傅里叶变换

不同于周期信号，非周期信号具有连续的频谱。

一个非周期信号\(x(t)\)的连续频谱（continuous-frequency spectrum），可以用其傅里叶变换\(X(f)\)来表示。这是傅里叶级数在\(T_p\rightarrow\infty,\ k\rightarrow\infty\)和\(k/T_p\rightarrow f\)的极限情况。

  • \(x(t)\)和\(X(f)\)的关系：

    \[
    \begin{split}
    \mbox{Forward | t-domain to f-domain} \quad X(f)=\int_{-\infty}^{\infty}x(t)\exp(-j2\pi ft)\mathrm{d}t \\
    \mbox{Inverse | f-domain to t-domain} \quad x(t)=\int_{-\infty}^{\infty}X(f)\exp(j2\pi ft)\mathrm{d}f
    \end{split}
    \]

  • Dirichlet’s Conditions

    要使得\(x(t)\)的傅里叶变换存在，以下几个条件需要被满足：

    • On any finite interval

      • \(x(t)\) is bounded

      • \(x(t)\) has a finite number of minima and maxima

      • \(x(t)\) has a finite number of discontinuities

    • \(x(t)\) is absolutely integrable

傅里叶变换的性质

规定

\[
\begin{cases}
X(f) = \Im \{x(t)\}\ &\mbox{denote the Fourier transform of}\ x(t)
\newline
x(t) \rightleftarrows X(f)\ &\mbox{denote a Fourier transform pair}
\end{cases}
\]

  • Linearity

    \[
    \alpha x_1(t) + \beta x_2(t) \rightleftarrows \alpha X_1(f) + \beta X_2(f)
    \]

  • Time Scaling

    \[
    x(\beta t) \rightleftarrows \frac{1}{\lvert \beta \rvert}X(\frac{f}{\beta})
    \]

  • Duality

    \[
    X(t) \rightleftarrows x(-f)\ \mbox{or}\ X(-t) \rightleftarrows x(f)
    \]

  • Time Shifting

    \[
    x(t-t_0) \rightleftarrows X(f)\exp(-j2\pi ft_0)
    \]

  • Frequency Shifting (Modulation)

    \[
    x(t)\exp(j2\pi f_0t) \rightleftarrows X(f-f_0)
    \]

  • Differentiation in the Time Domain

    \[
    \frac{\mathrm{d}}{\mathrm{d} t}x(t) \rightleftarrows j2\pi f\cdot X(f)
    \]

  • Integration in the Time Domain

    \[
    \int_{-\infty}^tx(\tau)\mathrm{d}\tau \rightleftarrows \frac{1}{j2\pi f}X(f) + \frac{1}{2}X(0)\delta (f)
    \]

  • Convolution in the Time Domain (or Multiplication in the Frequency Domain)

    \[
    \underbrace{\int_{-\infty}^{\infty} x_1(\zeta) x_2(t-\zeta) \mathrm{d} \zeta}_{x_1(t) * x_2(t)} \rightleftarrows X_1(f) X_2(f)
    \]

  • Multiplication in the Time Domain (or Convolution in the Frequency Domain)

    \[
    x_1(t) x_2(t) \rightleftarrows \int_{-\infty}^{\infty} X_1(\zeta) X_2(f-\zeta)\mathrm{d} \zeta
    \]

REAL Signal 的频谱性质

  • \(x(t)\) is REAL:此时\(x^*(t) = x(t)\)，\[
    \underbrace{X^*(f) = X(-f)}_{X(f)\ \mbox{is Conjugate Symmetric}}\quad \underbrace{\lvert X(f) \rvert = \lvert X(-f) \rvert }_{\mbox{EVEN Symmetry}}\quad \underbrace{\angle X(f) = -\angle X(-f)}_{\mbox{ODD Symmetry}}
    \]

  • \(x(t)\) is REAL and EVEN:此时\(x^*(t) = x(t)\mbox{ and }x(t) = x(-t)\)，\[
    \underbrace{\overbrace{X^*(f) = X(f)}^{\mbox{Real}}\ \mbox{and}\ \overbrace{X(f) = X(-f)}^{\mbox{Even}}}_{X(f)\ \mbox{is REAL and EVEN}}
    \]

  • \(x(t)\) is REAL and ODD:此时\(x^*(t) = x(t)\mbox{ and }x(-t) = -x(t)\)，\[
    \underbrace{\overbrace{X^*(f) = -X(f)}^{\mbox{Imaginary}}\ \mbox{and}\ \overbrace{X(f) = -X(-f)}^{\mbox{Odd}}}_{X(f)\ \mbox{is IMAGINARY and ODD}}
    \]

上述这些结论同样适用于傅里叶级数的系数。

Dirac-δ 和周期信号的频谱

The Continuous-time Unit Impulse (Dirac-δ function)

定义：

\[
\delta (t) =
\begin{cases}
\infty;\ t=0
\newline
0;\ t\neq 0
\end{cases}
\ \mbox{and}\
\int_{-\varepsilon}^{\varepsilon}\delta (t) \mathrm{d}t = 1;\ \forall \varepsilon > 0
\]

\(\delta (t)\)的性质：

  1. Symmetry:

    \[
    \delta (t) = \delta (-t)
    \]

  2. Sampling:

    \[
    x(t)\delta (t-\lambda ) = x(\lambda)\delta (t-\lambda )
    \]

  1. Sifting:

    \[
    \int_{-\infty}^{\infty}x(t)\delta (t-\lambda )\mathrm{d} t = x(\lambda )\int_{-\infty}^{\infty}\delta (t-\lambda )\mathrm{d} t = x(\lambda)
    \]

  2. Replication:

    \[
    \begin{split}
    &x(t)*\delta (t-\xi ) = \int_{-\infty}^{\infty}x(\xi )\delta (t-\zeta -\xi )\mathrm{d}\zeta = \int_{-\infty}^{\infty}x(\zeta) \delta (\zeta - (t - \xi ))\mathrm{d}\zeta = x(t-\xi ) \\
    &\mbox{Note: }x(t)*\delta (t) = x(t)
    \end{split}
    \]

  1. White Spectrum: \[
    \Im \{\delta (t)\} = \int_{-\infty}^{\infty}\delta (t)\exp (-j2\pi ft) \mathrm{d} t = 1
    \]

周期信号的频谱

在 unit impulse function 的帮助下，可以在周期信号上使用傅里叶变换来获取它的连续频谱。

  • DC

    \[
    [x(t) = K] \rightleftarrows [X(f) = K\delta (f)]
    \]

  • Complex Exponential

    \[
    [x(t) = K\exp (j2\pi f_0 t)] \rightleftarrows [X(f) = K \delta (f - f_0)]
    \]

  • Cosine

    \[
    [x(t) = K\cos (2\pi f_0 t)] \rightleftarrows \left [X(f) = \frac{K}{2} \delta (f - f_0) + \frac{K}{2} \delta (f + f_0) \right ]
    \]

  • Sine

    \[
    [x(t) = K\sin (2\pi f_0 t)] \rightleftarrows \left [\begin{split}X(f) &= \frac{K}{j2}\delta (f - f_0) - \frac{K}{j2}\delta(f+f_0)]\\&=\frac{K}{2}\exp(-j\frac{\pi}{2})\delta (f - f_0) + \frac{K}{2}\exp(j\frac{\pi}{2}\delta (f + f_0))\end{split} \right ]
    \]

  • Arbitrary periodic signals

    \[
    \left [x_p(t) = \sum_{k=-\infty}^{\infty}c_k\exp(j2\pi\frac{k}{T_p}t) \right ] \rightleftarrows \left [X_p(f) = \sum_{k=-\infty}^{\infty}c_k\delta (f - \frac{k}{T_p}) \right ]
    \]

    要求得周期信号\(x_p(t)\)的傅里叶变换\(X_p(f)\)，可以先求得其傅里叶系数，然后代入上面的式子。