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The ambiguity function (AF) as an important tool for time-frequency analysis, has widely been used in radar signal processing, sonar technology and so on. It does well in analysing chirp signals. However, it fails in estimating the cubic phase (CP) signal, which is required in many applications. As the generalisation of the Fourier transform (FT), the linear canonical transform (LCT) has received much attention, and has found many applications in filter design, pattern recognition, optics and so on. In this study, the authors define a new kind of AF - the AF based on the LCT (LCTAF), which is proposed to estimate the CP signal. Some important properties of LCTAF are discussed, such as symmetry and conjugation property, shifting property and Moyal formula. The relationships between the LCTAF and other time-frequency analysis distributions are derived, including the classical AF, the Wigner distribution function (WDF) based on LCT, the short-FT and the wavelet transform. The linear canonical AF (LCAF) is another kind of AF. The authors also discuss its relation with AF and WDF and give some new properties of the LCAF. At last, the LCTAF is applied for estimating the CP signal. The simulation indicates that the LCTAF is useful and effective.