The Cook-Toom algorithm (or Toom-Cook) is a high-speed, divide-and-conquer algorithm used for multiplying large integers or polynomials by splitting them into \(k\) parts. It generalizes Karatsuba (\(k=2\)) to reduce complexity, commonly running at \(\Theta (n^{1.46})\) for \(k=3\). The process involves splitting, evaluating at specific points, multiplying, and interpolating. 

1. COOK-TOOM ALGORITHM

2. OVERVIEW
 CONVOLUTION
 FAST CONVOLUTION
 COOK-TOOM ALGORITHM
 EXAMPLE
 ALGORITHM
 MODIFIED COOK-TOOM ALGORITHM
 IMPLEMENTATION OF FILTERBANK
 ADVANTAGES
 REFERENCES

3. CONVOLUTION
 Mathematical operation on two functions to produce a
third variable, typically viewed as a modified version of
the original function.
 The convolution theorem allows one to mathematically
convolve in the time domain by simply multiplying in the
frequency domain.
 Two types: linear and cyclic
 Linear-(M-1)(N-1) additions and MN multiplications

4.  Cyclic-N(N-1) additions and N*N multiplications
 For speeding up the calculation-FFT
 Commonly used for fast computation of
convolutions
 Disadvantage-complex arithmetic
 Another method is to convert 1D convolution
into multidimensional convolution
 By using efficient short length algorithms

5. FAST CONVOLUTION
 Convolution using fewer number of operations
 Algorithmic strength reduction: Number of
strong operations is reduced at the expense of
an increase in the number of weak operations.
 Best suited for implementation using
programmable or dedicated hardware.

6.  Assume (a+jb)(c+jd) = e+jf ,
where (a+jb) is the signal sample
(c+jd) is a coefficient
implemented using 4 multiplications and 2
additions
 Using fast convolution algorithm arithmetic
complexity is reduced to 3 multiplications and 3
additions

7. COOK-TOOM ALGORITHM
 Minimum number of multiplications
 For non cyclic convolution
( 1)
 This requires 2N-1 multiplications
 Define the generating polynomial of a sequence
xi, by
(2)

8.  Then W(z)=X(z)H(z), (3)
where H(z) and W(z) are generating polynomials
of hi and wi.
 W(z) is a 2N-2 degree polynomial
 To determine the 2N- 1 wi's, select 2N- 1
distinct numbers αj, j = 0, 1, . . , 2N - 2, and
substitute them for z in (3) to obtain the 2N - 1
products
m =W(α )=H(α )X(α ), j=0,1,…..2N-2 (4)

9.  Using Lagrange interpolation formula
 (5)
 Cost is 2N-1 multiplications
 (4) can be written in matrix form as, m =
( Ah)x(Ax) where

10.  From (5) the coefficients of W(z) will be the
linear combinations of the mj’s and can be
written as
w=C*m
where C* is a 2N-1 by 2N-1 matrix
 To calculate cyclic convolution, compute
Y(z)=W(z)mod(zN
-1)
which leads to y=Cm, where C is an N by 2N-1
matrix obtained from C* by performing row

11.  General form is m=(Ah)x(Bx)
 Value of (Ah) is precomputed.
 B and C will have no multiplications
 Only multiplications are the element by element
multiplication of Ah by Bx.
 Cook-Toom algorithm yields large integer
coefficients in A,B,C matrices which is costly as
multiplication.

12. Example
 (Qn). Calculate the non cyclic 2 point convolution
Using (1)
wo=hoxo
w1=h0x1+h1x0
w2=h1x1
 In terms of z transforms, this is equivalent to
w0+w1z+w2z2
=hoxo +(h0x1+h1x0)z+h1x1z2
= ho(x0+x1z)+h1z(x0+x1z)
= (h0+h1z)(x0+x1z)
 Let αj=-1,0,1 for j=0,1,2 in (4)

13. m0=(h0-h1)(x0-x1)
m1=h0x0
m2=(ho+h1)(x0+x1)
[put z=-1,0,1 in the above eqn. to obtain m0,m1,m2)
 From (5)

14. So that
w0=m1
w1=(m2-m0)/2
w2=[(m0+m2)/2]-m1
 Transferring denominators from C to A matrix,
combine the factor ½ with hj’s and store the
precomputed constants
a0=(h0-h1)/2
a1=h0
a2=(h0+h1)/2

15.  Hence
m0=a0(x0-x1)
m1=a1x0
m2=a2(x0-x1)
w0= m1
w1=m2-m0
w2=m0+m2-m1
 Only 3 multiplications and 5 additions are
required instead of 4 multiplications and 3
addition

16. Algorithm
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) using the equation

17.  Reduction of operation count occurs if the
numbers β0, β1,…. Βn are carefully chosen
 A better algorithm using Chinese remainder
theorem to compute the convolution as:
S(x)=[D(x)G(x)mod

18. Modified Cook-Toom Algorithm
 Choose L+N-2 different real numbers β1,… β0, βL+N-
2
 Compute h(βi) and x(βi),for i=0,1,…L+N-3
 Compute s(βi)=h(βi)x(βi), for i=0,1,…L+N-3
 Compute s’(βi)= s(βi)-sL+N-2 βiL+N-2, for i=0,1,…L+N-3
 Compute s’(p) using the equation
 Compute s(p)= s’(p)+sL+N-2pL+N-2

19. IMPLEMENTATION OF FILTER BANK
 Consider two polynomials, g(z-1
)=g0+g1z-1
+….+gL-1z-L+1
and
u(z-1
)=u0+u1z-1
+…uN-1z-N+1
 By computing in normal form it’s product y(z-1
) requires
NL multiplications
 Cook-toom algorithm will reduce it to M>=N+L-1
 First, choose a set of interpolation points{ρi}i=0:M-1 that
are the roots of r(z-1
) =
 Evaluate y(ρi)=g(ρi)u(ρi)

20.  Perform Lagrange interpolation to restore
y(z-1
)= with
Li(z-1
)=
 Eg: Consider g(z-1
)=g0+g1z-1
(L=2), and u(z-1
)=u0+u1z-1
(N=2)
 Choose interpolation points{0,1,-1}
 Then y(0)=u0g0
y(1)=(u0+u1)(g0+g1)
y(-1)=(u0-u1)(g0-g1)

21.  Lagrange polynomials are calculated as
L0(z-1
)=(1-z-2
)
L1(z-1
)=(z-1
+z-2
)/2
L-1(z-1
)=(-z-1
+z-2
)/2
ie, y(z-1
)=y(0)L0(z-1
)+y(1)L1(z-1
)+y(-1)L-1(z-1
) is
reconstructed

22.  Multiplication can be written as y=Gu, then algorithm can
be represented as the matrix decomposition G=CDA, with
D=diag(Bg)
 G is the (N+L-1) x N Toeplitz matrix defining the filter g(z-1
)
 A is the Vandermonde matrix with Am,n= (n=0:N-1)
 B is also the MxL Vandermonde matrix with Bm,l= (l=0:L-1)
 C is the (N+L-1) x M matrix whose ith column contains the
first N+L-1 coefficients of Li(z-1
)

23. ADVANTAGES
 The number of multiplications have been reduced to
L+N-1 at the expense of an increase in the number of
additions
 Adder has much smaller area and computation time
than multiplier
 Low hardware complexity
 Pre addition and post addition matrices are not simple

24. REFERENCES
[1]. Zdenka Babic,Danilo P.Mandic, “A Fast Algorithm for
Linear Convolution of Discrete Time signals” , TELSIKS,
pp.no-595-598,September 2001.
[2]. Yuke Wang,Keshab Parhi, “Explicit Cook-Toom
Algorithm for linear convolution” ,IEEE,2000.
[3]. Geert Van Meerbergen, Marc Moonen, Hugo De Man,
“Critically Subsampled filterbanks implementing Reed-
Solomon codes” ,vol 2, pp.no-989-992,IEEE,2004

25. [4].Keshab K. Parhi, “VLSI digital signal Processing
Systems, Design and Implementation”, pp.no:227-
237,New Delhi,1999.
[5].Ivan W.Selesnick,C. Sidney Burrus, “Fast Convolution
and Filtering” , Digital Signal Processing Handbook, CRC
Press LLC, 1999.
[6]. R. Meyer, R. Reng and K. Schwarz, Convolution
Algorithms On DSP Processors, IEEE,1991.
[7].R.E.Blahut, “Fast Convolution Algorithms for Digital
Signal Processing” , Addison-Wesley,1985.

26. [8]. H.J.Nussbaumer, “Fast Fourier Transform and Convolution
Algorithms” , Springer-Verlag, Berlin, Heidelberg, and New
York,1981
[9]. Ramesh C Agarwal,James W. Cooley, ”New Algorithms for
Digital Convolution”, IEEE Transactions on Accoustics,
Speech and Signal Processing, Vol. ASSP-25, No.5,October
1997.
[10]. Alberto Zanoni, Toom-”Cook 8-way For Long Integers
Multiplication”, 11th International Symposium on Symbolic
and Numeric Algorithms for Scientific Computing,2009

27. THANK YOU