01 EC 7311-Module IV.pptx

01 EC 7311
VLSI Structures for Digital
Signal Processing
.

Outline
 Fast convolution.
 Cook-Toom Algorithm.
 Modified Cook-Toom Algorithm.
 Winograd Algorithm.
 Cyclic convolution.

Introduction
 Fast convolution:
 Implementation of convolution algorithm using fewer multiplication operations by
algorithmic strength reduction.
 Algorithmic Strength Reduction: Reducing the number of strong operations (such as
multiplication operations) at the expense of an increase in the number of weak
operations (such as addition operations).
 Reduce computation complexity.
 Best suited for implementation using either programmable or dedicated hardware.

Introduction
 Example: Consider the complex number multiplication: (a+jb)(c+dj)=e+jf, where {a, b, c,
d, e, f } ∈ R. (R: set of real numbers).
 The direct implementation requires
four multiplications and two additions:
 The number of multiplications can be reduced to 3 at the expense of 3 extra additions
by using the identities:
Three Multiplications
Five Additions

Introduction
 Example: (contd…)
 In matrix form, its coefficient matrix can be decomposed as the product of a 2X3(C), a
3X3(H)and a 3X2(D) matrix:
 Where C is a post-addition matrix (requires two additions), D is a pre-addition matrix
(requires one addition), and H is a diagonal matrix (requires two additions to get its
diagonal elements).

Introduction
 Example: (contd…)
 The arithmetic complexity is reduced to three multiplications and three additions (not
including the additions in H matrix).
 Two well-known approaches for the design of fast short-length convolution algorithms:
1. Cook-Toom algorithm (based on Lagrange Interpolation) and
2. Winograd Algorithm (based on the Chinese remainder theorem)

 A linear convolution algorithm for polynomial multiplication based on the
Lagrange Interpolation Theorem.
 Lagrange Interpolation Theorem:
 Let β0, β1, ...., βn be a set of (n +1) distinct points, and let f(βi) , for i = 0, 1, …, n be
given. There is exactly one polynomial f ( p) of degree “n” or less that has value f(βi)
when evaluated at βi for i = 0, 1, …, n.
 It is given by:
Cook-Toom algorithm:

 Lagrange Interpolation Theorem: (contd…)
 Interpolation is a special case of data fitting, in which the goal is to find a linear
combination of “n” known functions to fit a set of data that imposes “n” constraints,
thus guaranteeing a unique solution that fits the data exactly, rather than
approximately.
 Data fitting: Constructing a continuous function from a set of discrete data points,
where the data is obtained from measurements.
Cook-Toom algorithm: (contd…)

 Compute the unique polynomial pn(x) of degree “n” that satisfies pn(xi) = yi, i = 0, . . . , n,
where the points (xi, yi) are given.
 The points x0, x1, . . . , xn are called interpolation points.
 The polynomial pn(x) is called the interpolating polynomial of the data (x0, y0), (x1, y1),
. . ., (xn, yn).
 Form the system Ax = b, where bi = yi , i = 0, . . . , n for computing the interpolation
polynomial

 The entries of A are defined by aij = pj (xi); i, j = 0, . . . , n, where x0, x1, . . . , xn are the
points at which the data y0, y1, . . . , yn are obtained, and pj (x) = xj , j = 0, 1, . . . , n.
 The basis {1, x, . . . , xn} of the space of polynomials of degree (n+1) is called the
monomial basis.
 The corresponding matrix A is called the Vandermonde matrix for the points x0, x1, . . . ,
xn.

 In Lagrange interpolation, the matrix A is simply the identity matrix.
 The interpolating polynomial is written as:
 Where the polynomials have the property that:

 The polynomials {£n,j}, j = 0, … , n are called the Lagrange polynomials for the
interpolation points x0, x1, … , xn.
 They are defined by:

 Convolution basics:
 Given two sequences:
o Data sequence di, 0 ≤ i≤ N-1, of length N.
o Filter sequence gi, 0 ≤ i≤ L-1, of length L
 The Linear convolution these two sequences is given by:
(NL multiplications)

 Convolution basics: (contd…)
 Express the convolution in the notation of polynomials:
o Let and , then
where

 An algorithm for linear convolution by using multiplying polynomials.
 Consider the following system:
 The application of Lagrange interpolation theorem into linear convolution.
 Consider an N-point sequence h = {h0, h1, ... , hN-1} and an L-point sequence x = {x0, x1, ...
, xL-1}.

 The linear convolution of h and x can be expressed in terms of polynomial
multiplication as follows: s( p) = h( p) . x( p), where
 The output polynomial s(p) has degree (L+N-2) and has (L+N-1) different points.

 s(p) can be uniquely determined by its values at (L+N-1) different points.
 Let {β0, β1, ... , βL+N-2} be (L+N-1) different real numbers.
 If s(βi) for i = {0,1, ... , (L+N-2)} are known, then s(p) can be computed using the
Lagrange interpolation theorem as:
 This equation is the unique solution to compute linear convolution for s(p) given the
values of s(βi ) , for i = {0,1, ..., (L+N-2)}.

 Algorithm Description:
1. Choose (L+N-1) different real numbers β0, β1, … , βL+N -2.
2. Compute h(βi ) and x(βi ), for i = {0,1, … , (L+N-2)}.
3. Compute s(βi) = h(βi) . x(βi), for i = {0,1, … , (L+N-2)}.
4. Compute s(p) by using:

 Algorithm Complexity:
 The goal of the fast-convolution algorithm is to reduce the multiplication complexity.
 So, if βi ‘s (where i=0, 1, …, L+N-2) are chosen properly, the computation in step-2
involves some additions and multiplications by small constants.
 The multiplications are only used in step-3 to compute s(βi).
 So, only (L+N-1) multiplications are needed.

 Advantages :
1. The number of multiplications is reduced from (LN) to (L+N-1) at the expense of an
increase in the number of additions.
2. An adder has much less area and computation time than a multiplier.
3. Lead to large savings in hardware (VLSI) complexity and generate computationally
efficient implementation.

 Example: Construct a (2X2) convolution algorithm using Cook-Toom
algorithm with β={0,1,-1}.
 Solution:
 Write the (2X2) convolution in polynomial multiplication form as:
s (p) = h (p) . x (p), where h (p) = h0 + h1p . x (p)
i.e., s (p) = s0 + s1p + s2p2
 Direct implementation requires four multiplications and one addition.
can be expressed in matrix form as follows:

 Solution: (contd…)
 The direct implementation can be expressed in matrix form as follows:

 Use Cook-Toom algorithm to get an efficient convolution implementation with reduced
multiplication number:
1. Step 1: Choose β0=0, β1=1, β2=-1
2. Step 2 :

3. Step 3 : Calculate s(β0), s(β1), s(β2) by using 3 multiplications.

4. Step 4: Compute s(p) by using Lagrange interpolation theorem:
i.e., s(β0) = h(β0) . x(β0), s(β1) = h(β1) . x(β1) and s(β 2) = h(β2 ) . x(β2 )

 In matrix form:

 The computation is carried out as follows (5 additions, 3 multiplications):
 Hence the Cook-Toom algorithm needs three multiplications and five additions
(ignoring the additions in the pre-computation ), i.e., the number of multiplications is
reduced by one at the expense of four extra additions.

 Conclusions:
 Some additions in the pre-addition or post-addition matrices can be shared.
 So, when we count the number of additions, we count only one, instead of two or three.
 If we take h0, h1 as the FIR filter coefficients and take x0, x1 as the signal (data)
sequence, then the terms H0, H1 need not be recomputed each time the filter is used.
 They can be precomputed once offline and stored. (So, we ignore these computations
when counting the number of operations).

 Conclusions:
 The Cook-Toom algorithm is a matrix decomposition.
 A convolution can be expressed in matrix-vector forms as:
 Generally, the equation can be expressed as:
 Where C is the post-addition matrix, D is the pre-addition matrix, and H is a diagonal
matrix with Hi, i = 0, 1, …, (L+N-2) on the main diagonal.

 Conclusions:
 Since T=CHD, it implies that the Cook-Toom algorithm provides a way to factorize the
convolution matrix T into multiplication of 1 post-addition matrix C, 1 diagonal
matrix H and 1 pre-addition matrix D, such that the total number of multiplications is
determined only by the non-zero elements on the main diagonal of the diagonal
matrix H (note matrices C and D contain only small integers).
 Limitations:
Although the number of multiplications is reduced, the number of additions has
increased.

 The Cook-Toom algorithm is used to further reduce the number of addition
operations in linear convolutions.
 Define s' (p) = s(p) - SL+N-2 . pL+N-2.
 Notice that the degree of s(p) is L+N-2 and SL+N-2 is its highest order coefficient.
 Therefore the degree of s' (p) is L+N-3.
 This modified polynomial s’ (p) is used in modified Cook-Toom algorithm.
Modified Cook-Toom algorithm:

1. Choose L+N-2 different real numbers β0, β1, … , βL+N-3
2. Compute h(βi) and x(βi) , for i ={0,1, … , L+N-3}
3. Compute s(βi) = h(βi) . x(βi), for i ={0,1, … , L+N-3}
4. Compute s'(βi) = s(βi)-(sL+N-2).(βi)L+N-2, for i ={0,1, … , L+N-3}
Modified Cook-Toom algorithm: (contd…)

5. Compute s'(p) by using:
6. Compute s(p) = s'(p) + (sL+N-2 ) . (pL+N-2)

 Example: Derive a (2X2) convolution algorithm using the modified
Cook-Toom algorithm with β={0,-1}.
 Solution:
 Consider the Lagrange interpolation for s'(p) = s(p) - h1x1p2 at {β0 = 0, β1 = -1}
 Find, s'(βi) = h(βi)x(βi) - h1x1βi
2
 Find,

 Solution: (contd…)
 Find,
 This computation requires 2 multiplications {one for h0x0 and another for (h0-h1)(x0-
x1); not counting that for h1x1 multiplication}.
 Apply the Lagrange interpolation
algorithm, we get:

 Therefore, s( p) = s'( p) + h1x1 p2 = s0 + s1p + s2p2
 The corresponding matrix-form expression expressions are:

 The computation is carried out as follows:
 The total number of operations are 3 multiplications (three in step-3) and 3 additions
(one in step-2 and two in step-4).
 Compared with the linear convolution using Cook-Toom algorithm, the number of
addition operations has been reduced by 2 while the number of multiplications
remains the same

 Conclusions:
 Limitations:
1. Although the number of multiplications is reduced, the number of additions has
increased (As the size of the problem increases, the number of additions increase
rapidly).
2. The choices of βi=0, ±1 are good, while the choices of ±2, ±4 (or other small integers)
result in complicated pre-addition and post-addition matrices.
3. For larger problems, Cook-Toom algorithm becomes cumbersome.
Cook-Toom algorithms:

 The Winograd short convolution algorithm:
 Based on the CRT (Chinese Remainder Theorem).
 Chinese Remainder Theorem:
 It’s possible to uniquely determine a nonnegative integer given its remainder with respect
to the given moduli, provided that the moduli are relatively prime and the integer is
known to be smaller than the product of the moduli.
Winograd Algorithm:

 The Winograd short convolution algorithm: ( contd…)
 Chinese Remainder Theorem for Integers:
 Review of Integer Ring:
 For every integer c and positive integer d, there is a unique pair of integer Q, called the
quotient, and integer s, called the remainder; such that c=dQ + s, where 0 ≤ s ≤ d-1.
 i.e., Q= ⎣ c/d ⎦ and s=Rd[c]
 Euclidean Algorithm: Given two positive integers s and t, where (t < s), then their
Greatest Common Divisor (GCD) can be computed by an iterative application of the
division algorithm.
Winograd Algorithm: (contd…)

 Chinese Remainder Theorem for Integers: (contd…)
 Euclidean Algorithm:
1. The process stops when a remainder of zero is
obtained.
2. The last nonzero remainder t(n) is the GCD(s,t).

 In matrix notation:
 For any integer s and t, there exists integers a and b such that GCD[s, t]= as + bt
 It is possible to uniquely determine a nonnegative integer given its moduli with
respect to each of several integers, provided that the integer is known to be smaller
than the product of the moduli.

 Example: {m1=3, m2=5} and M=3×5=15

 Euclidean Algorithm: Example: {m1, m2, m3} = {3, 4, 5} and M = 3× 4 × 5 = 60

 Given a set of integers m0, m1, … , mk that are pairwise relatively prime (co-prime),
then for each integer c, 0 ≤ c < M= m0m1…mk, there is a one-to-one map between c and
the vector of residues: (Rm0 [c], Rm1[c], … , Rmk [c])
 Conversely, given a set of co-prime integers m0, m1, …, mk and a set of integers c0, c1, …,
ck with ci < mi . Then the system of equations ci =c (mod mi), i=0,1,…,k has at most one
solution for 0 ≤ c < M.

 Define Mi=M/mi, then GCD[Mi, mi]=1. So there exists integers Ni and ni with GCD[Mi,
mi]= 1 = NiMi + nimi, i=0,1,…,k.
 The system of equations ci=c (mod mi), 0 ≤ I ≤ k, is uniquely solved by:

 Example: GCD

 Given ci = Rmi[c] (represents the remainder when cis divided by mi), for i = 0,1,...,k ,
where miare moduli and are relatively prime, then c = 𝑖=0
𝑘
𝑐𝑖 NiMi mod M, where
M = 𝑖=0
𝑘
𝑚𝑖, Mi = M/mi, and Niis the solution of NiMi + nimi = GCD(Mi, mi)=1,
provided that 0 ≤ c < M

 If NiMi + nimi = GCD (Mi, mi) = 1, then we have NiMi= 1 (mod mi).
 C (mod mi) = 𝑖=0
𝑘
𝑐𝑖𝑁𝑖𝑀𝑖 (𝑚𝑜𝑑 𝑚𝑖)
= ci NiMi (mod mi)
= ci(mod mi)

 Chinese Remainder Theorem (CRT) for Integers: (contd…)
 Example: Using the CRT for integer, Choose moduli m0=3, m1=4, m2=5.
 Then M = m0m1m2 = 60 and M i = M/mi
 By Euclidean theorem, we have:
 Where and are obtained using the Euclidean GCD algorithm.
 Given that the integer c satisfy: 0 ≤ c < M, Let ci = Rmi[c]

 Example:
 The integer c can be calculated as:
 For c=17:

 By taking residues, large integers are broken down into small pieces (that may be
easy to add and multiply).
 Examples:

 Chinese Remainder Theorem for Polynomials:

 Chinese Remainder Theorem for Polynomials: (contd…)
 Given a set of polynomials m(0)(x), m(1)(x), …, m(k)(x), that are pair-wise relatively
prime (co-prime), then for each polynomial c(x), deg(c(x)) < deg(M(x)), M(x)=
m(0)(x)m(1)(x)…m(k)(x), there is a one-to-one map between c(x) and the vector of
residues:

 Conversely, given a set of co-prime polynomials m(0)(x), m(1)(x), …, m(k)(x) and a
set of polynomials c(0)(x), c(1)(x), …, c(k)(x) with deg(c(i)(x)) < deg(m(i)(x)).
Then the system of equations c(i)(x) =c(x) (mod m(i)(x)), i=0,1,…,k has at most
one solution for deg(c(x)) < deg(M(x)).

 Define M(i)(x)=M(x)/m(i)(x), then GCD[M(i)(x),m(i)(x)]=1.
 So there exists polynomials N(i)(x) and n(i)(x) with GCD[M(i)(x), m(i)(x)] = 1 =
N(i)(x)M(i)(x) + n(i)(x)m(i)(x), i=0,1,…,k.
 The system of equations c(i)(x) =c(x) (mod m(i)(x) ), 0 ≤ I ≤ k, is uniquely solved
by:

 The remainder of a polynomial with regard to modulus xi + f(x), where deg(f(x)) <
i, can be evaluated. by substituting xi by –f(x) in the polynomial
 Example:

 Algorithm Description:
1. Choose a polynomial with degree higher than the degree of h(p) x(p ) and factor it
into (k+1) relatively prime polynomials with real coefficients,
i.e., m(p)= m(0)(p)m(1)(p) … m(k )(p).
2. Let M(i)( p) = m(p)/m(i) (p). Use the Euclidean GCD algorithm to solve N(i)(p)M(i)(p) +
n(i)(p)m(i)(p) = 1 for N(i)(p)

 Algorithm Description: (contd…)
3. Compute: h(i)(p) = h(p) mod m(i)(p) and x(i)(p) = x(p) mod m(i)(p) for i= 0, 1,
…, k.
4. Compute: S(i)(p) = h(i)(p) x(i)(P) mod m(i)(p) for i =0, 1, …, k.
5. Compute s(p) by using:

 Example: Consider a (2X3) linear convolution S(p) = h(p)x(p) where h(p) = h0 +
h1(p) and x(p) = x0 + x1p + x2p2. Construct an efficient realization using Winograd
algorithm with m(p) = p(p-1)(p2+1).
 Solution:
 Let m(0)(p) = p, m(1)(p) = (p-1) and m(2)(p) = (p2+1).
 Construct the following table using the relationships: M(i)(p) = m(p)/m(i)(p) and
N(i)(p)M(i)(p) + n(i)(p)m(i)(p) = 1 for i =0, 1, 2.

 Compute residues from h(p) = h0 + h1 p and x(p) = x0 + x1p + x2p2:
i.e., h(0)(p) = h0 and x(0)(p) = x0.
h(1)(p) = h0 + h1 and x(1)(p) = x0 + x1 + x2.
h(2)(p) = h0 + h1p and x(2)(p) = (x0 – x2) + x1p.
No multiplication
operations required

s(0)(p) = h0x0 = s0
(0)
s(1)(p) = (h0 + h1)(x0 + x1 + x2) = s0
(1)
s(2)(p) = (h0 + h1p)((x0-x2)+x1p) mod(p2+1)
= h0(x0-x2)-h1x1+(h0x1+h1(x0-x2))p
= s0(2) + s1
(2)p
 Algorithm complexity: One multiplication for s(0)(p), one for s(1)(p) and four for s(2)(p)

 The computational complexity can be reduced to three multiplications as shown
below:
 Then:

 Substitute s(0)(p), s(1)(p) and s(2)(p) in the relation for s(p) to obtain the following
table:

 Therefore, we have:
 Notice that:

 Finally, we have:
 Here the Winograd convolution algorithm requires five multiplications and 11
additions compared with six multiplications and two additions for direct
implementation.

 Conclusions:
 The number of multiplications in Winograd algorithm is highly dependent
on the degree of each m(i )( p).
 Therefore, the degree of m(p) should be as small as possible.
 More efficient form (or a modified version) of the Winograd algorithm can be
obtained by letting deg[m(p)]= deg[s(p)] and applying the CRT to:
s’(p) = s(p)-hN-1 . xL-1.m(p)

 Known as circular convolution.
 Let the filter coefficients be h = {h0, h1, … , hn-1}, and the data sequence be x ={x0,
x1, … , xn-1}.
 The cyclic convolution can be expressed as:
 The output samples are given by:
where ((i-k)) denotes (i-k) mod n
 The cyclic convolution can be computed as a linear convolution reduced by modulo
pn – 1 (Notice that there are 2n-1 different output samples for this linear convolution)
Cyclic Convolution:

 In circular convolution, coefficients with indices larger than (n-1) are folded back into
terms with indices small than n.
 We can express the cyclic convolution by polynomial product :
 The circular convolution system is shown below:
Cyclic Convolution: (contd…)

 The direct computation of cyclic convolution:

 Alternatively, the cyclic convolution can be computed using CRT with
m( p) = pn -1, which is much simpler.
 Example: Construct a (4X4) cyclic convolution algorithm using CRT with m( p) = p4-1
= (p - 1)(p + 1)(p2 + 1)
 Solution:
 Let h( p) = h0 + h1 p + h2 p2 + h p3 and x( p) = x0 + x1 p + x2 p2 + x3 p3
 Let m(0 )( p) = p -1, m (1)( p) = p +1 and m(2 )( p) = p2 +1.

 Get the following table using the relationships M (i)(p) = m(p)/m(i)(p) and
N (i)(p)M (i)(p) + n (i)(p)m(i)(p) = 1.

 Compute the residues:

 i.e., Individual convolution sum need five multiplications.

 In matrix form:
 Then:

 So we have:
 i.e.,

 Therefore, we have:
 This algorithm requires five multiplications and 15 additions.
 The direct implementation requires 16 multiplications and 12 additions.
 An efficient cyclic convolution algorithm can often be easily extended to construct
efficient linear convolution.

 The direct implementation requires 16 multiplications and 12 additions as shown in
the following matrix-form:
 Notice that the cyclic convolution matrix is a circulant matrix.

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