energy of a signal in t and f domain

The energy of a signal is expected to be the same in t and f domain.

n  = 1e4;
dx = 0.25;
x  = rand(n,1) -0.5;
ex = sum(x.^2) *dx; % energy in t domain
y  = fft(x);
fs = 1/dx;
df = fs/n;
ya = abs(y);
ey = sum(ya.^2) *df; % energy in f domain

but from the code, ey/ex=16, exactly the squared fs.

what's the problem?

Answer 1

[Star Strider]

You need to normalise the fft by dividing it by the length of the signal:

y  = fft(x)/length(x);

See the documentation for fft for details.

[Ray Lee]

Before, I got ey/ex=16

after normalization, I got ey/ex=1.6000e-07

[Ray Lee]

it seems y = fft(x) / fs

Answer 2

[Ray Lee]

I found the solution myself.

Normalizing spectral amplitude by fs will work.

But I don't know why to do this.