# An Overview of FIR Filter Design in Future Multicarrier Communication Systems
## 2 Multicarrier Modulations and FIR Filter
### FIR Filter  
In this subsection, we introduce the basic principle of digital FIR filters as well as the evaluation criteria of a designed filter. It is known that the condition of distortionless transmission and filtering is that the amplitude response of the system should be constant in the effective spectral range of the signal, and the phase response should be a linear function of the frequency (i.e., linear phase). The prototype filter is required to have linear phase characteristic in wireless communication applications. In contrast to the IIR filter, the FIR filter has the ability to achieve linear phase filtering [117]. In addition, as an all-zero filter, the hardware and software structures of the FIR filter could be established without considering the stability problem. Therefore, FIR filters have been widely used in the field of wireless communications [53]. Assuming the length of the impulse response \(h(n)\) of a linear phase FIR filter is \(N\), then the frequency response function is defined as  
\[H(e^{i\omega})=\sum_{n=0}^{N-1}h(n)e^{-i\omega n}, \tag{1}\]  
where \(\omega\) is the angular frequency. According to the parity of \(N\) and the symmetry of \(h(n)\), there are four types of FIR transfer functions. The condition for the most widely used low-pass FIR filter is given by  
\[h(n)=h(N-1-n),\quad 0\leq n\leq N-1. \tag{2}\]  
The ripples are observed in both the passband and stopband, and the maximum approximation error \(\delta_{p}\) is described by  
\[1-\delta_{p}\leq\mid H(e^{i\omega})\mid\leq 1,\quad\mid\omega\mid\leq\omega_{p}. \tag{3}\]  
The amplitude of the stopband is approximated by the maximum error \(\delta_{s}\), that is,  
\[\mid H(e^{i\omega})\mid\leq\delta_{s},\quad\omega_{s}\leq\mid\omega\mid\leq\pi, \tag{4}\]  
where \(\omega_{p}\) and \(\omega_{s}\) are passband and stopband boundary frequencies, \(\delta_{p}\) and \(\delta_{s}\) are passband and stopband ripples, respectively (Gibbs effect), and \(\omega_{s}-\omega_{p}\) is the transition bandwidth. In order to make designed filters close to the ideal low-pass filter, several evaluation criteria are given in [50; 51], including least-squares (LS), minimax, peak-constrained least-squares (PCLS), minimum total interference (ICI and ISI), and and so forth. * **LS criterion** The goal of the least-squares criterion is to minimize the stopband energy of the filter, whose objective function is \[J=\int_{\omega_{s}}^{\pi}\mid H(e^{i\omega})\mid^{2}d\omega.\] (5)
* **Minimax criterion** The goal of the minimax criterion is to minimize the maximum stopband ripple, and its objective function can be written as \[J=\max_{\omega\in[\omega_{s},\pi]}\mid H(e^{i\omega})\mid.\] (6)
* **PCLS criterion** The PCLS criterion establishes a trade-off between the LS and the minimax criteria. The PCLS criterion can be described as below \[ \begin{array}{l}J=\int_{\omega_{s}}^{\pi}\mid H(e^{i\omega})\mid^{2}d\omega\\ \text{s.t.}\quad\mid H(je^{i\omega})\mid\leq\delta,\end{array} \] (7) where \(\delta\) is a prescribed value. If \(\delta\) is close to zero, the PCLS criterion approaches the minimax criterion, while in the limit of \(\delta\), that is, \(\delta\) is up to infinite, the criterion turns out the LS criterion. * **Minimum total interference criterion** This criterion is to minimize the total interference of ICI and ISI for filter bank structure. Its objective function is defined as \[J=\text{ISI}+\text{ICI},\] (8) where \[\text{ISI}=\max_{k}\left(\sum_{n}([T_{\text{TMUX}}(n)]_{k,k}-\delta(n-\triangle) )^{2}\right),\] (9)\[\text{ICI}=\max_{k}\left(\sum_{l=0,l\
eq k}^{N-1}\sum_{n}([T_{\text{TMUX}}(n)]_{k,l })^{2}\right), \tag{10}\]  
where \(T_{\text{TMUX}}(n)\) is the transfer matrix, the element \([T_{\text{TMUX}}(n)]_{a,b}\) represents the relationship between the input signal \(X(z^{N})\) and the output signal \(Y(z^{N})\), where \(z\) is the complex argument of z-transform. \(\delta(n)\) is the ideal impulse, and \(\triangle\) denotes the delay of the TMUX system.