## 3 Theoretical Background
### Multicarrier Modulations
#### 3.1.2 Ufmc  
The UFMC technique overcomes the limitation of the OFDM by adding generalized pulse-shaping filters that ensure the frequency- and time-domain spectral localization. Its main characteristics are (1) the use of sub-band filtering instead of subchannel filtering in FBMC, (2) the whole band filtering in F-OFDM, and (3) the exploitation of a complex data symbol as in OFDM.  
Figure 1: CP-OFDM transceiver block diagram.  
Figure 2: CP insertion.  
* Transceiver description The UFMC transceiver scheme is illustrated in Figure 3. As shown, QAM data symbols are divided into sub-groups that undergo IFFT processing separately and are filtered using a sub-band filtering technique. The filter prototype used is the Finite Impulse Response (FIR) Dolph-Chebyshev filter of length \(L\). The aim of introducing an FIR filter to filter each sub-band is to reduce the Out-Of-Band (OOB) spectrum. All results from the sub-group data processing are summed up and therefore constitute the UFMC symbol. This symbol is transmitted through the fading channel, and AWGN is added to the signal. At the receiver side, a zero-padding technique is used to fill the data of length \(N_{FFT}+L-1\) with \(N_{FFT}-L+1\) zeros samples to reach a length of \(2\times N_{FFT}\). The purpose of this zero-padding technique is to exploit the FFT processing and filter properties to recover data [9]. Therefore, the FFT technique is applied to pass data from the time domain to the frequency domain. After that, a downsampling technique is applied to recover one sample out of two on all \(2\times N_{FFT}\) samples. Finally, the same subchannel equalizer (as applied in OFDM) is also applied on the recovered data to retrieve QAM data symbols.  
* Numerical signal expression Suppose that \(B*n_{i}\) represents complex data symbols resulting from QAM baseband modulation. As previously presented, the full band of \(N\) subcarriers is partitioned into \(B\) sub-bands. The number of subcarriers in each sub-band is fixed (\(n_{i}\) complex QAM symbols). In the transmitter section, to cancel sub-band interference, IFFT processing is employed to transform the frequency-domain signal into a time-domain signal. At each \(N_{FFT}\) point, sub-bands are computed, and zeros are attributed to unreserved subcarriers. The filtering operation is applied by performing a linear convolution between data IFFT processing and the filter components. The time-domain transmission \(x\) vector (at the input of the channel) for a particular multicarrier symbol is the superposition of the sub-band-wise filtered components, with filter length \(L\) and FFT length \(N_{FFT}\) (for simplicity, the time index m is dropped) [30]: \[x_{(N_{FFT}+L-1)\times 1}=\sum_{i=1}^{B}F_{i_{(N_{FFT}+L-1)\times N_{FFT}}} \times V_{i_{N_{FFT}\times n_{i}}}\times S_{i_{n_{i}\times 1}};\] (3) For each of the \(B\) sub-bands, indexed \(i\), the \(n_{i}\) complex QAM symbols gathered in \(S_{i}\) representing a frequency-domain signal are converted to a time-domain signal by the IFFT-matrix \(V_{i}\). \(V_{i}\) includes the relevant columns of the inverse Fourier matrix in accordance with the respective sub-band position within the overall available frequency range. \(F_{i}\) is  
Figure 3: UFMC block diagram.  
a Toeplitz matrix, composed of the filter impulse response which is used to perform the linear convolution.