DIGITAL MATCHED FILTERS
AND WAVEFORM DIVERSITY
A Four-Day Intensive Course
The matched filter has, since its conception in 1943, proven itself a remarkably potent, attractive type of filter for aiding in the detection of pulses. First seen as vital for radar/sonar time-of-arrival determination, matched filtering’s scope has grown into diverse fields (such as pattern recognition, x-ray event detection, biomedical signal identification and tracking, and Multi-User Detection in CDMA communications). The matched filter is the key element in understanding and pursuing high-caliber pulse transmission and detection systems, and furnishes the cornerstone element in this course.
This four-day intensive course takes up matched filtering philosophy in the context of “Waveform Adaptive Matching”, which is currently of topical interest in the Waveform Diversity community for enhancement of radar/sonar operability or improvement of multi-function digital communications in hostile environments. This interest has been sparked by recent advances in agile digital hardware operating at very high clock rates, offering unprecedented flexibility for the “smart signalling” techniques which are pushing ever farther into the realm of RF transmission and reception.
A digest of the requisite background digital filter and random noise theory is given prior to experimenting with the signal-to-noise criterion used for pulse detection and the optimal matched filter transfer function. It is shown that the noise colouration environment has a crucial effect on the degree to which optimality can be approached in practice.
Coverage is also provided of the selection strategy needed for optimally selecting the signal to undergo matched filtering. Meanwhile, the noise contamination of the signaling environment may - through natural or manmade means- prove severely hostile, and here the conditions for optimal disruption of pulse detection are set out. Finally, the dramatic self-interference effects such as those arising from clutter/reverberation can have on pulse detection capability are treated, along with optimal signal and detection filtering strategies for use in such contested detection environments.
2.Structure of the Four Days of Study
Each day of study is built around three core lecture sessions which are peppered with frequent hands-on computer-based confirmation and extension activities. These are followed by small design tasks aimed at consolidation of each session’s material. At least one Laboratory Investigation is undertaken daily to promote both individual reflection and small-group discussion of design issues. In total there is structured study time of about 8 hours per day.
(a) DAY 1: The Matched Filter
Review of digital filters: characterization by impulse response and transfer functions; arbitrary signal generation implemented by impulse-driven digital filters; attractions of FIR realizations; relationships between z-transform and DTFT representations; common stylized signal types such as tonebursts and chirps; convolution, correlation, energy density and passage of noise through digital filters; shaped power spectra; inverse and whitening filters.
Scalar products; energy and unit noise-power normalization of impulse responses with cascaded filters; SNR definition for pulse-event detection; Schwarz’s Inequality; the matched filter in white noise; threshold effect on detection probability with Gaussian noise; synchronization matters and improvement of detection through multiple pulse processing.
Derivation of the matched filter in coloured noise; closed-form filter determination when noise is pole-only; increased SNR potential with noise colouration; modification of the flip-template filter by filtered-reference and noise- matching structures; optimal transmit/receive filters and Root-Nyquist designs for pulse communication.
(b) DAY 2: Length-Deficient Matched Filters and Signal Selection
Parameter optimization in structurally-constrained variable filters; SNR degradation brought about by unknown parameters such as phase; performance of quadrature receivers; Toeplitz matrices and formation of the noise matrix; matrix formulation of the matched filter; design options when inflicting length deficiency on an FIR matched filter; perpetual availability of full-optimality SNR metric for benchmarking any compromise matched filter approximations; relative merits of the matrix-prepending and pole/zero whitener truncated-ACF design styles.
Maximizing Signal-to-Noise ratios by Rayleigh Quotient formulations; maximum whitener eigenvalue method for designing a signal to undergo matched filtering in pole-only noise; reciprocal noise eigenvalue method for general signal design; suboptimality of wideband signals such as chirps; eigenvector approximation as a windowed toneburst; tendency to narrowband nature for optimal signals encountering dominant single-peaked whitener filters.
Spectral fragmentation of optimal signals for multiple maximal whitener eigenvalues; formation trajectories of signal gain concentrations for multi-component whitener gains; spectral economy and high SNR potential offered by long signals.
(c) DAY 3: Optimal Noise Shaping and Contested Pulse-by-Pulse Signal Shaping
Noise shaping by signal eigenvector heuristics; evaluation of simple noise-shaping strategies such as agile centre-frequency, fixed-width noise bands; effectiveness of provoking signal gain fragmentation by multi-peak noise spectrum design.
Use of the Calculus of Variations in filter matching problems; derivation of the Shalvi filter for optimal-disruption noise shaping; practical design of Shalvi digital filters; effective disablement of matched filtering; detection/disruption performance of simple flip-template receivers.
Pulse-by-pulse jamming and signal dynamic redesign engagements; SNR excursions and gaming strategies to favour best-signalling and most-disruptive jamming iterations; the merits of multiple parallel detection filters; appeal of quasi-inverse filters as detection filters.
(d) DAY 4: Nonlinear Design Influences of Clutter
Embracing self-interference in a SINR metric; Kay’s iterative method of signal design in clutter; necessity of bandwidth spreading and associated nonlinear design influences; impact of clutter spectral shaping.
Pulse-by-pulse jamming and signal dynamic redesign engagements when jammer operates with no knowledge of clutter; clutter inference opportunities and effect of its utilization on SINR excursions.
Optimum noise shaping if clutter spectral shape is known to jammer; worst-case SINR achievement; examination of effects of peak-to-average signal energy constraints on optimal signal selection; consolidation exercises for pairwise optimization of signal, noise, detector, and clutter combinations.
3. Makeup of Course Notes
Course notes are PowerPoint slide images, with consistent notation and careful referencing of the relevant DSP literature. These notes are self-contained; however, students wishing to gain an alternative perspective will be encouraged to access the course reference text: Principles of Waveform Diversity and Design, SciTech: in press.
Students receive a DVD containing all course exercises and videoclips of indicative solution approaches.
4. Computation Support
This course has an exceptionally strong hands-on flavour. All student work is reliant on the ready availability of the DSP Creations’ toolset (Slifer, Sketch-a-Filt and DSP_Speedster) which are built around the MATLAB/Simulink environment from The MathWorks Ltd. Learning motivation is sharpened by a rich blend of user-friendly graphics display and interactive instrument control features and numerous illustrations of the issues in digital filter design which permeate this course. Students leave the course with their own personal copies of Slifer Lite and Sketch-a-Filt Lite for fuller absorption of, and later reflection upon, their course experience.
5. Course Pre-requisites
Engineers, mathematicians, physicists and other technical personnel with a ready grasp of mathematical analysis (including integral calculus and fundamentals of complex variables) are the intended participants. Familiarity with filtering terminology, DSP concepts, the fields of radar or sonar or basics of MATLAB - while not required - would be an added advantage.