# Impedance 1: Fourier Analysis

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 $g(t)$ be a complex-valued function of time, measured in seconds.  In Fourier analysis, we say $g$ is a time-domain function because, well, the domain of $g$ is time.  The basic idea of Fourier analysis is that $g$ can be expressed as a sum of complex exponentials $e^{2\pi ift}$ oscillating at frequency $f$ in Hertz.  Namely,

$\displaystyle g(t) = \int G(f)e^{2\pi i f t} df$.

The function $G(f)$, which denotes the contribution to $g$ of the exponential oscillating at frequency $f$, is called the Fourier transform of $g$.  We will also write $\mathcal{F}\{g(t)\}(f)$ for the Fourier transform $G(f)$.  It is given explicitly by the formula

$\displaystyle G(f) = \int g(t)e^{-2\pi i f t}\,dt$.

When $g(t)$ is periodic with minimal period $T$, that is, $g(t) = g(t + T)$ for all $t$, then the situation is simpler.  The Fourier transform is non-zero only for integer multiples of the fundamental frequency $f_0 = 1/T$.  The coefficients $G_k = G(k f_0)$ are called the Fourier series of $g$, and we can write

$\displaystyle g(t) = \sum_k G_k e^{2\pi i f_0 k t}$.

Finally, the Fourier transform enjoys some nice properties.  First, by linearity of the integral, the Fourier transform is also linear:

$\displaystyle\mathcal{F}\{ag(t) + bh(t)\}(f) = a\mathcal{F}\{g(t)\}(f) + b\mathcal{F}\{h(t)\}(f)$,

where $g, h$ are complex-valued functions and $a, b$ are complex numbers.  Second, it behaves nicely with respect to the derivative:

$\displaystyle\mathcal{F}\{f'(t)\}(f) = 2\pi i f \mathcal{F}\{f(t)\}(f)$.

What does this have to do with electronics?  More on that next time.