Confused about the similarities between Fourier Series and Fourier Transform

Question

My understanding of Fourier Series was, it is a method that decomposes a periodic signal into sum of signal given by infinite number of sines and cosines. And in case of Fourier Transform it was, that it gives the function producing a signal in frequency-domain using its function in time-domain. I thought of them as two different unrelated things.

But then I came to know, Fourier Series and Fourier Transform are similar in many ways. It seems like Fourier transform decomposes the given signal too, but instead of the decomposed signals' frequency being integral multiple of fundamental frequencies, they are continuous values in a given range. And, Fourier Series is also giving us the value of the signal in frequency domain but only at certain discrete steps, and values in the middle are missing.

Are the statements that I wrote in my second paragraph correct? If not, what is it that I understood wrong?

Also, how is a method of taking a signal from time domain to frequency domain similar/same as the method that decomposes a signal into small parts? i.e. I'm confused by alternate explanations of both Fourier series and Transform that are coming up when relating them with one another. I'm seeing Fourier Series being explained in terms of method that takes gives values of a signal in frequency domain, and Fourier Transform as the method that decomposes a signal. And that is not making any sense.

Answer 1

Fourier Series: only periodic functions can be transformed
Fourier Transform: every (natural) Signal can be transformed

---

in a FS, periodic signals superpose to reconstruct a given periodic signal. these signals range from minus inifinity to ininity, have unlimited resolution on the amplitude and the signals are multiples of the basic frequency called harmonics
So FS gets you Amplitudes at the harmonic frequencies - thats "discrete frequency steps"

in a FT sine and cosine signals are generated for every (unlimited resolution) frequency from -infinity to infinity. The "signal to be transformed" (which needs to be from -infinity to infinity in the time domain) is then "multiplied" with the generated signals. The result is, when the frequency of that generated sine and or cosine is a component of the "signal to be transformed", we get amplitude and phase.
So FT: signal in unlimited time and unlimited amplitude resolution is converted to unlimited frequencies and amplitude/phase pairs with unlimited resolution

FFT is then discrete in time and in Amplitude, which causes also discrete frequency steps and amplitude/phase pairs.

Edit: when the same periodic signal is calculated with FT and FS, the same amplitudes for the same harmonic frequencies are calculated.

Answer 2

If you make a function periodic by adding shifted copies of it (and dividing by the numbers of copies for normalisation) while increasing the number of added copies to infinity, its continuous spectrum condenses to a line spectrum with lines (effectively dirac pulses, so again there is a normalisation issue) apart by the inverse of the shift width. If you sample a function with a fixed sampling interval making everything in between zero (to get a proper transform at all, you again need to turn every sample into a dirac pulse with finite area), the transform becomes periodic.

Essentially, a dirac pulse comb transforms into a dirac pulse comb, and multiplication with it (sampling) turns into periodicity (convolution with it) and vice versa.

So a Fourier series is what results when you make a function periodic, sample it at an equally-spaced grid, and then only look at the possibly non-zero dirac-sized) points of one period of what is now a periodic line spectrum, because those completely represent the original sampled periodic function.

As this kind of special case of a continuous Fourier transform, it is not surprising that it obeys the rules of it and some additional ones due to it being a special case.