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.

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