The Wigner-Ville Distribution Based on the Linear Canonical Transform and Its Applications for QFM Signal Parameters Estimation

Table 1

WDLs of some common signals.

Signals	WDLs
1	√𝑎^−1𝑒^(𝑗𝑐/2𝑎)𝑢2
𝛿(𝑡−𝑡0)	𝛿(2(𝑡−𝑡0))𝐾𝐴(𝑢,2𝑡−2𝑡0)
𝑒^𝑗𝜆𝑡	2𝜋𝑏𝑗𝑒^𝑗(𝑑/2𝑏)𝑢2𝛿(𝑢−𝜆𝑏),𝑎=01𝑎𝑒^𝑗(𝑑/2𝑏)𝑢2−𝑗((𝑢−𝜔1𝑏)2/2𝑎𝑏),𝑎≠0
𝑒^𝑗𝜇𝑡2/2	2𝜋𝑏𝑗𝑒^𝑗(𝑑/2𝑏)𝑢2𝛿(𝑢−2𝜇𝑏𝑡),𝑎=01𝑎𝑒^𝑗(𝑑/2𝑏)𝑢2−𝑗((𝑢−2𝑏𝜇𝑡)2/2𝑎𝑏),𝑎≠0
𝑒^𝑗𝜇𝑡2/2+𝑗𝜆𝑡+𝑗𝛾	2𝜋𝑏𝑗𝑒^𝑗(𝑑/2𝑏)𝑢2𝛿(𝑢−𝜆𝑏−2𝜇𝑏𝑡),𝑎=01𝑎𝑒^𝑗(𝑑/2𝑏)𝑢2−𝑗((𝑢−𝜆𝑏−2𝑏𝜇𝑡)2/2𝑎𝑏),𝑎≠0
𝑒^−𝑡2/2	2𝜋𝑏𝑗𝑒^𝑗(𝑑/2𝑏)𝑢2𝛿(𝑗𝑏𝑡−𝑢),𝑎=01𝑎𝑒^(𝑗𝑐/2𝑎)𝑢2−(𝑡𝑢/𝑎)+𝑗(𝑡2𝑏/2𝑎),𝑎≠0