Part1: plural form Fourier series
Set \ (f (x) \) is the period of \ (L \) periodic function, if
\[ f(x)\sim \frac{a_0}2+\sum_{n=1}^{\infty}(a_n\cos\frac{n\pi x}l+b_n\sin \frac{n\pi x}l),\\ a_n=\frac1l\int_{-l}^lf(x)\cos \frac{n\pi x}l\mathrm dx,(n=0,1,2,\dots)\\ b_n=\frac1l\int_{-l}^lf(x)\sin \frac{n\pi x}l\mathrm dx.(n=1,2,\dots) \]
Note \ (\ Omega = \ FRAC {\ PI} L \) , the introduction of the plural forms:
\[ \cos n\omega x=\frac{\mathrm{e}^{\mathrm in\omega x}+\mathrm{e}^{-\mathrm in\omega x}}2,\sin n\omega x=\frac{\mathrm{e}^{\mathrm in\omega x}-\mathrm{e}^{-\mathrm in\omega x}}{2\mathrm i} \]
Progression into
\[ \begin{align} f(x)&\sim \frac{a_0}2+\sum_{n=1}^{\infty}(a_n\frac{\mathrm{e}^{\mathrm in\omega x}+\mathrm{e}^{-\mathrm in\omega x}}2+b_n\frac{\mathrm{e}^{\mathrm in\omega x}-\mathrm{e}^{-\mathrm in\omega x}}{2\mathrm i})\\ &=\frac{a_0}2+\sum_{n=1}^{\infty}(\frac{a_n-\mathrm ib_n}2\mathrm{e}^{\mathrm in\omega x}+\frac{a_n+\mathrm ib_n}2\mathrm{e}^{-\mathrm in\omega x}) \end{align} \]
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
\[ \begin{align} c_0&=\frac1{2l}\int_{-l}^lf(x)\mathrm dx,\\ c_n&=\frac1{2l}\int_{-l}^lf(x)\left(\cos n\omega x-\mathrm i\sin n\omega x\right)\mathrm dx=\frac1{2l}\int_{-l}^lf(x)\mathrm{e}^{-\mathrm in\omega x}\mathrm dx,\\ d_n&=\frac1{2l}\int_{-l}^lf(x)\left(\cos n\omega x+\mathrm i\sin n\omega x\right)\mathrm dx=\frac1{2l}\int_{-l}^lf(x)\mathrm{e}^{\mathrm in\omega x}\mathrm dx\\ &\triangleq c_{-n}=\bar{c_n},(n=1,2,\dots) \end{align} \]
Merged into
\[ c_n=\frac{1}{2l}=\int_{-l}^lf(x)\mathrm{e}^{-\mathrm in\omega x}\mathrm dx,(n\in \Z) \]
Progression into
\[ \sum_{n=-\infty}^{+\infty}c_n\mathrm{e}^{-\mathrm in\omega x}=\frac{1}{2l}\sum_{n=-\infty}^{+\infty}\left[\int_{-l}^lf(x)\mathrm{e}^{-\mathrm in\omega x}\mathrm dx\right]\mathrm{e}^{\mathrm in\omega x} \]
We call \ (c_n \) is \ (f (x) \) of the discrete spectrum (Discrete Spectrum) , \ (| c_n | \) is \ (f (x) \) of discrete amplitude spectrum (Discrete Amplitude Spectrum) , \ (\ arg c_n \) of \ (f (x) \) of a discrete phase spectrum (discrete phase spectrum) .
For any non-periodic function \ (f (t) \) can be seen as a certain period by an \ (L \) function \ (f (x) \) when \ (l \ to \ infty \) when come.
Part2: Fourier integral and Fourier Transform
Fourier integral formula
Set \ (f_T (t) \) is the period \ (T \) periodic function, in \ ([- \ frac T2, \ frac T2] \) on satisfies Dirichlet condition,
\[ f_T(t)=\frac1T\sum_{n=-\infty}^{\infty}\left[\int_{-\frac T2}^{\frac T2}f_T(t)\mathrm{e}^{-\mathrm jn\omega t}\mathrm dt\right]\mathrm{e}^{\mathrm{j}n\omega t},\omega=\frac{2\pi}T \]
(Wherein \ (\ mathrm j \) is an imaginary unit, in the Fourier analysis, we do not \ (\ mathrm i \) usually referred to as \ (\ mathrm j \) ) a \ (\ lim \ limits_ { T \ to \ infty} f_T ( t) = f (t) \) known,
\[ f(t)=\lim_{T\to\infty}\frac1T\sum_{n=-\infty}^{\infty}[\int_{-\frac T2}^{\frac T2}f_T(t)\mathrm{e}^{-\mathrm jn\omega t}\mathrm dt]\mathrm{e}^{\mathrm{j}n\omega t} \]
记\ (\ Delta \ omega = \ frac {2 \ pi} T \) , Provisions \ (\ Delta \ omega \-to 0 \ Leftrightarrow T \-to \ infty \) , Provisions
\[ \begin{align} f(t)&=\lim_{T\to\infty}\frac1T\sum_{n=-\infty}^{\infty}[\int_{-\frac T2}^{\frac T2}f_T(t)\mathrm{e}^{-\mathrm jn\omega t}\mathrm dt]\mathrm{e}^{\mathrm{j}n\omega t}\\ &=\lim_{\Delta \omega\to 0}\frac1{2\pi}\sum_{n=-\infty}^{+\infty}\left[\int_{\frac T2}^{\frac T2}f_T(t)\mathrm{e}^{-\mathrm{j}n\omega t}\mathrm dt\right]\mathrm{e}^{\mathrm jn\omega t}\Delta\omega \end{align} \]
令\(F_T(n\omega)=\int_{-\frac T2}^{\frac T2}f_T(t)\mathrm{e}^{-\mathrm jn\omega t}\mathrm dt\),则
\[ f(t)=\lim_{\Delta\omega\to 0}\frac1{2\pi}\sum_{n=-\infty}^{+\infty}F_T(n\omega)\mathrm{e}^{\mathrm jn\omega t}\Delta\omega,\\ F_T(t)\to \int_{-\infty}^{+\infty}f(t)\mathrm{e}^{-\mathrm j\omega t}\mathrm dt\triangleq F(\omega)(T\to\infty), \]
Defined by the definite integral \ (f (t) = \ frac1 {2 \ pi} \ int _ {- \ infty} ^ {+ \ infty} F (\ omega) \ mathrm {e} ^ {\ mathrm {j} \ omega } T \ mathrm D \ Omega \) , i.e.
\[ \boxed{f(t)=\frac1{2\pi}\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty}f(t)\mathrm{e}^{-\mathrm j\omega t}\mathrm dt\right]\mathrm{e}^{\mathrm j\omega t}\mathrm d\omega} \]
The above equation is called the Fourier integral formula .
Fourier integral existence theorem
If \ (f (t) \) meet at any finite interval Dirichlet conditions, and in the \ (\ R & lt \) absolute integrable, then
\ [\ Frac1 {2 \ pi} \ int _ {- \ infty} ^ {+ \ infty} \ left [\ int _ {- \ infty} ^ {+ \ infty} f (t) \ mathrm {e} ^ {- \ mathrm j \ omega t} \ mathrm dt \ right] \ mathrm {e} ^ {\ mathrm j \ omega t} \ mathrm d \ omega = \ begin {cases} f (t), t \ text {consecutive points }, \\ \ frac {f (t ^ -) + f (t ^ +)} 2, t \ text {intermittent point} \ end {cases} \].
Fourier transform
Set \ (f (t) \) satisfy the theorem of Fourier integral exists, defined
\[ F(\omega)=\int_{-\infty}^{+\infty}f(t)\mathrm{e}^{-\mathrm j\omega t}\mathrm dt \]
Is \ (f (t) \) is the Fourier transform (the Transform of Fourier) (actually a real variable from complex function ), referred to as
\[ F(\omega)=\mathcal{F}\left[f(t)\right] \]
Similarly, the definition of
\[ f(t)=\frac1{2\pi}\int_{-\infty}^{+\infty}F(\omega)\mathrm{e}^{-\mathrm j\omega t}\mathrm d\omega \]
As \ (F (\ omega) \ ) the inverse Fourier transform (Inverse Fourier the Transform) , denoted
\[ f(t)=\mathcal{F}^{-1}\left[F(\omega)\right] \]
Under certain conditions,
\[ \mathcal{F}\left[f(t)\right]=F(\omega)\Rightarrow\mathcal{F}^{-1}\left[F(\omega)\right]=f(t);\\ \mathcal{F}^{-1}\left[F(\omega)\right]=f(t)\Rightarrow\mathcal{F}\left[f(t)\right]=F(\omega). \]
\ (f (t) \) and \ (F (\ omega) \ ) is in the sense of a one-Fourier transform, called \ (f (t) \) and \ (F (\ omega) \ ) constitute a Fourier transform , denoted
\[ f(t)\overset{\underset{\mathcal{F}}{}}{\leftrightarrow}F(\omega) \]
Without confusingly, simply referred to as \ (F (T) \ leftrightarrow F. (\ Omega) \) . \ (F (T) \) referred to as the original image function (Original Image function) , \ (F. ( \ omega) \) is called as a function (Image function) .
In spectrum analysis, \ (F (\ Omega) \) , also known as \ (f (t) \) of the spectrum (density) function (Spectrum function) , \ (| F (\ Omega) | \) called \ (f (t) \) of the amplitude spectrum (amplitude spectrum) , \ (\ Arg F. (\ Omega) \) is called \ (f (t) \) of the phase spectrum (phase spectrum) .
Let's ask a few common signal as a function of the Fourier transform.
Example 1 seeking rectangular pulse function (rectangular pulse function)
\[ R(t)=\begin{cases} 1,|t|\le 1,\\ 0,|t|>1 \end{cases} \]
Fourier transform and the spectrum integral expression.
Solution :
\[ \begin{align} F(\omega)&=\mathcal{F}[R(t)]=\int_{-\infty}^{+\infty}R(t)\mathrm{e}^{-\mathrm j\omega t}\mathrm dt=\int_{-1}^1 R(t)\mathrm{e}^{-\mathrm j\omega t}\mathrm t\\ &=\left[\frac{\mathrm{e}^{-\mathrm j\omega t}}{-\mathrm j\omega}\right]^1_{-1}\\ &=-\frac{\mathrm{e}^{-\mathrm j\omega}-\mathrm{e}^{\mathrm j\omega}}{\mathrm j\omega}=\frac{2\sin\omega}{\omega};\\ R(t)&=\frac1{2\pi}\int_{-\infty}^{\infty}F(\omega)\mathrm{e}^{\mathrm j\omega t}\mathrm d\omega=\frac1{\pi}\int_0^{+\infty}F(\omega)\cos\omega t\mathrm d\omega\\ &=\frac1{\pi}\int_0^{+\infty}\frac{2\sin\omega}\omega\cos\omega t\mathrm d\omega=\frac2{\pi}\int_0^{+\infty}\frac{\sin\omega\cos\omega t}{\omega}\mathrm d\omega\\ &=\begin{cases} 1,|t|<1,\\ \frac12,|t|=1,\\ 0,|t|>1 \end{cases} \end{align} \]
Thus understood, when \ (t = 0 \) when there is
\[ \int_0^{+\infty}\frac{\sin t}x\mathrm dt=\frac{\pi}2 \]
Example 2 required an exponential decay function (exponential decay function)
\[ E(t)=\begin{cases} 0,t<0,\\ \mathrm{e}^{-\beta t},t\ge 0 \end{cases} \]
Fourier transform and the spectrum integral expression, where \ (\ beta> 0 \) is a constant.
Solution :
\[ \begin{align} F(\omega)&=\mathcal{F}[E(t)]=\int_{-\infty}^{+\infty}E(t)\mathrm{e}^{-\mathrm{j}\omega t}\mathrm dt\\ &=\int_0^{+\infty}\mathrm{e}^{-\beta t}\mathrm{e}^{-\mathrm j\omega t}\mathrm dt=\int_0^{+\infty}\mathrm{e}^{(\beta+\mathrm j\omega)t}\mathrm dt=\frac1{\beta+\mathrm j\omega}\frac{\beta-\mathrm j\omega}{\beta^2+\omega^2}\\ E(t)&=\frac1{2\pi}\int_{-\infty}^{+\infty}F(\omega)\mathrm{e}^{\mathrm j\omega t}\mathrm \omega=\frac1{2\pi}\int_{-\infty}^{+\infty}\frac{\beta-\mathrm j\omega}{\beta^2+\omega^2}\mathrm{e}^{\mathrm j\omega t}\mathrm \omega\\ &=\frac1{\pi}\int_{0}^{+\infty}\frac{\beta\cos\omega t+\omega\sin\omega t}{\beta^2+\omega^2}\mathrm d\omega=\begin{cases} 0,t<0,\\ \frac12,t=0,\\ \mathrm{e}^{-\beta t},t>0 \end{cases} \end{align} \]
Part3:单位脉冲函数
我们记电流脉冲函数
\[ q(t)=\begin{cases} 0,t\ne 0,\\ 1,t=0, \end{cases} \]
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
\[ q'(0)=\lim_{\Delta t\to 0}\frac{q(0+\Delta t)-q(0)}{\Delta t}=\lim_{\Delta t\to 0}-\frac1{\Delta t}=\infty \]
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
\[ \delta(t)=\begin{cases} 0,t\ne 0,\\ \infty,t=0, \end{cases} \]
一般地,给定一个函数序列
\[ \delta_{\varepsilon}(t)=\begin{cases} 0,t<0,\\ \frac1{\varepsilon},0\le t\le \varepsilon,\\ 0,t>\varepsilon \end{cases} \]
则有
\[ \delta(t)=\lim_{\varepsilon\to 0}\delta_{\varepsilon}(t)=\begin{cases} 0,t\ne 0,\\ \infty,t=0 \end{cases} \]
于是
\[ \boxed{ \int_{-\infty}^{+\infty}\delta(t)\mathrm dt=\lim_{\varepsilon\to0}\int_{-\infty}^{+\infty}\delta_{\varepsilon}\mathrm dt=\lim_{\varepsilon\to0}\int_{0}^{\varepsilon}\frac1{\varepsilon}\mathrm dt=1 } \]
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:
\[ \mathcal{F}[\delta(t)]=\int_{-\infty}^{+\infty}\delta(t)\mathrm{e}^{-\mathrm j\omega t}\mathrm t=\left.\mathrm{e}^{-\mathrm j\omega t}\right|_{t=0}=1 \]
于是\(\delta(t)\)与常数\(1\)构成了一对傅里叶变换对.
例3: 证明:
\[ \mathrm{e}^{\mathrm j\omega_0 t}\leftrightarrow 2\pi\delta(\omega-\omega_0) \]
其中\(\omega_0\)是常数.
证:
\[ \begin{align} f(t)&=\mathcal{F}^{-1}[F(\omega)]=\frac1{2\pi}\int_{-\infty}^{+\infty}2\pi\delta(\omega-\omega_0)\mathrm{e}^{\mathrm j\omega t}\mathrm d\omega\\ &=\left.\mathrm{e}^{\mathrm j\omega t}\right|_{\omega=\omega_0}=\mathrm{e}^{\mathrm j\omega_0 t} \end{align} \]
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\(f(t)=\sin\omega_0 t\)的傅氏变换.
解:
\[ \begin{align} F(\omega)&=\mathcal F[f(t)]=\int_{-\infty}^{+\infty}f(t)\mathrm{e}^{-\mathrm{j}\omega t}\mathrm dt\\ &=\int_{-\infty}^{+\infty}\mathrm{\mathrm{e}^{\mathrm j\omega_0} t-\mathrm{e}^{-\mathrm j\omega_0 t}}{2\mathrm j}\mathrm{e}^{-\mathrm j\omega t}\mathrm dt\\ &=\frac1{2\mathrm j}\int_{-\infty}^{+\infty}\left(\mathrm{e}^{-\mathrm j(\omega-\omega_0)t}-\mathrm{e}^{-\mathrm j(\omega+\omega_0)t}\right)\mathrm dt\\ &=\mathrm{j}\pi\left[\delta(\omega+\omega_0)-\delta(\omega-\omega_0)\right] \end{align} \]
同样我们易得
\[ \mathcal{F}(\cos \omega_0 t)=\pi\left[\delta(\omega+\omega_0)+\delta(\omega-\omega_0)\right] \]
例 证明:单位阶跃函数(unit step function)
\[ u(t)=\begin{cases} 0,t<0,\\ 1,t>0 \end{cases} \]
的傅氏变换为
\[ \mathcal F[u(t)]=\frac1{\mathrm j\omega}+\pi \delta(\omega) \]
证:
\[ \begin{align} \mathcal{F}^{-1}\left[\frac1{\mathrm j\omega}+\pi \delta(\omega)\right]&=\frac1{2\pi}\int_{-\infty}^{+\infty}\left[\frac1{\mathrm j\omega}+\pi\delta(\omega)\right]\mathrm{e}^{\mathrm j\omega t}\mathrm d\omega\\ &=\frac1{2\pi}\int_{-\infty}^{+\infty}\left[\pi\delta(\omega)\right]\mathrm{e}^{\mathrm j\omega t}\mathrm d\omega+\frac1{2\pi}\int_{-\infty}^{+\infty}\left[\frac1{\mathrm j\omega}\right]\mathrm{e}^{\mathrm j\omega t}\mathrm d\omega\\ &=\frac12+\frac1{2\pi}\int_{-\infty}^{+\infty}\left[\frac{\cos\omega t+\mathrm j\sin\omega t}{\mathrm j\omega}\right]\mathrm d\omega\\ &=\frac12+\frac1{2\pi}\int_{-\infty}^{+\infty}\left[\frac{\sin\omega t}{\omega}\right]\mathrm d\omega=\frac12+\frac1{\pi}\int_0^{+\infty}\left[\frac{\sin\omega t}{\omega}\right]\mathrm d\omega\\ \int_0^{+\infty}\frac{\sin \omega t}{\omega }\mathrm d\omega&=\begin{cases} \frac{\pi}2,t>0,\\ -\frac{\pi}2,t<0 \end{cases}\Rightarrow\\ \mathcal{F}^{-1}\left[\frac1{\mathrm j\omega}+\pi\delta(\omega)\right]&=\begin{cases} \frac12+\frac1{\pi}\left(-\frac{\pi}2\right)=0,t<0\\ \frac12,t=0,\\ \frac12+\frac1{\pi}\left(\frac{\pi}2\right)=1,t>0 \end{cases}=u(t). \end{align} \]