Dan P. Scholnik

Design of nonlinear-phase FIR filters with second-order cone programming

We extend classic linear-programming design of linear-phase FIR filters to second-order cone-programming design of nonlinear-phase FIR filters. Real and complex FIR filters with optional analog or digital cascade filters are designed using cones on discrete frequencies to approximate Chebychev, mean-absolute, and rms errors.

Optimal design of wideband array patterns

In radar systems, wideband array patterns are typically nothing but patterns designed in the conventional narrowband way and then time-delay steered. It is increasingly common to use digital filters to approximate the needed delays. We suggest, however, that approximating time delays is an inefficient use of the valuable resource represented by these filters and propose instead that their responses be jointly optimized to meet specifications on the array pattern as a function of angle and frequency. This frees the angle-dependence and frequency-dependence of the array function from the fixed relationship implied by time-delay steering and allows tremendous design flexibility, and it improves the tradeoff between filter length and ultimate array performance.

Superdirectivity and SNR constraints in wideband array-pattern design

Wide instantaneous and tunable-bandwidth arrays pose a challenging design problem. Since the elements are usually spaced close to one-half wavelength at the highest frequency of operation, at lower frequencies the array may be considerably oversampled spatially. In a conventional time-delay-steered array the result is a beamwidth that is proportional to frequency. With a FIR filter at each element, however, superdirectivity can be achieved at lower frequencies, improving the beam characteristics. Constraints on efficiency, sensitivity, and SNR are derived to limit the undesirable effects of superdirectivity.

Formulating wideband array-pattern optimizations

Custom design of wideband digital array patterns requires a systematic approach to mapping design specifications to a program understandable by optimization engines. We show that, as in the narrowband case, wideband array patterns are closely related to multidimensional FIR filter responses, suggesting the adaptation of powerful and efficient filter design techniques to the array problem. Previously reported FIR filter design techniques are then applied to an example array-pattern design.

Optimal Array-Pattern Synthesis for Wideband Digital Transmit Arrays

Some next-generation radio frequency systems are expected to share a common transmit aperture among multiple users across a wide range of frequencies and functions such as radar and communications. The requisite linear architectures and digital signal generation will permit far greater flexibility in the design of array patterns than traditional time-delay steered wideband transmit arrays. Merely replicating the traditional architecture in digital signal processing would generally represent an inefficient use of computational resources; thus, we propose instead to place a finite-impulse response filter per input signal at each element and to directly optimize the resulting wideband array pattern. For this architecture, we present a passband-equivalent transmit-array model and derive expressions for wideband directivity, efficiency, error sensitivity, gain, peak and mean-square sidelobes, mainlobe frequency-response flatness, and polarization. All can be constrained using second-order cone programming, a highly-efficient type of convex optimization. Several examples illustrate the design tradeoffs, including the need to limit undesirable superdirective effects in wideband arrays. The system model and the derivations are general enough to admit almost any array architecture, including arbitrary element locations, nonuniform element responses, and multiple polarizations.

A specification language for the optimal design of exotic FIR filters with second-order cone programs

Application-tailored individual and joint FIR-filter designs of remarkable complexity are elegantly coded using our MATLAB toolbox Opt, a research tool providing a DSP-oriented modeling language for driving ultra-efficient off-the-shelf numerical solvers of (linear and) second-order cone programs. Opt data types symbolically capture affine or (nonnegative definite) quadratic dependencies on optimization variables, which gain numeric values only later, when optimized. On those basic types it builds affine vector and complex-time-sequence types for specifying impulse response structures in 1D or multi-D, with sample spacing either uniform or not. Dependencies can be manipulated symbolically with arithmetic and DSP operations including convolution, filter match, and Fourier transform. Linear and MS errors in frequency and time domains can be constructed, constrained and optimized. MSE constructions include output powers of filter systems driven by symbolic random-process drive signals having user-specified PSDs.

Circuit Approaches to Nonlinear-ISI Mitigation in Noise-Shaped Bandpass D/A Conversion

This paper focuses on the analysis of nonlinear inter-symbol interference (ISI) in RF bandpass delta-sigma power digital-to-analog converters (DACs). Digital RF transmitters based on direct delta-sigma modulation have been proposed for efficient linear multi-functional radios. We show that a realistic one-bit DAC limits the linearity of such a transmitter. The linearity is discussed in terms of ISI, for both single-ended and differential DAC circuits. Theoretical and experimental comparisons are given with a three-tone delta-sigma test signal at a 1.5 GHz clock frequency with 60 MHz possible signal bandwidth centered at 375 MHz. The signal-to-noise-and-distortion (SINAD) is shown to be improved for the differential case. It is also shown that circuit design and bias voltages have a dramatic effect on signal linearity.

Range-ambiguous clutter suppression with Pulse-diverse waveforms

Conventional pulse-Doppler waveforms are the standard for radar detection in clutter due to their inherent robustness, especially to extended-range ambiguous clutter. The attendant drawbacks include coherent ambiguities and blind zones in range and/or Doppler, which require multiple coherent precessing intervals to resolve. Pulse-diverse waveforms, using techniques such as phase coding and nonuniform time offsets, can mitigate these drawbacks but at the cost of a significant reduction in clutter suppression using matched filtering. It is shown here that receive filtering optimized on a per-interval or per-range/Doppler-cell basis can provide these waveforms with clutter suppression over multiple range intervals, at the cost of significantly increased computation. Several examples illustrate the tradeoffs between different types of pulse diversity.

Space-time vector delta-sigma modulation

Classical delta-sigma modulation gains resolution through temporal oversampling by using a high-speed, low-resolution quantizer and shaping the resulting quantization errors out of the signal band. Likewise, spatial delta-sigma modulation is often used in image halftoning to create high-visual-quality binary images. We consider combining spatial and temporal delta-sigma modulation to reduce temporal oversampling requirements in high-SNR antenna transmit arrays using simple high-power switches as the output amplifiers. We introduce a vector delta-sigma architecture that allows noise shaping jointly in both temporal and spatial frequency. The two-dimensional pseudo-shift-invariant delta-sigma architecture commonly used for image halftoning is realized as a special case, and we consider when such a simplified model may be appropriate.

Vector delta-sigma modulation with integral shaping of hardware-mismatch errors

A common architecture here unifies ordinary scalar delta-sigma modulators, vector I/O delta-sigma modulators (including parallel banks of scalar ones), mismatch-shaping DACs, delta-sigma modulators incorporating mismatch-shaping DACs in the conventional way, and something new: scalar or vector delta-sigma modulators incorporating spectral shaping of hardware mismatches as a natural and integral part of the modulator structure. The common architecture resembles a conventional delta-sigma loop but using vector signals and a vector quantizer to operate in a space of higher dimension than the input/output, which are inserted/extracted from the loop along a subspace. The expansion of dimension provides the redundancy that permits the spectral shaping of hardware errors. This common view of these systems suggests that the multidimensional structure of the quantizer is key.