Fourier synthesis of periodic signals

Question

I was reading the Fourier synthesis of periodic signals

But I didn't understand the sentence i.e. "Although the calculation of $a_0, a_1, b_1, a_2, b_2$, is a mathematically straightforward process, it may become rather tedious depending on the complexity and the discontinuities of $f(x)$

1. So can anybody explain about what is mean by discontinuities in the $f(x)$ ?

2. Also, how Fourier synthesis takes care of factors like different phase values , different amplitude values and different frequency values of sinusoidal signals ?

![enter image description here][2]

Answer 1

factors like different phase values

$A\sin x +B\cos x = C \sin (x+\phi)$

Or complex coefficients for $e^{i\omega t}$ convention

different amplitude values

Fourier transform/series are a linear transformation. You can multiply the series by a constant and its transform (forward or reverse) will be multiplied by the same constant.

different frequency values

$a_0$ refers to constant ($0$ frequency)
$a_1$ refers to component with frequency equal to original signal's period
$a_5$ refers to component with frequency equal to original signal's period/5

discontinuities in the f(x)"

https://en.wikipedia.org/wiki/Gibbs_phenomenon

Other than that, it is not complicated at all.