线性卷积可以转化为循环卷积，循环卷积可以转化为频域的乘法，从而线性卷积可以采用基于FFT（(快速Fourier变换）的方法进行计算．本文给出了一种基于广义离散Fourier变换的线性卷积计算方法．本文首先分析了线性卷积和循环卷积的关系．然后，线性卷积的计算转化成一个特殊的Toeplitz矩阵与向量的乘积．然后，通过利用信号和滤波器的广义离散Fourier变换以及反变换，推导了这个乘积的快速算法．另外，本文推导方法还可以得到基于参数为$-1$的广义离散Fourier变换计算线性卷积的方法．

Linear convolutions can be converted to circular convolutions so that a fast trans-form with a convolution property can be used to implement the computation, which method is known as the FFT-based fast algorithm of linear convolution. In this paper, a novel proof of the computation of linear convolution based on Generalized Discrete Fourier Transform (GDFT) is constructed. Firstly, a relationship of linear convolution and circular convolution is thoroughly analyzed. Secondly, the computation of the linear convolution is translated to the multiplication of a special Toeplitz matrix and the signal. Lastly, this multiplication is accomplished by the inverse GDFT of the product of the GDFTs of the signal and the filter. Furthermore, a new method about the computation of complex linear convolution is constructed by using the GDFT with parameter $-1$, which is not considered in the previous literatures.