Part1:傅里叶级数的复数形式
设(f(x))是周期为(l)的周期函数,若
[f(x)sim frac{a_0}2+sum_{n=1}^{infty}(a_ncosfrac{npi x}l+b_nsin frac{npi x}l),\
a_n=frac1lint_{-l}^lf(x)cos frac{npi x}lmathrm dx,(n=0,1,2,dots)\
b_n=frac1lint_{-l}^lf(x)sin frac{npi x}lmathrm dx.(n=1,2,dots)
]
记(omega=frac{pi}l),引进复数形式:
[cos nomega x=frac{mathrm{e}^{mathrm inomega x}+mathrm{e}^{-mathrm inomega x}}2,sin nomega x=frac{mathrm{e}^{mathrm inomega x}-mathrm{e}^{-mathrm inomega x}}{2mathrm i}
]
级数化为
[egin{align}
f(x)&sim frac{a_0}2+sum_{n=1}^{infty}(a_nfrac{mathrm{e}^{mathrm inomega x}+mathrm{e}^{-mathrm inomega x}}2+b_nfrac{mathrm{e}^{mathrm inomega x}-mathrm{e}^{-mathrm inomega x}}{2mathrm i})\
&=frac{a_0}2+sum_{n=1}^{infty}(frac{a_n-mathrm ib_n}2mathrm{e}^{mathrm inomega x}+frac{a_n+mathrm ib_n}2mathrm{e}^{-mathrm inomega x})
end{align}
]
令(c_0=frac{a_0}2,c_n=frac{a_n-mathrm ib_n}2,d_n=frac{a_n+mathrm ib_n}2),则
[egin{align}
c_0&=frac1{2l}int_{-l}^lf(x)mathrm dx,\
c_n&=frac1{2l}int_{-l}^lf(x)left(cos nomega x-mathrm isin nomega x
ight)mathrm dx=frac1{2l}int_{-l}^lf(x)mathrm{e}^{-mathrm inomega x}mathrm dx,\
d_n&=frac1{2l}int_{-l}^lf(x)left(cos nomega x+mathrm isin nomega x
ight)mathrm dx=frac1{2l}int_{-l}^lf(x)mathrm{e}^{mathrm inomega x}mathrm dx\
& riangleq c_{-n}=ar{c_n},(n=1,2,dots)
end{align}
]
合并为
[c_n=frac{1}{2l}=int_{-l}^lf(x)mathrm{e}^{-mathrm inomega x}mathrm dx,(nin )
]
级数化为
[sum_{n=-infty}^{+infty}c_nmathrm{e}^{-mathrm inomega x}=frac{1}{2l}sum_{n=-infty}^{+infty}left[int_{-l}^lf(x)mathrm{e}^{-mathrm inomega x}mathrm dx
ight]mathrm{e}^{mathrm inomega x}
]
我们称(c_n)为(f(x))的离散频谱(discrete spectrum),(|c_n|)为(f(x))的离散振幅频谱(discrete amplitude spectrum),(arg c_n)为(f(x))的离散相位频谱(discrete phase spectrum).
对任何一个非周期函数(f(t))都可以看成是由某个由某个周期为(l)的函数(f(x))当(l oinfty)时得来的.
Part2:傅里叶积分和傅里叶变换
傅里叶积分公式
设(f_T(t))是周期为(T)的周期函数,在([-frac T2,frac T2])上满足狄利克雷条件,则
[f_T(t)=frac1Tsum_{n=-infty}^{infty}left[int_{-frac T2}^{frac T2}f_T(t)mathrm{e}^{-mathrm jnomega t}mathrm dt
ight]mathrm{e}^{mathrm{j}nomega t},omega=frac{2pi}T
]
(上式中(mathrm j)是虚数单位,在傅里叶分析中我们不用(mathrm i)而通常记作(mathrm j))由(limlimits_{T oinfty}f_T(t)=f(t))知,
[f(t)=lim_{T oinfty}frac1Tsum_{n=-infty}^{infty}[int_{-frac T2}^{frac T2}f_T(t)mathrm{e}^{-mathrm jnomega t}mathrm dt]mathrm{e}^{mathrm{j}nomega t}
]
记(Delta omega=frac{2pi}T),则(Deltaomega o 0Leftrightarrow T oinfty),则
[egin{align}
f(t)&=lim_{T oinfty}frac1Tsum_{n=-infty}^{infty}[int_{-frac T2}^{frac T2}f_T(t)mathrm{e}^{-mathrm jnomega t}mathrm dt]mathrm{e}^{mathrm{j}nomega t}\
&=lim_{Delta omega o 0}frac1{2pi}sum_{n=-infty}^{+infty}left[int_{frac T2}^{frac T2}f_T(t)mathrm{e}^{-mathrm{j}nomega t}mathrm dt
ight]mathrm{e}^{mathrm jnomega t}Deltaomega
end{align}
]
令(F_T(nomega)=int_{-frac T2}^{frac T2}f_T(t)mathrm{e}^{-mathrm jnomega t}mathrm dt),则
[f(t)=lim_{Deltaomega o 0}frac1{2pi}sum_{n=-infty}^{+infty}F_T(nomega)mathrm{e}^{mathrm jnomega t}Deltaomega,\
F_T(t) o int_{-infty}^{+infty}f(t)mathrm{e}^{-mathrm jomega t}mathrm dt riangleq F(omega)(T oinfty),
]
由定积分定义(f(t)=frac1{2pi}int_{-infty}^{+infty}F(omega)mathrm{e}^{mathrm{j}omega t}mathrm domega),即
[oxed{f(t)=frac1{2pi}int_{-infty}^{+infty}left[int_{-infty}^{+infty}f(t)mathrm{e}^{-mathrm jomega t}mathrm dt
ight]mathrm{e}^{mathrm jomega t}mathrm domega}
]
上述公式称为傅里叶积分公式.
傅里叶积分存在定理
若(f(t))在任何有限区间上满足狄利克雷条件,且在(R)上绝对可积,则
[frac1{2pi}int_{-infty}^{+infty}left[int_{-infty}^{+infty}f(t)mathrm{e}^{-mathrm jomega t}mathrm dt
ight]mathrm{e}^{mathrm jomega t}mathrm domega=
egin{cases}
f(t),t ext{为连续点},\
frac{f(t^-)+f(t^+)}2,t ext{为间断点}.
end{cases}
]
傅里叶变换
设(f(t))满足傅里叶积分存在定理,定义
[F(omega)=int_{-infty}^{+infty}f(t)mathrm{e}^{-mathrm jomega t}mathrm dt
]
为(f(t))的傅里叶变换(Fourier Transform)(实际上是一个实自变量的复值函数),记作
[F(omega)=mathcal{F}left[f(t)
ight]
]
类似地,定义
[f(t)=frac1{2pi}int_{-infty}^{+infty}F(omega)mathrm{e}^{-mathrm jomega t}mathrm domega
]
为(F(omega))的傅里叶逆变换(Inverse Fourier Transform),记作
[f(t)=mathcal{F}^{-1}left[F(omega)
ight]
]
在一定条件下,有
[mathcal{F}left[f(t)
ight]=F(omega)Rightarrowmathcal{F}^{-1}left[F(omega)
ight]=f(t);\
mathcal{F}^{-1}left[F(omega)
ight]=f(t)Rightarrowmathcal{F}left[f(t)
ight]=F(omega).
]
(f(t))与(F(omega))在傅氏变换意义下是一个一一对应,称(f(t))与(F(omega))构成一个傅氏变换对,记作
[f(t)overset{underset{mathcal{F}}{}}{leftrightarrow}F(omega)
]
在不引起混淆的情况下,简记为(f(t)leftrightarrow F(omega)).(f(t))称为原象函数(original image function),(F(omega))称为象函数(image function).
在频谱分析中,(F(omega))又称为(f(t))的频谱(密度)函数(spectrum function),(|F(omega)|)称为(f(t))的振幅频谱(amplitude spectrum),(arg F(omega))称为(f(t))的相位频谱(phase spectrum).
下面我们来求几个常见信号函数的傅氏变换.
例1 求矩形脉冲函数(rectangular pulse function)
[R(t)=egin{cases}
1,|t|le 1,\
0,|t|>1
end{cases}
]
的傅氏变换及其频谱积分表达式.
解:
[egin{align}
F(omega)&=mathcal{F}[R(t)]=int_{-infty}^{+infty}R(t)mathrm{e}^{-mathrm jomega t}mathrm dt=int_{-1}^1 R(t)mathrm{e}^{-mathrm jomega t}mathrm t\
&=left[frac{mathrm{e}^{-mathrm jomega t}}{-mathrm jomega}
ight]^1_{-1}\
&=-frac{mathrm{e}^{-mathrm jomega}-mathrm{e}^{mathrm jomega}}{mathrm jomega}=frac{2sinomega}{omega};\
R(t)&=frac1{2pi}int_{-infty}^{infty}F(omega)mathrm{e}^{mathrm jomega t}mathrm domega=frac1{pi}int_0^{+infty}F(omega)cosomega tmathrm domega\
&=frac1{pi}int_0^{+infty}frac{2sinomega}omegacosomega tmathrm domega=frac2{pi}int_0^{+infty}frac{sinomegacosomega t}{omega}mathrm domega\
&=egin{cases}
1,|t|<1,\
frac12,|t|=1,\
0,|t|>1
end{cases}
end{align}
]
因此可知,当(t=0)时,有
[int_0^{+infty}frac{sin t}xmathrm dt=frac{pi}2
]
例2 求指数衰减函数(exponential decay function)
[E(t)=egin{cases}
0,t<0,\
mathrm{e}^{-eta t},tge 0
end{cases}
]
的傅氏变换及其频谱积分表达式,其中(eta>0)为常数.
解:
[egin{align}
F(omega)&=mathcal{F}[E(t)]=int_{-infty}^{+infty}E(t)mathrm{e}^{-mathrm{j}omega t}mathrm dt\
&=int_0^{+infty}mathrm{e}^{-eta t}mathrm{e}^{-mathrm jomega t}mathrm dt=int_0^{+infty}mathrm{e}^{(eta+mathrm jomega)t}mathrm dt=frac1{eta+mathrm jomega}frac{eta-mathrm jomega}{eta^2+omega^2}\
E(t)&=frac1{2pi}int_{-infty}^{+infty}F(omega)mathrm{e}^{mathrm jomega t}mathrm omega=frac1{2pi}int_{-infty}^{+infty}frac{eta-mathrm jomega}{eta^2+omega^2}mathrm{e}^{mathrm jomega t}mathrm omega\
&=frac1{pi}int_{0}^{+infty}frac{etacosomega t+omegasinomega t}{eta^2+omega^2}mathrm domega=egin{cases}
0,t<0,\
frac12,t=0,\
mathrm{e}^{-eta t},t>0
end{cases}
end{align}
]
Part3:单位脉冲函数
我们记电流脉冲函数
[q(t)=egin{cases}
0,t
e 0,\
1,t=0,
end{cases}
]
严格地,由于(q(t))在(t=0)出不连续,所以(q(t))在(t=0)点是不可导的.但是,如果我们形式地计算这个导数,有
[q'(0)=lim_{Delta t o 0}frac{q(0+Delta t)-q(0)}{Delta t}=lim_{Delta t o 0}-frac1{Delta t}=infty
]
我们引进这样一个函数,称为单位脉冲函数(unit pulse function)或狄拉克(Dirac)函数,简记为(delta-)函数,即
[delta(t)=egin{cases}
0,t
e 0,\
infty,t=0,
end{cases}
]
一般地,给定一个函数序列
[delta_{varepsilon}(t)=egin{cases}
0,t<0,\
frac1{varepsilon},0le tle varepsilon,\
0,t>varepsilon
end{cases}
]
则有
[delta(t)=lim_{varepsilon o 0}delta_{varepsilon}(t)=egin{cases}
0,t
e 0,\
infty,t=0
end{cases}
]
于是
[oxed{
int_{-infty}^{+infty}delta(t)mathrm dt=lim_{varepsilon o0}int_{-infty}^{+infty}delta_{varepsilon}mathrm dt=lim_{varepsilon o0}int_{0}^{varepsilon}frac1{varepsilon}mathrm dt=1
}
]
若设(f(t))为连续函数,则(delta-)函数有以下性质:
[int_{-infty}^{+infty}delta(t)f(t)mathrm dt=f(0);\
int_{-infty}^{+infty}delta(t-t_0)f(t)mathrm dt=f(t_0)
]
于是我们可得:
[mathcal{F}[delta(t)]=int_{-infty}^{+infty}delta(t)mathrm{e}^{-mathrm jomega t}mathrm t=left.mathrm{e}^{-mathrm jomega t}
ight|_{t=0}=1
]
于是(delta(t))与常数(1)构成了一对傅里叶变换对.
例3: 证明:
[mathrm{e}^{mathrm jomega_0 t}leftrightarrow 2pidelta(omega-omega_0)
]
其中(omega_0)是常数.
证:
[egin{align}
f(t)&=mathcal{F}^{-1}[F(omega)]=frac1{2pi}int_{-infty}^{+infty}2pidelta(omega-omega_0)mathrm{e}^{mathrm jomega t}mathrm domega\
&=left.mathrm{e}^{mathrm jomega t}
ight|_{omega=omega_0}=mathrm{e}^{mathrm jomega_0 t}
end{align}
]
在物理学和工程技术中,有许多重要函数不满足傅氏积分定理中的绝对可积条件,即不满足条件
[int_{-infty}^{+infty}|f(t)|mathrm dt<infty
]
例如常数,符号函数,单位阶跃函数以及正,余弦函数等, 然而它们的广义傅氏变换也是存在的,利用单位脉冲函数及其傅氏变换就可以求出它们的傅氏变换.所谓广义是相对于古典意义而言的,在广义意义下,同样可以说,原象函数(f(t))和象函数(F(omega))构成一个傅氏变换对.
例 求正弦函数(f(t)=sinomega_0 t)的傅氏变换.
解:
[egin{align}
F(omega)&=mathcal F[f(t)]=int_{-infty}^{+infty}f(t)mathrm{e}^{-mathrm{j}omega t}mathrm dt\
&=int_{-infty}^{+infty}mathrm{mathrm{e}^{mathrm jomega_0} t-mathrm{e}^{-mathrm jomega_0 t}}{2mathrm j}mathrm{e}^{-mathrm jomega t}mathrm dt\
&=frac1{2mathrm j}int_{-infty}^{+infty}left(mathrm{e}^{-mathrm j(omega-omega_0)t}-mathrm{e}^{-mathrm j(omega+omega_0)t}
ight)mathrm dt\
&=mathrm{j}pileft[delta(omega+omega_0)-delta(omega-omega_0)
ight]
end{align}
]
同样我们易得
[mathcal{F}(cos omega_0 t)=pileft[delta(omega+omega_0)+delta(omega-omega_0)
ight]
]
例 证明:单位阶跃函数(unit step function)
[u(t)=egin{cases}
0,t<0,\
1,t>0
end{cases}
]
的傅氏变换为
[mathcal F[u(t)]=frac1{mathrm jomega}+pi delta(omega)
]
证:
[egin{align}
mathcal{F}^{-1}left[frac1{mathrm jomega}+pi delta(omega)
ight]&=frac1{2pi}int_{-infty}^{+infty}left[frac1{mathrm jomega}+pidelta(omega)
ight]mathrm{e}^{mathrm jomega t}mathrm domega\
&=frac1{2pi}int_{-infty}^{+infty}left[pidelta(omega)
ight]mathrm{e}^{mathrm jomega t}mathrm domega+frac1{2pi}int_{-infty}^{+infty}left[frac1{mathrm jomega}
ight]mathrm{e}^{mathrm jomega t}mathrm domega\
&=frac12+frac1{2pi}int_{-infty}^{+infty}left[frac{cosomega t+mathrm jsinomega t}{mathrm jomega}
ight]mathrm domega\
&=frac12+frac1{2pi}int_{-infty}^{+infty}left[frac{sinomega t}{omega}
ight]mathrm domega=frac12+frac1{pi}int_0^{+infty}left[frac{sinomega t}{omega}
ight]mathrm domega\
int_0^{+infty}frac{sin omega t}{omega }mathrm domega&=egin{cases}
frac{pi}2,t>0,\
-frac{pi}2,t<0
end{cases}Rightarrow\
mathcal{F}^{-1}left[frac1{mathrm jomega}+pidelta(omega)
ight]&=egin{cases}
frac12+frac1{pi}left(-frac{pi}2
ight)=0,t<0\
frac12,t=0,\
frac12+frac1{pi}left(frac{pi}2
ight)=1,t>0
end{cases}=u(t).
end{align}
]