一个讯号s(t)，自相关函数为
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{\displaystyle R(\tau )=\int _{-\infty }^{\infty }s(t)s^{*}(t-\tau )\,dt}
R
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{\displaystyle R(\tau )}
R
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{\displaystyle R(t,\tau )}
P
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{\displaystyle P(t,\omega )=\int _{-\infty }^{\infty }R(t,\tau )e^{-j\omega \tau }\,d\tau }
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{\displaystyle R(t,\tau )=s\left(t+{\frac {\tau }{2}}\right)s^{*}\left(t-{\frac {\tau }{2}}\right)}
R
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{\displaystyle R(t,\tau )}
对称模糊函数(symmetric ambiguity function,SAF)
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{\displaystyle SAF_{s}(\theta ,\tau )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }s\left(t+{\frac {\tau }{2}}\right)s^{*}\left(t-{\frac {\tau }{2}}\right)e^{j\theta t}\,dt}
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{\displaystyle SAF_{s}(\theta ,\tau )}
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{\displaystyle \int _{-\infty }^{\infty }SAF_{s}(\theta ,\tau )e^{-j\theta t}\,d\theta =s\left(t+{\frac {\tau }{2}}\right)s^{*}\left(t-{\frac {\tau }{2}}\right)}
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{\displaystyle WD_{s}(t,\omega )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }SAF_{s}(\theta ,\tau )e^{-j(\omega \tau +\theta t)}\,d\theta \,d\tau }
对称模糊函数 做两次傅立叶变换可以得到Wigner distribution
一个讯号为两个Gaussian函数的和:
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{\displaystyle s(t)=\sum _{i=1}^{2}s_{i}(t)=\sum _{i=1}^{2}{\sqrt[{4}]{\frac {\alpha }{\pi }}}e^{-{\tfrac {\alpha }{2}}(t-t_{i})^{2}+j\omega _{i}t}}
⇒
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{\displaystyle \Rightarrow SAF_{s}(\theta ,\tau )=\sum _{i=1}^{2}SAF_{si}(\theta ,\tau )+SAF_{s1,s2}(\theta ,\tau )+SAF_{s2,s1}(\theta ,\tau )}
其中
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{\displaystyle SAF_{s1}(\theta ,\tau ),SAF_{s2}(\theta ,\tau )}
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{\displaystyle SAF_{s1,s2}(\theta ,\tau )}
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{\displaystyle (t_{1}-t_{2},\omega _{1}-\omega _{2})}
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{\displaystyle SAF_{s2,s1}(\theta ,\tau )}
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{\displaystyle SAF_{s1,s2}(\theta ,\tau )}
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{\displaystyle (t_{2}-t_{1},\omega _{2}-\omega _{1})}
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{\displaystyle SAF_{s1,s2}(\theta ,\tau )=e^{-{\tfrac {1}{4\alpha }}(\theta -\omega _{d})^{2}+{\tfrac {\alpha }{4}}(\tau -t_{d})^{2}}e^{j(\omega _{u}\tau -\theta t_{u}+\omega _{d}t_{u})}}
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{\displaystyle t_{u}={\tfrac {t_{1}+t_{2}}{2}}}
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{\displaystyle \omega _{u}={\tfrac {\omega _{1}+\omega _{2}}{2}}}
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{\displaystyle t_{d}=t_{1}-t_{2}}
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{\displaystyle \omega _{d}=\omega _{1}-\omega _{2}}
从范例中得知一项重要事实，即为，在模糊域(ambiguity domain)中的auto-term总是集中在原点(0,0)，而cross-term总是在远离原点处，所以可以用一个2D lowpass filter在模糊域中抑制cross-term的干扰，如下:
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{\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }SAF_{s}(\theta ,\tau )\Phi (\theta ,\tau )e^{-j(\theta t+\omega \tau )}\,d\theta \,d\tau }
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{\displaystyle \Phi (\theta ,\tau )}
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{\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\Phi (\theta ,\tau )\,d\theta \,d\tau =\phi (t,\omega )}
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{\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }SAF_{s}(\theta ,\tau )\Phi (\theta ,\tau )e^{-j(\theta t+\omega \tau )}\,d\theta \,d\tau }
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{\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\phi (x,y)WD_{s}(t-x,\omega -y)\,dx\,dy=SWD(t,\omega )}
其中SWD为smoothed Wigner distribution 通常
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{\displaystyle \Phi (\theta ,\tau )}
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{\displaystyle \phi (t,\omega )}
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{\displaystyle WD_{s1,s2}(t,\omega )=A_{WD}(t,\omega )e^{j\varphi _{WD}(t,\omega )}}
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{\displaystyle SAF_{s1,s2}(\theta ,\tau )=A_{SAF}(\theta ,\tau )e^{j\varphi _{SAF}(\theta ,\tau )}}
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{\displaystyle {\frac {\partial }{\partial \theta }}\varphi _{SAF}(\theta ,\tau )=-t_{u}}
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{\displaystyle {\frac {\partial }{\partial \tau }}\varphi _{SAF}(\theta ,\tau )=\omega _{u}}
对称模糊函数 的相位做偏微分，会等于Wigner分布的时频(time-frequency)中心。
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{\displaystyle {\frac {\partial }{\partial \omega }}\varphi _{WD}(t,\omega )=t_{d}}
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{\displaystyle {\frac {\partial }{\partial t}}\varphi _{WD}(t,\omega )=\omega _{d}}
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{\displaystyle \omega _{1}=\omega _{2}=\omega _{0}}
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{\displaystyle SAF_{s1,s2}(\theta ,\tau )=e^{-[{\tfrac {1}{4\alpha }}\theta ^{2}+{\tfrac {\alpha }{4}}(\tau -t_{d})^{2}]}e^{j(\omega _{0}\tau -\theta t_{u})}}
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{\displaystyle \tau }
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{\displaystyle t_{1}=t_{2}=t_{0}}
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{\displaystyle SAF_{s1,s2}(\theta ,\tau )=e^{-[{\tfrac {1}{4\alpha }}(\theta -\omega _{d})^{2}+{\tfrac {\alpha }{4}}\tau ^{2}]}e^{j[\omega _{u}\tau -(\theta -\omega _{d})t_{0}]}}
θ
{\displaystyle \theta }
![{\displaystyle s(t)=\sum _{i=1}^{2}s_{i}(t)=\sum _{i=1}^{2}{\sqrt[{4}]{\frac {\alpha }{\pi }}}e^{-{\tfrac {\alpha }{2}}(t-t_{i})^{2}+j\omega _{i}t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/907a08b77bcea5e7266dbd0eb1d1de9bb5411f5e)
![{\displaystyle SAF_{s1,s2}(\theta ,\tau )=e^{-[{\tfrac {1}{4\alpha }}\theta ^{2}+{\tfrac {\alpha }{4}}(\tau -t_{d})^{2}]}e^{j(\omega _{0}\tau -\theta t_{u})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a0f47686d64b8b1e6d995dbb1a6371d6e6fcf7)
![{\displaystyle SAF_{s1,s2}(\theta ,\tau )=e^{-[{\tfrac {1}{4\alpha }}(\theta -\omega _{d})^{2}+{\tfrac {\alpha }{4}}\tau ^{2}]}e^{j[\omega _{u}\tau -(\theta -\omega _{d})t_{0}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e199399019265863b4e5fde588b8b946f148e07)
