Hardware Resource Optimized Detection of LFM Signals with Unknown Start Frequency and Frequency Rate

Abstract

Detection of very low-SNR LFM signals with unknown start frequency and frequency rate is of great interest both in electronic support measure (ESM), and radio astronomy. The direct method for LFM signal detection needs a bank of matched-filters which is a really hardware consuming solution. As another solution, a bank of de-ramping blocks, followed by FFT units, can be used with the same performance as matched-filters bank. In such an alternative solution, with no optimization constraint, it is quite likely to reach a hardware extensive solution with limited processing gain. In this paper, a novel method based on de-ramping bank is proposed. Also, an optimization problem is developed, which could determine the optimum values for detection structure’s parameters, e.g. number of channels, as well as FFT length. It is shown that, the optimized detector features better processing gain in comparison to the non-optimized versions. Furthermore, adding a moving average at the output of the FFT could make remarkable improvement on detection performance. Moreover, the proposed detector is compared against the conventional methods in terms of detection performance and computational complexity characteristic, which aptly prove the superiority of the proposed method.

Appendix

In this section, the procedure of calculation of minimum detectable SNR, as well as processing loss of the proposed detection structure is discussed in detail.

1.1 Minimum Detectable SNR for De-Ramping Structure

As we have two main blocks in algorithm structure, i.e. de-ramping-FFT and MA, the aggregate processing gain is equal to sum of these blocks’ processing gain (in dB). Thus, if SNR_in and SNR_fft denote input and FFT output SNRs, respectively, then we have:

$$ SN{R}_{fft}= SN{R}_{in}+10\log \left(\frac{4M{f}_s}{B_{max}}\right) $$

(32)

Where, the second term in the right side of the equation is the same processing gain which was resulted from the optimum choices of M and N_fft = N_optimum. Also, as the length of MA is $\xi =\frac{\frac{k_0}{2}N{N}_{fft}}{f_s^2}$, its processing gain would be equal to the square root of MA length, i.e. $\sqrt{\xi }$. Therefore, applying MA on the data, the output SNR shown by SNR_out could be calculated as:

$$ SN{R}_{out}= SN{R}_{fft}+10\log \left(\sqrt{\xi}\right) $$

(33)

Substituting (33) in (32), the output SNR in terms of input SNR would be equal to:

$$ SN{R}_{out}= SN{R}_{in}+10\log \left(\frac{4M{f}_s}{B_{max}}\right)+10\log \left(\sqrt{\xi}\right) $$

(34)

However as discussed before, number of effective cells is not always equal to its maximum, and could be less than that. So, a value equal to L should be considered as such a difference. Hence, output SNR is calculated as follows:

$$ SN{R}_{out}= SN{R}_{in}+10\log \left(\frac{4M{f}_s}{B_{max}}\right)+10\log \left(\sqrt{\xi}\right)-L $$

(35)

Any LFM detector requires a minimum SNR of 11 to 16 dB of SNR, here shown by SNR_min. Therefore, if an algorithm is able to provide SNR_min, the signal could be detected with P_d of 100%. Consequently, minimum detectable input SNR, SNR_{in, min} is as follows:

$$ SN{R}_{in,\mathit{\min}}= SN{R}_{min}-10\log \left(\frac{4M{f}_s}{B_{max}}\right)-10\log \left(\sqrt{\xi}\right)+L $$

(36)

Having considered the right choices of M and N_optimum and by considering N_DSP, minimum detectable input SNR could be approximated as (37).

$$ SN{R}_{in,\mathit{\min}}= SN{R}_{min}+L-10\log \left(\frac{4\sqrt{2}{f}_s}{B_{max}}{\left(\frac{N_{DSP}-{\log}_2\left({f}_s pw\right)}{2\left(1+{\log}_2\left({f}_s pw\right)\right)}\right)}^{3/2}\right) $$

(37)

where, pw = N/f_s. According to equation (37), the higher the amounts of DSP blocks is employed, the lower the minimum detectable SNR would be. This claim has visually proved in Fig. 9.

1.2 Processing Loss of De-Ramping Structure

Obviously, one approach to quantify the influence of approximations, proposed above, is to calculate the processing loss of the method comparing to the matched-filter performance, which directly corresponds to detection performance. According to the calculated processing gain of the method as (35), and the processing gain of matched-filter being equal to the number of received signal samples, the processing loss, L_p, can analytically be achieved as follows:

$$ {L}_p=10\log (N)-\left(10\log \left(\frac{4M{f}_s}{B_{max}}\right)+10\log \left(\sqrt{\xi}\right)-L\right) $$

(38)

where, the first term in the right side is related to matched-filter. By substituting the optimum choices in (38), the processing loss can be calculated as (39).

$$ {L}_p=10\log \left(\frac{pw{B}_{max}}{4\sqrt{2}}{\left(\frac{2\left(1+{\log}_2\left({f}_s pw\right)\right)}{N_{DSP}-{\log}_2\left({f}_s pw\right)}\right)}^{\frac{3}{2}}\right)+L $$

(39)

As can be seen in (39), the more number of DSP cores is used, the less processing is resulted. Also, the processing loss varies with pw, such that the lower value of pw leads to the lower loss, and better detection performance.