I Introduction
The fifth generation (5G) of mobile systems must handle diverse scenarios and meet the increasing demand for high data rates. To achieve an efficient use of limited spectrum, Cognitive Radio (CR) has gained a great deal of interest. The idea is to allow the exploitation of locally available frequency bands, on a temporary basis but under a noninterfering constraint. In order to avoid the interference from an opportunistic user to the primary user, a waveform used by the secondary user must have a low out of band radiation (OOB) [1]. Most modern digital communication standards use Orthogonal Frequency Division Multiplexing (OFDM) [2] as the air interface, because of its flexibility and robustness in frequencyselective channels. Nevertheless, OFDM presents some drawbacks such as the high OOB [3] that affects its application especially in CR systems. In contrast, GFDM can achieve a low OOB using a spectrallycontained pulse on each subcarrier, and it has higher spectrum efficiency, because it does not need to use virtual subcarriers to avoid adjacent channel interference, and because it reduces the ratio between the guard time interval and the total frame duration [1, 4]. However, the subcarrier filtering results in nonorthogonal subcarriers, leading to intersymbol interference (ISI) and intercarrier interference (ICI).
In this work, we develop a basis expansion model (BEM) approach for estimation of the channel parameters [6, 5, 7]
in OFDM and GFDM systems. We first consider classical leastsquares (LS) and linear minimum meansquare error (LMMSE), and then examine BEMtype channel estimation algorithms. We then use a strategy to estimate the covariance matrix of the channel vector containing the coefficients of the channel and incorporate it into a BEMtype channel estimation technique that approximates the LMMSE channel estimator. Simulations show that the BEM estimator presents results close to those of the LMMSE estimator, but there is no need to know the power delay profile (PDP) of the channel and the estimator complexity is feasible.
The paper is organized as follows: in Section II, a description of the signal model of the OFDM and GFDM systems is given; Section III describes estimation algorithms known in the literature; the BEM algorithms are described in Section IV; Section V describes the covariance matrix approach and aproximatedLMMSEBEM algorithm; the results of the simulations are presented in Section VI and in Section VII, some conclusions are drawn.
Ii Signal Model
The OFDM transmit signal is given by [2]
(1) 
where is the data symbol at the subcarrier and is the number of subcarriers in the OFDM symbol.
In GFDM, each data symbol is pulseshaped using a circularly time and frequency shifted version of a prototype filter whose energy over one period is normalized to one ( is the number of subsymbols per GFDM block and is the number of subcarriers per subsymbol). The GFDM transmit signal is given by [1]
(2) 
where is the data symbol at time index and subcarrier index .
In order to insert and extract interferencefree pilots, we rewrite the GFDM transmit signal by [8]
(3) 
where contains the symbols at data subcarriers with zeros at pilot subcarriers and carries information at only the pilot subcarriers with zeros at data subcarriers. is the unitary DFT matrix of size and . Here, can be any permutation matrix of compatible size which allocates the pilots to any frequency bin within the pilot subcarriers, and is a block diagonal matrix according to its matrix entries with being the topleft and being the bottomright blocks. The parameter is a scaling factor that normalizes the pilot energy to one, and
is an identity matrix of size
, where is the number of pilots in each . is fold repetition matrix which concatenates identity matrices of size ; is a matrix of ones, and is the Kronecker product. The value of is based on the number of nonzero values in the filter frequency response. The subcarrier filter is diagonal in the frequency domain. The circulant filter is the version of downsampled by factor . An upconversion of the subcarrier to its respective subcarrier frequency is performed by the shift matrix , where returns the circulant matrix based on the input vector, and is a column vector where the element is and all others are zeros. The K subcarriers are summed and transformed back to the timedomain with , where is the Hermitian transpose of (.).To ensure the interferencefree pilot insertion, in expression (3) the plus sign cannot superimpose the information, i.e., if is a pilot subcarrier becomes and if it belongs to data subcarriers , it would be . The pilots need to be located at the frequency bins where no intercarrier interference is present.
At the receiver, the received signal is described by
(4) 
where describes the channel, is the vector with addition of the cyclic prefix (CP), and is a noise vector containing additive white Gaussian noise (AWGN) samples.
The signal is converted from the analog to the digital domain, followed by removal of the CP. Then, we apply the Fourier transform to extract the pilots in the frequency domain, to estimate the channel
and to obtain .On the demodulator side, the recovered data symbols for the subcarrier in GFDM system are given by
(5) 
In order to mitigate the interference between adjacent data subcarriers, we use the interference cancellation (IC) algorithm proposed in [9]. Note that other interference cancellation techniques [16, 17, 18, 19, 20, 21, 22, 63, 58, 30, 25, 24, 26, 28, 30, 31, 32, 33, 34, 37, 36, 37, 38, 39, 40, 41]. can also be examined. In this case, the notation needs to be extended by an iteration index . The vector of received frequency domain samples then corresponds to , and the received data symbols to , respectively. With this notation, and denote vectors on which no interference cancellation has been performed. The IC algorithm is detailed in Algorithm 1.
Essentially, the process consists of detecting the data symbols and using them in the th iteration to compute the interfering signal , which is then subtracted from the original signal in order to obtain an interferencereduced version of the data estimate from .
The analysis of the possible error propagation issue in the interference cancellation performed by algorithm 1 is beyond the scope of this work. However, it was possible to observe in the simulations carried out that such error propagation can be considered negligible in the scenario evaluated.
The computational complexity of the systems (OFDM and GFDM) in terms of complex valued multiplications can be expressed as: , where is due to the fact that this operation occurs in the transmitter and receiver, is the number the symbol per block and originates from the points DFT (see Eq. (1)); and , where denotes the matched filtering of the subcarriers, and originates from the and points DFT, respectively (see Eqs. (3) and (5)).
Applying the IC algorithm times to all subcarriers introduces additional operations for GFDM system, where for transforming the estimated data symbols to the frequency domain, for applying the interference filter and another for transforming back to the time domain.
The block diagram of the proposed solution for the OFDM and the GFDM systems is shown in Fig. 1.
Iii Classical Channel Estimation Algorithms
In this section, we describe the channel estimation methods that are utilized for comparison with the BEM based technique.
At the receiver side, after performing the timefrequency conversion, we can rewrite (4) in the pilot positions, as:
(6) 
where , , and is the number of pilots per symbol.
Iiia Least Squares
Least Squares (LS) [10] provides the estimate of the channel frequency response at the pilot positions by
(7) 
where the matrix and the vector occupies the main diagonal of the matrix.
The LS algorithm has low computational complexity
, but the estimates have a high variance due to the noise presence.
IiiB Linear Minimum Mean Square Error
The Linear Minimum Mean Square Error (LMMSE) [6] can be written as
(8) 
where , with being the constellation points of the mapping, is the signal to noise ratio, and is an identity matrix. To compute an element of the channel covariance matrix , we need to know the PDP of the channel. The need to estimate the channel covariance together with the need to estimate the SNR periodically, makes the use of the LMMSE estimator computationally prohibitive; its computational complexity is .
Iv BEM Channel Estimation Algorithms
The BEM approach represents the channel frequency response using basis functions[11]. It should be noted that other subspace techniques [42, 43, 44, 45, 46, 47, 48, 49, 63, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 74, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93] exploit the eigenstructure of the input data matrix can be also considere. Using BEM, the vector of the channel frequency response can be represented as
(9) 
where is the matrix of basis functions, every column of is a basis function, and is the vector of BEM expansion coefficients.
When the channel estimation is transformed into estimation of the vector , its complexity is reduced because the size of is smaller than .
Iva LsBem
To get an estimate , we can rewrite (6) by replacing by and we compute that minimizes the cost function . This procedure is known as LeastSquares BEM (LSBEM) [12]. The minimum of is found by differentiating it with respect to and setting the derivative equal to a null vector. The resultant LS estimate of is
(10) 
where .
Since each pilot subcarrier has modulus equal to (), then is an identity matrix. As the matrix with the bases is known a priori, the matrix can be precomputed. Its inverse can also be precomputed, thus avoiding the realtime matrix inversion.
Taking into account that , the LSBEM channel estimate can be represented as
(11) 
where can also be precomputed to reduce the realtime computation.
The LSBEM algorithm has computational complexity , where is the length of the vector which is smaller than . However, the LSBEM channel estimates are sensitive to the additive noise.
IvB LmmseBem
To improve the performance, we take the level of the noise (i.e. the noise variance) into account [13]. This is usually done by applying the Bayesian GaussMarkov theorem [14] to minimize a cost function , with , and by obtaining the LMMSE estimator for as given by
(12) 
where is the covariance matrix of the expansion coefficients.
The covariance matrix of the expansion coefficients can be obtained from the channel covariance, assuming that , i.e., the modeling error is negligible. We can find . Then, we write .
By multiplying from the left by and from the right by , and also taking into account that , we obtain
(13) 
The LMMSEBEM algorithm has computational complexity , but the problem to find the channel PDP is still present.
IvC BEM designs
Many traditional BEM designs have been used to model the channels. In this work, we use the Complex Exponentials (CEBEM)[13] and Legendre Polynomials (LPBEM)[11].
The CEBEM is described by [13]
(14) 
where ’s are the normalized frequencies in the pilot positions, is the pilot separation and ’s are the basis function index.
V Covariance Matrix Approach and AproximatedLMMSEBEM Algorithm
As known in the literature, the LMMSE regularization requires the knowledge of the channel covariance matrix, which in turn is defined by the channel PDP. It is difficult to accurately estimate it in practice. In order to overcome this problem, we develop a BEMtype channel estimator for GFDM and other multicarrier systems that considers that the channel PDP is constant and that the coefficients are evenly distributed along the cyclic prefix (CP), whose length is . Therefore, we obtain the covariance matrix for the channel approximation as
(16) 
where indicates the position of the pilot subcarrier in the symbol of length . This matrix computation is done once and offline.
The aLMMSEBEM algorithm is detailed in Algorithm 2.
The computational complexity of the aLMMSEBEM algorithm is .
Vi simulations results
To validate the BEM based channel estimation technique in OFDM and GFDM systems, simulations were performed comparing its performance with that of the LS and LMMSE estimators. The complexity of the solution for each system is the sum of the computational complexity of the system (OFDM or GFDM) added to the computational complexity of the selected channel estimation algorithm. The simulations were performed with the parameters listed in Table I.
number of subcarriers [K]  128 
pilot separation []  4 
GFDM subsymbols [M]  5 
pulse shaping filter [G]  RRC 
rolloff factor []  
modulation subcarriers  QPSK 
number of CE basis functions []  18 
number of LP basis functions []  18 
number of iterations of the IC algorithm []  2 
The choice of the number of basis functions was performed in order to optimize the MSE in all range of the tested .
In the first example, we observe the mean square error (MSE) performance in a multipath channel[15]. This channel has been modeled as a tappeddelay line, whose power delay profile is () for the fractional delays (
) samples, each tap is modeled by a Gaussian distribution. In order to provide a simple way of equalization, the length of CP was chosen with 8 samples.
In Fig. 2, we can observe that the OFDM system (solid line) and the GFDM system (dotted line) show very similar performances, the small difference observed can be attributed to the presence of CP in all OFDM symbols, which degrades the . The approximated LMMSE BEM (aLMMSEBEM) shows a MSE dB lower than that of the LSBEM, and dB lower than that of the classic LS. The curves CEBEMaLMMSE and LPBEMaLMMSE show a similar performance for low SNR, but the CEBEMaLMMSE shows the best performance for high SNR (above dB). This behavior shows that the CE basis functions are a superior choice for the estimation of this channel model, while the LP would need a greater number of basis functions to avoid the floor observed in its MSE estimates. As already known in the literature, the best estimation is obtained with the LMMSE [10], which requires knowledge of channel PDP and whose complexity is high.
In the second example, shown in Fig. LABEL:ber_ofdm, we can observe that the Bit Error Rate (BER) performance for the aLMMSEBEM estimator is approximately dB better than that of LSBEM, dB better than that of LS, and very close to that of the LMMSE and when the channel state information (CSI) is known. The CE and LP basis functions show quite similar performance. The best performance of the BER in the GFDM system, when compared to the OFDM system, may be associated with the degradation of the due to the presence of the CP in all the OFDM symbols.
Vii Conclusions
This paper has investigated channel estimation techniques for 5G multicarrier systems. The classical LS and LMMSE criterionbased estimators have been briefly reviewed and a BEMtype channel estimator has been developed, which effectively approximates the linear MMSE channel estimator. The resulting MSE performance shows that the aLMMSEBEM algorithm presents results close to LMMSE, but its computational complexity is reduced. The MSE performance of the GFDM channel estimation shows that the performances are quite similar to OFDM, which shows that the pilots remained orthogonal to the data symbols in the frequency domain, suffering no interference in GFDM systems. The BER curves show that the performance of the proposed estimator is very close to the LMMSE, which requires knowledge of the channel PDP and whose complexity is high.
Viii Acknowledgement
The work of Y. Zakharov is partly supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through Grant EP/R003297/1.
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