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High Speed ASIC Design of Complex Multiplier Using Vedic Mathematics pdf
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High Speed ASIC Design of Complex Multiplier Using Vedic Mathematics



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Abstract

Vedic Mathematics is the ancient methodology of
Indian mathematics which has a unique technique of calculations
based on 16 Sutras (Formulae). A high speed complex multiplier
design (ASIC) using Vedic Mathematics is presented in this
paper. The idea for designing the multiplier and adderlsubtractor
unit is adopted from ancient Indian mathematics
"Vedas". On account of those formulas, the partial products
and sums are generated in one step which reduces the carry
propagation from LSB to MSB. The implementation of the Vedic
mathematics and their application to the complex multiplier
ensure substantial reduction of propagation delay in comparison
with DA based architecture and parallel adder based
implementation which are most commonly used architectures.
The functionality of these circuits was checked and performance
parameters like propagation delay and dynamic power
consumption were calculated by spice spectre using standard
90nm CMOS technology. The propagation delay of the resulting
(16, 16)x(16, 16) complex multiplier is only 4ns and consume 6.5
mW power. We achieved almost 25% improvement in speed
from earlier reported complex multipliers, e.g. parallel adder and
DA based architectures.

INTRODUCTION

Complex multiplication is of immense importance in
Digital Signal Processing (DSP) and Image Processing (IP).
To implement the hardware module of Discrete Fourier
Transformation (DFT), Discrete Cosine Transformation
(DCT), Discrete Sine Transformation (DST) and modem
broadband communications; large numbers of complex
multipliers are required. Complex number multiplication is
performed using four real number multiplications and two
additions/ subtractions. In real number processing, carry needs
to be propagated from the least significant bit (LSB) to the
most significant bit (MSB) when binary partial products are
added [1]. Therefore, the addition and subtraction after binary
multiplications limit the overall speed. Many alternative
method had so far been proposed for complex number
multiplication [2-7] like algebraic transformation based
implementation[2], bit-serial multiplication using offset binary
and distributed arithmetic [3], the CORDIC (coordinate
rotation digital computer) algorithm [4], the quadratic residue
number system (QRNS) [5], and recently, the redundant
complex number system (RCNS) [6].

MATHEMATICAL FORMULATION OF VEDIC SUTRAS

The gifts of the ancient Indian mathematics in the world
history of mathematical science are not well recognized. The
contributions of saint and mathematician in the field of
number theory, 'Sri Bharati Krsna Thirthaji Maharaja', in the
fonn of Vedic Sutras (fonnulas) [11] are significant for
calculations. He had explored the mathematical potentials
from Vedic primers and showed that the mathematical
operations can be carried out mentally to produce fast answers
using the Sutras. In this paper we are concentrating on
"Urdhva-tiryakbyham", and "Nikhilam Navatascaramam
Dasatah" fonnulas and other fonnulas are beyond the scope of
this paper.

A. "Urdhva-tiryakbyham " Sutra

The meaning of this sutra is "Vertically and crosswise"
and it is applicable to all the multiplication operations. Fig. 1
represents the general multiplication procedure of the 4x4
multiplication. This procedure is simply known as array
multiplication technique [12]. It is an efficient multiplication
technique when the multiplier and multiplicand lengths are
small, but for the larger length multiplication this technique is
not suitable because a large amount of carry propagation
delays are involved in these cases. To overcome this problem
we are describing Nikhilam sutra for calculating the
multiplication of two larger numbers.

Exponent Determinant

The hardware implementation of the exponent
determinant is shown in Fig. 4.The integer part or exponent of
the number from the binary fixed point number can be
obtained by the maximum power of the radix. For the nonzero
input, shifting operation is executed using parallel in
parallel out (PIPO) shift registers. The number of select lines
(in FigA it is denoted as S], So) of the PIPO shifter is chosen
as per the binary representation of the number (N-1)IO. 'Shift'
pin is assigned in PIPO shifter to check whether the number is
to be shifted or not (to initialize the operation 'Shift' pin is
initialized to low). A decrementer [13] has been integrated in
this architecture to follow the maximum power of the radix. A
sequential searching procedure has been implemented here to
search the first 'I' starting from the MSB side by using
shifting technique.

SIMULATION RESULT ANALYSIS

All the algorithm of this paper was simulated and their
functionality was examined by using Spice Spectre.
Performance parameters such as propagation delay and power
consumptions analysis of this paper using standard 90nm
CMOS technology. To evaluate the performance parameters,
we give the values of the computational effort using array
multiplier and Vedic multiplier. As shown, the application of
the Vedic method for mUltiplication cuts the amount of the
hardware as well as increases the performance parameters
such as propagation delay, dynamic switching power
consumptions, and dynamic leakage power consumptions. The
performance parameters analysis using array multiplication
and Vedic multiplication is shown in Table I. Input data is
taken as a regular fashion for experimental purpose. We have
kept our main concentration for reducing the propagation
delay, dynamic switching power and dynamic leakage power
consumption and energy delay product.

CONCLUSION

In this paper we report on a novel complex number
multiplier design based on the formulas of the ancient Indian
Vedic Mathematics, highly suitable for high speed complex
arithmetic circuits which are having wide application in VLSI
signal processing. The implementation was done in Spice spectre
and compared with the mostly used architecture like distributed
arithmetic, parallel adder based implementation, and algebraic
transformation based implementation. This novel architecture
combines the advantages of the Vedic mathematics for
multiplication which encounters the stages and partial product
reduction. The proposed complex number multiplier offered
20% and 19% improvement in terms of propagation delay and
power consumption respectively, in comparison with parallel
adder based implementation. Whereas, the corresponding
improvement in terms of delay and power was found to be
33% and 46% respectively, with reference to the algebraic
transformation based implementation.
 

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