\\ Centered vector for cyclic indices.
t_center (v) = {my (n=#v,d=n\2);
vector(n,i,v[(i+d)%n+1])}

\\ Fourier transform on a cyclic group.
t_ft (v,realpart=0) = {my (n,m,Omstar);
n=#v; w=exp(2*Pi*I/n);
Omstar=matrix(n,n,j,k,w^(-(j-1)*(k-1)));
t=(v*Omstar)/n; if (!realpart,t,apply(real,t))}

t_ftstrabs (v,d=2,center=1) = {my (w);
w=t_ft(v); if (center, w=t_center(w));
apply(x->Strprintf("%.*f",d,abs(x)),w)}

t_ftstrarg (v,d=2,center=1) = {my (w);
w=t_ft(v); if (center, w=t_center(w));
apply(x->Strprintf("%.*f",d,arg(x)),w)}

\\Inverse Fourier transform on a cyclic group.
t_fti (v,realpart=0) = {my (n,m,Om);
n=#v; w=exp(2*Pi*I/n);
Om=matrix(n,n,j,k,w^((j-1)*(k-1)));
t=v*Om; if (!realpart,t,apply(real,t))}

\\ Walsh transform.
t_ftw (v) = {my (k,W);
n=#v; k=#binary(n)-1;
W=matrix(1,1,i,j,1);
for (i=1,k,W=matconcat([W,W;W,-W]));
(v*W)/n}

\\ Inverse Walsh transform.
t_ftwi (v) = {my (k,W);
n=#v; k=#binary(n)-1;
W=matrix(1,1,i,j,1);
for (i=1,k,W=matconcat([W,W;W,-W]));
v*W}

t_ftwrite (F,X,name,G=F) = {my (u,v);
u=apply(G,X); v=apply(F,X);
t_writeplotpairs(Strprintf("%s.fun",name),u);
t_writeplotpairs(Strprintf("%s.abs",name),t_ftstrabs(v));
t_writeplotpairs(Strprintf("%s.arg",name),t_ftstrarg(v));}