# An Overview of FIR Filter Design in Future Multicarrier Communication Systems
## 3 FIR Prototype Filter Design
#### 3.3.5 Karaboga's Method
This implies that the sampling frequency can only be an integral multiple of \(2\pi/N\), resulting in that the cutoff frequency of filter is hard to control. If the cutoff frequency is freely selected, we must increase the number of sampling points \(N\), that is, increasing the filter length, which is not conducive to short uplink burst communication in 5G scenarios. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **OFDM** & **FBMC** & **GFDM** & **UFMC** & **F-OFDM** \\ \hline
**Spectral Efficiency** & Medium & High & Medium & High & Medium \\
**Out of Band** & High & Low & Low & Low & Low \\
**Cyclic Prefix** & Yes & No & Yes & No & Yes \\
**Synchronized Requirement** & High & Low & Medium & Low & Low \\
**Latency** & Short & Long & Short & Short & Short \\
**Effect of Frequency Offset** & Medium & Low & Medium & Medium & Medium \\
**PAPR** & High & High & Low & Medium & High \\
**Computational Complexity** & Low & High & High & High & Medium \\
**Short-Burst Traffic** & No & No & Yes & Yes & No \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Respective characteristics and applicability of different multicarrier modulation waveforms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **OFDM** & **FBMC** & **GFDM** & **UFMC** & **F-OFDM** \\ \hline Frequency Sampling Methods & — & [51; 118] & [148; 149] & — & — \\ Windowing based Methods & [150; 151] & [4] & [40; 152] & [42] & [45; 46; 145; 153] \\ Optimization based Methods & [154] & [140; 155; 156; 157] & [158; 159; 160; 161] & [162; 163] & — \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Summary of filter design methods used in existing multicarrier modulation waveforms. Figure 9: (**a**,**c**,**e**) The PSD performance comparison of FBMC based systems using different prototype filter design methods. (**b**,**d**,**f**) The BER performance comparison of FBMC based systems using different prototype filter design methods. ### Windowing Based Methods

The performances of PSD and BER about different prototype filters using windowing based techniques are given in Figure 9c,d, respectively. The aforementioned windowing based techniques have similar PSD and BER performances while the method in Reference [129] performs relatively poor compared with other methods. The windowing based FIR filters are designed in time domain, and the basic idea is that the linear phase FIR filter multiplies by a specific windowing function, thus the time complexity is proportional to \(\mathcal{O}(N)\). Generally, windowing based techniques can be seen as improved versions of classical windowing functions, nevertheless the performance improvement is not very remarkable compared with typical techniques such as Hamming window, Hanning window, Blackman window and so on. From the view of practical use, windowing based functions are easy to implement and bring little computational complexity burden on systems, therefore they are able to be used for the prototype filter design in almost all multicarrier modulation systems, as shown in Table 7. However, it is difficult to accurately control the passband cutoff frequency of the filter. The windowing process leads to a truncation effect and hence produces a transition band. Although increasing the length of the window can reduce the transition band, the amplitude of fluctuation cannot be suppressed due to Gibbs phenomenon. Changing the windowing shape can reduce the passband/stopband attenuation, but at the expense of increasing the width of transition band. ### Optimization Based Methods

The performances of PSD and BER about different prototype filters using optimization based techniques are given in Figure 9e,f, respectively. The evolutionary algorithms in References [131; 132; 133; 134; 135] are used for linear phase FIR filter design in recent years, and they could be taken into account in multicarrier modulation applications. In this paper, FBMC based systems using five different evolutionary algorithms are simulated, as shown in Figure 9e,f.