This is the content of an introductory lesson to Signals. Inmersive content about Fourier series and how to apply the formulae.
Time: Exposition time: about 10 minutes and then go through the practical examples.
Fourier series in action
Fourier Transform and Periodicity
Are Fourier transforms only useful for periodic functions? No. A Fourier transform completely defines a periodic function, but it can also precisely define a section of a non-periodic function. Although most real signals are not periodic, there are cases of periodic signals that can be fully defined, for instance in power.
Decomposition of a Periodic Function
If I have a periodic function, I can decompose it as a sum of coefficients multiplied by complex exponential functions.
\begin{equation} f(t) = \sum_{n=-\infty}^\infty d_n e^{j 2 \pi n f_0 n t} \end{equation}
Coefficients
The coefficients ‘d_n’ are obtained by integrating.
\begin{equation} d_n = \frac{1}{T_0} \int_{-T / 2}^{T/2} f(t) e^{-j 2 \pi n f_0 t} dt \end{equation}
Integration
The integral will be more or less complicated depending on the function we want to approximate and the period we take. Whenever possible, we will avoid integrating; by applying properties, we will be able to obtain the value of the integral through other means.
Fourier Series
The terms Fourier Transform and Fourier Series are related, but not equivalent. The Fourier series rewrites the original signal as a sum of sines and cosines.
Generating Function
When we have a periodic signal, there is always a series of values that repeats, and we can express the signal as the repetition of that period. The repeating function is called the generating function.
Period ‘T’
It is the time it takes for something to repeat; it is the same as the inverse of frequency ‘f_0’.
Hertz
If we measure frequency in Hertz, we are measuring cycles of repetition per second. The conversion to radians per second is done by multiplying the frequency by ‘2π’.
\begin{equation} ω = 2πf \end{equation}