We see in our example that the magnitude of X(1) is 8 while the amplitude of the 10Hz
signal it represents is 2 . Why the discrepancy ? The reason is that our definition of X(k)
does not give the correct normalization of the amplitudes. However it does give the
correct relative amplitudes and the correct phases. In our example the total 10Hz signal
from X(k) is :
apparently a factor of 8 (i.e. N) too great. If we had taken 16 samples over the same time span
rather than 8 the corresponding X(k) would be a factor of 2 greater again. The correct absolute
amplitudes then are (using the S&K indexing from 
:
But 
.