Nonlinear FM Waveform Design to Reduction of sidelobe level in Autocorrelation Function

In this paper, we analyze variation of peak sidelobe level with regard to mainlobe width for four different window functions. In order to reduce sidelobe level of autocorrelation function, shaping the power spectral density can be used, since autocorrelation function is the inverse Fourier transform of the PSD [8], [9]. To achieve this goal, Chebyshev, Kaiser, Taylor, and raised-cosine windows are considered as four desired PSDs. Afterwards, group time delay function in frequency domain is carried out using the stationary phase concept (SPC). This was done by integration of PSD function and applying boundary conditions [10]. Then group time delay inverse function is computed yielding frequency function in time domain. The results of autocorrelation function obtained from the simulation are presented for all the aforementioned window functions and comparison between sidelobe level and mainlobe width are made.

As the sidelobe level reduces, the mainlobe width increases [11], [12]. This can diminish target detection accuracy. In order to preserve the accuracy, an optimized condition should be considered which is discussed in the paper. Furthermore, the resulting PSD plot and desired PSD plot were compared in order to calculate error percentage. It should be mentioned that the SPC is the possible source for this error due in large to the fact that SPC is based on approximated relations.

The remainder of the paper is organized as follows. NLFM signal is designed for four different windows in section II. Section III discusses the simulation results; Section IV concludes the paper.

The first study on using SPC for NLFM was carried out in 1964 by Fowle [8]. To design NLFM signal, a low-pass signal denoted as 𝑥(𝑡) is considered.

(1) 𝑥(𝑡) = 𝑎(𝑡)exp (𝑗𝜑(𝑡))

Where 𝑎(𝑡) and 𝜑(𝑡) are amplitude and phase of the signal respectively. The instantaneous frequency 𝑓 𝑚 at time 𝑡 𝑚 is defined as time derivative of the phase:

(2) 𝑓 𝑚 = 1 2𝜋

We shall write the Fourier transform of 𝑥(𝑡) as 𝑋(𝑓). Based on SPC, PSD at a frequency is large if the rate of frequency change at that time is relatively small. Thus, the relation between PSD and frequency variation can be expressed as:

According to (3) it is clear that PSD is inversely proportional to second time derivative of phase and it is directly proportional with signal amplitude. Since frequency changes with regard to time is linear for LFM, the second time derivative of phase is constant and the PSD depends on signal amplitude. In NLFM approach, signal amplitude is assumed to be constant (𝑎(𝑡) = 𝐴, A is constant) and PSD is depended on the value of second order phase derivative. Let us consider 𝑋 2 (𝑓) with an desired PSD such as 𝑍 2 (𝑓) , where 𝑍 2 (𝑓) only depends on the second derivative of phase of 𝑥(𝑡) signal. Equation ( 3) is reproduced in frequency domain as:

⁄ (𝐵 is the bandwidth), then the first derivative of phase 𝛷′(𝑓) is resulted by integral of the second derivative 𝛷′′(𝑓) which can be expressed as:

The following relation defines group time delay function 𝑇 𝑔 (𝑓) as:

Substituting of ( 6) in (5) yields:

Where 𝑘 1 and 𝑘 2 are the integration constant which are determined using the boundary conditions of group time delay function. If the signal pulse width is equal to T, then the boundary conditions of group time delay function written in the form

. Moreover, instantaneous frequency as a function of time is given by ( 8)

Finally, the phase of the designed signal can be obtained from frequency function using the following equation.

NLFM waveforms can be designed using the raised cosine window. For reason of better results, second order raised cosine window is considered in this paper. Thus, PSD relation can be written in the following form Where 𝑘 is a constant value which can control sidelobe level and M is window length. Assume that 𝑓 = 𝑛𝐵 (𝑀 -1) ⁄ then, by integration and applying the boundary condition, group time delay function can be derived as:

The inverse function of 𝑇 𝑔 (𝑓) complicated and it does not have closed-form expression. To illustrate the result, a simulation of inverse function for 𝑇 𝑔 (𝑓) was performed.

The optimal Dolph-Chebyshev window transform in closedform is given by:

Where M and 𝛼 are window length and parameter for determining the Chebyshev norm of the sidelobes to be -20𝛼 dB, respectively. Afterwards, zero-phase Dolph-Chebyshev window was calculated using the inverse DFT of 𝑊(𝑚) [13]. ( 14)

The Kaiser window can be computed through approximation of the discrete prolate spheroidal sequence (DPSS) window using Bessel functions which is written in the following form:

In the above equation, 𝐼 0 and 𝛼 repr

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