二维傅里叶变换的推导过程
非周期傅里叶变换公式推导？
非周期傅里叶变换公式推导？
推导过程:
F m ∑ n 0 N 1 f n e 2 π i m n / N f n 1 N ∑ m 0 N 1 F m e 2 π i m n / N F_msum_{n0}^{N-1}f_ne^{-2pi imn/N}leftrightarrow f_nfrac{1}{N}sum_{m0}^{N-1}F_me^{2pi imn/N} Fmn0∑N1fne2πimn/NfnN1m0∑N1Fme2πimn/N
f ( x ) ∑ n 0 N 1 f n δ ( x x n ) f(x)sum_{n0}^{N-1}f_ndelta(x-x_n) f(x)n0∑N1fnδ(xxn)
F m ∫ T T ∑ n 0 N 1 f n δ ( x x n ) e i x k m d x ∑ n 0 N 1 ∫ f n δ ( x x n ) e i x k m d x ∑ n 0 N 1 f n e i x n k m egin{aligned} F_m int_{-T}^{T}sum_{n0}^{N-1}f_ndelta(x-x_n)e^{-ixk_m}dx sum_{n0}^{N-1}int f_ndelta(x-x_n)e^{-ixk_m}dx sum_{n0}^{N-1}f_ne^{-ix_nk_m} end{aligned} Fm∫TTn0∑N1fnδ(xxn)eixkmdxn0∑N1∫fnδ(xxn)eixkmdxn0∑N1fneixnkm
接下来我们假设 d x , d k dx,dk dx,dk分别是 { x n } {x_n} {xn}, { k n } {k_n} {kn}的间距，那么：
x n n d x , k m m d k x_nndx,qquad k_m mdk xnndx,kmmdk
代入上式：
F m ∑ n 0 N 1 f n e i x n k m ∑ n 0 N 1 f n e i m n d x d k egin{aligned} F_m sum_{n0}^{N-1}f_ne^{-ix_nk_m} sum_{n0}^{N-1}f_ne^{-imndxdk} end{aligned} Fmn0∑N1fneixnkmn0∑N1fneimndxdk