# 0.12 Appendix: a matlab program for generating prime length fft  (Page 2/3)

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The following programs print the program statements that carry out the operation $I\otimes {D}_{k}\otimes I$ and $I\otimes {D}_{k}^{t}\otimes I$ . They are modeled after kpi in the text.

function kpi(d,g,r,c,n,Y,X,fid) % kpi(d,g,r,c,n,Y,X,fid);% Kronecker Product : A(d(1)) kron ... kron A(d(n)) % g : permutation of 1,...,n% r : [r(1),...,r(n)] % c : [c(1),..,c(n)]% r(i) : rows of A(d(i)) % c(i) : columns of A(d(i))% n : number of terms for i = 1:na = 1; for k = 1:(g(i)-1)if i>find(g==k) a = a * r(k);else a = a * c(k);end endb = 1; for k = (g(i)+1):nif i>find(g==k) b = b * r(k);else b = b * c(k);end end% Y = (I(a) kron A(d(g(i))) kron I(b)) * X; if i == 1S1 = sprintf([Y,' = ID%dI(%d,%d,',X,'); '],d(g(i)),a,b);S2 = sprintf(['%% ',Y,' = (I(%d) kron D%d kron I(%d)) * ',X],a,d(g(i)),b);fprintf(fid,'%-35s%s\n',S1,S2); elseif d(g(i)) ~= 1S1 = sprintf([Y,' = ID%dI(%d,%d,',Y,'); '],d(g(i)),a,b);S2 = sprintf(['%% ',Y,' = (I(%d) kron D%d kron I(%d)) * ',Y],a,d(g(i)),b);fprintf(fid,'%-35s%s\n',S1,S2); endend function kpit(d,g,r,c,n,Y,X,fid) % kpit(g,r,c,n,Y,X,fid);% (transpose) % Kronecker Product : A(d(1))' kron ... kron A(d(n))'% g : permutation of 1,...,n % r : [r(1),...,r(n)]% c : [c(1),..,c(n)] % r(i) : rows of A(d(i))'% c(i) : columns of A(d(i))' % n : number of termsfor i = 1:n a = 1;for k = 1:(g(i)-1) if i>find(g==k) a = a * r(k);else a = a * c(k);end endb = 1; for k = (g(i)+1):nif i>find(g==k) b = b * r(k);else b = b * c(k);end end% x = (I(a) kron A(d(g(i)))'' kron I(b)) * x; if i == nS1 = sprintf([Y,' = ID%dtI(%d,%d,',X,'); '],d(g(i)),a,b);S2 = sprintf(['%% ',Y,' = (I(%d) kron D%d'' kron I(%d)) * ',X],a,d(g(i)),b);fprintf(fid,'%-35s%s\n',S1,S2); elseif d(g(i)) ~= 1S1 = sprintf([X,' = ID%dtI(%d,%d,',X,'); '],d(g(i)),a,b);S2 = sprintf(['%% ',X,' = (I(%d) kron D%d'' kron I(%d)) * ',X],a,d(g(i)),b);fprintf(fid,'%-35s%s\n',S1,S2); endend

## Programs for computing multiplicative constants

The following programs carry out the operation of ${F}_{{d}_{1}}\otimes \cdots \otimes {F}_{{d}_{K}}$ where $F$ is the reconstruction matrix in a linear convolution algorithm. See the appendix, Bilinear Forms for Linear Convolution.'

function u = KFt(f,r,c,u) % u = (F^t kron ... kron F^t)*u% (transpose) % f = [f(1),...,f(K)]% r : r(i) = rows of F(i) % c : c(i) = columns of F(i)% u : length(u) = prod(c); K = length(f);for i = 1:K m = prod(c(1:i-1));n = prod(r(i+1:K)); u = IFtI(f(i),r(i),c(i),m,n,u);end function y = IFtI(s,r,c,m,n,x); % y = (I(m) kron F(s)^t kron I(n))*x% (transpose) % r : rows of F(s)% c : columns of F(s) v = 0:n:n*(c-1);u = 0:n:n*(r-1); for i = 0:m-1for j = 0:n-1 y(v+i*c*n+j+1) = Ftop(s,x(u+i*r*n+j+1));end end function y = Ftop(k,x) if k == 1, y = x;elseif k == 2, y = F2t(x); elseif k == 3, y = F3t(x);elseif k == 4, y = F4t(x); elseif k == 6, y = F6t(x);elseif k == 8, y = F8t(x); elseif k == 18, y = F18t(x);end`

The following programs carry out the operation of ${G}_{{p}_{1}^{{e}_{1}}}\otimes \cdots \otimes {G}_{{p}_{K}^{{e}_{K}}}$ were $G$ is given by Equation 13 and Equation 14 from Bilinear Forms for Circular Convolution .

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
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