It is known that the total power of an OFDM signal equals to the sum of the powers on each individual subcarriers. I am a bit puzzled by the total transmit power of a generalized FDM system if subcarriers are not orthogonal and if it should remain the same or changed due to the non-orthogonality.
Let us consider the following OFDM transmit signal: $$s(t) = \sum_{k = 0}^{N-1} a_k e^{j2\pi f_k t} = \sum_{k = 0}^{N-1} a_k e^{j2\pi \frac{k \: \phi}{T} t}, \quad 0\leq t\leq T,$$
where $k$ is the subcarrier index, $a_k$ are i.i.d. data symbols with zero mean, $N$ is the total number of subcarriers, $T$ is the FDM symbol duration, and $\phi < 1$ is introduced to break the orthogonality.
The transmit power is represented as
$$ P = \frac{1}{T} \int_{0}^{T} s(t) \: s^*(t) dt,$$
which can be further simplified as
$$P = \frac{1}{T} \sum_{k = 0}^{N-1} \sum_{k' = 0}^{N-1} a_k a^*_{k'} \int_{0}^{T} e^{j2\pi \phi (k - k')t/T} dt$$
Now, the value of the integral $\int_{0}^{T} e^{j2\pi \phi (k - k')t/T} dt$ does not equal to $0$ if $k \neq k'$ (due to the non-orthogonality introduced by $\phi$). This means that the transmit power in case of non-orthogonal subcarriers is no longer equals the sum of the power on each subcarrier? However, it is higher than the sum!!!
I feel there is something wrong in this simple derivation, as by intuition, if a power source spends a certain amount of power on each subcarrier, then how come the total power will be greater than the sum of these individual powers?
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Update
The following update to make the modeling of non-orthogonality more clear. Let us consider the FDM signal written as
$$s(t) = \sum_{k = 0}^{N-1} a_k e^{j2\pi (f_k + \delta_k) t} = \sum_{k = 0}^{N-1} a_k e^{j2\pi (\frac{k \:}{T} + \delta_k) t}, \quad 0\leq t\leq T,$$
where $\delta_k$ is the frequency offset on subcarrier $k$. Following similar approach, the power $P$ can be found as
$$P = \frac{1}{T} \sum_{k = 0}^{N-1} \sum_{k' = 0}^{N-1} a_k a^*_{k'} \int_{0}^{T} e^{j2\pi (k/T + \delta_k - k'/T - \delta_{k'})t} dt$$
which does not equal to the sum of power on individual subcarriers.