On the transmitter side, the block of size N is composed by symbols belonging to an improper constellation (such as M-PAM or M2-OQAM). The transmitted signal will pass through a channel with an impulse response of size. Thus, the cyclic prefix appended to the block before transmission must have a length LCP of at least, resulting in of length N + LCP. Complex proper uncorrelated additional white Gaussian noise (AWGN) with zero mean and variance also contaminates the transmitted signal.
Due to the cyclic prefix, the N × N channel matrix H′ is circulant, with its first column containing the channel impulse response appended by () zeros. Since H′ is a circulant matrix, we can apply an eigendecomposition to this matrix to obtain W∗H W, where W is the normalized discrete Fourier transform (DFT) matrix of size N × N and H is a N × N diagonal matrix with its (k,k)th entry Hk,k corresponding to the k th coefficient of the N-sized DFT of the channel impulse response.
The signal with length N + LCP at the entry of the receiver has its cyclic prefix removed and passes to the frequency domain through a fast Fourier transform (FFT), so that equalization can be done in the frequency domain. This will result in the signal r of length N, expressed as
(1)
where H corresponds to the channel frequency response of a specific channel realization, is the transmitted signal in the frequency domain, and is the noise in the frequency domain. Equalization is performed by filters based on the MMSE criterion. However, since the equalizer is dealing with a signal from an improper constellation (which has non-zero pseudocorrelation), it has to employ widely linear processing to use all the second-order statistics made available by the received signal. In order to do that, the original version of the received signal in the frequency domain together with its conjugate version should be processed by the equalizer.
2.1 WL-MMSE equalizer
The system model for a SC-FDE system employing widely linear MMSE-based equalization is presented in Figure1. The signal at the output of the equalizer z, with size N, is given by
(2)
with of size N × 2N andof size 2N.
The cost function εWL to derive the equalizer A based on the WL-MMSE criterion is
(3)
where
(4)
(5)
(6)
with U expressed by
(7)
(8)
and
(9)
with E[n nT] = 0 (since the noise is proper), and W WH = I
N
. We obtain the optimal equalizer A by differentiating εWL with respect to A and equalling the result to zero, resulting in
(10)
Using blockwise matrix inversion, can be expressed by
(11)
with
(12)
(13)
(14)
(15)
and
(16)
Analyzing (16), it is possible to see that Hmod is a diagonal matrix with its diagonal equal to [2|H1|2, (|H2|2 + |H
N
|2), (|H3|2 + |HN-1|2),…2|HN/2+1|2,…(|H3|2 + |HN-1|2), (|H2|2 + |H
N
|2)].
This way, the widely linear equalizer A can be expressed as
with the filters A1 (which processes the received signal in the frequency domain) and A2 (which processes its conjugate version) being given by
(19)
and
(20)
When transmitting proper signals, A is reduced to the strictly linear MMSE one, since with proper signals E[s sT] = 0; thus, taking into account the conjugate version of the received signal in the equalization process does not lead to a performance improvement in this case. This process is very similar to the one done in[14], but better details A1 and A2.
After equalization, an inverse fast Fourier transform (IFFT) is done on z so that the symbol decision is realized in the time domain, resulting in (with size N). Due to the fact that widely linear processing is employed in the equalizer, the estimated symbols at the output of the receiver will be purely real.
2.2 WL-MMSE DFE equalizer
When using a WL-MMSE DFE equalizer, the system model is described in Figure2. This system employs a time-domain feedback filter in addition to a frequency-domain feedforward filter to obtain the symbol estimate.
Assuming that correct past decisions are passed along in the feedback filter, the frequency-domain representation q of the symbol estimate (both with size N) can be expressed as
(21)
where B is a 2N × N matrix corresponding to the feedforward filter and D is a N × N matrix with its main diagonal being the N × 1-sized frequency-domain representation of the real-valued time-domain feedback filter.
In q, the error (ISI plus noise) component eWL-DFE is given by
(22)
Differentiating the autocorrelation matrix of this error vector with respect to the feedforward filter B and setting the derivative to zero, we obtain the optimal value of B, expressed as
(23)
Replacing (23) in (22) and going to the time domain, we have
(24)
Since, eWL-DFE can be rewritten as
(25)
Thus, the error autocorrelation matrix C
ee
can be calculated as
(26)
Since our goal is to minimize the mean square error (MSE), the trace of the error covariance matrix C
ee
should be minimized. This trace is
(27)
Using the feedback filter in the time domain instead of its frequency domain version D in (27), we have
(28)
To minimize t r(C
ee
), we derive (28) with respect to the feedback filter coefficients and equal it to zero, obtaining the following linear system:
The matrix F and the column vector g are expressed, respectively, as
(30)
and
(31)
To initialize the feedback filter, the last symbols of can be used. Once is determined, B can be calculated by (23). The size of the feedback filter should be equal to the channel length to cancel all the ISI from the previous detected symbols.
2.3 WL-MMSE Tomlinson-Harashima precoder
A block diagram for the SC-FDE system using WL-MMSE Tomlinson-Harashima precoding is shown in Figure3.
In this system model, we consider a single-carrier block transmission, with the block to be transmitted of size composed by symbols belonging to an improper constellation (such as M-PAM or M2-OQAM) with unit energy. then goes to the Tomlinson-Harashima precoder, which consists of a-sized filter and a modulo operator, resulting in. We recall that the task of the Tomlinson-Harashima precoder is to use the available channel state information in the transmitter to cancel the interference caused by the channel before transmission. The modulo operator is present to reduce the transmitted signal to a prescribed range, since the precoding operation may increase a given constellation point to an out-of-range value.
The input to the modulo operator in the precoder is given by
(32)
This modulo operation to is done independently on the real and imaginary parts. The output of this modulo operation is given by
(33)
If the real (imaginary) part of is greater than M, 2M is (repeatedly) subtracted from it until the result is less than M. If this real (imaginary) part is less than -M, 2M is (repeatedly) added to it until the result is greater than or equal to -M. In other words, is reduced modulo 2M to the half-open interval [ -M,M), limiting the effective dynamic range of the transmitted signal to this interval. This modulo operation is represented by the sequence. After this operation, zeros are appended to to initialize the state of the precoding filter, resulting in the vector of size N. More power is necessary to transmit the precoded symbols when compared to non-precoded ones[15]; however, this penalty becomes negligible with an increase in constellation size.
follows the same path of a SC-FDE WL-MMSE-DFE up to the feedback filter (cyclic prefix insertion, channel, cyclic prefix removal, FFT, WL-MMSE equalization, and IFFT). The same modulo operation realized in the transmitter is done in the receiver to to map the received data to the interval (-M,M], resulting in the symbol estimate. Only the first elements of are used for the decision.
An equivalent linearized scheme of the system model presented in Figure3 is shown in Figure4, following the time-domain THP conversion process made in[16]. In this figure, K = [H H∗]T (with size N × 2N) and D′ is a N × N diagonal matrix with its main diagonal being the N-sized Fourier transform of the Tomlinson-Harashima precoder.
Figure4 shows that the symbol estimate of size is given by
(34)
where is the desired symbol vector, the filtered noise, and the remaining interference is expressed by, all of size. This way, the error vector is
(35)
Using (35), we obtain the mean square error E′, given by
(36)
Minimizing (36), we can find that B′ and are the same as the ones in a SC-FDE system employing a MMSE-based decision-feedback equalizer together with widely linear processing. Thus, the coefficients of the Tomlinson-Harashima precoder are given by (29), and the widely linear MMSE equalizer B′ = B is given by (23).