Multiplicative Mismatched Filters for Barker Codes Adly T. Fam, Indranil Sarkar Department of Electrical Engineering The State University of New York at Buffalo
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.
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.
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.
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:
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
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).
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).
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.
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.
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.
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)
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
Proposed mismatched filter • Instead of expanding H(z) in Taylor series, we use a multiplicative expansion as follows:.
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:
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.
Hardware requirements • The hardware requirement for the filter depends on the number of stages implemented. The variation is shown in the figure.
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.
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.