科恩系列分佈

科恩系列分佈（Cohen's class distribution）於1966年由L. Cohen首次提出，且其使用雙線性轉換亦是此種轉換形式中最通用的一種。在幾種常見的時頻分佈中，Cohen's class分佈是最強大的轉換之一。隨著近幾年來時頻分析發展，應用也越來越多元。Cohen's class分佈和短時距傅立葉變換比較起來有較高的清晰度，但也相對的有交叉項(cross-term)的問題，不過可選擇適當的遮罩函數(mask function)來將交叉項的問題降到最低。 C x ( t , f ) = ∫ − ∞ ∞ ∫ − ∞ ∞ A x ( η , τ ) Φ ( η , τ ) exp ⁡ ( j 2 π ( η t − τ f ) ) d η d τ , {\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,} ， 其中 A x ( η , τ ) = ∫ − ∞ ∞ x ( t + τ / 2 ) x ∗ ( t − τ / 2 ) e − j 2 π t η d t . {\displaystyle A_{x}\left(\eta ,\tau \right)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi t\eta }\,dt.} 為模糊函數(Ambiguity Function) ，且 Φ ( η , τ ) {\displaystyle \Phi \left(\eta ,\tau \right)} 為一遮罩函數，通常是低通函數用來濾除雜訊。 當Cohen's class分佈中的 Φ ( η , τ ) = 1 {\displaystyle \Phi \left(\eta ,\tau \right)=1} 時，Cohen's class分佈會成韋格納分布(Wigner distribution function) W x ( t , f ) = ∫ − ∞ ∞ x ( t + τ / 2 ) x ∗ ( t − τ / 2 ) e − j 2 π f τ d τ {\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi f\tau }\,d\tau } 。 利用韋格納分佈對函數 x ( t ) = e x p ( j 0.015 t 4 + j 0.06 t 3 − j 0.3 t 2 + j t ) {\displaystyle x(t)=exp(j0.015t^{4}+j0.06t^{3}-j0.3t^{2}+jt)} 作時頻分析的結果可見右圖。 當Cohen's class分佈中的 ϕ ( t , τ ) = 1 | τ | e x p ( − 2 π α τ 2 ) Π ( t τ ) {\displaystyle \phi (t,\tau )={\frac {1}{\left|\tau \right|}}exp(-2\pi \alpha \tau ^{2})\Pi ({\frac {t}{\tau }})} ，且 Φ ( η , τ ) = s i n c ( η τ ) e x p ( ( − 2 π α τ 2 ) {\displaystyle \Phi \left(\eta ,\tau \right)=sinc(\eta \tau )exp((-2\pi \alpha \tau ^{2})} 時， 其中 ϕ ( t , τ ) = ∫ − ∞ ∞ Φ ( η , τ ) e x p ( j 2 π η t ) d η {\displaystyle \phi (t,\tau )=\int _{-\infty }^{\infty }\Phi (\eta ,\tau )exp(j2\pi \eta t)d\eta } ，Cohen's class分佈會成錐狀分布。 右圖為不同的 α {\displaystyle \alpha } 值下的錐狀分佈時頻分析圖。 當Cohen's class分佈中的 Φ ( η , τ ) = e x p [ − α ( η τ ) 2 ] {\displaystyle \Phi \left(\eta ,\tau \right)=exp[-\alpha (\eta \tau )^{2}]} 時，Cohen's class分佈會成喬伊-威廉斯分布。 右圖為不同的 α {\displaystyle \alpha } 值下的錐狀分佈時頻分析圖。 優點: 1.可選擇適當的遮罩函數來避免掉交叉項問題 。 2.具有高清晰度。 缺點 1. 需要較高的計算量與時間。 2. 缺乏良好的數學特性。 C x ( t , f ) = ∫ − ∞ ∞ ∫ − ∞ ∞ A x ( η , τ ) Φ ( η , τ ) exp ⁡ ( j 2 π ( η t − τ f ) ) d η d τ , {\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,} = ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ x ( u + τ 2 ) x ∗ ( u − τ 2 ) Φ ( η , τ ) e − j 2 π u η + j 2 π ( η t − τ f ) d u d τ d η {\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{*}(u-{\frac {\tau }{2}})\Phi (\eta ,\tau )e^{-j2\pi u\eta +j2\pi (\eta t-\tau f)}dud\tau d\eta } 對   | η | > B   {\displaystyle \ \left|\eta \right|>B\ } 或   | τ | > C {\displaystyle \ \left|\tau \right|>C} ，若 Φ ( η , τ ) = 0 {\displaystyle \Phi (\eta ,\tau )=0} ，則 C x ( t , f ) = ∫ − C C ∫ − B B ∫ − ∞ ∞ x ( u + τ 2 ) x ∗ ( u − τ 2 ) Φ ( η , τ ) e − j 2 π u η + j 2 π ( η t − τ f ) d u d τ d η {\displaystyle C_{x}(t,f)=\int _{-C}^{C}\int _{-B}^{B}\int _{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{*}(u-{\frac {\tau }{2}})\Phi (\eta ,\tau )e^{-j2\pi u\eta +j2\pi (\eta t-\tau f)}dud\tau d\eta } C x ( t , f ) = ∫ − C C ∫ − ∞ ∞ x ( u + τ 2 ) x ∗ ( u − τ 2 ) [ ∫ − B B Φ ( η , τ ) e − j 2 π , η ( t − u ) d η ] e − j 2 π τ , f d u d τ {\displaystyle C_{x}(t,f)=\int _{-C}^{C}\int _{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{*}(u-{\frac {\tau }{2}})[\int _{-B}^{B}\Phi (\eta ,\tau )e^{-j2\pi ,\eta (t-u)}d\eta ]e^{-j2\pi \tau ,f}dud\tau } = ∫ − C C ∫ − ∞ ∞ x ( u + τ 2 ) x ∗ ( u − τ 2 ) Φ ( τ , t − u ) e − j 2 π τ , f d u d τ {\displaystyle =\int _{-C}^{C}\int _{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{*}(u-{\frac {\tau }{2}})\Phi (\tau ,t-u)e^{-j2\pi \tau ,f}dud\tau } ，其中 Φ ( τ , t − u ) = ∫ − B B Φ ( η , τ ) e − j 2 π , η ( t − u ) d η {\displaystyle \Phi (\tau ,t-u)=\int _{-B}^{B}\Phi (\eta ,\tau )e^{-j2\pi ,\eta (t-u)}d\eta } ，由於 Φ ( τ , t − u ) {\displaystyle \Phi (\tau ,t-u)} 和輸入無關，可事先算出，因此可簡化成兩個積分式。 C x ( t , f ) = ∫ − ∞ ∞ ∫ − ∞ ∞ x ( u + τ 2 ) x ∗ ( u − τ 2 ) ϕ ( t − u , τ ) d u e − j 2 π f τ d τ {\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{*}(u-{\frac {\tau }{2}})\phi (t-u,\tau )due^{-j2\pi f\tau }d\tau } ，其中 ϕ ( t , τ ) = ∫ − ∞ ∞ Φ ( η , τ ) e x p ( j 2 π η t ) d η {\displaystyle \phi (t,\tau )=\int _{-\infty }^{\infty }\Phi (\eta ,\tau )exp(j2\pi \eta t)d\eta } 。對 | t | > b {\displaystyle \left|t\right|>b} 或是 | τ | > c {\displaystyle \left|\tau \right|>c} ，則 C x ( t , f ) = ∫ − c c ∫ t − b t + b x ( u + τ 2 ) x ∗ ( u − τ 2 ) ϕ ( t − u , τ ) d u e − j 2 π f τ d τ {\displaystyle C_{x}(t,f)=\int _{-c}^{c}\int _{t-b}^{t+b}x(u+{\frac {\tau }{2}})x^{*}(u-{\frac {\tau }{2}})\phi (t-u,\tau )due^{-j2\pi f\tau }d\tau } ，上式為一摺積式。 模糊函數的定義為： A x ( η , τ ) = ∫ − ∞ ∞ x ( t + τ 2 ) x ∗ ( t − τ 2 ) e − j 2 π t η d t {\displaystyle A_{x}\left(\eta ,\tau \right)=\int _{-\infty }^{\infty }x(t+{\tfrac {\tau }{2}})x^{*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} 我們來看一下 x ( t ) {\displaystyle x(t)} 對於模糊函數的影響 (1) 假設 x 1 ( t ) {\displaystyle x_{1}(t)} 是一個高斯函數: a e − ( t − b ) 2 / 2 c 2 {\displaystyle ae^{-(t-b)^{2}/2c^{2}}} , 其中 a = 1 , b = 0 , c = 1 2 α {\displaystyle a=1,b=0,c={\sqrt {\tfrac {1}{2\alpha }}}} 那麼我們可以得到 x 1 ( t ) = e − α π t 2 {\displaystyle x_{1}(t)=e^{-\alpha \pi t^{2}}} , 代入模糊函數 A x ( η , τ ) {\displaystyle A_{x}\left(\eta ,\tau \right)} 中： A x 1 ( η , τ ) = ∫ − ∞ ∞ e − α π ( t + τ 2 ) 2   e − α π ( t − τ 2 ) 2   e − j 2 π t η   d t {\displaystyle A_{x_{1}}\left(\eta ,\tau \right)=\textstyle \int \limits _{-\infty }^{\infty }\displaystyle e^{-\alpha \pi {(t+{\tfrac {\tau }{2}})}^{2}}\ e^{-\alpha \pi {(t-{\tfrac {\tau }{2}})}^{2}}\ e^{-j2\pi t\eta }\ dt} = ∫ − ∞ ∞ e − α π ( 2 t 2 + τ 2 2 )   e − j 2 π t η   d t {\displaystyle =\textstyle \int \limits _{-\infty }^{\infty }\displaystyle e^{-\alpha \pi (2t^{2}+{\tfrac {\tau }{2}}^{2})}\ e^{-j2\pi t\eta }\ dt} (2) 假設 x 2 ( t ) {\displaystyle x_{2}(t)} 是一個經過 shifting 和 modulation 的高斯函數: 那麼我們可以得到 x 2 ( t ) = e − α π ( t − t 0 ) 2 + j 2 π f 0 t {\displaystyle x_{2}(t)=e^{-\alpha \pi (t-t_{0})^{2}+j2\pi f_{0}t}} , 代入模糊函數 A x ( η , τ ) {\displaystyle A_{x}\left(\eta ,\tau \right)} 中： A x 2 ( η , τ ) = ∫ − ∞ ∞ e − α π ( t + τ 2 − t 0 ) 2 + j 2 π f 0 ( t + τ 2 )   e − α π ( t − τ 2 − t 0 ) 2 − j 2 π f 0 ( t − τ 2 )   e − j 2 π t η   d t {\textstyle A_{x_{2}}\left(\eta ,\tau \right)=\textstyle \int \limits _{-\infty }^{\infty }\displaystyle e^{-\alpha \pi {(t+{\tfrac {\tau }{2}}-t_{0})}^{2}+j2\pi f_{0}(t+{\tfrac {\tau }{2}})}\ e^{-\alpha \pi {(t-{\tfrac {\tau }{2}}-t_{0})}^{2}-j2\pi f_{0}(t-{\tfrac {\tau }{2}})}\ e^{-j2\pi t\eta }\ dt} = ∫ − ∞ ∞ e − α π ( 2 ( t − t 0 ) 2 + τ / 2 2 ) + j 2 π f 0 τ   e − j 2 π t η   d t {\displaystyle =\textstyle \int \limits _{-\infty }^{\infty }\displaystyle e^{-\alpha \pi (2(t-t_{0})^{2}+{\tau /2}^{2})+j2\pi f_{0}\tau }\ e^{-j2\pi t\eta }\ dt} = ∫ − ∞ ∞ e − α π ( 2 t 2 + τ 2 2 )   e j 2 π f 0 τ   e − j 2 π t η   e − j 2 π t 0 η   d t {\displaystyle =\textstyle \int \limits _{-\infty }^{\infty }\displaystyle e^{-\alpha \pi (2t^{2}+{\tfrac {\tau ^{2}}{2}})}\ e^{j2\pi f_{0}\tau }\ e^{-j2\pi t\eta }\ e^{-j2\pi t_{0}\eta }\ dt} 我們可以看到 | A x 1 ( τ , η ) | = | A x 2 ( τ , η ) | {\displaystyle |A_{x_{1}}\left(\tau ,\eta \right)|=|A_{x_{2}}\left(\tau ,\eta \right)|} , 因此我們可以得出 time shifting t 0 {\displaystyle t_{0}} 和 modulation f 0 {\displaystyle f_{0}} 並不會影響 | A x ( τ , η ) | {\displaystyle |A_{x}\left(\tau ,\eta \right)|} 積分後， A x ( τ , η ) = 1 2 α e − π ( α τ 2 2 + η 2 2 α ) e j 2 π ( f 0 τ − t 0 η ) {\displaystyle A_{x}\left(\tau ,\eta \right)={\sqrt {\tfrac {1}{2\alpha }}}e^{-\pi ({\tfrac {\alpha \tau ^{2}}{2}}+{\tfrac {\eta ^{2}}{2\alpha }})}e^{j2\pi (f_{0}\tau -t_{0}\eta )}} 所以 A x ( τ , η ) {\displaystyle A_{x}\left(\tau ,\eta \right)} 在 τ = 0 , η = 0 {\displaystyle \tau =0,\eta =0} 的地方會有最大的 | A x ( τ , η ) | {\displaystyle |A_{x}\left(\tau ,\eta \right)|} 上述所列出來的是當 x ( t ) {\displaystyle x(t)} 只有一項而已 (one term only)，如果 x ( t ) {\displaystyle x(t)} 有兩項以上的元素構成 (more than two terms), x ( t ) = x 1 ( t ) + x 2 ( t ) + ⋅ ⋅ ⋅ + x n ( t ) {\displaystyle x(t)=x_{1}(t)+x_{2}(t)+\cdot \cdot \cdot +x_{n}(t)} ，依然會有交叉項 (cross-term) 的問題存在。 假設 x ( t ) = x 1 ( t ) + x 2 ( t ) {\displaystyle x(t)=x_{1}(t)+x_{2}(t)} , 其中 { x 1 ( t ) = e − α 1 π ( t − t 1 ) 2 + j 2 π f 1 t x 2 ( t ) = e − α 2 π ( t − t 2 ) 2 + j 2 π f 2 t {\displaystyle {\begin{cases}x_{1}(t)=e^{-\alpha _{1}\pi (t-t_{1})^{2}+j2\pi f_{1}t}\\x_{2}(t)=e^{-\alpha _{2}\pi (t-t_{2})^{2}+j2\pi f_{2}t}\end{cases}}} 將 x ( t ) {\displaystyle x(t)} 代入模糊函數 A x ( η , τ ) = ∫ − ∞ ∞ x ( t + τ 2 ) x ∗ ( t − τ 2 ) e − j 2 π t η d t {\displaystyle A_{x}\left(\eta ,\tau \right)=\int _{-\infty }^{\infty }x(t+{\tfrac {\tau }{2}})x^{*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} 中： A x ( η , τ ) = ∫ − ∞ ∞ x 1 ( t + τ 2 ) x 1 ∗ ( t − τ 2 ) e − j 2 π t η d t ⏟ A x 1 ( τ , η ) + ∫ − ∞ ∞ x 2 ( t + τ 2 ) x 2 ∗ ( t − τ 2 ) e − j 2 π t η d t ⏟ A x 2 ( τ , η ) {\displaystyle A_{x}\left(\eta ,\tau \right)=\underbrace {\int _{-\infty }^{\infty }x_{1}(t+{\tfrac {\tau }{2}})x_{1}^{*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} _{A_{x_{1}}(\tau ,\eta )}+\underbrace {\int _{-\infty }^{\infty }x_{2}(t+{\tfrac {\tau }{2}})x_{2}^{*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} _{A_{x_{2}}(\tau ,\eta )}} + ∫ − ∞ ∞ x 1 ( t + τ 2 ) x 2 ∗ ( t − τ 2 ) e − j 2 π t η d t ⏟ A x 1 x 2 ( τ , η ) + ∫ − ∞ ∞ x 2 ( t + τ 2 ) x 1 ∗ ( t − τ 2 ) e − j 2 π t η d t ⏟ A x 2 x 1 ( τ , η ) {\displaystyle +\underbrace {\int _{-\infty }^{\infty }x_{1}(t+{\tfrac {\tau }{2}})x_{2}^{*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} _{A_{{x_{1}}{x_{2}}}(\tau ,\eta )}+\underbrace {\int _{-\infty }^{\infty }x_{2}(t+{\tfrac {\tau }{2}})x_{1}^{*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} _{A_{{x_{2}}{x_{1}}}(\tau ,\eta )}} 其中 { A u t o − t e r m s : A x 1 ( τ , η ) ,   A x 2 ( τ , η ) C r o s s − t e r m s :   A x 1 x 2 ( τ , η ) ,   A x 2 x 1 ( τ , η ) {\displaystyle {\begin{cases}Auto-terms:\quad A_{x_{1}}(\tau ,\eta ),\ A_{x_{2}}(\tau ,\eta )\\Cross-terms:\ A_{{x_{1}}{x_{2}}}(\tau ,\eta ),\ A_{{x_{2}}{x_{1}}}(\tau ,\eta )\end{cases}}} A x 1 ( τ , η ) = 1 2 α 1   e − π ( α 1 τ 2 2 + η 2 2 α 1 )   e j 2 π ( f 1 τ − t 1 η ) {\displaystyle A_{x_{1}}\left(\tau ,\eta \right)={\sqrt {\tfrac {1}{2\alpha _{1}}}}\ e^{-\pi ({\tfrac {\alpha _{1}\tau ^{2}}{2}}+{\tfrac {\eta ^{2}}{2\alpha _{1}}})}\ e^{j2\pi (f_{1}\tau -t_{1}\eta )}} A x 2 ( τ , η ) = 1 2 α 2   e − π ( α 2 τ 2 2 + η 2 2 α 2 )   e j 2 π ( f 2 τ − t 2 η ) {\displaystyle A_{x_{2}}\left(\tau ,\eta \right)={\sqrt {\tfrac {1}{2\alpha _{2}}}}\ e^{-\pi ({\tfrac {\alpha _{2}\tau ^{2}}{2}}+{\tfrac {\eta ^{2}}{2\alpha _{2}}})}\ e^{j2\pi (f_{2}\tau -t_{2}\eta )}} (1) α 1 ≠ α 2 {\displaystyle \alpha _{1}\neq \alpha _{2}} A x 1 x 2 ( τ , η ) = 1 ( α 1 + α 2 )   e − π ( ( α 1 + α 2 ) ( τ − t 1 + t 2 ) 2 4   +   [ ( α 1 − α 2 ) ( τ − t 1 + t 2 ) − j 2 ( η − f 1 + f 2 ) ] 2 4 ( α 1 + α 2 ) )   e j 2 π [ ( f 1 + f 2 2 ) τ − t 1 + t 2 2 η + ( f 1 − f 2 ) ( t 1 + t 2 ) 2 ] {\displaystyle A_{{x_{1}}{x_{2}}}(\tau ,\eta )={\sqrt {\tfrac {1}{(\alpha _{1}+\alpha _{2})}}}\ e^{-\pi ({\tfrac {(\alpha _{1}+\alpha _{2})(\tau -t_{1}+t_{2})^{2}}{4}}\ +\ {\tfrac {[(\alpha _{1}-\alpha _{2})(\tau -t_{1}+t_{2})-j2(\eta -f_{1}+f_{2})]^{2}}{4(\alpha _{1}+\alpha _{2})}})}\ e^{j2\pi [({\tfrac {f_{1}+f_{2}}{2}})\tau -{\tfrac {t_{1}+t_{2}}{2}}\eta +{\tfrac {(f_{1}-f_{2})(t_{1}+t_{2})}{2}}]}} = 1 2 α u   e − π ( α u ( τ − t d ) 2 2   +   [ α d ( τ − t d ) − j 2 ( η − f d ) ] 2 8 α u )   e j 2 π ( f u τ − t n η + f d t u ) {\displaystyle ={\sqrt {\tfrac {1}{2\alpha _{u}}}}\ e^{-\pi ({\tfrac {\alpha _{u}(\tau -t_{d})^{2}}{2}}\ +\ {\tfrac {[\alpha _{d}(\tau -t_{d})-j2(\eta -f_{d})]^{2}}{8\alpha _{u}}})}\ e^{j2\pi (f_{u}\tau -t_{n}\eta +f_{d}t_{u})}} A x 2 x 1 ( τ , η ) = A x 1 x 2 ∗ ( − τ , − η ) {\displaystyle A_{{x_{2}}{x_{1}}}(\tau ,\eta )=A_{{x_{1}}{x_{2}}}^{*}(-\tau ,-\eta )} { t u = t 1 + t 2 2 ,   f u = f 1 + f 2 2 ,   α u = α 1 + α 2 2 t d = t 1 − t 2 ,   f d = f 1 − f 2 ,   α d = α 1 − α 2 {\displaystyle {\begin{cases}t_{u}={\tfrac {t_{1}+t_{2}}{2}},\ f_{u}={\tfrac {f_{1}+f_{2}}{2}},\ \alpha _{u}={\tfrac {\alpha _{1}+\alpha _{2}}{2}}\\t_{d}=t_{1}-t_{2},\ f_{d}=f_{1}-f_{2},\ \alpha _{d}=\alpha _{1}-\alpha _{2}\end{cases}}} (2) α 1 = α 2 {\displaystyle \alpha _{1}=\alpha _{2}} A x 1 x 2 ( τ , η ) = 1 2 α u   e − π ( α u ( τ − t d ) 2 2   +   ( η − f d ) 2 2 α u )   e j 2 π ( f u τ − t n η + f d t u ) {\displaystyle A_{{x_{1}}{x_{2}}}(\tau ,\eta )={\sqrt {\tfrac {1}{2\alpha _{u}}}}\ e^{-\pi ({\tfrac {\alpha _{u}(\tau -t_{d})^{2}}{2}}\ +\ {\tfrac {(\eta -f_{d})^{2}}{2\alpha _{u}}})}\ e^{j2\pi (f_{u}\tau -t_{n}\eta +f_{d}t_{u})}} A x 2 x 1 ( τ , η ) = A x 1 x 2 ∗ ( − τ , − η ) {\displaystyle A_{{x_{2}}{x_{1}}}(\tau ,\eta )=A_{{x_{1}}{x_{2}}}^{*}(-\tau ,-\eta )} 因此，我們目前得到 A x 1 ( τ , η ) , A x 2 ( τ , η ) {\displaystyle A_{x_{1}}\left(\tau ,\eta \right),A_{x_{2}}\left(\tau ,\eta \right)} (auto-terms) 和 A x 1 x 2 ( τ , η ) , A x 2 x 1 ( τ , η ) {\displaystyle A_{{x_{1}}{x_{2}}}(\tau ,\eta ),A_{{x_{2}}{x_{1}}}(\tau ,\eta )} (cross-terms) 的公式，我們再仔細的分析 auto-terms 和 cross-terms 分別發生最大值的位置。 首先，先看 Auto-terms： | A x 1 ( τ , η ) | {\displaystyle |A_{x_{1}}\left(\tau ,\eta \right)|} 最大值發生在 τ = 0 , η = 0 {\displaystyle \tau =0,\eta =0} 的地方 | A x 2 ( τ , η ) | {\displaystyle |A_{x_{2}}\left(\tau ,\eta \right)|} 最大值發生在 τ = 0 , η = 0 {\displaystyle \tau =0,\eta =0} 的地方 而 Cross-terms： | A x 1 x 2 ( τ , η ) | {\displaystyle |A_{{x_{1}}{x_{2}}}(\tau ,\eta )|} 最大值發生在 τ = t d , η = f d {\displaystyle \tau =t_{d},\eta =f_{d}} 的地方 | A x 2 x 1 ( τ , η ) | {\displaystyle |A_{{x_{2}}{x_{1}}}(\tau ,\eta )|} 最大值發生在 τ = − t d , η = − f d {\displaystyle \tau =-t_{d},\eta =-f_{d}} 的地方 換句話說，如果我們繪製一個 x軸為 τ {\displaystyle \tau } , y軸為 η {\displaystyle \eta } 的座標圖，Auto-terms發生在原點 ( 0 , 0 ) {\displaystyle (0,0)} 的位置，而 Cross-terms 則是以原點為對稱中心，在第一象限和第三象限的位置， 這也是為什麼可以透過一個低通函數來濾除雜訊，把主成分 Auto-terms 分離出來，避免交叉項的問題。 維格納分布是由尤金·維格納於 1932 年提出的新的時頻分析方法，對於非穩態的訊號有不錯的表現。 相較於傅立葉轉換或是短時距傅立葉轉換，維格納分布能有比較好的解析能力。 維格納分布的定義為: W x ( t , f ) = ∫ − ∞ ∞ x ( t + τ 2 ) x ∗ ( t − τ 2 ) e − j 2 π τ f d τ {\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+{\frac {\tau }{2}})x^{*}(t-{\frac {\tau }{2}})e^{-j2\pi \tau f}\,d\tau } 如果我們假設 x ( t ) {\displaystyle x(t)} 是一個具有弦波特性的訊號, x ( t ) = e j 2 π f 0 t {\displaystyle x(t)=e^{j2\pi f_{0}t}} 那麼將此 x ( t ) {\displaystyle x(t)} 代入維格納分布中， W x ( t , f ) = ∫ − ∞ ∞ e j 2 π f 0 ( t + τ 2 ) e − j 2 π f 0 ( t − τ 2 )   e − j 2 π τ f   d τ {\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }e^{j2\pi f_{0}(t+{\tfrac {\tau }{2}})}e^{-j2\pi f_{0}(t-{\tfrac {\tau }{2}})}\ e^{-j2\pi \tau f}\ d\tau } = ∫ − ∞ ∞ e j 2 π f 0 τ   e − j 2 π τ f   d τ {\displaystyle =\int _{-\infty }^{\infty }e^{j2\pi f_{0}\tau }\ e^{-j2\pi \tau f}\ d\tau } = ∫ − ∞ ∞ e − j 2 π τ ( f − f 0 ) d τ {\displaystyle =\int _{-\infty }^{\infty }e^{-j2\pi \tau (f-f_{0})}d\tau } = δ ( f − f 0 ) {\displaystyle =\delta (f-f_{0})} 所以當 x ( t ) = e j 2 π f 0 t {\displaystyle x(t)=e^{j2\pi f_{0}t}} 時， W x ( t , f ) {\displaystyle W_{x}(t,f)} 在 f = f 0 {\displaystyle f=f_{0}} 的地方會有最大值。 換句話說，當 x ( t ) {\displaystyle x(t)} 有 modulation f 0 {\displaystyle f_{0}} 或是有 time shifting t 0 {\displaystyle t_{0}} 的情況發生時，會影響維格納分布 (Wigner Distribution Function) 最大值 | W x ( t , f ) | {\displaystyle |W_{x}(t,f)|} 的位置 然而，對於科恩系列分布 (Cohen's class distribution)而言，time shifting t 0 {\displaystyle t_{0}} 和 modulation f 0 {\displaystyle f_{0}} 並不會影響 | A x ( τ , η ) | {\displaystyle |A_{x}\left(\tau ,\eta \right)|} Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007. Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2018.