# An Overview of FIR Filter Design in Future Multicarrier Communication Systems
## 3 FIR Prototype Filter Design
### Windowing Based Methods  
The idea of windowing based methods is to use the digital FIR filter to approximate the desired filtering characteristics. Assuming that the desired filter frequency response function is \(H_{d}(e^{j\omega})\), the impulse response is denoted as \(h_{d}(n)\). Considering the windowing based methods focus on the perspective of time domain, the ideal \(h_{d}(n)\) with a certain shape of window function is intercepted as \(h(n)\) with finite length, whose frequency response \(H(e^{j\omega})\) approximates the desired frequency response \(H_{d}(e^{j\omega})\). The specific design steps of windowing based methods are as follows:  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Prototype Filter Design** & **Amplitude Distortion Peak** & **Aliasing Error** & **Minimum Stopband Attenuation** \\ \hline Ref. [120] & \(7.0449\times 10^{-4}\) & \(-139.48\) dB & 78 dB \\ Ref. [123] & \(3.5001\times 10^{-5}\) & \(-151.39\) dB & 108 dB \\ Ref. [124] & \(1.2848\times 10^{-4}\) & \(-93.27\) dB & 69 dB \\ \hline \hline \end{tabular}
\end{table}
Table 1: Analysis of prototype filter design in [120; 123; 124] by amplitude distortion peak, aliasing error, minimum stopband attenuation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Comments** & **Pros and Cons** \\ \hline [PERSON]’s method [118] & Design the filter parameter under the constraints of controlling stopband performance and satisfying Nyquist criterion. & **Pros:** Favorable stopband attenuation. \\ \hline [PERSON]’s method [51] & Optimize the filter parameters according to different evaluation criteria under the condition of Nyquist criterion. & **Pros:** Good stopband attenuation; Flexibly select suitable filter parameters according to different evaluation criteria. \\ \hline [PERSON]’s method I [120] & An optimization scheme based on frequency sampling is proposed to obtain the filter parameters. & **Pros:** Low computational complexity; Design a filter with arbitrary length. \\ \hline [PERSON]’s method I [120] & Based on the optimization algorithm proposed in [120], the objective function is improved to achieve multi-objective optimization. & **Pros:** The stopband energy and stopband attenuation are minimized simultaneously. \\ \hline [PERSON]’s method [124] & The VLEP algorithm [124] is proposed based on the classical and fast evolutionary programming algorithm. & **Pros:** Robust to the initial conditions; The minimum of the objective function can be reliably obtained. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of frequency sampling methods.  
**step 1:**: Taking linear phase low-pass FIR filter as an example, the general selection of \(H_{d}(e^{j\omega})\) is  
\[H_{d}(e^{j\omega})=\begin{cases}e^{-j\omega\tau},&|\omega|\leq\omega_{c},\\ 0,&\omega_{c}\leq|\omega|\leq\pi,\end{cases} \tag{39}\]  
where \(\tau\) is a constant.
**step 2:**: Determine \(h_{d}(n)\) via IDFT  
\[h_{d}(n)=\frac{1}{2\pi}\int_{-\omega_{c}}^{\omega_{c}}H_{d}(e^{j\omega})e^{j \omega n}d\omega=\frac{\sin[\omega_{c}(n-\tau)]}{\pi(n-\tau)}. \tag{40}\]
**step 3:**: The impulse response \(h(n)\) of the linear phase FIR filter is obtained by multiplying a specific window \(w(n)\), as below  
\[h(n)=h_{d}(n)w(n). \tag{41}\]  
Thus the corresponding \(H(e^{j\omega})\) is obtained by DFT.