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I have learned that the Fourier transform of a continuous-time unit-periodic stochastic process is:

$$x(t) = \sum\limits_{k=-\infty}^{\infty} a_k e^{i2\pi kt} \quad \quad \text{ where } \quad \quad a_k = \int \limits_{0}^{1} x(t) e^{-i 2\pi kt} dt.$$

I have also seen another representation (see equation 9, page 183 of this article) as:

$$x(t) = \text{Re} \left\{\sum\limits_{k=1}^{N} b_k e^{i2\pi kt}\right\},$$

I am mainly interested in the second representation. My question is, what is $N$ in the second representation? If it is an integer which approximates the function conveniently well, how can I find this integer in real life?

You can also share your thoughts like whether the representation seems incorrect to you, or whether the representation is valid under "certain" assumptions, etc.

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  • $\begingroup$ Can you share where you saw these formulas? Because, they seem like Fourier Series representations with errors in it. $\endgroup$ – gunes Mar 26 '19 at 11:15
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    $\begingroup$ I think the first one is well-known. I have added a link for the second one. Thanks @gunes! $\endgroup$ – Joy Mar 26 '19 at 13:20
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    $\begingroup$ Now, the first two formula make sense for me (when you corrected int. limits, I've also realized that $x(t)$ is actually CT and unit-periodic ($T=1$), not continuous-time-unit and periodic with some unknown $T$). Thanks for the link by the way. Though, I still have doubts about the terminology. This is exactly the FS representation of the signal, and $a_k$ are the FS coefficients, although it is closely related to Fourier transform, which can be obtained using $a_k$ via some algebra. For the reference, they say $b_k=\int_{0}^T x(t)e^{i\omega_j t}$, w/o $1/T$. I've never seen this formula. $\endgroup$ – gunes Mar 26 '19 at 14:29
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Since no-one else has posted an answer, I'm going to give one here, even though I find the equations in the paper confusing (and they appear to me to be wrong). Your question does not state the equation quite the same way as the linked paper. Adjusting the notation of the paper to your notation, they assert that:

$$x(t) = \text{Re} \left\{\sum\limits_{k=1}^{N} b_k e^{-i \omega_k t}\right\} \quad \quad \quad \quad \quad b_k = \int \limits_0^T x(t) e^{i \omega_k t} \ dt.$$

This is very hard to understand, even within the context of the previous information in the paper, so I can see why you're having trouble with it. It is a strange set of equations because they mix continuous representation with discrete representation. If you have a look at the paragraph that precedes Equation (9) of that paper, you will see that they introduce this by noting that "In practical data analysis, the data consists of a string of real numbers...". Although it is not stated explicitly, I assume that the intention in the equations is to talk about a practical situation like this, where you have a finite sample of time-series data. In view of that, it appears that they are assuming that you have observed $N \in \mathbb{N}$ observations in the time-series.

If I understand correctly, their intention here is to use a continuous function $x: \mathbb{R} \rightarrow \mathbb{R}$ for the time series, but they assume that the observed data from this series comes from $N$ discrete points, and the continuous function is a continuous extension of that discrete set of observable values (defined according to the inverse-Fourier transform from the frequency coefficients). Presumably the first of these two equations is the inverse Fourier transform that returns the (continuous) time-series from the Fourier coefficients,$^\dagger$ and the second is the Fourier transformation that obtains the Fourier coefficients from the time-series.

When dealing with a discrete time series containing $N$ data points, it is true that you only need $N$ values of the Fourier coefficients at different frequencies $\omega_1,...,\omega_N$ to fully determine the series. The authors initially define $\omega = 1/T$ but then they never give definitions of $\omega_1,...,\omega_N$, so it is not really clear if these are supposed to be particular frequencies, or just arbitrary (non-degenerate) frequencies. The equations for the Fourier transform and its inverse include the complex exponential and its inverse, so that part makes sense. However, the equations do not appear to me to be correct, since they lack the unitary adjustment $1/\sqrt{N}$ that would appear in the discrete Fourier transform and its inverse, and they lack an appropriate scaling factor to deal with the integral up to $T$ in the Fourier transformation.

It is possible that I have misunderstood these equations somehow, and perhaps the authors intended for there to be some (unstated) relationship between $N$ and $T$. In some applied papers there is a lot of background knowledge of meanings of particular variables, and sometimes this is unstated. As it is written, with the context they give, there are a number of things that are undefined and unclear, and so the result looks like it is wrong. In cases like this, you can write to the authors seeking clarification, and this might reveal some aspect of the definition that was unstated, which fixes up the apparent problems. In any case, if you are just looking for a general explanation of Fourier transforms with discrete data, there are much better and simpler papers on that topic that I would recommend over this one.


$^\dagger$ The $\text{Re}$ operator in the first equation appears to be superfluous, since the inverse-Fourier transform ---if it is stated correctly--- should already return the original time-series, which is a real function.

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  • $\begingroup$ Thank you very much @Ben! I was wondering if the adjustment $1/\sqrt{N}$ is needed in $b_k$? Does not it correspond to the continuous representation of $x(t)$? Now you said, so I think that the authors only provided a real-data counterpart for $x(t)$, but not for the $b_k$, right? (However, I agree that $1/T$ should be there in $b_k$.) $\endgroup$ – Joy Apr 4 '19 at 10:15
  • $\begingroup$ @Joy: Unless there is some unstated connection between $N$ and $T$, I don't see any adjustment that would make it fit. As to the second part, providing $N$ values of $b_k$ is equivalent to providing $N$ data points. $\endgroup$ – Ben - Reinstate Monica Apr 5 '19 at 21:35

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