模糊函數與韋格納分佈的關係

模糊函數(Ambiguity function,AF): A F s ( θ , τ ) = ∫ − ∞ ∞ s ( t + τ 2 ) s ∗ ( t − τ 2 ) e − j θ t d t {\displaystyle AF_{s}(\theta ,\tau )=\int _{-\infty }^{\infty }s\left(t+{\frac {\tau }{2}}\right)s^{*}\left(t-{\frac {\tau }{2}}\right)e^{-j\theta t}\,dt} 韋格納分佈(Wigner distribution,WD): W D s ( t , ω ) = ∫ − ∞ ∞ s ( t + τ 2 ) s ∗ ( t − τ 2 ) e − j ω τ d τ {\displaystyle WD_{s}(t,\omega )=\int _{-\infty }^{\infty }s\left(t+{\frac {\tau }{2}}\right)s^{*}\left(t-{\frac {\tau }{2}}\right)e^{-j\omega \tau }\,d\tau } 一個訊號s(t)，自相關函數為 R ( τ ) = ∫ − ∞ ∞ s ( t ) s ∗ ( t − τ ) d t {\displaystyle R(\tau )=\int _{-\infty }^{\infty }s(t)s^{*}(t-\tau )\,dt} 如果 R ( τ ) {\displaystyle R(\tau )} 為時間相依性(time-dependent)，則時間相依自相關(time-dependent auto-correlation)為 R ( t , τ ) {\displaystyle R(t,\tau )} ， 時間相依(時變)頻譜(time-dependent spectrum)可以表示的形式類似於傳統的功率譜，即對時間相依自相關函數做傅立葉變換。 P ( t , ω ) = ∫ − ∞ ∞ R ( t , τ ) e − j ω τ d τ {\displaystyle P(t,\omega )=\int _{-\infty }^{\infty }R(t,\tau )e^{-j\omega \tau }\,d\tau } 不同的時間相依自相關會導致不同的時間相依功率譜。 如果 R ( t , τ ) = s ( t + τ 2 ) s ∗ ( t − τ 2 ) {\displaystyle R(t,\tau )=s\left(t+{\frac {\tau }{2}}\right)s^{*}\left(t-{\frac {\tau }{2}}\right)} ，則時間相依功率譜變成為Wigner distribution 若對 R ( t , τ ) {\displaystyle R(t,\tau )} 中的t做傅立葉逆轉換，得到另一個時頻表示，對稱模糊函數(symmetric ambiguity function,SAF) S A F s ( θ , τ ) = 1 2 π ∫ − ∞ ∞ s ( t + τ 2 ) s ∗ ( t − τ 2 ) e j θ t d t {\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} 模糊函數反映信號在時間和相位的相關性，並已廣泛應用在雷達和聲納系統上。 給一個對稱模糊函數 S A F s ( θ , τ ) {\displaystyle SAF_{s}(\theta ,\tau )} ，透過傅立葉變換可以得到時間相依自相關: ∫ − ∞ ∞ S A F s ( θ , τ ) e − j θ t d θ = s ( t + τ 2 ) s ∗ ( t − τ 2 ) {\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)} 由上式可以推得 W D s ( t , ω ) = ∫ − ∞ ∞ ∫ − ∞ ∞ S A F s ( θ , τ ) e − j ( ω τ + θ t ) d θ d τ {\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 } 也就是對對稱模糊函數做兩次傅立葉變換可以得到Wigner distribution 一個訊號為兩個Gaussian函數的和: s ( t ) = ∑ i = 1 2 s i ( t ) = ∑ i = 1 2 α π 4 e − α 2 ( t − t i ) 2 + j ω i t {\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}} ⇒ S A F s ( θ , τ ) = ∑ i = 1 2 S A F s i ( θ , τ ) + S A F s 1 , s 2 ( θ , τ ) + S A F s 2 , s 1 ( θ , τ ) {\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 S A F s 1 ( θ , τ ) , S A F s 2 ( θ , τ ) {\displaystyle SAF_{s1}(\theta ,\tau ),SAF_{s2}(\theta ,\tau )} 集中在原點(0,0)，而 S A F s 1 , s 2 ( θ , τ ) {\displaystyle SAF_{s1,s2}(\theta ,\tau )} 集中在 ( t 1 − t 2 , ω 1 − ω 2 ) {\displaystyle (t_{1}-t_{2},\omega _{1}-\omega _{2})} ，而 S A F s 2 , s 1 ( θ , τ ) {\displaystyle SAF_{s2,s1}(\theta ,\tau )} 相似於 S A F s 1 , s 2 ( θ , τ ) {\displaystyle SAF_{s1,s2}(\theta ,\tau )} ，除了中心點在 ( t 2 − t 1 , ω 2 − ω 1 ) {\displaystyle (t_{2}-t_{1},\omega _{2}-\omega _{1})} S A F s 1 , s 2 ( θ , τ ) = e − 1 4 α ( θ − ω d ) 2 + α 4 ( τ − t d ) 2 e j ( ω u τ − θ t u + ω d t u ) {\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})}} , t u = t 1 + t 2 2 {\displaystyle t_{u}={\tfrac {t_{1}+t_{2}}{2}}} , ω u = ω 1 + ω 2 2 {\displaystyle \omega _{u}={\tfrac {\omega _{1}+\omega _{2}}{2}}} , t d = t 1 − t 2 {\displaystyle t_{d}=t_{1}-t_{2}} , ω d = ω 1 − ω 2 {\displaystyle \omega _{d}=\omega _{1}-\omega _{2}} 從範例中得知一項重要事實，即為，在模糊域(ambiguity domain)中的auto-term總是集中在原點(0,0)，而cross-term總是在遠離原點處，所以可以用一個2D lowpass filter在模糊域中抑制cross-term的干擾，如下: ∫ − ∞ ∞ ∫ − ∞ ∞ S A F s ( θ , τ ) Φ ( θ , τ ) e − j ( θ t + ω τ ) d θ d τ {\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }SAF_{s}(\theta ,\tau )\Phi (\theta ,\tau )e^{-j(\theta t+\omega \tau )}\,d\theta \,d\tau } ,其中 Φ ( θ , τ ) {\displaystyle \Phi (\theta ,\tau )} 為2D lowpass filter 如果 ∫ − ∞ ∞ ∫ − ∞ ∞ Φ ( θ , τ ) d θ d τ = ϕ ( t , ω ) {\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\Phi (\theta ,\tau )\,d\theta \,d\tau =\phi (t,\omega )} ，則 ∫ − ∞ ∞ ∫ − ∞ ∞ S A F s ( θ , τ ) Φ ( θ , τ ) e − j ( θ t + ω τ ) d θ d τ {\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }SAF_{s}(\theta ,\tau )\Phi (\theta ,\tau )e^{-j(\theta t+\omega \tau )}\,d\theta \,d\tau } = ∫ − ∞ ∞ ∫ − ∞ ∞ ϕ ( x , y ) W D s ( t − x , ω − y ) d x d y = S W D ( t , ω ) {\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\phi (x,y)WD_{s}(t-x,\omega -y)\,dx\,dy=SWD(t,\omega )} 其中SWD為smoothed Wigner distribution 通常 Φ ( θ , τ ) {\displaystyle \Phi (\theta ,\tau )} ( 和 ϕ ( t , ω ) {\displaystyle \phi (t,\omega )} )當作kernal function，用來控制SWD的特性。 若Wigner分佈和對稱模糊函數用大小(magnitude)及相位(phase)表示，如下: W D s 1 , s 2 ( t , ω ) = A W D ( t , ω ) e j φ W D ( t , ω ) {\displaystyle WD_{s1,s2}(t,\omega )=A_{WD}(t,\omega )e^{j\varphi _{WD}(t,\omega )}} S A F s 1 , s 2 ( θ , τ ) = A S A F ( θ , τ ) e j φ S A F ( θ , τ ) {\displaystyle SAF_{s1,s2}(\theta ,\tau )=A_{SAF}(\theta ,\tau )e^{j\varphi _{SAF}(\theta ,\tau )}} 而 ∂ ∂ θ φ S A F ( θ , τ ) = − t u {\displaystyle {\frac {\partial }{\partial \theta }}\varphi _{SAF}(\theta ,\tau )=-t_{u}} , ∂ ∂ τ φ S A F ( θ , τ ) = ω u {\displaystyle {\frac {\partial }{\partial \tau }}\varphi _{SAF}(\theta ,\tau )=\omega _{u}} 也就是說對對稱模糊函數的相位做偏微分，會等於Wigner分佈的時頻(time-frequency)中心。 相反地， ∂ ∂ ω φ W D ( t , ω ) = t d {\displaystyle {\frac {\partial }{\partial \omega }}\varphi _{WD}(t,\omega )=t_{d}} , ∂ ∂ t φ W D ( t , ω ) = ω d {\displaystyle {\frac {\partial }{\partial t}}\varphi _{WD}(t,\omega )=\omega _{d}} 則為對Wigner分佈的相位做偏微分，會等於對稱模糊函數的中心。 如果 ω 1 = ω 2 = ω 0 {\displaystyle \omega _{1}=\omega _{2}=\omega _{0}} ，則 S A F s 1 , s 2 ( θ , τ ) = e − [ 1 4 α θ 2 + α 4 ( τ − t d ) 2 ] e j ( ω 0 τ − θ t u ) {\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})}} 會集中在 τ {\displaystyle \tau } 軸上。 如果 t 1 = t 2 = t 0 {\displaystyle t_{1}=t_{2}=t_{0}} ，則 S A F s 1 , s 2 ( θ , τ ) = e − [ 1 4 α ( θ − ω d ) 2 + α 4 τ 2 ] e j [ ω u τ − ( θ − ω d ) t 0 ] {\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}]}} 會集中在 θ {\displaystyle \theta } 軸上。 Weiss, Lora G. 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