write[45, 46]
where and is a zero-mean complex-valued Gaussian white process with variance , which is independent of .