# An Overview of FIR Filter Design in Future Multicarrier Communication Systems
## 3 FIR Prototype Filter Design
### Optimization Based Methods
#### 3.3.5 Karaboga's Method
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**Methods** & **Comments** & **Pros and Cons** \\ \hline [PERSON]’s method [131] & Design the filter using the PSO algorithm. & **Pros:** Simple and intuitive. **Cons:** Poor stopband attenuation performance of the filter under low-order conditions. \\ \hline [PERSON]’s method [132] & Design the filter using the DEPSO algorithm. & **Pros:** Compared with PSO algorithm, the filter has better convergence consistency. **Cons:** Improvement of the stopband attenuation is not obvious. \\ \hline [PERSON]’s method [133] & Design the filter using the RPSO algorithm. & **Pros:** Compared with PSO algorithm, the filter has better convergence and can avoid falling into local optimum. **Cons:** Improving the performance of the stopband attenuation is not obvious. \\ \hline [PERSON]’s method [134] & Design the filter using the GA algorithm. & **Pros:** Obtain near global optimum solutions. **Cons:** Slow convergence rate and poor stopband attenuation performance. \\ \hline [PERSON]’s method [135] & Design the filter using the DE algorithm. & **Pros:** Better convergence and acceptable computational complexity compared to GA. **Cons:** Improvement of the stopband attenuation is not obvious. \\ \hline [PERSON]’s method [98] & Directly optimize filter coefficients. & **Pros:** Minimize the ISI/ICI and the stopband energy for FBMC modulation. **Cons:** High computational complexity. **Pros:** Minimize the TF resolution. **Cons:** High first sidelobe. \\ \hline [PERSON]’s method [139] & Design the filter by convex optimization using DIRR algorithm. & **Pros:** Robust design under the phase and group delay constraints. **Cons:** Approximated solution. \\ \hline [PERSON]’s method [140] & Minimize the out of band pulse energy through a relaxed QCQP. & **Pros:** High symbol reconstruction performance and desirable spectral features. **Cons:** Approximated solution. \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of optimization based methods. ### Frequency Sampling Methods

The performances of PSD and BER about different prototype filters using frequency sampling techniques are given in Figure 9,b, respectively. From the Figure 9, it can be seen that the frequency sampling methods can achieve good PSD performance. The prototype filter designed by [PERSON]'s method has the best performance compared to other frequency sampling techniques, on the other hand, the prototype filter designed in Reference [123] has the worst BER performance while the others have the similar performance, as shown in Figure 9.

Generally, a small number of parameters are simply designed in frequency domain, thus frequency sampling methods are inclined to optimal design and narrowband filter design where only a few non-zero values are needed. Furthermore, it is obvious that there are few side-lobes which result in better spectrum utilization and improving spectral efficiency. For time complexity, [PERSON]'s method achieves the order of magnitude \(\mathcal{O}(N\log N)\) as most of other methods are its improved versions, and \(N\) denotes the filter length. When the value of \(N\) increases, low side-lobes may appear in the frequency domain and adjacent channel interference can be reduced, thus frequency sampling methods can be taken into consideration while low interference is required. In addition, frequency sampling techniques can satisfy the NPR condition, the research projects of 5G related networks, such as PHYDYAS and 5 GNOW, consider the frequency sampling technique for FBMC modulation. Frequency sampling methods are also applied for GFDM modulation [148; 149] as concluded in Table 7. However, frequency sampling methods generally have long filter length, which results in long latency, that is, there is a trade-off between high spectrum efficiency and low latency. Moreover, the position of the frequency control point is limited by \(N\)-sampling points on the frequency domain.