You won’t get very far in electronics without understanding the concept of impedance. Loosely speaking, impedance is generalized frequency-dependent resistance. To understand what that means, we need to review the basic ideas of Fourier analysis.
Let be a complex-valued function of time, measured in seconds. In Fourier analysis, we say is a time-domain function because, well, the domain of is time. The basic idea of Fourier analysis is that can be expressed as a sum of complex exponentials oscillating at frequency in Hertz. Namely,
The function , which denotes the contribution to of the exponential oscillating at frequency , is called the Fourier transform of . We will also write for the Fourier transform . It is given explicitly by the formula
When is periodic with minimal period , that is, for all , then the situation is simpler. The Fourier transform is non-zero only for integer multiples of the fundamental frequency . The coefficients are called the Fourier series of , and we can write
Finally, the Fourier transform enjoys some nice properties. First, by linearity of the integral, the Fourier transform is also linear:
where are complex-valued functions and are complex numbers. Second, it behaves nicely with respect to the derivative:
What does this have to do with electronics? More on that next time.