restart:N:=8:
a:=1: #0.0001 to 10f:=array(0..N-1):for n from 0 to N-1 do f[n]:=(1+a^2)/(1+a*n^2);od:W:=evalf(exp(-2*Pi*I/N)):for a1 from 0 to 1 dofor a0 from 0 to 1 dofor b0 from 0 to 1 doadd(f[4*a2+2*a1+a0]*W^(4*a2*b0),a2=0..1);X0[4*a1+2*a0+b0]:=%;od: od: od:
for i from 1 to N-1 do X0[i] od:for a0 from 0 to 1 dofor b0 from 0 to 1 dofor b1 from 0 to 1 doadd(X0[4*a1+2*a0+b0]*W^(2*a1*(2*b1+b0)),a1=0..1);X1[4*a0+2*b0+b1]:=%;od: od: od:QyQ/KEkiaUc2IiIiIkYmLCZJIk5HRiVGJiEiIkYmSSV0cnVlRyUqcHJvdGVjdGVkRyZJI1gxR0YlNiNGJEYpfor b2 from 0 to 1 dofor b1 from 0 to 1 dofor b0 from 0 to 1 doadd(X1[4*a0+2*b0+b1]*W^(a0*(4*b2+2*b1+b0)),a0=0..1);F[4*b2+2*b1+b0]:=%;od: od: od:
for i to N-1 do F[i] od:u:=[seq([i,abs(f[i])],i=0..N-1)]:
v:=[seq([i,abs(F[i])],i=0..N-1)]:plot(u,colour=blue,thickness=2);
plot(v,colour=blue,thickness=2);
for j from 0 to N-1 do
add(F[k]*W^(-j*k),k=0..N-1);inv[j]:=%/N; od:for i to N-1 do inv[i] od:u:=[seq([i,abs(inv[i])],i=0..N-1)]:plot(u,colour=blue,thickness=2);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LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlZzpGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiRTJkYqLSUpQ0hJTERSRU5HNiI=