Fourier Transform

目录

• Fourier Transform
  • 基本的定义
  • 性质
    • 对称性
  • 卷积
  • FFT

Fourier Transform

基本的定义

严格来说, 傅里叶变换是Schwartz space 上的一一映射, 对于(L^1), 即可积函数我们都可以找到其对应的傅里叶变换.

符号	定义
傅里叶变换: (hat{f}(u))	(int_{-infty}^{+infty} f(t) e^{-j2pi u t} mathrm{d}t)
逆傅里叶变换: (check{F}(t))	(int_{-infty}^{+infty} F(u) e^{j2pi tu} mathrm{d}u)​
离散傅里叶变换: (F_m=hat{f}_m)	(sum_{n=0}^{N-1} f_n e^{-j2pi nm/N})​
离散逆傅里叶变换: (f_n = check{F}_n)	(frac{1}{N}sum_{m=0}^{N-1} F_m e^{j2pi mn /N})
卷积: (f star g (t))	(int_{-infty}^{+infty} f( au) g(t- au) mathrm{d} au)
correlation: (f ullet g(t))​	(int_{-infty}^{+infty}f^*( au)g( au + t)mathrm{d} au)
二元傅里叶变换: (hat{f}(u, v))	(int_{-infty}^{+infty}int_{-infty}^{+infty} f(t, z) e^{-j2pi ut} e^{-j2pi vz} mathrm{d}t mathrm{d}z)
二元傅里叶逆变换: (check{F}(t, z))	(int_{-infty}^{+infty}int_{-infty}^{+infty} F(u, v) e^{j2pi tu} e^{-j2pi zv} mathrm{d}t mathrm{d}z)
二元卷积: (f star g (t, z))	(int_{-infty}^{+infty}int_{-infty}^{+infty} f( au, omega) g(t - au, z - omega) mathrm{d} au mathrm{d}omega)
共轭	(f^*(t) = [r(t) + j i(t)]^* = r(t) - ji(t), quad r(t), i(t) in mathbb{R})

注: 不同版本的傅里叶变换的定义存在系数上的差异.

性质

	(hat{})	(mathop{Leftrightarrow} limits^{mathcal{F}})	(check{})
1	(f(t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u))
2	(f star g (t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) cdot G (u))
3	(f ullet g(t))​	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F^*(u) cdot G(u))
4	$f(t) cdot g(t) $	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F star G (u))
5	(f(t) + g(t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) + G(u))
6	(f(t + t_0))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(e^{j2pi t_0 u} F(u))
7	(f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(-u))
8	(F(u)=hat{f}(u))​	(mathop{Leftrightarrow} limits^{mathcal{F}})	(f(-t) = hat{hat{f}}(t))
以上	对于	离散	成立
9	(f(at))	(mathop{Leftrightarrow} limits^{mathcal{F}})	$frac{1}{
10	(f'(t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(j2pi u F(u))
11	(-j2pi tf(t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F'(u))
12	(f(R mathbf{t})), (R^TR=I)	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(Rmathbf{u}))

注: 虽然以上性质除11外都是用了标量, 但是实际上对于一般的傅里叶变换也是成立的.

1. pass

2. 卷积:

   [egin{array}{ll} (f star g)^{hat{}}(u) &= int f star g (t) e^{-j2pi u t} mathrm{d}t \ &= int int f(t- au) g ( au) mathrm{d} au e^{-j2pi u t} mathrm{d}t \ &= int int f(t- au) g ( au) e^{-j2pi u t} mathrm{d}t mathrm{d} au quad ightarrow mathrm{Fubini}\ &= int int f(t- au) e^{-j2pi u (t- au)} mathrm{d}t : g( au)e^{-j2pi u au} mathrm{d} au quad \ &= F(u) cdot G(u). end{array} ]

   (Leftarrow)是类似的证明.

3. correlation:

   [egin{array}{ll} (f ullet g)^{hat{}}(u) &= int f*g(t) e^{-j2pi ut} mathrm{d}t \ &= int int f^*( au) g( au + t) mathrm{d} au e^{-j2pi ut} mathrm{d}t \ &= int f^*( au) int g( au + t) e^{-j2pi ut} mathrm{d}t mathrm{d} au \ &= G(u)int f^*( au) e^{j2pi u au} mathrm{d} au \ &= G(u)[int f( au) e^{-j2pi u au} mathrm{d} au]^* \ &= F^*(u)G(u) end{array} ]

4. 乘积的证明和上面是一样的, 无非是作用于(check{})上.

5. pass

6. 平移:

   [(f(t+t_0))^{hat{}} = int f(t+t_0) e^{-j2pi ut} mathrm{d}t =e^{j2pi u t_0}int f(t+t_0) e^{-j2pi u(t+t_0)} mathrm{d}t = e^{j2pi t_0u} F(u). ]

7. 逆:

   [(f(-t))^{hat{}} = int f(-t) e^{-j2pi u t} mathrm{d}t = int f(t) e^{j2pi u t} mathrm{d}t = F(-u). ]

8. (hat{F}(t) = int F(u) e^{j2pi t u} mathrm{d}u = (F(-u))^{check{}}(t) = f(-t)).

9. scaling:

   [(f(at))^{hat{}}(u) = int_{-infty}^{+infty} f(at) e^{-j2pi ut} mathrm{d}t = frac{1}{|a|}int_{-infty}^{+infty} f(z) e^{-j2pi frac{u}{a}z} dz = frac{1}{|a|} F(frac{u}{a}). ]

10. 导函数:

    [egin{array}{ll} f'(t) &= frac{mathrm{d}}{mathrm{d}t} int_{-infty}^{+infty} F(u) e^{j2pi t u} mathrm{d}u \ &= int_{-infty}^{+infty} F(u) frac{mathrm{d}}{mathrm{d}t} e^{j2pi t u} mathrm{d}u \ &= int_{-infty}^{+infty} F(u) (j2pi u) e^{j2pi t u} mathrm{d}u \ &= (j2pi u F(u))^{check{}}. end{array} ]

11. 同上

12. 这个性质说明了, 对原图像加以旋转, 等价地在频域上加以同样的旋转.

    [egin{array}{ll} G(mathbf{u}) &=int f(Rmathbf{t}) e^{-j2 pi mathbf{u}^T mathbf{t}} mathrm{d}mathbf{t} \ &=int f(mathbf{x}) e^{-j2 pi (Rmathbf{u})^T mathbf{x}} |R^{-1}|mathrm{d}mathbf{x} \ &=int f(mathbf{x}) e^{-j2 pi (Rmathbf{u})^T mathbf{x}} mathrm{d}mathbf{x} \ &=F(Rmathbf{u}) end{array} ]

对称性

主要指的:

1. 奇函数

   [f(x) = -f(-x). ]

2. 偶函数

   [f(x) = f(-x). ]

3. 共轭对称

   [f^*(x) = f(-x). ]

考查(f(t)=r(t) + j i(t))与(F(u) = R(u) + j I(u))之间的关系:

	(hat{ })	(Leftrightarrow)	(check{ })
1	(f(t) = r(t))	(Leftrightarrow)	(F^*(u) = F(-u))
2	(f(t)=r(t))	(Leftrightarrow)	(R(u)=R(-u), I(u)=-I(-u))
3	(f(t) = ji(t))	(Leftrightarrow)	(F^*(-u) = -F(u))
4	(f(t) = ji(t))	(Leftrightarrow)	(R(u)=-R(-u), I(u)=I(-u))

	(hat{})	(mathop{Leftrightarrow} limits^{mathcal{F}})	(check{})
5	(f(-t)in mathbb{R})	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F^*(u) in mathbb{C})
6	(f(-t) in mathbb{C})	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(-u) in mathbb{C})
7	(f^*(t) in mathbb{C})	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F^*(-u) in mathbb{C})
8	(f(t) in mathbb{R}), 且(f(t) = f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) in mathbb{R}), 且满足(F(u)=F(-u))
9	(f(t) in mathbb{R}), 且(f(t) = -f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) = jI(u) in mathbb{C}), 且(F(u)=-F(-u))
10	(f(t) = ji(t) in mathbb{C}), 且(f(t) = f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) = jI(u) in mathbb{C}), 且(F(u)=F(-u))
11	(f(t)=ji(t) in mathbb{C}), 且(f(t)=-f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) in mathbb{R}), 且满足(F(u)=-F(-u))
12	(f(t) in mathbb{C}), 且(f(t) = f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) in mathbb{C}), 且满足(F(u) = F(-u))
13	(f(t) in mathbb{C}), 且(f(t) = -f(-t))	(mathop{Leftrightarrow} limits^{mathcal{F}})	(F(u) in mathbb{C}), 且满足(F(u) = -F(-u))

1. real ->((Leftarrow)采用反证法即可)

   [egin{array}{ll} F^*(u) &=[int f(t) e^{-j2pi ut} mathrm{d}t]^* \ &=int f^*(t) e^{j2pi ut} mathrm{d}t \ &=int f(t) e^{-j2pi (-u)t} mathrm{d}t ightarrow f*(t) = r^*(t) = r(t)\ &=F(-u). end{array} ]

2. 由1易证.

3. imaginary ->

   [egin{array}{ll} F^*(-u) &=[int f(t) e^{j2pi ut} mathrm{d}t]^* \ &=int f^*(t) e^{-j2pi ut} mathrm{d}t \ &=int -f(t) e^{-j2pi ut} mathrm{d}t ightarrow f^*(t) = -ji(t)=-f(t) \ &=-F(u). end{array} ]

4. 由3易证.

5. 由性质6以及上述的对称性3可知((f(-t))^{hat{}}= F(-u) = F^*(u)).

6. 由性质6可知((f(-t))^{hat{}} = F(-u)).

7. [int f^*(t) e^{-j2pi ut} mathrm{d}t = [int f(t) e^{j2pi ut} mathrm{d}t]^* = F^*(-u). ]

8. [F(u) = (f(t))^{hat{}} = (f(-t))^{hat{}} = F(-u) = F^*(u). ]

9. [F(u) = (f(t))^{hat{}} = -(f(-t))^{hat{}} = -F(-u) = -F^*(u). ]

10. [F(u) = (f(t))^{hat{}} = (f(-t))^{hat{}} = F(-u) = -F^*(u). ]

11. [F(u) = (f(t))^{hat{}} = -(f(-t))^{hat{}} = -F(-u) = F^*(u). ]

12. [F(u) = (f(t))^{hat{}} = (f(-t))^{hat{}} = F(-u). ]

13. [F(u) = (f(t))^{hat{}} = -(f(-t))^{hat{}} = -F(-u). ]

卷积

这里特别说明离散卷积的一个重要性质:
注意到

[h star f (n) = sum_{k=0}^{N-1} h(n-k) f(k) = a^T_n f, \ a(n) = [h(0), h(n-1), cdots, h(n-N+1)], ]

则

[A = [a_0, a_1, cdots, a_{N-1}] = left [ egin{array}{cccc} h(0) & h(1) &cdots & h(N-1) \ h(-1) & h(0) & cdots & h(N-2)\ vdots & vdots & ddots & vdots \ h(1-N) & h(2-N) & cdots & h(0), end{array} ight ] [h star f] = A^T f. ]

定义

[h'(n) = h(-n), ]

则有

[[h' star f] = Af. ]

FFT

考虑离散傅里叶变换:

[F_m = sum_{n=0}^{N-1} f_n e^{-j2pi nm/N}, m=0,1,cdots, N-1. ]

每次计算包含(N)个乘法, 故傅里叶变换的计算量是(N^2)级别的.

下面假设(N = P * Q, P, Q in mathbb{N}^+), 则

[m = uQ + v, quad u=0,1,cdots, P-1, v=0,1,cdots, Q-1, \ n = sP + t, quad s=0,1,cdots, Q-1, t=0,1,cdots, P-1. \ ]

注意到:

[nm = (sP + t)(uQ + v) = su N + svP + tm. ]

此时, (m, n)与唯一的((u,v), (s, t))匹配, 则傅里叶变换可以表示为:

[egin{align} F_m &= sum_{n=0}^{N-1}f_n e^{-j2pi nm / N} \ &= sum_{n=0}^{N-1}f_n e^{-j2pi (sP+t)(uP+v)/ N} \ &= sum_{n=0}^{N-1}f_n e^{-j2pi (suN + svP + tm)/ N} \ &= sum_{n=0}^{N-1}f_n e^{-j2pi (svP + tm)/ N} \ &= sum_{t=0}^{P-1}e^{-j2pi tm / N}sum_{s=0}^{Q-1} f_{sP+t} e^{-j2pi sv/ Q} \ end{align} ]

令

[F_v^t := sum_{s=0}^{Q-1} f_{sP+t} e^{-j2pi sv /Q}, quad v = 0,1, cdots, Q-1, t=0,1,cdots, P-1. ]

显然总共需要(QN)次乘法, 而

[F_m = sum_{t=0}^{P-1} e^{-j2pi tm / N} F_v^t, quad m=0,1,cdots, N-1, \ F_{uQ+v} = sum_{t=0}^{P-1} e^{-j2pi tv /N}F_v^t e^{-j2pi tu / P} . ]

总共有(PN)次乘法, 故总共有((P+Q) N)次乘法运算.
由于每个(F^t)都是一个独立的傅里叶变换过程, 故倘若(Q)能够进一步分解, 计算量可以进一步降低.
甚至, 若假设(N = P^J), 则可以证明最后的计算量可以分解为

[PJN = P Nlog_P N, ]

特别的, 常常(N=2^J), 则计算量是(Nlog_2 N)量级的.

注: 卷积运算([fstar g]_n, n=0,cdots, N-1), 计算量也是(N^2)级别的, 但是通过FFT, 应当是((PNlog_P + 1)N)​乘法次数, 也是可以降低计算量的.

逆傅里叶变换实际上就是

[hat{F^*}^* = [sum_{m=0}^{N-1} F_m^* e^{-j2pi mn/N}]^*, ]

虽然下面的代码我并没有采取这种策略.

from typing import Iterable, Union
import numpy as np



def factorization(N: int):
    for P in range(2, N + 1):
        if N % P is 0:
            return P

def dft(arr: Iterable, m: int):
    N = len(arr)
    basis = np.arange(N) * m * 2 * np.pi * 1j * -1
    basis = np.exp(basis / N)
    return arr @ basis

def idft(arr: Iterable, m: int):
    N = len(arr)
    basis = np.arange(N) * m * 2 * np.pi * 1j
    basis = np.exp(basis / N) / N
    return arr @ basis

def basis_dft(N: int):
    basis = np.outer(np.arange(N), np.arange(N))
    basis = basis * 2 * np.pi * 1j * -1
    return np.exp(basis / N)

def basis_idft(N: int):
    basis = np.outer(np.arange(N), np.arange(N))
    basis = basis * 2 * np.pi * 1j
    return np.exp(basis / N) / N

def fft(arr: Iterable) -> np.ndarray:
    N = len(arr)
    P = factorization(N)
    if P is N:
        return arr @ basis_dft(N)
    Q = N // P
    Fvt = np.array([fft(arr[t::P]) for t in range(P)]).T # Q x P
    dummy = np.outer(np.arange(Q), np.arange(P))
    dummy = np.exp((dummy * 2 * np.pi * 1j * -1) / N)
    Fvt_ext = Fvt * dummy
    return (Fvt_ext @ basis_dft(P)).T.flatten()

def ifft(arr: Iterable) -> np.ndarray:
    N = len(arr)
    P = factorization(N)
    if P is N:
        return arr @ basis_idft(N)
    Q = N // P
    Fvt = np.array([ifft(arr[t::P]) for t in range(P)]).T # Q x P
    dummy = np.outer(np.arange(Q), np.arange(P))
    dummy = np.exp((dummy * 2 * np.pi * 1j) / N)
    Fvt_ext = Fvt * dummy
    return (Fvt_ext @ basis_idft(P)).T.flatten()