## 2 Background
### Equalization and Channel Estimation  
The key receiver tasks are to perform equalization and unformat the received block. The latter operation is established by executing the opposite of the transmitter scheme, i.e., performing block time deinterleaving, matched filtering, demodulation, and channel decoding. The received TIBWB-OFDM block is converted into the frequency-domain through a DFT operation, and, assuming that the used ZP is long enough to avoid ISI, frequency-domain equalization (FDE) is executed: either linear FDE such as zero-forcing (ZF) or minimum mean square error (MMSE) methods or nonlinear iterative equalization block-based methods such as the IB-DFE [7] may be employed at this stage. When it comes to the FDE of the TIBWB-OFDM symbol, this has several advantages over time-domain equalization (TDE) in outdoor high-mobility propagation environments and for channels with severe delay spreads, since the receiver complexity can be kept low [27]. In channels of which the impulse responses remain constant within one transmitted symbol period, frequency-domain received TIBWB-OFDM signal \(Y_{k}\) at each subcarrier \(k=0,\ldots,N_{b}-1\) (with \(N_{b}=N_{s}\times N(1+\beta)\)) can be expressed as  
\[Y_{k}=H_{k}X_{k}+N_{k}, \tag{12}\]  
where \(H_{k}\) represents the channel frequency response (CFR). By employing linear FDE, the estimated signal is given by  
\[\hat{X}_{k}=F_{k}Y_{k}, \tag{13}\]where \(F_{k}\) is the frequency response of the feedforward equalization filter. The two most popular linear equalization schemes are the ZF and MMSE equalizers. The ZF equalizer simply uses the inverse of the CFR, i.e.,  
\[F_{k}=\frac{H_{k}^{*}}{|H_{k}|^{2}}=\frac{1}{H_{k}}, \tag{14}\]  
with the received signal given by  
\[\hat{X}_{k}=\frac{H_{k}X_{k}}{H_{k}}+\frac{N_{k}}{H_{k}}=X_{k}+\frac{N_{k}}{H_{ k}}. \tag{15}\]  
Although its computational simplicity, this equalization technique results in noise enhancement caused on the term \(\frac{N_{k}}{H_{k}}\), especially in the carriers that suffer deep fading. An MMSE receiver tries to minimize \(E\{|\hat{X}_{k}-X_{k}|^{2}\}\) by taking SNR component, \(\gamma\), into account. Therefore, the equalization weight on the subcarrier \(k\) is given as  
\[F_{k}=\frac{H_{k}^{*}}{|H_{k}|^{2}+\frac{1}{\gamma}}, \tag{16}\]  
resulting in  
\[\hat{X}_{k}=\frac{H_{k}^{*}H_{k}X_{k}}{|H_{k}|^{2}+\frac{1}{\gamma}}+\frac{N_ {k}H_{k}^{*}}{|H_{k}|^{2}+\frac{1}{\gamma}}=\frac{|H_{k}|^{2}X_{k}}{|H_{k}|^{2 }+\frac{1}{\gamma}}+\frac{N_{k}H_{k}^{*}}{|H_{k}|^{2}+\frac{1}{\gamma}}. \tag{17}\]  
This equalizer has the advantage of minimizing the noise-enhancement problem in low SNR regimes, although it does not make a perfect inversion of the channel. When the SNR is high enough, it is also clear from (17) that the MMSE equalizer approaches the zero-forcing equalizer [19]. However, performance is far from that of the matched filter bound (MFB), so the alternative approach arrives by adopting nonlinear schemes such as decision feedback equalizers [28]. The IB-DFE equalizer is particularly suited for block-based single-carrier (SC) transmissions, and can deal with ISI and interblock interference (IBI) [7] by means of both a feedforward filter (\(F_{k}^{l}\)) that acts as a conventional FDE in order to decrease precursors created by the wireless channel, and a feedback filter (\(B_{k}^{l}\)) that attempts to cancel the remaining interferences through information from previous precursors. Note that in the case of a TIBWB-OFDM transmission, the whole received block is first processed as being a type of a block-based SC transmission [6]. By employing IB-DFE, the equalized TIBWB-OFDM symbol is given by  
\[\hat{X}_{k}^{(l)}=F_{k}^{(l)}Y_{k}-B_{k}^{(l)}\hat{X}_{k}^{(l-1)}, \tag{18}\]  
and \(\hat{X}_{k}^{(l-1)}\) denotes the frequency-domain signal estimated at the previous iteration by implementing either a _hard_ or _soft_ feedback filter decision [7].