# (b) Principle of the PFT

Time-frequency analysis is a useful tool in ground moving target parameter estimation, and it is widely adopted in many research articles. Time-frequency analysis can achieve high precision estimation of Doppler parameters of LFM signals, while it will cost a lot of calculation and limit its usage in real-time GMTI project at the same time.

WVD, STFT, FrFT and PFT are all time-frequency analysis algorithms. WVD is the most commonly used algorithm, but it suffers from the interference of cross-terms since it is bilinear transform. FrFT is free cross-term because it is a linear transform, but its estimation precision depends on the resolution of angle division. Both algorithms above are only able to estimate first- and second- order phase, which is not suitable for fast moving target imaging.

PFT is also a linear transform and shares the advantages of FrFT. Besides, PFT is capable of estimating higher order phase parameters of moving targets, which makes it perfect for fast moving target imaging [12]. Suppose x(n) is a discrete signal, its standard DFT is expressed as

If x(n) is a monochromatic signal, energy of *x(n)* is congregated in frequency domain. If x(n) has higher order phase terms, its energy will spread in frequency domain after DFT.

Consider a kth-order PPS x(n), let its expression be
where *a* are the coefficients of each order. PFT is defined as

According to the definition of (3.5.4), PFT is an expansion of DFT into kth-order. If coefficients a satisfy condition

Energy of PPS x(n) will congregated into a peak in k-order coefficient plane through PFT. By searching the value of a, phase coefficient *a* can be estimated as

The remaining problem of PFT is that higher order estimation introduces more calculation requirements. However, since the Doppler centroid has already been estimated by Hough transform, only the second- and third-order Doppler parameters are estimated in this step.