Jeffrey O. Coleman

Chebyshev Stopbands for CIC Decimation Filters and CIC-Implemented Array Tapers in 1D and 2D

The stopbands of a cascaded integrator-comb (CIC) decimation filter are ordinarily very narrow, as each results from a single multiple zero. Here response sharpening with a Chebyshev polynomial, using a previously reported CIC variant, separates each such multiple zero into an equiripple stopband. By trading unneeded depth at stopband center for improved depth at the stopband edge, the latter depth improves by some 6(N-1) dB in an Nth-order system. Increased computational complexity is modest: a few low-speed additions and multiplications by small integer coefficients that can often be chosen as powers of two. Alternatively, parameters can be configured to replace the many small stopbands with one large one, and this is demonstrated here with example spatial-processing CIC designs that create pencil beams for 1D and 2D receive antenna arrays.

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.

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.

Cascaded coefficient number systems lead to FIR filters of striking computational efficiency

Multiplierless FIR filters (or other fixed linear combiners) are built as add/subtract networks operating on bit-shifted input data. Classically, the computational structure required is determined by simply expressing the coefficients in canonical-signed-digit (CSD) form. In this paper, expressing coefficients in a higher-radix number system instead results in a computational structure for a partial solution, one that reduces a large linear-combination problem to a smaller one. A well-chosen sequence of such number systems then leads to a cascade of these problem-reducing networks that together solve the original problem with remarkable overall computational efficiency, especially for larger filters. An example FIR filter with a real chirp impulse response 3000 samples in length (a matched filter for a pulse-compression radar) was easily realized with -95 dB rms approximation error using less than two add or subtract operations per coefficient. This is a reduction of approximately 60% relative to the usual CSD method.

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.

Low-complexity hybrid form FIR filters using matrix multiple constant multiplication

Hybrid form FIR filters have been shown to provide a trade-off between the direct form and transposed direct form FIR filters resulting in a low power implementation. However, the use of multiple constant multiplication (MCM) techniques is less advantageous as it results in several MCM blocks. In this work a method of implementing low-complexity hybrid form FIR filters using matrix multiple constant multiplication blocks is proposed. The utilized filter structure can be seen as a polyphase decomposition with common delay lines for the subfilters.

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.