Nh-1

线性卷积: y(n) = sum h(k)x(n-k),        0<=n<=Nh+Nx-1

                 k=0

                 Nh-1

循环卷积: y(n) = sum h(k) x((n-k)mod Nx), 0<=n<=max(Nh,Nx)

                 k=0

频域相乘等价于时域的循环卷积.当对x补0构成Nx+Nh-1的序列x_pad时,则循环卷积的结果与线性卷积一致, 当对h补0构成Nx+Nh-1的序列h_pad后有: C = IFFT( FFT(x_pad).*FFT(h_pad) ), 结果与线性卷积一致.

令h = [1,2,1], x = [1,2,3]

线性卷积                                                循环卷积

-----------------------------------      ----------------------------------------

                        [1,2,1]                                     [1,2,1]

                    [3,2,1]                                         [1,3,2]

                    ---------------                                 --------

y(0)=sum h(n)x(0-n)  0+0+1+0+0 = 1       y(0)=sum h(n)x((0-n)mod3)   1+6+2 = 9

                        [1,2,1]                                     [1,2,1]

                      [3,2,1]                                       [2,1,3]

                    ---------------                                 --------

y(1)=sum h(n)x(1-n)    0+2+2+0 = 4       y(1)=sum h(n)x((1-n)mod3)   2+2+3 = 7

                        [1,2,1]                                     [1,2,1]

                        [3,2,1]                                     [3,2,1]

                    ---------------                                 --------

y(2)=sum h(n)x(2-n)      3+4+1 = 8       y(2)=sum h(n)x((2-n)mod3)   2+2+3 = 8

                        [1,2,1]             y_circular = [9,7,8]

                          [3,2,1]          

                    ---------------      

y(3)=sum h(n)x(3-n)      0+6+2+0= 8

                        [1,2,1] 

                            [3,2,1]          

                    ---------------      

y(3)=sum h(n)x(4-n)      0+0+3+0+0= 3

y_linear = [1,4,8,8,3];

如果对x补0构成 x_pad = [1,2,3,0,0], 则与h的循环卷积等于[1,4,8,8,3].