Babak Ayazifar

Circulant structures and graph signal processing

Linear shift-invariant processing of graph signals rests on circulant graphs and filters. The spatial features of circulant structures also permit shift-varying operations such as sampling. Their spectral features-as described by their Graph Fourier Transform profiles-enable novel multiscale signal processing systems and methods. To extend the reach of circulant structures, we present a method to decompose an arbitrary graph or filter into a combination of circulant structures. Our decomposition is analogous to resolving a linear time-varying system into a bank of linear time-invariant systems. As an application, we perform multiscale decomposition on temperature data spanning the continental United States.

Critically-sampled perfect-reconstruction spline-wavelet filterbanks for graph signals

Inspired by first-order spline wavelets in classical signal processing, we introduce two-channel (low-pass and high-pass), critically-sampled, perfect-reconstruction filterbanks for signals defined on circulant graphs, which accommodate linear shift-invariant filtering. We then generalize to filters that process signals defined on noncirculant graphs. We apply these filters, which can be tuned to approximate desired frequency responses, to signals defined on synthetic graphs and examine their performance.

Multiresolution graph signal processing via circulant structures

We use circulant structures to present a new framework for multiresolution analysis and processing of graph signals. Among the essential features of circulant graphs is that they accommodate fundamental signal processing operations, such as linear shift-invariant filtering, downsampling, upsampling, and reconstruction-features that offer substantial advantage. We design two-channel, critically-sampled, perfect-reconstruction, orthogonal lattice-filter structures to process signals defined on circulant graphs. To extend our reach to noncirculant graphs, we present a method to decompose a connected, undirected graph into a combination of circulant graphs. To evaluate our proposed framework, we offer examples of synthetic and real-world graph signal data and their multiscale decompositions.

Can we make signals and systems intelligible, interesting, and relevant?

rently a teacher assistant in Analog Circuits Design. He is the GOLD Representative of Region 9 and member of the IEEE CASS Board of Governors for 2009–2011. Currently at Universidad Nacional del Sur he is working in the development of CNN 3D pixel processors. His current field of research is the applications of VLSI circuits for image sensing and processing. His research interests include digital image processing, CNNs, neuromorphic circuits, and bio-electronic implants.

Spline-Like Wavelet Filterbanks for Multiresolution Analysis of Graph-Structured Data

Multiresolution analysis is important for understanding graph signals, which represent graph-structured data. Wavelet filterbanks permit multiscale analysis and processing of graph signals-particularly, useful for harvesting large-scale data. Inspired by first-order spline wavelets in classical signal processing, we introduce two-channel (low-pass and high-pass) wavelet filterbanks for graph signals. This class of filterbanks boasts several useful properties, such as critical sampling, perfect reconstruction, and graph invariance. We consider an application in graph semi-supervised learning and propose a wavelet-regularized semi-supervised learning algorithm that is competitive for certain synthetic and real-world data.

Wavelet-regularized graph semi-supervised learning

Graph semi-supervised learning (GSSL) is a technique that uses a combination of labeled and unlabeled nodes on a graph to determine a classifier for new, incoming data. This problem can be analyzed through the lens of graph signal processing. In particular, the penalty functions used in the optimization formulation of standard GSSL algorithms can be interpreted as appropriately-defined filters in the Graph Fourier domain. We propose a wavelet-regularized semi-supervised learning algorithm using suitably-defined spline-like graph wavelets. These wavelets are critically-sampled, perfect-reconstruction basis representations, in contrast to much of the existing work proposing overcomplete representations. Critical sampling is essential for controlling the complexity in applications dealing with large scale datasets. We are also interested in understanding when wavelet-regularized approaches perform better than traditional Fourier-based regularizers. We compare the performance of our proposed spline-like, wavelet-regularized learning algorithm (as well as other existing graph wavelet designs) to some standard graph semi-supervised learning techniques on synthetic and real-world datasets.

A first lab in filter design: Power line hum suppression in an ECG signal

Our introductory course in Signals and Systems at UC Berkeley assumes only basic physics and moderate fluency with calculus. We outline a computer laboratory project to illustrate how we bring the principles and techniques of design-oriented analysis-as well as meaningful, interesting applications-within the reach of our students. In particular, we describe a guided filter design problem to remove a 60 Hz power line hum from an electrocardiogram (ECG) signal.

The elegant geometry of fourier analysis

We outline a method that brings the elegant, unifying geometry of orthogonal function expansions to the teaching of Fourier Analysis in our gateway course on Signals and Systems at UC Berkeley. Our approach starts with discrete-time periodic signals. Their straightforward representation as finite-dimensional Cartesian vectors provides a gentle ingress into the more abstract Euclidean vector spaces that inform the Fourier decompositions of richer signal types. As we describe how a signal fragments into its elemental frequencies, we are careful with the mathematics but we do not let rigor eclipse clarity; plausible reasoning often suffices. We sequence the topics and develop the theory to reduce algebraic clutter and promote geometric insight into the progressively nuanced world of frequency decompositions nestled in the beautiful heart of Fourier Analysis.

Rethinking Fourier's legacy in signals and systems education

In our gateway Signals and Systems courses at UC Berkeley we teach design-oriented modeling and analysis of linear, time-invariant systems in the frequency domain immediately after the convolution sum and well before the full-scale infrastructure of Fourier analysis. This exposes students to refined engineering techniques and interesting applications early. Two features are integral to our approach. First, we recast certain advanced concepts in a vernacular thats accessible even to recent high school graduates. We prefer to invoke elementary geometry, algebra, and trigonometry before college mathematics. Second, we begin with rudimentary, oft-repeated motifs, which we then perturb and embellish to create more elaborate systems. Throughout, we keep the students mindful of how certain design, modeling, and analytical tools enable a graceful progression from the simple to the more complex structures.

ISCAS 2011 special sessions on education innovations and experiences

We discuss certain aspects of the state of core education in circuits, signals, and systems and list a few current problems and issues. The colleagues who have contributed to the two special sessions on education not only share their valuable experiences and innovations, but also express their opinions on how to address a number of salient problems. In other subjects, such as mathematics, some educators have taken to drastic approaches—one so unusual that it was dubbed “street-fighting mathematics”. In a similar vein, we launch a debate on a collection of items for inclusion in, or exclusion from, basic circuits, signals, and systems education. We hope these ideas and novel practices will penetrate the diverse curricula around the globe.