I-Hung Khoo

2-D symmetry: theory and filter design applications

In this comprehensive review article, we present the theory of symmetry in two-dimensional (2-D) filter functions and in 2-D Fourier transforms. It is shown that when a filter frequency response possesses symmetry, the realization problem becomes relatively simple. Further, when the frequency response has no symmetry, there is a technique to decompose that frequency response into components each of which has the desired symmetry. This again reduces the complexity of two-dimensional filter design. A number of filter design examples are illustrated.

Symmetry Study for Delta- Operator-Based 2-D Digital Filters

The complexity in the design and implementation of two-dimensional (2-D) filters can be considerably reduced if we utilize the symmetries that might be present in the frequency response of these filters. As the delta-operator formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrowband filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in gamma-domain which utilize the various symmetries in filter specifications. With this motivation, we comprehensively establish the theory of constraints for delta-operator formulated discrete-time real-coefficient polynomials and functions, arising out of the many types of symmetries in their magnitude responses. We also show that as sampling time tends to zero, the gamma-domain symmetry constraints merge with those of s-domain symmetry constraints. We then present a least square error criterion based procedure to design 2-D digital filters in gamma-domain that utilizes the symmetry properties of the magnitude specification. A design example is provided to illustrate the savings in computational complexity resulting from the use of the gamma-domain symmetry constraints

Power-Efficient and Cost-Effective 2-D Symmetry Filter Architectures

This paper presents two-dimensional (2-D) VLSI digital filter structures possessing various symmetries in the filter magnitude response. For this purpose, four Type-1 and four Type-2 power-efficient and cost-effective 2-D magnitude symmetry filter architectures possessing diagonal, fourfold rotational, quadrantal, and octagonal symmetries with reduced number of multipliers and one power-efficient and cost-effective multimode 2-D symmetry filter are given. By combining the identities of the four Type-1 symmetry filter structures, the proposed multimode 2-D symmetry filter is capable of providing four different operation modes: diagonal symmetry mode (DSM), fourfold rotational symmetry mode (FRSM), quadrantal symmetry mode (QSM), and octagonal symmetry mode (OSM). The proposed diagonal, fourfold rotational, quadrantal, and octagonal symmetry filter structures can attain power savings of 16.77%, 36.30%, 22.90%, and 37.73% with respect to that of the conventional 2-D filter design without symmetry. On the other hand, the proposed DSM, FRSM, QSM, and OSM modes can reduce power consumption by 11.01%, 31.42%, 17.53%, and 35.26% compared with that of the conventional 2-D filter design. The proposed multimode filter can result in a 63.25% area reduction compared with the sum of the areas of the four individual Type-1 symmetry filter structures.

2-D digital filter architectures without global broadcast and some symmetry applications

Four new 2-D filter VLSI architectures without global broadcast are presented. The first is a transposed systolic structure which requires fewer delay elements compared to the original systolic structure in [1]. By combining the sub-blocks of the original with the new transposed structure, two additional systolic structures are obtained to realize transfer functions with separable denominators, which require fewer multipliers. These separable denominator structures have important symmetry applications. A structure which possesses diagonal symmetry is then shown which requires even fewer multipliers.

Generalized formulation of 2-D filter structures without global broadcast for VLSI implementation

A generalized formulation is developed that allows the derivation of various new 2-D VLSI filter structures, without global broadcast, using different filter sub-blocks and their interconnections (frameworks). With this formulation, lattice-type and direct-form structures realizing general 2-D IIR and FIR transfer functions, IIR transfer functions with separable denominators, and transfer functions with quadrantal magnitude symmetry are easily obtained. The separable denominator and quadrantal symmetry structures have the advantage of reduced number of multipliers.

Theory and test procedure for symmetries in the frequency response of complex two-dimensional delta operator formulated discrete-time systems

This paper provides the theory and an efficient tabular algorithm to test for various symmetries in the magnitude response of two-dimensional (2-D) complex-coefficient delta operator formulated discrete-time systems. In general, centro symmetry is not preserved in the complex case. The conditions under which this is preserved is discussed in the paper. It is to be noted that as the sampling period (/spl Delta/) goes to zero, the symmetry conditions merge with that of 2-D continuous-time case.

Vibrotactile device for rehabilitative training of persons with lower-limb amputation

Individuals with prostheses use sensations experienced through the residual limb during ambulation to perceive the state of the prosthesis. For lower-limb prosthetic users, the loss of proprioception can affect stability and contribute to falls. In this work, we developed a vibrotactile device to enhance residual limb proprioception, which can be used for rehabilitation training of persons with transtibial amputation. The system consists of two vibrating motors, a solenoid, a goniometer, and a control interface. Two types of vibrotactile feedback can be generated: a short vibration via the motors and a single knock via the solenoid. The device is universally designed to fit standard prosthetic components and is inserted in line between the pylon and the socket adapter allowing the vibration to propagate from the unit to the residual limb. The design and the results of the initial functionality test of the vibrotactile unit are reported.

Area-efficient 2-D digital filter architectures possessing diagonal and four-fold rotational symmetries

In this paper, two area-efficient two-dimensional (2-D) IIR filter architectures for 2-D transfer function with diagonal and four-fold rotational symmetries are proposed for image processing. For this purpose, two different intermediate transfer functions are applied and the corresponding two new filter architectures are obtained. Under satisfactory average error performance, we determine that the new filter structure requires less multiplier wordlength than the previously published Type-1 filter structure. With the features of less numerator multiplier wordlength and shorter critical path, the proposed diagonal and four-fold rotational symmetry filter architectures can result in 10.17% and 3.35% area reduction with respect to the published Type-1 IIR filter structures with diagonal and four-fold rotational symmetries in terms of synthesis results.

Delta operator based 2-D VLSI filter structures without global broadcast and incorporation of the quadrantal symmetry

Having local data communication (without global broadcast of signals) among the elements is important in VLSI designs. Recently, 2-D systolic digital filter architectures were presented which eliminated the global broadcast of the input and output signals. The delta discrete-time operator based 1-D and 2-D digital filters (in γ-domain) were shown to offer better numerical accuracy and lower coefficient sensitivity in narrowband filter designs when compared to the traditional shiftoperator formulation. Further, the complexity in the design and implementation of 2-D filters can be reduced considerably if the symmetries that might be present in the frequency responses of these filters are utilized. With this motivation we present new 2-D VLSI filter structures, without global broadcast, using delta discrete-time operator for the first time. We also present frame works in γ-domain that realizes 2-D filters possessing quadrantal symmetry in its magnitude response. The separable denominator and quadrantal symmetry structures have the advantage of reduced number of multipliers while ensuring the 2-D filter stability.

Delta discrete-time operator based realization procedure for low sensitivity sampled-data and digital ladder filters

A delta operator based ladder synthesis procedure is presented for the realization of three different types of transfer functions. The procedure overcomes the numerical problem of the conventional z-domain LDI approach when the filter poles are clustered near z=1 and does not require any s-domain polynomial transformation to achieve spectral factorization [4]. Also, the entire operation involves only rational polynomials.