Incorporation of Reduced Full Adder and Half Adder into Wallace Multiplier and Improved Carry-Save Adder for Digital FIR Filter

Improvement of digital FIR filter is vital in the field of Digital Signal Processing in order to reduce the area, delay and power. Multiplication and Accumulation (MAC) unit of Finite Impulse Response (FIR) filter has been designed using efficient multiplier and adder circuits for optimized APT (Area,Power and Timing) product. In this paper, the design of direct form FIR filter with efficient MAC unit has been presented. Initially, full adder and half adder structures are shrunk down by reducing number of gates. These compact full adder and half adder structures are incorporated into Wallace Multiplier and Improved Carry-Save Adder. The proposed 16-bit Carry-Save Adder has been improved by splitting into four parallel phases. Consequently the delay of enhanced Carry- Save Adder is reduced. Generation of carry output is performed using number of OR gates in a sequential manner. All these enhanced architectures are incorporated into the Digital FIR Filter to reduce the area, delay and power utilization.

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.