单位冲激抽样序列频谱与傅里叶级数收敛性分析

Analysis of frequency spectrum and the Fourier series convergence of the unit impulse train

由傅里叶变换的时移和频移特性,单位冲激抽样序列有两种频谱函数:周期型频谱和级数型频谱,其中周期型频谱函数的推导并不严谨,缺少傅里叶级数收敛性分析。对此提出通过证明两频谱函数等价来验证周期冲激信号傅里叶级数的收敛性。根据脉冲函数定义,运用极限和积分思想,利用抽样函数性质,证明了级数型频谱函数本质是强度和周期均为圆频率的频域冲激序列,验证了冲激抽样序列傅里叶级数的收敛性,周期型频谱函数的傅里叶级数与级数型频谱函数的分析也再次验证了级数的收敛性,但不能验证冲激点不存在吉布斯现象的观点。

According to the shift theorem in the time and frequency domain,the unit impulse train has two frequency spectrum functions which were called the periodic spectrum and the series spectrum.The periodic spectrum is not rigorous for the lack of the analysis of the Fourier series convergence.According to the definition of the Dirac function,it proved that the series spectrum is in fact an impulse train in the frequency domain whose energy and period are equal to the angular frequency value.The sampling function characteristic,the limit and integral theory were made full use of.It shows that the Fourier series of the unit impulse train is convergent.The convergence was verified again by the analyses about the difference between the series spectrum and the Fourier series of the periodic spectrum.It failed to verify the miss of the Gibbs phenomenon.