Power Spectrum of a sinusoid

We can define the energy of a signal as the sum of the squared signal. We begin with 20 seconds of a sinusoid at 9.3 Hz sampled at 100 Hz

 
-->inct = 0.01;
 
-->N = 2000;
 
-->t = (0:N-1)*inct;
 
-->frec = 9.3;
 
-->x = sin(2*%pi*frec*t);
 

We are going to calculate the squared sum of its values but instead of calculating the whole vector we are going to calculate by groups of 100 samples (1 s). then, we calculate the Fourier Transform and add the squared values of its coefficients. Both values are coincident when we divide the sum of the squared coefficients of the Fourier Transform by the number of samples

-->x1 = x(1:100);
 
-->ssumx1 = x1 * x1'
 ssumx1  =
 
    49.769926  
 
-->ffx1 = fft(x1,-1);
 
-->ssumffx1 = sum(abs(ffx1)^2) / 100
 ssumffx1  =
 
    49.769926

If we repeat the process with the values of the next second, we get a very similar (although not identical) result

-->x2 = x(101:200);
 
-->ssumx2 = x2 * x2'
 ssumx2  =
 
    49.697833  
 
-->ffx2 = fft(x2,-1);
 
-->ssumffx2 = sum(abs(ffx2)^2) / 100
 ssumffx2  =
 
    49.697833

If the amplitude of the signal is measured in $V$, energy will be measured in $V^2$; if the amplitude of the signal is measured in $mV$, energy will be in $mV^2$.