Fourier expansion of periodic signal proof

Question

Definitions

• Period: The period is the smallest value of T satisfying $$g(t + T) = g(t)\label{0}\tag{0}$$ for all t. The period is defined so because if g(t + T) = g(t) for all t, it can be verified that g(t + T') = g(t) for all t where T' = 2T, 3T, 4T, ... In essence, it's the smallest amount of time it takes for the function to repeat itself. If the period of a function is finite, the function is called "periodic". Functions that never repeat themselves have an infinite period, and are known as "aperiodic functions". The period of a periodic waveform will be denoted with a capital T. The period is measured in seconds.

• Kronecker delta function: $$ \delta_{ij}= \begin{cases} 0&\text{if}\, i\neq j\\ 1&\text{if}\, i= j \end{cases} \label{1}\tag{1} $$

• Discrete-time signal x: $x(n)=\sum_{k=-\infty}^{\infty}x(k)\delta(n-k)\label{2}\tag{2}$

• Continuous-time signal x: $$x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau)\delta{\tau}\label{3}\tag{3}$$
• Euler formula derivations:

$$e^{j\theta}=\cos(\theta)+j\sin(\theta)\label{4}\tag{4}$$

$$e^{-j\theta}=\cos(\theta)-j\sin(\theta)\label{5}\tag{5}$$

$$\cos(\theta) = \frac{1}{2}(e^{-j\theta}+e^{j\theta})\label{6}\tag{6}$$

$$\sin(\theta) = \frac{1}{2j}(e^{-j\theta}-e^{j\theta})\label{7}\tag{7}$$

Problem

The fourier expansion of a periodic signal $x_T(t)=x_T(t+T)$ is

$$x_T(t)=F^{-1}[X[k]]=\sum_{k=-\infty}^{+\infty}X[k]e^{jkw_0t}\label{8}\tag{8}$$

where $X[k]$ is the fourier coefficient

$$X[k]=F[x_T(t)]=\frac{1}{T}\int_Tx_T(t)e^{-jkw_0t}dt \label{9}\tag{9}$$

where $k=0, \pm1,\pm2,...$

Question

I'd like to prove this:

$$x_T(t)=\sum_{k=-\infty}^{+\infty}X[k]e^{jkw_0t}\label{10}\tag{10}$$

I believe by considering the above definitions it should be possible to get a nice simple proof but I don't know how to proceed here.

Answer 1

Very roughly (i.e. with little rigour) the identity follows from the fact that one can write the Dirac Delta function as $$ \delta(x)=\frac1{2\pi}\sum_{n=-\infty}^\infty e^{inx}. $$ (cf https://en.wikipedia.org/wiki/Dirac_delta_function#Fourier_kernels) This means that we have \begin{equation}\begin{aligned} \frac1T\sum_{k=-\infty}^\infty\int_0^Tdt'x(t')e^{2\pi ik\frac{t-t'}T}&=\frac1T\int_0^Tdt'x(t')\sum_{k=-\infty}^\infty e^{2\pi ik\frac{t-t'}T}\\ &=x(t). \end{aligned}\end{equation} Again, this is very sketchy and not rigorous at all. I'm swapping limits and messing around with Dirac deltas without much thought.

Answer 2

I think the problem of convergence of the Fourier series of a periodic function $f$ to $f$ is very wide, when we don’t specify conditions on $f$, like continuity and smoothness and a type of convergence of the series, like pointwise or uniform. Thus I think that Wikipedia article on convergence of Fourier series provides an excellent brief survey of the state of the art.

In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.

Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, $L^p$ spaces, summability methods and the Cesàro mean.

Thus, when the problem conditions are not specified I think there is no much sense to argue about convergence proofs. On the other hand, I have a few theorems providing sufficient conditions for pointwise and uniform convergence of the Fourier series of a periodic function $f$ to $f$ in my student analysis book (in Ukrainian). But my working experience with application people like programmers or engineers shows that they usually don’t need a proof, but at most a reference to a theorem providing a needed result.