# Analysis of the Nonlinearity Problem.

The ideal transmitted signal, received signal and beat signal without nonlinear phase error can be expressed respectively as

where s is the time delay. As shown in Eq. 5.3.3, the signal after the deramping operation is a signal-frequency signal of t_{r}, and the last term is the RVP term.

If the nonlinear phase error exists in the system, the transmitted signal is added with a nonlinear phase error. Suppose the nonlinearity error is *e(t),* the transmitted signal with nonlinear phase error can be expressed as

The received signal can be expressed as

And the beat signal can be expressed as

It can be noted that the transmitted and received nonlinearities are overlapped in beat signal, and the nonlinearity term in Eq. 5.3.6 differs as time delay changes. If the time delay is small, most of the nonlinearities from the transmitted and received signals are eliminated, while the nonlinear phase error increases with the time delay in the beat signal.

The phase in Eq. 5.3.7 can be considered as the difference of the transmitted nonlinearity and its attenuated copy. Hence, to correct nonlinearity of beat signal, the transmitted nonlinearity must be estimated first. The estimation precision of transmitted nonlinearity determines the performance of nonlinearity correction of FMCW SAR.