Multiplicative Mismatched Filters for Barker Codes

1. Multiplicative Mismatched Filters for Barker Codes Adly T. Fam, Indranil Sarkar Department of Electrical Engineering The State University of New York at Buffalo

2. Outline of presentation • Introduction to Barker codes and compound Barker codes. • Introduction to mismatched filtering for sidelobe reduction. • Effect of the filtering process on the SNR. • Proposed mismatched filter and its performance.

3. Introduction to Barker codes • Monopulse radar is not suitable for military applications. • One of the earliest and most popular methods of pulse compression is phase coding. • The phase coded waveform is matched filtered to recover the pulse.

4. Introduction to Barker codes • The matched filter produces the aperiodic autocorrelation function for the code. • Due to the inherent nature of the codes we get both a mainlobe and sidelobes in the aperiodic autocorrelation. • Barker codes have the smallest sidelobes possible for bi-phase codes.

5. Introduction to Barker codes

6. Barker code of length 13[-1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1] • Barker codes produce the best known sidelobe to mainlobe ratio. • The aperiodic autocorrelation is given by:

7. Compound Barker codes • Since Barker codes are not available for lengths greater than 13, compound Barker codes are used to obtain greater lengths. • Compound Barker codes are obtained by nesting existing codes. Two Barker codes u and v can be used to generate a compound code as : where the operator denotes the Kronecker product. • In the frequency domain, a Barker code nested within itself is given as: where N = length of the original code

8. Compound Barker codes • Higher order compounding can be achieved by repeating the process • The aperiodic autocorrelation of a compound sequence of length 132 is shown. • Even though these sequences achieve better SNR performance, the sidelobe to mainlobe ratio is not improved and remains (1/13).

9. Sidelobe to mainlobe ratio of (1/13) X(z) or X(z)X(zN) X(z-1) or X(z-1)X(z-N) Conventional matched filter • The conventional matched filtering approach for both length 13 Barker codes and compound Barker codes of length 132 produces a sidelobe to mainlobe ratio of (1/13).

10. Conventional matched filter • The mainlobe to sidelobe ratio produced at the output of the conventional matched filter corresponds to 22.2789 dB. • This is not sufficient for most radar applications which need at least 30 dB of sidelobe suppression. • It has been the goal of researchers for along time to improve this mainlobe to sidelobe ratio.

11. Prior art in the field • Mismatched filters have been proposed in the literature to suppress the sidelobes of the Barker codes. • Methodologies can be broadly classified into two categories: • Method I : A matched filter is first used to perform the pulse compression correlation. The mismatched filter is then used in cascade with the MF to improve the mainlobe to sidelobe ratio. • Method II : The mismatched filter is designed from the scratch without using a MF at the front end.

12. Effect of filtering on the SNR • Matched filters can be proved to be optimum in the SNR sense. • Any further processing of the matched filter output degrades the SNR. This is also called the mismatch loss or the loss in SNR (LSNR). • Hence, for any filter, the LSNR performance must be evaluated.

13. X(z-1) Rationale behind the proposed filter • The idea of the proposed filter stems from the concept of inverse filtering. In absence of the proposed filter, the matched filter produces the autocorrelation function at the output: • Ideally, we would like the autocorrelation function to have a peak of N in the middle and zero sidelobes elsewhere. For that, we could pass the matched filter output through an inverse matched filter as shown below: Y(z) R(z)

14. Proposed mismatched filter • The MF output consists of a mainlobe of height N and sidelobes of height 1. In general it can be denoted by: • Evidently, the mismatched filter should be the inverse filter of R(z) and is given by

15. Proposed mismatched filter • Instead of expanding H(z) in Taylor series, we use a multiplicative expansion as follows:.

16. Proposed mismatched filter • Ignoring the denominator, the filter transfer function is approximated as: • In order to make the filter more efficient, we introduce some parameters and the parameterized transfer function becomes:

17. Proposed mismatched filter • The filter is implemented cascading the three filters corresponding to the three terms. For even more flexibility, three multipliers were introduced in the filter structure.

18. Hardware requirements • The hardware requirement for the filter depends on the number of stages implemented. The variation is shown in the figure.

19. Final output

20. Comparison with other filters • Rihaczek and Golden proposed the R-G filters and they were improved by Hua and Oskman. The filters by Hua and Oskman remained the best till date.

21. Ongoing work and further scope • Filter design for length 11 Barker codes. • Bandwidth considerations. • Optimum filter designs for other compound Barker codes. • Examining other classes of random codes and polyphase codes.