Fourier coefficients, Fourier spectrum and signal energy

I was wondering if the equation 1) is correct in this example. The author described the difference between equations for $$X(w)$$ and $$X[k]$$. The only difference I see is the $$\frac{1}{T}$$ coefficient. Thus I can't see where $$w$$ comes from in eq. 1) denominator, or small $$t$$ in its middle part. Shouldn't it be a big $$T$$?

Now assuming that indeed $${|X[k]|}^2$$ equals the energy contained at that frequency of periodic signal. How could we use the relation between equations for $$X(w)$$ and $$X[k]$$ to derive eq. 2)? I'm not sure how we know that $${|X[w]|}^2$$ represents density of a non-periodic signal. Isn't it just equal to energy contained at frequency $$w$$ multiplied by $$T^{2}$$, where $$T \to \infty$$?

The notation is really confusing. It took a while for me to understand. But he is talking about units in 1).

So [XX] refers to the unit of XX.

Essentially he is saying, the unit of the continuous spectrum (X[w]) is the unit of the discrete spectrum (X[k]) times the unit of time ([t]), or divided by the unit of frequency ([w]). That is a rather obvious statement, and has nothing to do with 2).

So 1) and 2) are in my opinion totally unrelated. Probably this is where the confusion is coming from.

• Any idea where 2) comes from, knowing that ${|X[k]|}^2$ equals signal energy at k-th frequency? The equation just appears there with no justification. – user4205580 Dec 27 '14 at 12:04
• There is no formal mathematical derivation of 2), so formally it is a definition. You can motivate it from the unit (i.e. |X(w)|^2 is energy per frequency) and the fact that the total signal energy is obtained by integrating over -inf .. inf (This is the Plancherel theorem). I dont see any benefit from relating it to |X(k)|^2 - maybe just that it looks similar. – Andreas H. Dec 27 '14 at 12:31
• Thanks. Just making sure - in eq.1) X(w) is not equal to X[k] * w, but it's just about the difference in units of X(w) and X[k]? Secondly, is there any easy explanation of why we are squaring X[k] or X(w) in the energy formula? – user4205580 Dec 27 '14 at 12:45
• 1) Yes. 2) Hmm, I think of it as X(w) being kind an "amplitude" so X(w)^2 is a "power/energy". – Andreas H. Dec 27 '14 at 14:17
• You mentioned Plancherel theorem - didn't you mean Parseval's theorem? (quote: The interpretation of this form of the theorem is that the total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f.) – user4205580 Dec 27 '14 at 17:14