An interesting phenomena, that the effective detector is not only limited to the integration of the 4th-power modulus of the fractional Fourier transform, but can also be generalized to *n* th-power modulus for , is found in our simulation. In this section, the mathematical derivation is implemented to prove it.

The generalized detector is written as follows:

The effectiveness of the detector defined by (13) will be demonstrated, if the condition given as follows is satisfied:

where *K* is the chirp rate of the LFM-signal.

Thus the differential of the detector with respect to the variable *α* is necessary to work out,

Since the kernel function *K* _{
α
}(*t*,*u*) is the only item that contains the variable *α* in the definition formula of FRFT expressed by (1), the differential of the FRFT *X* _{
α
}(*u*) with respect to the variable *α* can be written as

From (2), the differential of the kernel function *K* _{
α
}(*t*,*u*) with respect to the variable *α* is

Therefore

Substituting the property of FRFT expressed by (3) into (18), we can obtain

Considering the real part of (19) and omitting the imaginary part, the following equation can be obtained:

For LFM-signal, the first- and second-order differential in time domain is expressed as follows, respectively,

Implementing FRFT operation to both sides of (21) and (22), we get

Substituting the property of FRFT expressed by (3) into the right side of (23) and (24), the FRFT of the second-order differential of the LFM-signal can be obtained as:

Meanwhile, another form of the FRFT of the differential of the LFM-signal can be derived using the temporal differential property of FRFT expressed by (4), respectively,

Combining (23), (25), and (26) the second order differential of FRFT of LFM-signal in fractional Fourier domain is obtained and expressed by

Substituting (27) into (20), we can obtain

By making the following change: , (28) can be rewritten as the following form:

Substituting (29) into (15), an ordinary differential equation of the detector function can be obtained.

Combining the initial condition

The solution of the ordinary differential equation is obtained.

It can be seen that, for LFM-signal, the generalized detector will reach its maximum corresponding to the chirp rate. Therefore, the generalized detector can be regarded as an effective detector for LFM-signal. However, the 4th-power modulus form detector is most frequently used in practice for its lower computational complexity compared with other generalized forms.