function y = fft_rec(x,p)
n = length(x);
if n == 1
y = x;
else
m = n/2;
y_top = fft_rec(x(1:2:(n-1)),p(1:2:(n-1)));
y_bottom = fft_rec(x(2:2:n),p(2:2:n));
d = exp(2 * pi * 1i / n) .^ p(1:m);
z = d .* y_bottom;
y = [ y_top + z , y_top - z ];
end
function y = fft_rec(x)
n = length(x);
if n == 1
y = x;
else
m = n/2;
y_top = fft_rec(x(1:2:(n-1)));
y_bottom = fft_rec(x(2:2:n));
d = exp(-2 * pi * i / n) .^ (0:m-1);
z = d .* y_bottom;
y = [ y_top + z , y_top - z ];
end
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