A novel approach for high speed convolution of finite

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 11 | Nov-2013, Available @ http://www.ijret.org 654
A NOVEL APPROACH FOR HIGH SPEED CONVOLUTION OF FINITE
AND INFINITE LENGTH SEQUENCES USING VEDIC MATHEMATICS
M. Bharathi1
, D. Leela Rani2
1
Assistant Professor, 2
Associate Professor, Department of ECE, Sree Vidyanikethan Engineering College, Tirupati, India,
bharathi891@gmail.com, dlrani79@gmail.com
Abstract
Digital signal processing, Digital control systems, Telecommunication, Audio and Video processing are important applications in
VLSI. Design and implementation of DSP systems with advances in VLSI demands low power, efficiency in energy, portability,
reliability and miniaturization. In digital signal processing, linear-time invariant systems are important sub-class of systems and are
the heart and soul of DSP.
In many application areas, linear and circular convolution are fundamental computations. Convolution with very long sequences is
often required. Discrete linear convolution of two finite-length and infinite length sequences using circular convolution on for
Overlap-Add and Overlap-Save methods can be computed. In real-time signal processing, circular convolution is much more
effective than linear convolution. Circular convolution is simpler to compute and produces less output samples compared to linear
convolution. Also linear convolution can be computed from circular convolution. In this paper, both linear, circular convolutions are
performed using vedic multiplier architecture based on vertical and cross wise algorithm of Urdhva-Tiryabhyam. The implementation
uses hierarchical design approach which leads to improvement in computational speed, power reduction, minimization in hardware
resources and area. Coding is done using Verilog HDL. Simulation and synthesis are performed using Xilinx FPGA.
Keywords: Linear and Circular convolution, Urdhva - Tiryagbhyam, carry save multiplier, Overlap –Add/ Save Verilog
HDL.
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1. INTRODUCTION
Systems are classified in accordance with a no. of characteristic
properties or categories, namely: linearity, causality, stability
and time variance. Linear, time-invariant systems are important
sub-class of systems. Urdhva-Tiryagbhyam sutra is used in
developing carry save multiplier architecture to perform
convolution of two finite and infinite length sequences [1].
Linear and circular convolutions, which are fundamental
computations in Linear time-invariant (LTI) systems are
implemented in Verilog HDL. Simulation and Synthesis are
verified in Xilinx 10.1 ISE.
Multiplications, in general are complex and slow in operation.
The overall speed in multiplication depends on number of
partial products generated, shifting the partial products based
on bit position and summation of partial products. In carry save
multiplier, the carry bits are passed diagonally downwards,
which requires a vector merging adder to obtain final sum of all
the partial products. In convolution, fundamental computations
includes multiplication and addition of input and impulse
signals or samples[2],[3].
2. CIRCULAR CONVOLUTION
Let x1(n) and x2(n) be two finite- duration sequences of length
N. Their respective N-point DFT’s are
( ) ( )
1
2 /
1 1
0
N
j nk N
n
X K x n e π
−
−
=
= ∑ k= 0, 1… N-1 (1)
( ) ( )
1
2 /
2 2
0
N
j nk N
n
X K x n e π
−
−
=
= ∑ k= 0, 1… N-1 (2)
If two DFT’s a multiplied together, the result is a DFT, X3(k) of
a sequence x3(n) of length N.
The relationship between X3(K) and sequences X1(k) and X2(k)
is
X3(k)=X1(k)X2(k) k=0,1,……N-1 (3)

__________________________________________________________________________________________
The DFT of {x3(k)} is
( ) ( ) ( )( )
1
3 1 2
0
N
N
n
x n x n x m n
−
=
= −∑ m=0,1… N-1 (4)
Here
( )( ) ( )N
x m n x m n N− = − +
(5)
The above expression has the form of a convolution sum. It
involves the index ((m-n))N and is called circular
convolution[4].
It is not the ordinary linear convolution which relates the output
sequence y(n) of a linear system to the input sequence x(n) and
the impulse response h(n). Thus it can be concluded that the
multiplication of the DFT’s of two sequences is equivalent to
circular convolution of two sequences in the time domain.
The methods that are used to find the circular convolution of
two sequences are
a. Concentric circle method
b. Matrix multiplication method
Let x1(n) and x2(n) be two sequences of length L and M
respectively.
Let x3(n) be the output sequence. The length N, of the output
sequence, N= Max (L, M).
2.1. Concentric circle method
The length of x1(n) should be equal to length of x2(n) in order
to perform circular convolution using concentric circle method.
We have three cases here.
• The length L of sequence x1(n) is equal to length M of
sequence x2(n). then the procedure explained below
can be followed directly.
• If L>M then M is made equal to L by adding L-M
number of zero samples to the sequence, x2(n)
• If M>L, then L is made equal to M by adding M-L
number of zero samples to the sequence x1(n).
• After making the lengths of two sequences equal to N
samples the circular convolution using concentric
circle method between two sequences is performed
using following steps. The N samples of sequence
x1(n) are graphed as equally spaced points around an
outer circle in counter clockwise direction.
• Starting at the same point as x1(n) the N samples of
x2(n) are graphed as equally spaced points in
clockwise direction around an inner circle.
• The corresponding samples are multiplied on two
circles and the products are added to produce first
sample of output sequence, x3(n).
• The samples on the inner circle are rotated one
position in counter clock wise direction successively
and step 3 is repeated to obtain the next sample of
output sequence x3(n).
• Step 4 is repeated until the first sample of inner circle
lines up with the first sample of outer circle once
again. Hence all the samples of output sequence x3(n)
are collected.
2.2. Matrix Multiplication Method
Circular convolution of two sequences x1(n) and x2(n) is
obtained by representing the sequences in matrix form as
shown below
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
2 2 2 1 3
2 2 2 1 3
2 2 2 1 3
0 1 ...... 1 0 0
1 0 ....... 2 1 1
...... ... ....
1 2 ....... 0 1 1
x x N x x x
x x x x x
x N x N x x N x N
−     
     
     =
     
     
− − − −           (6)
The columns of NxN matrix is formed by repeating the samples
of x2(n) via circular shift. The elements of column matrix are
the samples of sequence x1(n). The circular convolution of two
sequences, x3(n), is obtained by multiplying NxN matrix of
samples of x2(n) and column matrix which consists of samples
of x1(n).
3. LINEAR CONVOLUTION OF SHORT
DURATION SEQUENCE
In discrete time, the output sequence y[n] of a linear time
invariant system, with impulse response h[n] due to any input
sequence x[n] is the convolution sum of x[n] with h[n] and is
given as
[ ] [ ] [ ] [ ] [ ]*y n x n h n x k h n k
∞
−∞
= = −∑ (7)
h[n] is the response of the system to impulse sequence, δ[n].
To implement discrete time convolution, the two sequence x[k]
and h[n-k] are multiplied together for -∞ < k < ∞ and the
products are summed to compute output samples of y[n].
Convolution sum serves as an explicit realization of a discrete-
time linear system. The above equation expresses each sample
of output sequence in terms of all samples of input and impulse
response sequence.

__________________________________________________________________________________________
Fig. 1 Block diagram for computation of linear convolution
Let the length of input and impulse sequences, x[n] and h[n] be
L and M. The starting time of input and impulse sequences are
represented by n1 and n2 respectively.
Therefore, the length N, of output sequence y[n]= L+M-1 and
the starting time n = n1 + n2
The samples of output sequence is computed using convolution
sum
[ ] [ ] [ ]y n x k h n k
∞
−∞
= −∑ (8)
4. LINEAR CONVOLUTION OF LONG
DURATION SEQUENCE
In real time signal processing applications concerned with
signal monitoring and analysis linear filtering of signals is
involved. The input sequence x(n) is often a very long
sequence[5].
Practically, it is difficult to store a long duration input
sequence. So, in order to perform linear convolution of such a
long duration input sequence with the impulse response of a
system, the input sequence is divided into blocks. The
successive blocks are processed one at a time and the results are
combined to obtain the output sequence. The blocks are filtered
separately and results are combined using overlap save method
or overlap adds method [6].
Linear filtering performed via the DFT involves operations on a
block of data, which by necessity must be limited in size due to
limited memory of digital computers. A long input signal
sequence must be segmented to fixed-size blocks prior to
processing.
4.1 Overlap-Save Method
Let the length of long duration input sequence be LL. The
length of impulse response = M
The input sequence is divided into blocks of data. The length of
each block is N= L+M-1
Each block consists of last (M-1) data points of previous block
followed by L new data points for first block of data. The first
M-1 points are set to zero.
Therefore blocks of input sequence are
x1(n)= {0,0….0, x(0), x(n)…. x(L-1)}
The first (M-1) samples are zeros.
x2(n)= {x(L-M+1)… x(L-1), x(L)… x(2L-1)}
x(L-M+1)… x(L-1) are the last (M-1) samples and from x1(n)
and x(L0… x(2L-1) are L new samples
x3(n)= {x(2L-M+1)… x(2L-1), x(2L)… x(3L-1)}
x(2L-M+1)… x(2L-1) are the last (M-1) samples from x2(n)
x(2L)…. x(3L-1) are the L new samples
The length of impulse response is increased by appending L-1
zeros
Circular convolution of xi(n) and h(n) is computed for each
block, which leads to blocks of output sequences yi(n)
Because of aliasing the first (M-1) samples of each output
sequence yi(n) is discarded.
The final output sequence after discarding first (M-1) samples
of each output sequence yi(n) consists of samples of all blocks
arranged in sequential order.
4.2. Overlap-Add Method
In this method also the long direction input sequence is divided
into blocks of data.
The length of each block is L+M-1
The first L samples are new samples taken from long duration
input sequence and the last M-1 samples are zero appended to
have total length of samples as L+M-1
The data blocks are represented as
x1(n)= {x(0), x(1)… x(L-1), 0,0…}
x2(n)= {x(L), x(L+1)… x(2L-1), 0,0…}
x3(n)= {x(2L), x(2L+1)… x(3L-1), 0,0…}
The last M-1 samples in each sequence are zeros appended to
have total length as L+M-1

__________________________________________________________________________________________
Similarly the length of impulse response is increased to L+M-1
by appending L-1 zeros to it.
Circular convolution is performed on each block of input
sequence with the impulse response to have blocks of output
sequences.
The last M-1 samples of each block of output sequence is
overlapped and added to the first M-1 samples of succeeding
block. The samples thus obtained are arranged in sequential
order to have the final output sequence y(n).
So this method is called as Overlap-Add method.
5. MULTIPLICATION TECHNIQUE
Jagadguru Swami Sri Bharati-Krishna Swamiji introduced his
research on mathematics based on sixteen sutras for
multiplication. A multiplier is the key block in Digital Signal
processing. In the increasing technology, researchers are trying
to design multipliers which offer high computational speed, less
delay, low power and area efficient arithmetic building blocks
[7].
In Linear Convolution, the multiplication is performed using
Urdhva-Tiryagbhyam Sutra of Vedic mathematics[8]. The
Comparison between number of multiplications and additions
in Conventional Mathematical approach and vedic mathematics
is shown. [9]
Table 1: Comparison between normal multiplication and vedic
mathematics multiplication
Normal multiplier Vedic multiplier
For 2 bit multiplication
No. of multiplications : 4
No. of additions :2
No. of additions :1
No. of additions :7
No. of additions :5
No. of multiplications :
16
No. of additions
:15
No. of additions :9
No. of multiplications :
64
No. of additions
:77
No. of additions :53
Example
Multiplication of 1234 and 2116
Adder
Step1:
4x6=24, 2, Sthe carry is placed below the second digit
Step2:
(3x6) + (4x1) = 22. 2, the carry is placed below the third digit.
Step3:
(2x6) + (4x1) + (3x1) = 19. 1, the carry is placed below the
fourth digit.
Step4:
(1x6) + (2x4) + (2x1) + (3x1) = 19. The carry 1 is placed
below the fifth digit.
Step5:
1 2 3 4
2 1 1 6
1 2 3 4
2 1 1 6
2 5 9 9 9 2
0 0 1 1 2 2
4
2 6 1 1 1 4 4
1 2 3 4
2 1 1 6
1 2 3 4
2 1 1 6
1 2 3 4
2 1 1 6
1 2 3 4
2 1 1 6

__________________________________________________________________________________________
(1x1) + (3x2) + (2x1) = 9. The carry 0 is placed below the
sixth digit.
Step6:
(1x1) + (2x2) = 5. The carry 0 is placed below seventh digit.
Step7:
(1x 2)=2.
6. SIMULATION RESULTS
6.1. Circular Convolution
Fig. 2 Circular convolution output
Here input sequence is a(n)= [a3,a2,a1,a0]
Impulse sequence is b(n)= [b3,b2,b1,b0]
In this each value is of 4 bit length.
The given inputs are a(n)= [ 1, 2, 3, 4 ]
Impulse sequence is b(n)= [1, 1, 1, 0 ]
Output in hexadecimal format is (Each of 8 bit length)
Y(n)= [8’h 06,8’h06,8’h04,8’h05 ]
6.2. Linear Convolution for Short Duration Sequence
Fig. 3 Linear convolution for short duration sequence
Here input sequence is x(n)= [x3,x2,x1,x0]
Impulse sequence is h(n)= [h3,h2,h1,h0]
In this each value is of 4 bit length.
The given inputs are x(n)= [ 1, 2, 3, 4 ]
Impulse sequence is h(n)= [2, 3, 4, 5 ]
Output in hexadecimal format is(Each of 8 bit length)
Y(n)= [8’h 14,8’1f,8’22,8’h1E,8’h 10,8’h 07,8’h02 ]
6.3. Linear Convolution for Long Duration Sequence
Overlap-Add Method
Fig. 4 Linear convolution for long duration sequence Overlap-
Add method
1 2 3 4
2 1 1 6
1 2 3 4
2 1 1 6

__________________________________________________________________________________________
Here, convolution is applied between sequences of lengths 12
and 2 respectively.
Input sequence x(n)= [ i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11 ]
Impulse sequence h(n)= [ h0,h1 ]
Convolved sequence y(n)= [g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,
g10,g11,g12 ]
The example taken here is
x(n)= [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] (Decimal format)
h(n)= [4,5 ] (Decimal format)
y(n)= [8’h 3C,8’h67,8’h5E,8’h55,8’h4C,8’h43,8’h3A,8’h31,
8’h28,8’h1F,8’h16,8’h0D,8’h04 ]
6.4. Linear Convolution for Long Duration Sequence
Overlap- Save Method
Fig. 5 Linear convolution for long duration sequence Overlap-
Save method
Here, convolution is applied between sequences of lengths 12
and 2 respectively.
Input sequence x(n)= [ i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11 ]
Impulse sequence h(n)= [ h0,h1 ]
Convolved sequence y(n)= [g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,
g10,g11,g12 ]
The example taken here is
x(n)= [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] (Decimal format)
h(n)= [4,5 ] (Decimal format)
y(n)= [8’h 3C,8’h67,8’h5E,8’h55,8’h4C,8’h43,8’h3A,8’h31,
8’h28,8’h1F,8’h16,8’h0D,8’h04 ]
CONCLUSIONS
Circular and Linear convolution of discrete finite and infinite
length sequences are performed using carry save multiplier
based on Vedic multiplication. The multiplier proposed in
this paper, using Vedic mathematics results in high
computation speed and minimum critical path, which results in
less delay, when compared to normal multiplier. Speed can be
further optimized by high performance adders.
REFERENCES
[1] M.Bharathi, D.Leela Rani, & S.Varadrajan “High speed
carry save multiplier based linear convolution using
Vedi mathematics”, International journal of computer
and techonolgy, volume 4,no.2, March- April 2013.ISSN
no.277-3061.
[2] Jan M. Rabaey, Anantha Chandrasasan Borivoje
Nikdic,2003, Digital Integrated Circuits-A Design
perspective, Prentice-Hall.
[3] Purushotam D. Chidgupkar and Mangesh T. Karad,
2004, “The Implementation of vedic Algorithms in
Digital Signal Processing”, Global J. of Engng. Educ.,
Vol.8, No.2, UICEE Published in Australia.
[4] J.G. Proakis and D.G. Monalkies,1988, Digital Signal
Processing. Macmillian.
[5] A.V. Oppenheim and R. Schafer, 1975, Discrete-Time
Signal Processing Englewood Cliffs, NJ:Prentice-Hall.
[6] Asmita Haveliya, Kamlesh Kumar Singh,2011, “A
Novel Approach For High Speed Block Convolution
Algorithm”, proc. Of the International Conference on
Advanced Computing and Vommunication Technologies
(ACCT).
[7] Jagadguru Swami Sri Bharati Krishna Tirthji
Maharaja,1986, “Vedic Mathematics”, Motilal
Banarsidas, Varanasi, India.
[8] Human Tharafu M.C. Jayalaxmi. H. Renuka R.K.,
Ravishankar. M.,2007, “A. high speed block convolution
using Ancient Indian Vedic Mathematics”, IEEE
International conference on computational intelligence
and multimedia applications.
[9] A.P. Nicholas, K.R Willaiams, J. Pickles, 2003,
Vertically and Crosswise applications of the Vedic
Mathematics Sutra, Motilal Banarsidass Publishers,
Delhi.

__________________________________________________________________________________________
BIOGRAPHIES:
Ms. M. Bharathi, M.tech, is currently
working as an Assistant Professor in ECE
department of Sree Vidyanikethan
Engineering College, Tirupati. She has
completed M.tech in VLSI Design, in
Satyabhama University. Her research areas
are Digital System Design, VLSI Signal Processing.
Ms. D.Leela Rani received the M.Tech.
Degree from Sri Venkateswara University,
Tirupati. She is currently working towards
the Ph.D. degree in the Department of
Electronics and communication Engineering,
SVU College of Engineering,Tirupati.
Currently she is working as an Associate professor in Sree
Vidyanikethan Engineering College (Autonomous). Her
research areas include, Atmospheric Radar Signal Processing
and VLSI Signal Processing.

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