# 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
In order to satisfy both constraints on magnitude and phase responses of the filter, a non-convex quadratic constrained quadratic programming (QCQP) is constructed. Then, by relaxing the QCQP, a convex SDP is obtained, where the variables of optimization are limited to rank-1. Finally the global optimum can be found by using a convex solver. To obtain the rank-1 solution, a novel directed iterative rank refinement (DIRR) algorithm providing monotonic improvement is proposed, and a sequence of convex optimization problems are solved to minimize an adaptively chosen cost function. The performance of this design method is extensively illustrated over design cases under a variety of constraints. #### 3.3.9. [PERSON]'s Method

A relatively new method of convex optimization is proposed to obtain the filter coefficients for FBMC, in pursuit of superior spectrum features while maintaining high symbol reconstruction quality [140]. The objective function is constructed as the minimization of out-of-band pulse energy. At the meanwhile the constraints include a maximum tolerable self-interference level and a fast spectrum decay. The main idea of the work in Reference [140] locates in transforming the formulated problem, which belongs to a non-convex QCQP, into a convex QCQP. In order to circumvent the non-convexity, a relaxation is utilized to transform the norm-2 equality constraint into a norm-1 equality constraint. Thus the transformed problem benefits from existent optimization tools. The comments, pros and cons of the above optimization based methods are summarized in Table 5.

## 4 Discussion

Future wireless communication networks will impose strict requirements on multicarrier modulation schemes in terms of spectral efficiency, latency, computational complexity, and so forth. As shown in Table 6, crucial characteristics and applicability of five promising modulation waveforms are clearly reflected, including spectral efficiency, out of band, CP, synchronization requirement, latency, effect of frequency offset, PAPR, computational complexity, and short-burst traffic, which implicate a concrete limit to the real use of different MCM schemes. To further learn about the merits and defects of various MCM waveforms, interested readers are referred to the related works in Reference [141] for more detailed information. Since prototype filter design directly affects most characteristics of modulation waveforms such as spectral efficiency [142; 143], out of band [144], it has attracted extensive research attention in the field of MCM communication. In recent years, various prototype filter design methods are designed in accordance with the above practical requirements of modulation waveforms in Table 6, and some commonly used FIR design methods for different multicarrier modulation waveforms are concluded in Table 7. The contents of this table will be continuously enriched as time goes on. In addition, we analyze the MCM system performance using three types of FIR design methods in terms of PSD and BER, which are two another important evaluation criteria [81; 82; 145; 146; 147]. The simulation results of PSD and BER performances are presented in Figure 9, where the FBMC is selected as the system modulation type, the number of subcarriers for FBMC is 1024, and the offset quadrature amplitude modulation (OQAM) technology is adopted. In the following, some critical characteristics and application scope of the three categories of FIR design methods are discussed and summarized, respectively.